Verification in computational structural mechanics: recovery-based a posteriori error estimation by Giovanni Castellazzi Advisor: Prof. Ing. Francesco Ubertini co-Advisor: Ing. Stefano de Miranda Dottorato di Ricerca in Meccanica delle Strutture - XIX Ciclo Coordinatore: Prof. Ing. Erasmo Viola Settore scientifico disciplinare: Area 08 - Ingegneria civile e Architettura ICAR/08 - Scienza delle Costruzioni ALMA MATER STUDIORUM - University of Bologna Bologna, March 12th , 2007 “Verification in computational structural mechanics: recovery-based a posteriori error estimation,” a dissertation prepared by Giovanni Castellazziin partial fulfillment of the requirements for the degree of Doctor of Philosophy, has been approved and accepted by the following: Prof. Elio Sacco Prof. Alessandro De Stefano Prof. Francesco Ubertini April 2007 Abstract Verification in computational structural mechanics: recovery-based a posteriori error estimation by Giovanni Castellazzi Dottorato di Ricerca in Meccanica delle Strutture - XIX Ciclo Coordinatore: Prof. Ing. Erasmo Viola Prof. Francesco Ubertini Ing. Stefano de Miranda ALMA MATER STUDIORUM - University of Bologna Computational engineering, the discipline concerned with the use of computational methods and devices to simulate physical events and engineering systems, is being heralded by many as one of the most important developments in recorded history. Computer predictions of physical events, it is argued, can be of enormous importance in making critical decisions that aﬀect every facet of human existence. As the speed and capacity of computer systems continue to grow, the expectations of users of computer models in decision making continues to grow in kind. Today, some look toward computer-based predictions as a means to obtain vital quantitative information on events that influence the security, health, and well being of much of mankind and many nations and that influence the success of major businesses and other enterprises worldwide. "Homo qui erranti comiter monstrat viam Quasi lumen de suo lumine accendat facit. Nihilo minus ipsi lucet cum illi accenderit." [maxim ascribed to Ennio by Cicerone in "De Oﬃciis", I, chapter XVI] Aknowledgements I owe a debt of gratitude to Professor Francesco Ubertini who gave me the opportunity to write the present work and to Ing. Stefano de Miranda who was always helpful. I would say that I feel very lucky to have worked in this group because they embody the maxim reported on this page. I wish that it would never change, because there are no better research mates for me. I would like to thank Professor Erasmo Viola for giving me the opportunity to attend the Ph.D. program in Structural Mechanics, and for his enthusiasm, passion and greatest care to improve at the highest level the school that I’m leaving in reaching this goal. I would like to thank all the guys of the LAMC laboratory for the friendship and the support during these years. I would like to thank John Barlow for his kindly encouragement and suggestion for historical information about pioneers in FE, and Richard Haigh of Rolce Royce Heritage Trust who put me in contact with John Barlow. I would like to thank Giuseppe Miccoli of NAFEMS Italian Division, Trevor Hellen and Tim Morris of NAFEMS Ltd for some interesting advice about the bibliography of geometry distortions topic. I would like to thank my family: my sister Sara and my parents Rosanna and Dario who always supported me in all situations and in all my choices. Bologna, March 12th , 2007 Chapter 0 ix Contents Sommario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 What is the error? 3 1.1 A point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Computational engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 1.2.1 Idealization of a physical event . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Analysis process by the finite element method . . . . . . . . . . . . . . 12 1.2.3 Interpretations of the finite element method . . . . . . . . . . . . . . . . 13 1.2.4 Historical sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 The main focus of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Finite element analysis 2.1 23 The basic ideas of the finite element method . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Variants of the finite element method . . . . . . . . . . . . . . . . . . . . . 25 2.1.2 Discretization of the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.3 Identification of the state variable(s) . . . . . . . . . . . . . . . . . . . . . . 27 2.1.4 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 The model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Definitions of error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 A-priori error estimation 37 3.1 Analysis of the finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Convergence of the finite element method . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 3.3 3.4 Richardson’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Completeness requirements of shape functions . . . . . . . . . . . . . . . . . . . . 43 3.3.1 Constructing approximate functions for the elements . . . . . . . . 45 3.3.2 Isoparametric quadrilateral elements . . . . . . . . . . . . . . . . . . . . . . 47 Numerical integration and optimal sampling points . . . . . . . . . . . . . . . . 54 x 0.0- Contents 3.5 3.4.1 A one-dimensional example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4.2 Optimal sampling points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Eﬀects of geometry distortion on the element performance . . . . . . . . . 60 3.5.1 3.6 3.7 3.8 A simple distortion test for quadratic elements . . . . . . . . . . . . . 64 A new procedure to evaluate element distortions . . . . . . . . . . . . . . . . . . 66 3.6.1 Some basic distortions of plane elements . . . . . . . . . . . . . . . . . . . 69 3.6.2 A simple rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.3 A generic linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6.4 A bilinear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6.5 Higher degree transformations: quadratic curved-edge . . . . . . . 76 Some numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.7.1 Bilinear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.7.2 Higher order transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Distortion measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.8.1 Distortion measures for four node elements . . . . . . . . . . . . . . . . . 85 3.8.2 Test No.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.8.3 Test No.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4 A posteriori error estimation 4.1 4.2 4.3 89 Aims and concepts of a posteriori error estimation . . . . . . . . . . . . . . . . 89 4.1.1 Residual based approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.1.2 Recovery based approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Some recovery methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.1 Superconvergent Patch Recovery: SPR . . . . . . . . . . . . . . . . . . . . 95 4.2.2 Recovery by Equilibrium in Patches: REP . . . . . . . . . . . . . . . . . 99 Recovery by Compatibility in Patches: RCP . . . . . . . . . . . . . . . . . . . . . 105 4.3.1 Formulation of the RCP procedure . . . . . . . . . . . . . . . . . . . . . . . 105 4.3.2 Implementation of the RCP procedure . . . . . . . . . . . . . . . . . . . . 107 4.3.3 Consistency of the RCP procedure . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.4 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3.5 An alternative version of RCP recovery . . . . . . . . . . . . . . . . . . . 111 Chapter 0 xi 4.4 Recovery on the boundary of the domain . . . . . . . . . . . . . . . . . . . . . . . . 112 4.5 The quality of an error estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.6 4.7 4.5.1 The benchmarking approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.5.2 The robustness test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.6.1 Asymptotic behaviour and robustness . . . . . . . . . . . . . . . . . . . . 126 4.6.2 Benchmark tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Recovery based error estimators can fail? . . . . . . . . . . . . . . . . . . . . . . . 147 5 Adaptivity 155 5.1 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 Adaptive finite element techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.3 5.4 5.2.1 Adapt elements or functions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.2.2 Structured or unstructured mesh? . . . . . . . . . . . . . . . . . . . . . . . . 161 5.2.3 h-refinement on structured mesh . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.2.4 h-refinement on unstructured mesh . . . . . . . . . . . . . . . . . . . . . . . 164 Predicting the required element size in h-adaptivity . . . . . . . . . . . . . . 164 5.3.1 The h−refinement strategy with RCP . . . . . . . . . . . . . . . . . . . . 166 5.3.2 The hp−refinement strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.4.1 Class I problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.4.2 Class II problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6 Some advanced applications in structural mechanics 187 6.1 Recovery for assumed stress finite elements . . . . . . . . . . . . . . . . . . . . . . 188 6.2 Plate structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.3 Some numerical tests on bending response . . . . . . . . . . . . . . . . . . . . . . . 192 6.4 Laminated composite plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.5 Some numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7 Concluding remarks: looking toward Validation 7.1 219 Assessing the numerical error in the finite element solution . . . . . . . . 220 xii 0.0- Contents 7.2 Validate the model, not the codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.3 An overview of the PTC 60 / V&V 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.3.1 7.4 The guide and the present work . . . . . . . . . . . . . . . . . . . . . . . . . 226 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Appendix A Structural models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Appendix B More benchmark tests . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Appendix C The hybrid stress element for plate structures (9βQ4) . . . 261 Appendix D Formulation of a transition element for plane problems . . 263 Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Sommario Questa tesi riguarda la stima a posteriori dell’errore commesso nella discretizzazione in elementi finiti di problemi di analisi strutturale e sulle tecniche di adattività ad essa correlate. L’attenzione è rivolta ai metodi di ricostruzione, di recovery in dizione anglosassone, che stimano l’errore per confronto con una soluzione più accurata che viene ricostruita con opportune procedure a partire da quella per elementi finiti. In questo contesto è stata sviluppata una nuova procedura di ricostruzione locale degli sforzi, ottenuta rilassando la compatibilità cinematica, attraverso la minimizzazione di un funzionale tipo energia complementare, appositamente definito su un piccolo insieme di elementi. La procedura è stata impiegata con risultati molto soddisfacenti su vari modelli strutturali, discretizzati sia con elementi compatibili che con elementi finiti agli sforzi. In particolare, essa è stata applicata con successo alle piastre laminate, descritte attraverso la cosiddetta "First order Shear Deformation Theory", con l’obiettivo di ricostruire la distribuzione delle tensioni tangenziali lungo lo spessore della piastra. In aggiunta a ciò, infine, è stato analizzato il ruolo dello stimatore d’errore quale parametro guida per la ricostruzione della discretizzazione in analisi adattative, mostrando l’eﬃcacia della procedura sviluppata anche in questo contesto. La tesi si articola in sei capitoli. Nel primo capitolo, introduttivo, vengono illustrate le motivazioni che hanno spinto e spingono gli analisti alla ricerca di strumenti di verifica che consentano di attribuire alle analisi numeriche un determinato grado di aﬃdabilità. Nel secondo capitolo, viene presentato il metodo degli elementi finiti, definito il problema modello ed introdotto il concetto di errore commesso nella discretizzazione per elementi finiti, inquadrando la tesi nello stato dell’arte sull’argomento. Nel terzo capitolo, vengono illustrati alcuni risultati classici sulla stima a priori dell’errore e presentati alcuni aspetti originali inerenti gli eﬀetti delle distorsioni di geometria sulle prestazioni degli elementi finiti isoparametrici. Nel quarto capitolo, viene discussa la stima a posteriori dell’errore. Vengono analizzati e confrontati i principali metodi di stima a posteriori, ponendo l’attenzione sui metodi basati sulla ricostruzione del campo di sforzi (ovvero i cosiddetti recovery based methods). In questo contesto, viene sviluppata una nuova procedura che si dimostra essere versatile, robusta e superconvergente. Alcune xiv 0.0- Sommario applicazioni numeriche sono presentate a conclusione di tale capitolo. Nel quinto capitolo, viene discusso il ruolo dello stimatore d’errore quale parametro di controllo per la ricostruzione della discretizzazione in analisi adattative. Alcune applicazioni numeriche mostrano l’eﬃcacia della procedura proposta come guida del processo di analisi. Nel sesto capitolo, conclusivo, vengono presentate alcune applicazioni avanzate, quali l’impiego della tecnica proposta con elementi finiti di tipo misto agli sforzi, l’applicazione alle piastre e l’estensione alla ricostruzione del profilo delle tensioni tangenziali lungo lo spessore di compositi laminati. Chapter 0 Pagina Bianca da togliere nel documento PDF una volta fatta la conversione. 1 Chapter 1 What is the error? Questo capitolo introduce il concetto di verifiche in analisi strutturali per elementi finiti. A tale scopo, vengono illustrati i passaggi fondamentali che conducono ad una analisi agli elementi finiti, mettendo n luce i punti in cui vengono introdotte le approssimazioni e gli errori che ne conseguono. In particolare, vengono illustrati gli errori di modellazione, discretizzazione e soluzione, e viene definito il processo di verifica come quello che conduce alla quantificazione della somma degli ultimi due. Nella parte conclusiva del capitolo, inoltre, viene riportato un piccolo estratto storico sull’evoluzione del metodo degli elementi finiti, cercando di evidenziare la crescente sensibilità degli analisti verso i procedimenti di verifica e controllo dell’errore in analisi strutturali agli elementi finiti. 1.1 A point of view The word error has diﬀerent meanings in diﬀerent fields. The concrete meaning of the Latin word error means "wandering" or "straying", although the metaphorical meaning "mistake, misapprehension" is actually more common. To the contrary of an illusion, an error or a mistake can be dispelled through knowledge (knowing that one is looking at a mirage and not at real water doesn’t make the mirage disappear). Looking at the Figure 1.1 is simple to understand that something was wrong wen the train broke down the safety structure, passing through the restaurant railway, destroying a newspaper kiosk and concluding its trip on the 4 1.1- A point of view Figure 1.1: A famous image about error: train wreck at Montparnasse, Paris, France, 1895, (photgrapher unknown). From Wikipedia, the free encyclopedia, www.wikipedia.org road. Let us start with some point of view, in some disciplines. In statistics an error is a diﬀerence between a computed, estimated, or measured value and the true, specified, or theoretically correct value. In experimental science an error is a bound on the precision and accuracy of the result of a measurement. These can be classified into two types: statistical error (see above) and systematic error. Statistical error is caused by random (and therefore inherently unpredictable) fluctuations in the measurement apparatus, whereas systematic error is caused by an unknown but nonrandom fluctuation. If the cause of the systematic error can be identified, then it can usually be eliminated. Such errors can also be referred to as uncertainties. In engineering an error is a diﬀerence between desired and actual performance. Engineers often seek to design systems in such a way as to mitigate or preferably avoid the eﬀects of error, whether unintentional or not. One type of error is the human error which includes cognitive bias. Human factors engineering is often applied to designs in an attempt to minimize this type of error by making systems more forgiving or error-tolerant. Errors in a system can also be latent design errors that may go unnoticed for years, until the right set of circumstances arises that cause them to become active. Chapter 1 5 In computer programming (i.e. in software engineering), the term error refers to an incorrect action or calculation performed by software as a result of a fault (a fault being incorrect code or data in the software or hardware). If, as a result of the error, the software performs an undesired action or fails to perform a desired action, then this is referred to as a failure. Note that a fault can exist in software and not cause an error, and an error can occur without causing a failure. Looking again at Figure 1.1 is clear that the desired performance was not reproduced in the practice and that the safety structures and the restaurant furniture were not forgiving system as described above. In science, the word error does not carry the usual connotations of the terms mistake or blunder. Error in a scientific measurement means the inevitable uncertainty that attends all measurements. As such, errors are not mistakes, you cannot eliminate them by being very careful. The best you can hope to do is to ensure that errors are as small as reasonably possible and to have a reliable estimate of how large they are. For now, error is used exclusively in the sense of uncertainty, and the two words are used interchangeably. 1.2 Computational engineering Computational engineering, the discipline concerned with the use of computational methods and devices to simulate physical events and engineering systems, is being heralded by many as one of the most important developments in recorded history. Computer predictions of physical events, it is argued, can be of enormous importance in making critical decisions that aﬀect every facet of human existence. As the speed and capacity of computer systems continue to grow, the expectations of users of computer models in decision making continues to grow in kind. Today, some look toward computer-based predictions to obtain vital quantitative information on events that influence the security, health, and well being of much of mankind and of many nations and that influence the success of major businesses and other enterprises worldwide. Somebody speaks of the coming crisis in computational engineering arising from three major challenges: a) performance, b) programming, and c) prediction. The performance and programming challenges have been met or will be met soon, but the prediction challenge will require considerable advancement and maturity in the way that simulation is done and interpreted. The prediction challenge, which is at the heart of computational engineering, is viewed as the most diﬃcult challenge to be met in the future, and stands as a major bottleneck, perhaps a crisis, in computational engineering. Again, the 6 1.2- Computational engineering major issue is the reliability of computer predictions and their use as a basis for important decisions [Babuška & Oden, 2005]. It is understandable that the reliability of computer predictions has been an issue of growing concern for several decades. Major questions that arise with increasing frequency are "can computer predictions be used as a reliable base for crucial decisions? How can one assess the accuracy or validity of a computer prediction? What confidence can be assigned to a computer prediction of a complex event?". Professor E. Wilson, who has been responsible for the development of several computer programs, talking about structural engineering, collected all these questions giving a nice definition: Structural Engineering is: The art of using materials That have properties which can only be estimated To build real structures That can only be approximately analyzed To withstand forces That are not accurately known So that our responsibility with respect to Public safety is satisfied This little aphorism collects all the common greater uncertain about structural engineering. It is possible to extend it by including the idea that the computer nowadays is an essential powerful tool to use in structural design and modelling. So Computational Engineering is: The art of simulate physical events That have properties which can only be estimated To develop mathematical models That can only be approximately analyzed To improve the solution reliability That are not accurately known So that our responsibility with respect to Public safety is satisfied The collection of processes, philosophical issues, and procedures connected with answering these questions has become known as the subject of Verification & Validation (V&V), the verification process addressing the quality of the numerical treatment of the model used in the prediction and the validation process addressing the quality of the model. V&V has emerged in recent years as the intellectual and technological discipline that addresses the prediction challenge. Both are processes, verification being the processes addressing the quality of the numerical approximation of the mathematical model used as the basis for a prediction, and validation being the process Chapter 1 7 addressing the reliability of the mathematical model as a faithful abstraction of reality. V&V has been the focus of much study and debate in recent years and a relatively large literature exists and is expanding. Verification: the process of determining if a computational model obtained by discretizing a mathematical model of a physical event and the code implementing the computational model can be used to represent the mathematical model of the event with suﬃcient accuracy. Validation: the process of determining if a mathematical model of a physical event represents the actual physical event with suﬃcient accuracy. A typical example of a mathematical model is the set of equations and conditions characterizing a boundary value problem involving deterministic or stochastic diﬀerential equations together with functionals defining quantities of interest. These quantities of interest are the goals of the computer predictions, which, in turn, are the basis for decisions. Let us start with some primitive notions, depicted by Babuška and Oden in [Babuška & Oden, 2004]. Physical event: an occurrence in nature or in a physical system; a fundamental entity of a physical reality; a physical phenomenon. The dictionary (MerriamWebster Collegiate Dictionary, 10th edition) indicates that an event is "something that happens". Thus, we are interested in something that happens as a physical reality; not for example, in behavioral aspects or trends in, for instance, sociological or economical systems. Simulation: a process that build a likeness; in our case, a likeness produced by an interpretation of output from a computer or computational device. Mathematical model (of a physical event): a collection of mathematical constructions that provide abstractions of a physical event consistent with a scientific theory proposed to cover that event. Data of a mathematical model (of a physical event): factual information that defines the values or ranges of values of parameters in the mathematical model of a physical event. Discretization: the process that transforms a mathematical model into a finite number of discrete components that can be processed by a digital computer. Computational model: the discretized version of a mathematical model that has been designed to be implemented on (or to be processed by) a computer or computational device. Code: A computer program designed (in the present context) to implement a computational model. Prediction: something that is predicted, declared or indicated in advance; fore- 8 1.2- Computational engineering told on the basis of observation, experience, or scientific reason. A prediction is not simply a deduction or a consequence of a theory of something that mayor may not be known. It is the indication of an event not already known. The mathematical problem is described by its structure and its input data. The structure of the mathematical problem comprises functional relations between the input and the output. For example, the structure can be expressed by a system of conservation laws. The input data is the set of all admissible data needed for solving the mathematical problem. For example, input data includes the classic boundary conditions and parameters used in the structure of the problem. The data also include the characterization of the uncertainty as when it is part of the mathematical problem. In the literature the term mathematical problem and mathematical model are not distinguished, although the term model is often used in a more generic sense than mathematical, which involves all input data. We will not distinguish between the problem and the model either. Verification therefore is the process of determining if a computational problem and the code implementing the computational problem leads to a prediction of sufficient accuracy, i.e., the diﬀerence between the exact and computed quantity of interest is suﬃciently small. Hence, verification has two aspects, the approximation aspect and the verification of the correctness of the code, i.e., the program developed to implement the computational model can faithfully produce the intended results. Although code verification is obviously essential, we will not address it in the present thesis. The first part of verification, so-called solution verification, is essentially a problem of a-posteriori error estimation. It addresses not only classical methods of error estimation of standard approximation methods (such as finite elements) but also errors due to simplification of the problem. For example, errors due to the linearization of a nonlinear problem or of dimensional reduction. A posteriori error estimation is a fairly mature subject and many techniques for developing a-posteriori error estimates have been proposed in the literature. A posteriori error estimation is a purely mathematical process and, while many open problems remain, eﬀective methods exist for addressing solution verification for a large class of computational models. Validation is the process of determining if the mathematical model of a physical event (the prediction) represents the actual physical event with suﬃcient reliability. In contrast to the verification, validation addresses the problem how well the theory describes reality. This question is related to a major problem in philosophy, especially in the philosophy of science. A first question is what is actually meant by validation and whether validation is even possible. In the validation procedure, Chapter 1 9 we consider a set of validation problems. These problems are specific mathematical problems for which some of the input data are the same as those in the prediction problem, but others may be diﬀerent. For example, in the elasticity problem, the domain and the boundary conditions are diﬀerent for the validation and the prediction problem, but the constitutive law is the same. These validation problems are simpler than the prediction problems and can, in general, be experimentally studied. 1.2.1 Idealization of a physical event The goal of computer simulation is thus to make predictions of physical events using computational models implemented with appropriate codes. It is important to underscore the relationship between a mathematical model of an event and the scientific theory or theories used to characterize the event. As mathematics is, in a fundamental way, the language of science and technology, the mathematical model can be viewed as a transcription into a concrete and precise format of the theoretical framework in which the modeler intends the event to be depicted. A mathematical model may not necessarily be equivalent to a complete theory covering events in nature. Mathematical models may be the mathematical constructions representing the result of collections of assumptions and constraints on theoretical arguments and, therefore, may not describe all physical processes responsible for an event we are interested in predicting. This relationship between the mathematical model and the scientific theory provides an indirect connection between the philosophical underpinnings of V&V and major tenants of contemporary philosophy of science. If the computational model describes the mathematical model well and the mathematical model relates to the theory well, then the computational model also relates well to the theory. The models referred to in the definitions of verification and those in validation are, in general, quite diﬀerent, as are, in some respects, the events they seek to emulate. Note that both the verification and validation processes involve determining if the respective processes lead to results of "suﬃcient accuracy", leaving open both the meaning of "suﬃcient" and of how "accuracy" is to be quantified. 1.2.1.1 Discrete model: some implications The limitations of the human mind are such that it cannot grasp the behaviour of its complex surroundings and creations in one operation. Thus the process of subdividing all systems into their individual components or elements, whose behaviour is readily understood, and then rebuilding the original system from such components to study its behaviour is a natural way in which the engineer, or even the scientist proceeds. 10 1.2- Discrete model: some implications Figure 1.2: Verification and Validation processes In many situations an adequate model is obtained using a finite number of well-defined components. We shall term such models discrete. In other situations the subdivision is continued indefinitely and the problem can only be defined using the mathematical fiction of an infinitesimal. This leads to diﬀerential equations or equivalent statements which imply an infinite number of elements. We shall term such models continuous. With the advent of digital computers, discrete problems can generally be solved readily even if the number of elements is very large. As the capacity of all computers is finite, continuous problems can only be solved exactly by mathematical manipulation. To overcome the intractability of realistic types of continuous problems, various methods of discretization have from time to time been proposed both by engineers and mathematicians. All involve an approximation which, hopefully, approaches in the limit the true continuous solution as the number of discrete variables increases. The discretization of continuous problems has been approached diﬀerently by mathematicians and engineers. Mathematicians have developed general techniques applicable directly to abstract diﬀerential equations, such as finite diﬀerence approximations, various weighted residual procedures or approximate techniques for determining the stationarity of properly defined functionals. Engineers, on the other hand, often approach the problem more intuitively by creating analogies between real discrete elements and finite portions of a continuum domain. Figure 1.3 shows an interesting discretization study of a car, which is a very common physi- Chapter 1 11 cal system, for performing crashworthiness simulations. The funny thing for this case study is that for first the discretization has begun directly on the car, by a preliminary vehicle tear-down. Actually in this study the non-structural trim and interior components have been removed. Tape on vehicle is a coordinate system used to measure the positions of removed components. The resulting measured surfaces were then modified to an appropriate format for use in the vehicle model mesh generation program. This is also an attempt of model validation using both the component tests and full vehicle crash tests as shown in [Kirkpatrick et al. , 1998]. 1.2.1.2 Discretization methods A typical classification of computational analysis is based on the discretization method by which the continuum mathematical model is discretized (in space or/and in time), i.e., converted to a discrete model with a finite number of degrees of freedom. Some of the most famous methods are: (a) Finite Element Method (FEM), (b) Boundary Element Method (BEM), (c) Finite Diﬀerence Method (FDM), (d) Finite Volume Method (FVM), (e) Spectral Method, (f) Meshfree Method. In computational solid and structural mechanics the finite element method currently dominates the scene as regards space discretization of linear problems. Boundary element method posts a strong second choice in specific application areas. For nonlinear problems the dominance of the finite element method is overwhelming. Finite diﬀerence methods for space discretization in solid and structural mechanics have virtually disappeared from practical use. This statement is not true, however, for fluid mechanics, where finite diﬀerence discretization methods are still important. Finite volume methods, which directly address the discretization of conservation laws, are important in diﬃcult problems of fluid mechanics, for example high-Reynolds gas dynamics. Spectral methods are based on transforms that map space and/or time dimensions to domains (for example, the frequency domain) where the problem is easier to solve. A recent newcomer to the scene is the brood class of meshfree methods. These combine techniques and tools of finite element methods, such as variational formulation and interpolation, with finite diﬀerence features, such as non-local support. 12 1.2- Analysis process by the finite element method Figure 1.3: Model of a Ford Crown Victoria for performing crashworthiness simulations: (a) physical system; (b) discretization process; (c) discrete model; (d) subdomain subdivision (http://www.arasvo.com/crown_victoria/crown_vic.htm) 1.2.2 Analysis process by the finite element method Processes using the Finite Element Method involve carrying out a sequence of steps in some way. This sequence takes two canonical configurations, depending on (i) the environment in which FEM is used and (ii) the main objective of the analysis: model-based simulation of physical systems, or numerical approximation to mathematical problems. Both are reviewed below to introduce terminology used in the sequel. Historically the model based was the first one to be developed to model complex physical systems. The second came later and, among other things, provided the necessary theoretical underpinnings to extend the method beyond structural analysis. A glance at the schematics of a train wreck shown in Figure 1.1 makes obvious the reasons behind the necessity of simplifications. There is no simple diﬀerential equation that captures, at a continuum mechanics level, the train structure, coal, cargo, and passengers eating dinner [Felippa, 2004]. There is no reason for despair, however. The time honored divide and conquer strategy 1 , coupled with abstrac1 Divide et impera : the phrase is attributed to Philip II, king of Macedon (382-336 BC), describing his policy toward the Greek city-states. In computer science, divide and conquer (D&C) is an important algorithm design paradigm. It works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. A divide and conquer algorithm is closely tied to a type of recurrence relation between functions of the data in question; data is "divided" into smaller portions and the result calculated thence. Chapter 1 13 tion, comes to the rescue. First, separate the structure out and view the rest as masses and forces, most of which are time-varying and nondeterministic. Second, consider the train structure as built of substructures (a part of a structure devoted to a specific function): wheel, wheel bars, steam boiler, coal box and so on. Take each substructure, and continue to break it down into components: rings, ribs, spars, cover plates, etc., continuing through as many levels as necessary. Figure 1.4: Divide and conquer (divide et impera) strategy: (a) physical system; (b) idealization process Eventually those components become suﬃciently simple in geometry and connectivity that they can be reasonably well described by the continuum mathematical models provided, for instance, by Mechanics of Materials or the Theory of Elasticity. At that point, stop. The component level discrete equations are obtained from a finite element method library based on the mathematical model. The system model is obtained by going through the reverse process: from component equations to substructure equations, and from those to the equations of the complete locomotive. This system assembly process is governed by the classical principles of Newtonian mechanics, which provide the necessary "component glue." The multilevel decomposition process is diagramed in Figure 1.4. 1.2.3 Interpretations of the finite element method As shown in the previous section there are two practicable way to study and use the finite element method. Model based simulation has been shaped by the 14 1.2- Interpretations of the finite element method discovery and extensive use of the method in the field of structural mechanics. This historical connection is reflected in the use of structural terms such as "stiﬀness matrix", "force vector" and "degrees of freedom". This terminology carries over to non-structural applications. The basic concept in the physical interpretation is the breakdown (that mean disassembly, tearing, partition, separation, decomposition) of a complex mechanical system into simpler, disjoint components called finite elements, or simply elements. The mechanical response of an element is characterized in terms of a finite number of degrees of freedom. These degrees of freedoms are represented by the values of the unknown functions in a set of node points. The element response is defined by algebraic equations constructed from mathematical or experimental arguments. The response of the original system is considered to be approximated by that of the discrete model constructed by connecting or assembling the collection of all elements. The breakdown-assembly concept occurs naturally when an engineer considers many artificial and natural systems. For example, it is easy and natural to visualize a building, bridge, or a metallic frame as being fabricated from simpler parts. As discussed in the previous paragraph, the underlying theme is divide and conquer. If the behavior of a system is too complex, the recipe is to divide it into more manageable subsystems. If these subsystems are still too complex the subdivision process is continued until the behavior of each subsystem is simple enough to fit a mathematical model that represents well the knowledge level the analyst is interested in. In the finite element method such "primitive pieces" are called elements. The behavior of the total system is that of the individual elements plus their interaction. A key factor in the initial acceptance of the FEM was that the element interaction can be physically interpreted and understood in terms that were eminently familiar to structural engineers. On the other hand there are the mathematical interpretations. The method is viewed as a procedure for obtaining numerical approximations to the solution of boundary value problems (BVPs) posed over a domain. This domain is replaced by the union of disjoint subdomains called finite elements. The unknown function (or functions) is locally approximated over each element by an interpolation formula expressed in terms of values taken by the function(s), and possibly their derivatives, at a set of node points generally located on the element boundaries. The states of the assumed unknown function(s) determined by unit node values are called shape functions. The union of shape functions "patched" over adjacent elements form a trial function basis for which the node values represent the generalized coordinates. The trial function space may be Chapter 1 15 inserted into the governing equations and the unknown node values determined by the Ritz method (if the solution extremizes a variational principle) or by the Galerkin, least-squares or other weighted-residual minimization methods if the problem cannot be expressed in a standard variational form. 1.2.4 Historical sketch This historical sketch is intended to outline an ideal common way about the finite elements method, by thinking back the experience of the most famous author. A first idea about finite element method may be associate to method of exhaustion developed by Eudoxus from Cnido (408—355 A.C.) (a very far-oﬀ start point) calculating an area by approximating it by the areas of a sequence of polygons. J.W. Strutt (Lord Rayleigh) Walter Ritz Lately Archimedes from Syracuse (287-212 A.C.) generalized the method and went on to use Eudoxus’s method of exhaustion to prove a remarkable collection of theorems. A first important contribution was brought by Lord Rayleigh, whose real name was John William Strutt, after revised by W. Ritz in 1909. He developed an eﬀective method [Ritz, 1909] for the approximate solution of problems in the mechanics of deformable solids. It includes an approximation of energy functional by the known functions with unknown coeﬃcients. Minimization of functional in relation to each unknown leads to the system of equations from which the unknown coeﬃcients may be determined. One of the main restrictions in the Ritz method is that functions used should satisfy to the boundary conditions of the problem. In 1910 Richardson [Richardson, 1910] published the first paper for practical computations utilizing finite diﬀerences and which presents a general procedure by which errors can be estimated. The original Richardson method starts from the fact that errors in the solution of any finite diﬀerence scheme usually depend of some power of the mesh size used (or today the size of finite elements). The order of the error is thus usually given and depends as well on the solution method adopted. In 1915 Boris Grigorievich Galerkin published an article, in which he put forward an idea of diﬀerential equations boundary problems approximate solution method. He has applied his method to a big number of pivots and plates analysis problems. Some time before I.G. Bubnov developed a similar approach for the variational problems solution, which he interpreted as a variant of the Ritz method algorithm. The Galerkin method (or Bubnov-Galerkin method) with Galerkin’s (or "weak") 16 1.2- Historical sketch diﬀerential equations problem statement form are known all over the world. Boris Grigorievich Galerkin Richard Courant In 1943 Richard Courant considerably increased the possibilities of the Ritz method by introducing of the special linear functions defined over triangular regions and applied the method for the solution of torsion problems [Courant, 1943]. The values of functions in the node points of triangular regions were chosen as unknowns. Thus, the main restriction of the Ritz functions - a satisfaction to the boundary conditions was eliminated. The name "finite element method" was not due to Courant, however, but appears only in the 1960s. The Ritz method together with the Courant modification is similar with the method proposed independently by Clough many years later introducing for the first time in 1960 the term "finite element" in the paper "The finite element method in plane stress analysis" [Clough, 1960]. Clough baptized the method and went on to form at Berkeley the first research group that expanded the idea into Civil Engineering applications. John H. Argyris Ray Clough Edward L. Wilson Now, for exposition convenience, structural "finitelementology" may be divided into four generations that span 10 to 15 years each, as suggested by C. Felippa [Felippa, 2004]. The First generation, which spans 1950 through 1962, formed by the pioneers, starts from the coming of word finite element method. The 1956 paper by Turner, Clough, Martin and Topp [Turner et al. , 1956], henceforth abbreviated to TCMT, is recognized as the start of the current FEM, as used in the overwhelming majority of commercial codes. Indeed the finite element procedure as we know it today has its origin in this work, which diﬀers from the previous concepts in establishing arbitrary-shaped elements directly as pieces of the original domain. The original process suggested by them assumed specific stress distribution in each element equilibrating nodal forces. The main reason of wide spreading of FEM in 1960 is the possibility to use computers for the big volume of computations required by FEM. However, Courant Chapter 1 17 did not have such possibility in 1943, in fact he employ his students to compute the calculus of its methods: by disposing the students on a mesh, so that each students working as a node can spread the calculus, he can compute the simple example presented in his work. It interesting to note that each student was paid proportionally to each calculus. Finally, it should be remembered that an important contribution was brought into FEM development by the papers of Argyris [Argyris, 1954]-[Argyris, 1955]. The pioneers were structural engineers, schooled in classical mechanics. They followed a century of tradition in regarding structural elements as a device to transmit forces. This "element as force transducer" was the standard view in precomputer structural analysis. It explains the use of flux assumptions to derive stiﬀness equations in TCMT. Element developers worked in, or interacted closely with, the aircraft industry. Accordingly they focused on thin structures built up with bars, ribs, spars, stiﬀeners and panels. Although the Classical Force Method dominated stress analysis during the 1950s, stiﬀness methods were kept alive by use in dynamics and vibration. It is a pity that, not being an engineer, Courant did not link the idea with networks of discrete elements and that, for this reason maybe, his work had to lie in obscurity for so many years. Olgierd C. Zienkiewicz Robert L. Taylor The next period, second generation, spans the golden age of FEM: 1962-1972. This is the variational generation. Melosh showed that conforming displacement models are a form of Rayleigh-Ritz based on the minimum potential energy principle. This influential paper marks the confluence of three lines of research: Argyris’s dual formulation of energy methods, the Direct Stiﬀness Method (DSM) of Turner, and early ideas of interelement compatibility as basis for error bounding and convergence. From 1962 onward a two-step interpretation emerges: discrete elements approximate continuum models, which in turn approximate real structures. In 1963 E. L. Wilson and R. Clough developed, using the FORTRAN language, the SMIS software (Symbolic Matrix Interpretive System) based on the matrix analysis, for static and dynamic structural analysis. After, in 1969, Wilson was responsible for the development of several computer programs which are extensively used by professionals in the Civil, Mechanical and Aerospace engineering. The general three-dimensional finite element analysis program SAP and the TABS (after called ETABS) series of programs for the static and dynamic analysis of 18 1.2- Historical sketch three-dimensional building systems are examples of programs initially developed by Wilson. He was the first to develop computational methods and practical computer programs based on matrix notation. Bruce Irons John Barlow Olek Zienkiewicz, originally an expert in finite diﬀerence methods who learned the trade from Southwell, was convinced in 1964 by Clough to try FEM. He went on to write with Cheung the first textbook [Zienkiewicz & Cheung, 1967] on the subject and called "The finite element method in structural and continuum mechanics". This book presents the broad interpretation of the method and its applicability to any general field problem. Zienkiewicz organized another important Civil Engineering research group at the University of Wales at Swansea. In 1967 John Barlow at Rolls-Royce Aero Engine Division, discovered the optimal stress sampling points, after usually called Barlow points, but he published his work only in 1976, (as reported by his dear friend Bruce Irons in [Irons & Ahmadn, 1980]). This was the first UK FE system to exploit features such as isoparametric elements, the front solution, eigenvalue economisation, etc. After, in 1968, B. Irons and O.C. Zienkiewicz presented the isoparametric formulation of finite element stiﬀness matrices and this work had an immediate and significant impact on the finite element research being conducted at Berkeley. Thomas J. Hughes Klaus J. Bathe Professor Taylor was the first to program this new formulation at Berkeley and to demonstrate the power of this new type of element formulation. In a short time software and structural code were available: STRESS (1964) [Fenves et al. , 1964], STRUDL (1966), ASTRAN (1969), SAP (1970) in the United States of America, SAMCEF (1965) and ASKA (1969) in Europe. Low order displacement models often exhibit disappointing performance. Thus there was a frenzy to develop higher order elements. Other variational formulations, notably hybrids, mixed and equilibrium models emerged. In this second generation can be placed the monograph of Strang and Fix [Strang & Fix, 1973], the first book to focus on the mathematical Chapter 1 19 foundations. A third generation, based on consolidation of the method, put substantial eﬀort into improving the stock of second generation displacement elements by tools initially labeled variational crimes, but later justified. Textbooks by Hughes [Hughes, 1987] and Bathe [Bathe, 1996] reflect the technology of this period. Hybrid and mixed formulations record steady progress. Assumed strain formulations appear. The residual based error estimators were introduced by the pioneering work of Babuška and Rheinboldt in 1978 [Babuška & Rheinboldt, 1978], and further developed by many other authors. In fact, during the early 1980s the search for eﬀective adaptive methods led to a wide variety of ad hoc error estimators. Many of these were based on a priori or interpolation estimates, that provided crude but eﬀective indications of features of error suﬃcient to drive adaptive processes. A large activity in error estimation and mesh adaptivity is fostered by better understanding of the mathematical foundations. Ivo M. Babuška J. Tinsley Oden Commercial FEM codes gradually gain importance. They provide a reality check on what works in the real world and what does not. By the mid-1980s there was gathering evidence that complex and high order elements were commercial flops. Exotic gadgetry interweaved amidst millions of lines of code easily breaks down in new releases. Complexity is particularly dangerous in nonlinear and dynamic analyses conducted by novice users. A trend back toward simplicity starts. The fourth generation begins by the early 1980s. More approaches come on the scene, notably the Free Formulation, orthogonal hourglass control, Assumed Natural Strain methods, stress hybrid models in natural coordinates, as well as variants and derivatives of those approaches. Although technically diverse the fourth generation approaches share two common objectives: 1. elements must fit into DSM-based programs since that includes the vast majority of production codes, commercial or otherwise; 2. elements are kept simple but should provide answers of engineering accuracy with relatively coarse meshes (these were collectively labeled "high performance elements" in 1989). Zienkiewicz and Zhu in 1987 developed a simple error estimation technique that is eﬀective for some classes of problems and types of finite element approximations. 20 1.2- Historical sketch Gradients of solutions obtained on a particular partition are smoothed and then compared with the gradients of the original solution to assess error. This four generations reflects so the main steps done by scientist until 1990s. After, several develops have been done. In particular the dedication posed in adaptive procedures is to be noted. Actually, with the coming of faster computers, the adaptive finite element methods have become an important tool for the analysis of engineering structures,.and a very important feature of any adaptive code is a posteriori error estimation, i.e. the ability of the code to estimate the error in the computed engineering quantity of interest. By the early 1990s the basic techniques of a posteriori error estimation were established. Briefly two types of procedures are currently well-known, by engineering point of view, for deriving error estimators. They are either Residual based or Recovery based. Here, substantial progress was made in 1993 with the introduction of so called residual equilibration by Ainsworth and Oden [Ainsworth & Oden, 1993]. B. Boroomand Mark Ainsworth On the other hand, the recovery based error estimators are more recent, having been first introduced by Zienkiewicz and Zhu in 1987 [Zienkiewicz & Zhu, 1987]. Again, these were extensively improved by the introduction of new recovery processes. Here, in particular, the so called SPR (or Superconvergent Patch Recovery) method introduced in 1992 by the same authors [Zienkiewicz & Zhu, 1992c]-[Zienkiewicz & Zhu, 1992a]-[Zienkiewicz & Zhu, 1992b] has produced a very significant improvement of performance of the Recovery based methods. Many others attempted further improvement [Wiberg & Abdulwahab, 1993]-[Blacker & Belytschko, 1994] but the simple procedure originally introduced remains still most eﬀective. A viable recovery alternative, known by the acronym REP (Recovery by Equilibrium in Patches), was first presented by Boroomand and Zienkiewicz [Boroomand & Zienkiewicz, 1997b] and later improved by same authors [Boroomand & Zienkiewicz, 1997a]. In 2002 a new eﬀective stress recovery procedure, called RCP (Recovery by Compatibility in Patches), and based on complementary energy has been presented by Ubertini in [Ubertini, 2004]. This procedure can be viewed as dual to REP procedure in a certain sense. So we reached now the present days and the error estimation topic. We have quickly flown over the later improvements of finite element procedures, because Chapter 1 21 this will be the main topic of the present work, and will be discussed widely in the next chapters. 1.3 The main focus of this work Nowadays the combination of this powerful modern computers with eﬀective numerical procedures, particularly finite element method, have transformed what were once purely qualitative theories of mechanics and physics into eﬀective bases for simulation of physical phenomena important in countless engineering applications and scientific predictions. No matter how sophisticated and appropriate the mathematical models as an event, all computational results obtained using them involve numerical error. So the presence of numerical error in calculations has been a principal source of concern since the beginning of computer simulations of physical phenomena. The discretization process of transforming a continuum model of physical phenomena into one manageable by digital computers cannot capture all of the information embodied in models characterized by partial diﬀerential equations or integral equations. What is the approximation error in such simulations? How can the error be measured, controlled, and eﬀectively minimized? These questions have confronted computational mechanicians, practitioners, and theorists alike since the earliest applications of numerical methods to problems in engineering, and science. The reliability of computer simulations has become one of the most critical subjects in computational science. Lately, much attention has been posed on procedures aimed at improving accuracy of finite element solutions. Recovered solutions can be directly used for most practical purposes and, in particular, for evaluating the finite element discretization errors, that is a verification of solution. The subject of a posteriori error estimation for finite element approximation has now reached maturity. The emphasis has now shifted from the development of new techniques to the study of robustness of existing estimators and identifying limits on their performance. This work is intended to provide a contribution to give an answer to these questions through a study of error estimation for finite element approximations of boundary value problems, from a priori to a posteriori error estimation.In particular the work put particular care on a recent a posteriori error estimator, RCP procedure, and the role that this can assume as an independent measure of the quality of the simulation under study. Actually solution verification of a computational model must, in general, be based on a posteriori estimates of the error. By definition, methods of a posteriori error estimation are post-processing techniques 22 1.3- The main focus of this work that attempt to determine quantitative information on the actual numerical error in computed approximations of the target quantities of interest. Eﬀective methods for such calculations exist and are used with increasing frequency in solution verification processes. F. Ubertini S. de Miranda G. Castellazzi We expect to obtain quantitative information on system performance and outputs, and this expectation put stringent demands on the sophistication and accuracy of computer simulations. Chapter 2 Finite element analysis In questo capitolo vengono illustrati gli aspetti fondamentali del metodo degli elementi finiti, presentato il problema modello (assunto lineare) e introdotta la definizione di errore di discretizzazione. Tutto ciò getta le basi per la stima dell’errore, la quale può essere aﬀrontata a priori, ovvero prima di eseguire l’analisi per elementi finiti, oppure a posteriori, ovvero dopo aver eseguito l’analisi. Tali approcci verranno aﬀrontati separatamente nei capitoli successivi. 2.1 The basic ideas of the finite element method The finite element method is a computational technique for obtaining approximate solutions to the partial diﬀerential equations that arise in scientific and engineering applications. Rather than approximating the partial diﬀerential equation directly as with, e.g., the finite diﬀerence methods, the finite element method utilizes a variational problem that involves an integral of the diﬀerential equation over the problem domain. The finite element method can be described in a few words. Suppose that the problem to be solved is in variational form, it may be required to find the function which minimizes a given expression of potential energy. This minimizing property leads to a diﬀerential equation (the Euler equation), but normally an exact solution is impossible and some approximation is necessary. The Rayleigh-RitzGalerkin idea is to choose a finite number of trial functions and among all their linear combinations to find the one which is minimizing. This is the Ritz approxi- 24 2.1- The basic ideas of the finite element method mation. The unknown weights are determined, not by a diﬀerential equation, but by a system of discrete algebraic equations which the computer can handle. The theoretical justification for this method is simple, and compelling: the minimizing process automatically seeks out the combination which is closest to the solution. Therefore, the goal is to choose trial functions which are convenient enough for the potential energy to be computed and minimized, and at the same time general enough to approximate closely the unknown solution. The real diﬃculty is the first one, to achieve convenience and computability. In theory there always exists a set of trial functions which is complete, their linear combinations fill the space of all possible solutions as the trial functions increases, and therefore the Ritz approximations converge, but to be able to compute with them is another matter. This is what finite elements have accomplished. The underlying idea is simple. It starts by a subdivision of the structure, or the region of physical interest, into smaller pieces. These pieces must be easy for the computer to record and identify; they may be triangles or quadrilateral. Then within each piece the trial functions are given an extremely simple form (normally they are polynomials). Boundary conditions are infinitely easier to impose locally, along the edge of a triangle or quadrilateral, than globally along a more complicated boundary. The accuracy of the approximation can be increased, if that is necessary, but not by the classical Ritz method of including more and more complex trial functions. Instead, the same polynomials are retained, and the subdivision is refined. The computer follows a nearly identical set of instructions, and just takes longer to finish. In fact, a large scale finite element system can use the power of the computer, for the formulation of approximate equations as well as their solution, to a degree never before achieved in complicated physical problems. The method was created by structural engineers, and it was not recognized at the start as an instance of the Rayleigh-Ritz principle. The subdivision into simpler pieces, and the equations of equilibrium and compatibility between the pieces, were initially constructed on the basis of physical reasoning. The later development of more accurate elements happened in a similar way; it was recognized that increasing the degree of the polynomials would greatly improve the accuracy, but the unknowns computed in the discrete approximation have always retained a physical significance. In this respect the computer output is much easier to interpret than the weights produced by the classical method. The whole procedure became mathematically respectable at the moment when the unknowns were identified as the coeﬃcients in a Ritz approximation and the discrete equations were seen to be exactly the conditions for minimizing the po- Chapter 2 25 tential energy. The eﬀect was instantly to provide a sound theoretical basis for the method. As the techniques of constructing more refined elements have matured, the underlying theory has also begun to take shape [Strang & Fix, 1973]. The fundamental problem is to discover how closely piecewise polynomials can approximate an unknown solution. In other words, we must determine how well finite elements, which were developed on the basis of computational simplicity, satisfy the second requirement of good trial functions, to be eﬀective in approximation. Intuitively, any reasonable function can be approached to arbitrary accuracy by piecewise linear functions. The mathematical task is to estimate the error as closely as possible and to determine how rapidly the error decreases as the number of pieces (or the degree of the polynomial within each piece) is increased. 2.1.1 Variants of the finite element method In the analysis of problems of a discrete nature, a standard methodology has been developed over the years. The civil engineer, dealing with structures, first calculates force-displacement relationships for each element of the structure and then proceeds to assemble the whole by following a well-defined procedure of establishing local equilibrium at each node or connecting point of the structure. The resulting equations can be solved for the unknown displacements. Similarly, the electrical or hydraulic engineer, dealing with a network of electrical components (resistors, capacitances, etc.) or hydraulic conduits, first establishes a relationship between currents (flows) and potentials for individual elements and then proceeds to assemble the system by ensuring continuity of flows. The term Finite Element Method actually identifies a broad spectrum of techniques that share common features. Two subclassifications that fit well applications to structural mechanics are: 1. FEM Formulation: (a) Displacement; (b) Equilibrium; (c) Mixed; (d) Hybrid; 2. FEM Solution: (a) Stiﬀness; (b) Flexibility; (c) Mixed; The existence of a unified treatment of standard discrete problems leads us to the first definition of the finite element process as a method of approximation to continuum problems such that 1. the continuum is divided into a finite number of parts (elements), the behaviour of which is specified by a finite number of parameters, 2. the solution of the complete system as an assembly of its elements follows precisely the same rules as those applicable to standard discrete problems. It will be found that most classical mathematical approximation procedures as 26 2.1- Discretization error well as the various direct approximations used in engineering fall into this category. 2.1.2 Discretization of the domain Since the problem is usually defined over a continuous domain, the governing equations, with the exception of the essential boundary conditions, are valid for the entirety of, as well as for any portion of, that domain. This allows idealization of the domain in the form of interconnected finite-sized domains (elements) of diﬀerent size and shape (see Figure 2.1). By doing this, certain approximations are introduced (e.g., cutting the corners, making curved lines straight and curved elements flat). Putting enough numbers of nodes between the elements (higherorder elements, etc.) also comes into the picture at this stage of the method. Here, one should be concerned with how well the idealized discrete domain represents the actual continuous domain. To a certain extent, it is true that the smaller elements (finer mesh) produce better results. But it is also true that the finer mesh results in a larger number of equations to be solved. The question then arises: What is the most eﬃcient element type, size, and pattern? A partial answer to this question is given in the literature under the key word modeling. Adaptive processes or mesh refinements and automatic mesh generation are also techniques relevant to the discretization of the domain. Figure 2.1: Discretization of a domain Ω 2.1.2.1 Discretization error The real issue for us to grapple with now is that the computational model prepared to simulate the mathematical model may be faulty and can lead to errors. In the process of replacing the continuum region by finite elements, errors originate in Chapter 2 27 many ways. From physical intuition, we can argue that this will depend on the type, shape and number of elements that we use, the grading or density of the mesh used, the way distributed loads are assigned to nodes, the manner in which boundary conditions are modelled by specification of nodal degrees of freedom etc. These are the discretization errors that can occur. We can assume that the approximation for the displacement u will yield the exact solution in the limit as the size h of elements decreases. The arguments for this are simple: if the expansion is capable, in the limit, of exactly reproducing any displacement form conceivable in the continuum, then, as the solution of each approximation is unique, it must approach, in the limit of h → 0, the unique exact solution. In some cases the exact solution is indeed obtained with a finite number of subdivisions (or even with one element only) if the approximation used in that element fits exactly the correct solution. Thus, for instance, if the exact solution is of the form of a quadratic polynomial and the shape functions include all the polynomials of that order, the approximation will yield the exact answer. The last argument helps in determining the order of convergence of the finite element procedure as the exact solution can always be expanded in the vicinity of any point (or node) i of coordinates x0 as a polynomial: u (x) = u (x0 ) + ∇u|x=x0 (x − x0 ) + . . . (2.1) If within an element of size h, a polynomial expansion of degree p is employed for u, this can fit locally the Taylor expansion up to that degree and, as (x − x0 ) is of the order of magnitude h, the error in u will be of the order hp+1 . 2.1.3 Identification of the state variable(s) Until this step, no reference has been made to the physical nature of the problem. Whether it is a heat-transfer problem, fluid- or solid-mechanics problem, etc., comes into the picture at this stage. The mathematical description of steadystate physical phenomena, for instance, leads to an elliptic boundary-value problem in which the formula contains the state variable and the flux. These variables are related to each other by a constitutive equation representing a mathematical expression of a particular physical law. Table 2.1 presents various physical problems with associated state variables and constitutive equations. Once the state variable and the flux have been identified, the formulation can take place containing either or both. The choice is usually dictated by the problem. 2.1.4 Formulation of the problem Posing the problem to be solved in its most general terms we find that we seek an 28 2.1- Formulation of the problem Table 2.1: Classification of various physical problems Physical Problem State variable Flux Constitutive Equation Elasticity Torsion Heat Transfer Fluid Flow Flow through porous media Electrostatics Magnetostatic Displacements or forces Warping function Temperature Velocity Hydraulic head Electric potential Magnetic potential Stress or strain Rate of twist Heat Flux Shear stress Flow rate Electric flux Magnetic Flux Hooke’s law Hooke’s law Fourier’s law Stokes’s law Darcy’s law Couloumb’s law Maxwell’s law unknown function u such that it satisfies a certain diﬀerential equation set ⎫ ⎧ ⎪ ⎬ ⎨ A1 (u) ⎪ A2 (u) = 0, A (u) = ⎪ ⎪ .. ⎭ ⎩ . (2.2) in a domain (volume, area, etc.) Ω (Figure 2.2), together with certain boundary conditions ⎫ ⎧ ⎪ ⎬ ⎨ B1 (u) ⎪ B2 (u) = 0, (2.3) B (u) = ⎪ ⎪ .. ⎭ ⎩ . on the boundaries ∂Ω of the domain. The function sought may be a scalar quantity or may represent a vector of several variables. Similarly, the diﬀerential equation may be a single one or a set of simultaneous equations and does not need to be linear. It is for this reason that we have resorted to matrix notation in the above. The finite element process, being one of approximation, will seek the solution in the approximate form n X u ≈ uh = Ui di = Ud, (2.4) i=1 where Ui are shape functions prescribed in terms of independent variables (such as the coordinates x1 , x2 , etc.) and all or most of the parameters di are unknown. As the set of diﬀerential equations (2.2) has to be zero at each point of the domain Ω, it follows that Z Z vT A (u) dV ≡ [v1 A1 (u) + v2 A2 (u) + . . .] dV = 0, (2.5) Ω Ω Chapter 2 29 Figure 2.2: Problem domain Ω and boundary ∂Ω where ⎫ ⎧ ⎪ ⎬ ⎨ v1 ⎪ v2 , v= ⎪ ⎭ ⎩ .. ⎪ . (2.6) is a set of arbitrary functions equal in number to the number of equations (or components of u) involved. The statement is, however, more powerful. We can assert that if (2.5) is satisfied for all v, then the diﬀerential equations (2.2) must be satisfied at all points of the domain. The proof of the validity of this statement is obvious if we consider the possibility that A(u) 6= 0 at any point or part of the domain. Immediately, a function v can be found which makes the integral of (2.5) non-zero, and hence the point is proved. If the boundary conditions are to be simultaneously satisfied, then we require that Z Z v̄T B (u) dS ≡ [v̄1 B1 (u) + v̄2 B2 (u) + . . .] dS = 0, (2.7) Ω Ω for any set of functions v̄. Indeed, the integral statement that Z Z vT A (u) dV + v̄T B (u) dS = 0, Ω (2.8) ∂Ω is satisfied for all v and v̄ is equivalent to the satisfaction of the diﬀerential equations (2.2) and their boundary conditions (2.3). In the above discussion it was implicitly assumed that integrals such as those in Equation (2.8) are capable of being evaluated. This places certain restrictions on the possible families to which the functions v or u must belong. In general we shall seek to avoid functions which result in any term in the integrals becoming infinite. 30 2.1- Formulation of the problem Thus, in Equation (2.8) we generally limit the choice of v and v̄ to bounded functions without restricting the validity of previous statements. What restrictions need to be placed on the functions? The answer obviously depends on the order of diﬀerentiation implied in the equations A(u) [or B(u)]. Consider, for instance, a function u which is continuous but has a discontinuous slope in a particular direction. We imagine this discontinuity to be replaced by a continuous variation in a very small distance. It is easy to see that although the first derivative is not defined here, it has finite value and can be integrated easily, but the second derivative tends to infinity. This therefore presents diﬃculties if integrals are to be evaluated numerically by simple means, even though the integral is finite. If such derivatives are multiplied by each other the integral does not exist and the function is known as non-square integrable. Such a function is said to be C 0 continuous. In a similar way it is easy to see that if nth-order derivatives occur in any term of A or B then the function has to be such that its n − 1 derivatives are continuous (C n−1 continuity). On many occasions it is possible to perform an integration by parts on Equation (2.8) and replace it by an alternative statement of the form Z Z T T A (v) A (u) dV + B (v̄) B (u) dS = 0. (2.9) Ω ∂Ω In this, the operators A, A, B and B usually contain lower order derivatives than those occurring in operators A or B. Now a lower order of continuity is required in the choice of the function u at a price of higher continuity for v and v̄. The statement (2.9) is now more permissive than the original problem posed by Equations (2.2), (2.3), or (2.8) and is called a weak form of these equations. It is a somewhat surprising fact that often this weak form is more realistic physically than the original diﬀerential equation which implied an excessive smoothness of the true solution. Integral statements of the form of (2.8) and (2.9) will form the basis of finite element approximations. The finite element method is one particular Galerkin method, named after the Russian engineer Galerkin. It is a method for finding approximate solutions to partial diﬀerential equations. It is closely related to the Rayleigh-Ritz method which involves choosing functions (a basis) for the solution and finding the amplitude of each function by minimizing the energy. The Galerkin method is however more general, being able to solve a greater range of problems. The essence of the Galerkin method involves taking the weak form of the governing Equation (2.8), and finding the best solution to a problem given a collection of functions. First, consider the approximate solution to some problem, uh ∈ S h , where Chapter 2 31 S h is a finite-dimensional space. This means that there is a limited number of possibilities. For example, uh could be a combination of low order polynomial functions. The Galerkin problem for Equation (2.9) involves: find uh ∈ S h such thatZ Z ¡ ¢T ¡ ¢ ¡ ¢T ¡ ¢ ∀vh ∈ V h , A vh A uh dV + B vh B uh dS = 0, (2.10) Ω ∂Ω where V h is a finite dimensional space. The Galerkin method (more specifically, the Bubnov-Galerkin method) requires that the weight and trial functions, vh and uh respectively, come from the same finite-dimensional space, taking into account the special requirements on the weight and trial functions where Dirichlet boundary conditions are applied. In PetrovGalerkin method, the weight functions come from a diﬀerent function space than the trial functions. This method is used for special applications, often in fluid mechanics. Diﬀerent Galerkin-based methods are defined by how the unknown field uh is represented. In the finite element method, uh and vh will be simple continuous, piecewise low-order polynomials defined on finite elements. Spectral Galerkin methods for example use a truncated Fourier series as the basis. A basic question which arises when computing an approximate solution is how uh relates to the exact solution u. Given a finite number of possibilities in S h , which solution does the method seek? Understanding this requires some basic error analysis. The error analysis will tell how the computed solution uh diﬀers from the actual solution u and why the Galerkin method works (or for problems not considered here, why it does not work). This is examined in the following chapter. 2.2 The model problem Consider the model problem governed by the following equations: D∗ σ = b, (2.11) = C², (2.12) ² = Du, (2.13) σ in Ω, together with the following boundary conditions u = ū on ∂Ωu , NT σ (2.14) = t̄ on ∂Ωt . (2.15) ∗ In the above equations, u, ² and σ are the unknown vectors, D and D are linear diﬀerential operators, C is a symmetric positive definite matrix of physical 32 2.2- The model problem parameters, b is a prescribed source term, N is a matrix which contains the direction cosines of the outward unit normal vector on ∂Ω, Ω is the domain and ∂Ω is the domain boundary. Operators D and D∗ are assumed to be related by the Gauss-Green formula Z Z Z vT D∗ w dV + wT Dv dV = Ω Ω vT NT w dS, (2.16) ∂Ω where v and w are vectors of suﬃciently regular functions defined on Ω. Combining Equations (2.11)-(2.15), the boundary value problem can be put in the form A (u) ≡ D∗ CDu − b = 0, B (u) ≡ NT CDu − t̄ = 0, in Ω, (2.17) on ∂Ωt . (2.18) The model problem stated above is typical of many self-adjoint problems in engineering. To fix the ideas, we may refer to a problem of elasticity so that u can be interpreted as the displacement vector, ² as the strain vector, σ as the stress vector, D and D∗ as the compatibility and equilibrium operators and C as the elasticity matrix. Then, Equations (2.11)-(2.13) are the equilibrium equation, the constitutive equation and the compatibility equation, respectively. More details about the structural model used in the present work are given in Appendix A. Figure 2.3: Problem domain Ω and boundary ∂Ωt and ∂Ωu The model problem admits weak formulations in terms of stationary of single or multifield functionals. In elasticity, the variational formulation in terms of displacements stems from the principle of minimum potential energy Z Z Z 1 Π (u) = (Du)T CDudV − uT bdV − uT tdS, (2.19) 2 Ω Ω ∂Ωu Chapter 2 33 whose domain is the set of the functions u continuous in Ω, which satisfy the displacement boundary condition (2.14). Assuming the stress as an additional variable leads to the Hellinger-Reissner functional ¶ Z µ Z 1 T −1 T T ΠHR (u, σ) = σ Du − σ C σ + b u dV − uT tdS, (2.20) 2 Ω ∂Ωt whose domain is the set of functions u continuous in Ω, and functions σ fulfilling a priori the stress-strain relationship. If σ satisfies a priori the equilibrium equation (2.11), then the Hellinger-Reissner functional reduces to the hybrid stress functional Z Z 1 ΠHY (u, σ) = − σ T C−1 σdV − uT tdS. (2.21) 2 Ω ∂Ωt Functionals (2.19) and (2.20) are the variational supports in developing compatible and mixed finite element models, respectively. While functional (2.21) is the variational support in developing hybrid stress model as well as the support of the so called Equilibrium Model II [de Veubeke, 1965]—[Pian & Tong, 1972]. Let’s focus attention on the potential energy functional (2.19). Displacement u, is approximated over each element Ωe as follows u ' uh = U (x) ūh , (2.22) where U is the matrix of shape functions and d is the vector of nodal displacement values. Once nodal displacements are determined, stresses (or gradients) are usually computed by means of Equations (2.12) and (2.13): σ h = CBd, (2.23) where B = DU. (2.24) h It is well known that finite element stresses σ are in general discontinuous at inter-elements and have a low accuracy at nodes and element boundaries. The objective of a stress recovery procedure is to compute improved stresses by postprocessing the finite element solution. In the case of the mixed and hybrid finite element models based on the HellingerReissner’s functional (2.20) and the hybrid functional (2.21), respectively, u is represented, over each element, according to Equation (2.22), and stress σ is represented as σ h (x) = S (x) s, where S is a matrix of square integrable functions and s a vector of unknown stress parameters. 34 2.3 2.3- Definitions of error Definitions of error Before proceeding further it is necessary to define what we mean by error. In principle, the error is the diﬀerence between the exact solution and the approximate one. This can apply to the basic function, such as displacement which we have called u and can be given as e = u − uh . (2.25) In a similar way, however, we could focus on the error in the strains ² or stresses σ (i.e., gradients of the solution), and define an error in those quantities as eε eσ = ² − ²h , (2.26) = σ−σ . (2.27) h The specification of local errors in the manner given in Equations (2.25)-(2.27) is not always convenient and occasionally misleading. For instance, under a point load both errors in displacements and stresses will be locally infinite but the overall solution may well be acceptable. Similar situations will exist near re-entrant corners where, as is well known, stress singularities exist in elastic analysis and gradient singularities develop in field problems. For this reason various norms representing some integral scalar quantity are often introduced to measure the error. We can define an energy norm written for the error as ∙Z ¸ 12 ¡ ¢ ¡ ¢ h T h kek = ²−² σ − σ dV , (2.28) Ω = ∙Z Ω = ∙Z Ω = ∙Z Ω ¡ ¢T ¡ ¢ σ − σ h C−1 σ − σ h dV ¸ 12 ¡ ¢T ¢ ¡ u − uh D∗ CD u − uh dV ¡ ¢T ¡ ¢ ² − ²h C ² − ²h dV ¸ 12 , ¸ 12 , (2.29) . (2.30) (2.31) and its relation to strain energy is evident. Other scalar norms can easily be devised. For instance, the L2 norm of displacement and stress error can be written as ¸ 12 ∙Z ¡ ¢ ¡ ¢ h T h u−u u − u dV kekL2 = , (2.32) Ω keσ kL2 = ∙Z Ω ¡ ¢T ¡ ¢ σ − σh σ − σ h dV ¸ 12 . (2.33) It is possible to evaluate root mean square (RMS) values of error. For instance, Chapter 2 35 the RMS error in displacement, ∆u, |∆u| = Ã 2 kekL2 Ω ! 12 . Similarly, the RMS error in stress, ∆σ, becomes for the domain Ω Ã !1 keσ k2L2 2 . |∆σ| = Ω (2.34) (2.35) Any of the above norms can be evaluated over the whole domain or over subdomains or even individual elements. We note that nelem X 2 kek = kek2e , (2.36) e=1 where e refers to individual elements Ωe such that their sum (union) is Ω. We note further that the energy norm given in terms of the stresses, the L2 stress norm and the RMS stress error have a very similar structure and that these are similarly approximated. To describe the behaviour of stress analysis problems we define the variation of the relative energy norm error (percentage) as kek × 100 (%) , η= (2.37) E where ¶ 12 µZ ²T C²dV , (2.38) E= Ω is the (exact) energy norm of the solution. At the end of this chapter is clear that the analysis of the finite element method lead to a natural definition of error, by the attempt to compare the approximated quantity to the exact one. It appears by this definition, that this comparison is not always possible, due on the fact that the exact or true solution does not exist for all problems. Still actually the necessity to quantify this error by using diﬀerent tools. Commonly two are the practicable ways: the a priori error estimation and the a posteriori error estimation. In the next two chapters this two complementary ways, that can sometime interact, will be shown. Chapter 3 A-priori error estimation Nella prima parte di questo capitolo vengono brevemente illustrati alcuni risultati classici sulla stima a priori dell’errore di discretizzazione in analisi agli elementi finiti. La seconda parte del capitolo si incentra, invece, sullo studio delle prestazioni che diversi elementi finiti mostrano in funzione del loro grado di approssimazione. In particolare, viene presentato un nuovo metodo di indagine [Castellazzi & Ubertini, 2004] per la valutazione del ruolo che le distorsioni di geometria hanno sulla prestazioni degli elementi finiti. L’indagine prende in esame elementi finiti parametrici piani, e ne quantifica la sensibilità alle distorsioni di geometria. L’indagine é inoltre accompagnata da un ’ampia sperimentazione numerica in cui si evidenzia come la progressiva distorsione della geometria influisca sulle prestazioni dell’elemento. Alcune considerazioni conclusive terminano il capitolo. 3.1 Analysis of the finite element method Most of the errors introduced by the finite element methods are diﬃcult to quantify analytically or determine in a logically coherent way. We can only rely on heuristic judgement (based on intuition and experience) to understand how best to minimize errors. However, we shall now look only at that category of discretization error that appears because the computational or discretized model uses trial functions which are an approximation of the true solution to the mathematical model. It seems possible that to some extent, analytical quantification of these errors is possible. 38 3.1- Analysis of the finite element method At the same time, a large body of literature has been created by mathematical analysts using the concepts of variational calculus and functional analysis to derive projection theorems and energy error rules on an a priori basis. These were first reported more than thirty years ago by Strang and Fix in [Strang & Fix, 1973]. Unfortunately, these statements on error analysis of the finite element method are derived using rigorous mathematical abstractions which are sometimes diﬃcult for the engineer to grasp. It is indeed possible to re-derive these using the energy, virtual work and least action principles that the engineer or physicist is more familiar with, especially for linear elastostatics. As shown in the previous chapter, the finite element method is one particular Galerkin method, and finds approximate solutions for partial diﬀerential equations. The essence of the Galerkin method involves taking the weak form of the governing equation, as developed in the previous chapter, and finding the best solution to a problem given a collection of functions. It is interesting to check how the solution computed using the Galerkin procedure compares to the exact solution. Following the nomenclature used in Strang and Fix [Strang & Fix, 1973], we write the weak form in terms of the energy inner product for the exact solution u to the problem. Let be V a space of functions, which has an infinite number of members, and u ∈ V, then consider the following: Z Z T (Du) CDudV = uT bdV ∀u ∈ V, (3.1) Ω Ω where the first term of Equation (3.1) is a measure of the total elastic strain energy. The vector u describes the degrees of freedom and the second term of the Equation (3.1) describes the potential of the applied loads. This virtual work statement refers to the exact solution of the elastostatic problem: the trial function and test function are taken as u ∈ V and the virtual work argument establishes that (3.1) is truly satisfied only when u is the exact solution at the point of equilibrium. Consider now a subspace V h ⊂ V which has a finite number of members, then we can rewrite the Equation (3.1) as Z Z ³ ´T ¡ h ¢T u bdV ∀uh ∈ V h . (3.2) Duh CDudV = Ω Ω In (3.2), we take note of the fact that the test function uh ∈ V h need not be the exact displacement function for the virtual work principle to be true. For convenience, we take this to be the discrete finite element displacement field, as long as it is admissible (i.e. it satisfies all the geometric boundary conditions). By using uh for both the trial and test function, we get the actual finite element Chapter 3 39 equations, or we pose the Galerkin formulation, finding uh ∈ V h such that Z Z ³ ´T ¡ h ¢T u bdV ∀uh ∈ V h , Duh CDuh dV = Ω (3.3) Ω with the right hand side leading to the consistent load vector and the left hand side representing the stiﬀness matrix. This equation will now reflect the error due to the finite element discretization. We are now in a position to see how the error e = u − uh can be assessed. Comparing (3.2) and (3.3) and noting that the energy inner product is bilinear, we can arrive at Z ³ Z ³ ´T ´T h CDudV = ∀uh ∈ V h , (3.4) Du Duh CDuh dV Ω Ω and from this we obtain the projection, or Cea’s, theorem: Z ³ ´T ¢ ¡ Duh CD u − uh dV = 0 ∀uh ∈ V h . (3.5) Ω This important result is commonly known also as Galerkin orthogonality. This means that the approximate solution uh is a projection of the exact solution u onto the space of the weight functions. In the Bubnov-Galerkin, the weight functions vh come from the same space as the trial functions uh , hence the solution uh is the projection of the exact solution onto the finite dimensional space of trial functions. In simple linear elastostatics cases, this would imply that the strains or stresses are obtained in a best-fit sense and that there would be points in the element domain where these stresses or strains are very accurately computed. Consider the following: Z Z £ ¡ ¢¤T ¢ ¡ T D u − uh CD u − uh dV = (Du) CDudV + Ω Z ³ Ω ´T Z ³ ´T h h CDu dV − 2 Du Duh CDudV + Ω Ω Z ³ Z ´T T (Du) CDudV − Duh CDuh dV + = Ω ¸ ∙ZΩ ³ Z ³ ´T ´T h h h CDudV − CDu dV Du Du −2 Z ³Ω ZΩ ´T T (Du) CDudV − Duh CDuh dV + = Ω Ω Z ³ ´T ¢ ¡ h CD u − uh dV , Du − | Ω {z } =0 for any uh ∈ V h . So we get an energy error theorem which can be expressed as: Energy of the error = Error of the energy, ° ° ° ° °u − uh ° = kukE − °uh °E , E (3.6) (3.7) 40 3.1- Analysis of the finite element method where the subscript E indicates that the type of norm is in energy sense: Z ° ° £ ¡ ¢¤T ¢ ¡ °u − uh ° D u − uh = CD u − uh dV, E ZΩ T kukE = (Du) CDudV, ZΩ ³ ´T ° h° °u ° = Duh CDuh dV. E (3.8) (3.9) (3.10) Ω As the left hand side of (3.6) is always positive definite, we get the useful statement that: Z ³ Z ´T (Du)T CDudV. (3.11) Duh CDuh dV < Ω Ω Thus, in a variationally correct approach, the energy inner product of the approximate (Ritz or finite element) solution will always be a lower bound of the exact energy. From these statements, we can set simple estimates for the error in a finite element discretization and, therefore, for the rate of convergence of the solution. It can be shown that the Galerkin finite element method is optimal in terms of the energy. The finite element solution is therefore seen to be a best-fit or best approximation solution. This error analysis tells something of what the Galerkin method calculates. Given some approximate functions, the Galerkin method will yield the best fit to the exact solution in terms of energy. Consider the following: Z ° ° £ ¡ ¢¤T ¢ ¡ °u − uh + vh ° D u − uh + vh = CD u − uh + vh dV E ZΩ £ ¡ ¢¤T ¢ ¡ D u − uh + vh = CD u − uh dV + Ω Z £ ¡ ¢¤T ¢ ¡ D u − uh + CD u − uh + vh dV Z Ω £ ¡ ¢¤T ¢ ¡ D u − uh = CD u − uh dV + Ω Z Z ³ ´T £ ¡ ¢¤T D u − uh +2 CDvh dV + Dvh CDvh dV Ω | Ω {z } =0 h h for any v ∈ V . As seen before this last Equation was simplified by the Equation (3.5). Denoting now wh = uh − vh , the above result leads to the conclusion that: Z ° ° £ ¡ ¢¤T ¢ ¡ °u − uh ° D u − uh = CD u − uh dV, (3.12) E Ω Z ° ° £ ¡ ¢¤T ¢ ¡ °u − wh ° D u − wh = CD u − wh dV, (3.13) E Ω ° ° ° ° °u − uh ° ≤ °u − wh ° E E ∀wh ∈ V h . (3.14) Chapter 3 41 This implies that the solution uh is closer to u than any other element of V h in terms of energy. Given a choice of functions, the Galerkin method therefore chooses those which minimize the error in terms of the strain energy. There is a close relationship between the Rayleigh-Ritz method and the Galerkin method. It has been shown that the Galerkin method minimizes the error in terms of the energy, which is the principle behind the Rayleigh-Ritz method. For many problems in solid mechanics, the two are equivalent. 3.2 Convergence of the finite element method A priori error estimation does not provide quantitative information about error in a finite element simulation. What it does provide is the order of convergence. The order of convergence indicates how quickly the error reduces upon mesh refinement. The order depends on several factors. For finite element analysis, the order of convergence depends heavily on the type of element. We want to know how much smaller the error will be if we halve the element size (leading to at least a doubling in computational eﬀort). To give an indication of the size of the error e = u − uh , it is necessary to consider a norm of the error, kekn . The subscript n indicates the type of norm. The energy norm, which is the type of norm which will be used here, is given by: Ã n Z !1/2 X (Dα e) C (Dα e) dV , (3.15) kekn = α=0 Ω α where D denotes the α derivative. This scalar value is a measure of the magnitude of the error. Two norms of particular interest involve n = 0 and n = 1. For example, setting n = 0 gives: µZ ¶1/2 T kek0 = e CedV , (3.16) Ω which is a measure of the error in the displacements. Setting n = 1, µZ ³ ¶1/2 ´ T kek1 = , eT Ce + (De) CDe dV (3.17) Ω which now includes the operator D on e, thereby introducing a measure of the error in the strain. Consider the nodal interpolate uI , X uI = Ui ai . (3.18) i The nodal interpolate is equal to the exact solution at the nodes (u−uI = 0 at the nodes), hence it interpolates the exact solution using the provided shape functions. 42 3.2- Convergence of the finite element method Given that a finite solution is the best possible solution from the finite-dimensional space V h , it holds that ° ° kek ≤ °u − uI ° . E E This means that the finite element solution is at least as accurate as the nodal interpolate in terms of energy. In this way, the question of determining an error estimate is no longer a finite element question, rather a question which can be answered from interpolation theory. Using the Equation (3.5) we can state that: ° ° kekr ≤ α °u − uI °r , (3.19) where α is a positive constant. Interpolation theory provides the inequality ° ° °u − uI ° ≤ chk+1−m kuk , (3.20) E m where k is the polynomial order, c is an unknown constant, and h is a measure of the element size. Note that to ensure convergence it should be: k + 1 > m. Inserting now Equation (3.19) into the above expression, kekr ≤ Chk+1−r kukE , (3.21) where C is a constant, independent of h. For an elasticity problem, r = 1, hence kek1 ≤ Chk kukE . (3.22) This implies that the solution converges in this norm at a rate equal to the polynomial order. For practical purposes, this norm is dominated by the error in the strain, hence it represents the error in the energy. If the exact solution is not suﬃciently smooth, there maybe no gain in accuracy through increasing the polynomial order. It would be useful to find an estimate in terms of lower norms, such as the error in the displacements, kek0 . An estimate can be obtained using Nitsche’s trick (see [Strang & Fix, 1973] for details). It leads to the estimate: keks ≤ Chβ kukE , (3.23) where β = min {k + 1 − s, 2 (k + 1 − r)}. In elasticity, the two quantities of special interest are the displacements and the strains (and hence the stresses). For k ≥ 1, the displacement error is: kek0 ≤ Chk+1 kukE , (3.24) kek1 ≤ Chk kukE , (3.25) and the strain error is: which implies that the approximate strains converge to the exact result slower than Chapter 3 43 the approximate displacements. For linear elements k = 1. Therefore: kek0 ≤ Ch2 kukE , (3.26) kek1 ≤ Ch kukE . (3.27) ¡ 2¢ Hence linear elements are known as being O h in terms of displacements, and O (h) in terms of strains. The notation O (ha ) means that the error is of the order ha . Of course, as h becomes small, the greater a the smaller the error. Larger a implies faster convergence. 3.2.1 Richardson’s theorem This procedure may be applied to a sequence of displacement models only. Thus, for instance, if the displacement converges at O(h2 ) and we have two approximate solutions u1 and u2 obtained with meshes of size h and h2 , we can write, with u being the exact solution, ¡ ¢ O h2 u1 − u = ¡ ¢2 = 4 (3.28) u2 − u O h 2 From the above, an (almost) exact solution u can be predicted and, then, an estimation kuR kE of the exact strain energy can be obtained.. Another method to obtain an estimation of the exact strain energy of the structure is to perform dual analyses [de Veubeke, 1965] in which the same problem is solved by using both displacement models and equilibrium models. Then if the displacement boundary conditions are homogeneous and consistent loads are applied, a displacement model gives a lower bound to the exact strain energy while an equilibrium model gives an upper bound. Richardson’s extrapolation may then be applied to both sequences of displacement and equilibrium models so as to obtain two estimates which bound the exact strain energy. 3.3 Completeness requirements of shape functions Convergence of finite element results with the mesh refinement is an important requirement to be satisfied by any successful element. The continuity and completeness requirements (e.g., see, Zienkiewicz [Zienkiewicz & Taylor, 1989], Bathe [Bathe, 1996]) must be satisfied so as to ensure convergence. The continuity requirement demands that the shape functions must be chosen so that the assumed interpolation function of the field variable, the displacement in structural analysis, is suﬃciently continuous within and across the boundary of the elements. The completeness requirement demands that the shape functions must be such that the interpolation function is capable of representing rigid body motions and constant 44 3.3- Completeness requirements of shape functions strain states correctly. The continuity requirement is usually satisfied by ensuring that the shape functions (and hence the displacement model) are polynomials of an appropriate order. Typically if the stiﬀness integrands involve derivatives of order m, the shape functions must be at least C m continuous within the element and C m−1 continuous across the element interface. The completeness requirements are satisfied for a two dimensional isoparametric element, for examples if the element shape functions, Ui satisfy the conditions X Ui = 1, (3.29) i X Ui x1,i = x1 , (3.30) Ui x2,i = x2 , (3.31) i X i In the literature, the completeness requirements are frequently associated with the ability of the element to represent an arbitrary linear polynomial displacement field. For this reason, we will refer to these completeness requirements as the linear completeness requirements. In the limit of mesh refinement, the element size becomes so small that the displacement field within each element can be considered to be linear, and,. hence an element satisfying the linear completeness requirements is expected to converge to an exact solution without any problem. It was once believed that elements satisfying the continuity and completeness requirements would indeed, in the limit of mesh refinement, converge to the actual solution without any diﬃculty. However, there have been instances which have contradicted this belief. Locking problems (e.g. see, [Prathap, 1993] and [Wilson et al. , 1973]) and the deterioration of solution accuracy with geometric distortion are two examples. To avoid the problem of locking, in addition to the continuity and completeness requirements, certain other conditions have to be satisfied by the displacement model [Prathap, 1993]. Nonconforming elements (e.g. see, [Zienkiewicz & Taylor, 1989], [Wilson et al. , 1973] and [Taylor et al. , 1976]) which violate the inter-element continuity requirements have been used successfully. However, such formulations are sensitive to geometric distortion of the element, and the patch test (see [Irons & Razzaque, 1972]—[Irons & Ahmadn, 1980]) bas been introduced to cheek the ill eﬀects of the violation of inter-element continuity and completeness requirements in a global sense, i.e., for a patch. These observations are the source of motivation for several investigations into the continuity and completeness requirements. It is frequently inconvenient to refine the mesh successively in order to check Chapter 3 45 for convergence. As an alternative to successive mesh refinement, higher order elements are sometimes used. In the context of the higher order elements, however, the completeness requirements need to be interpreted carefully. These elements use a higher order interpolation model, and attempt to obtain convergence using fewer elements. With fewer elements, the elements are larger and. hence, the linear completeness requirements are no longer suﬃcient as they ensure convergence only in the limit of mesh refinement. Thus, for the eﬀective use of higher order elements, it is necessary to investigate the higher order completeness requirements of shape functions. 3.3.1 Constructing approximate functions for the elements Once the state variable(s) and the local coordinate system have been chosen, the state variable(s) can be approximated in numerous ways, i.e. diﬀerent choices for expression (2.4) are possibile. We mention here only the approximation in terms of algebraic functions. There are two entities that need to be approximated. The first is physical (the state variable) and the second is geometrical (the shape of the element). If the element is actually made of straight lines or planes, the coordinates of primary nodes (those at the extremes of the elements) will define the element shape accurately. In this case, the geometric approximation does not enter into the picture. Because of this, discretization of the entire domain is most often made by straight-line (linear) elements. For some problems, however, linear elements (i.e., flat elements in shells) may introduce unacceptable errors, and discretization must be done by using isoparametric elements (see Figure 3.1). A similar argument is, of course, valid for the approximation of the state variable. It can be approximated in the form of a linear function or a higher-order function (i.e., quadratic, cubic, etc.). The analyst then must decide whether to approximate physics (state variable) and geometry (element shape) equally or to give preference to one or the other in various regions of the domain. This leads to three diﬀerent categories of elements with m and n representing the degree of approximation for element shape and for the state variable, respectively. Figure 3.1 illustrates examples, related to plane case, for (a) superametric (m > n), (b) isoparametric (m = n), and (c) subparametric (m < n) elements. Approximation of any function can be made in the form of a family of algebraic equations, using the well-known Pascal’s Triangle. At this point one is tempted to say that the finite element solution may converge to the exact solution either by increasing the degrees of polynomials (these often correspond to the number of nodes in the element) or by decreasing the element size. Each has advantages and disadvantages. Since, however, the finite element method is a procedure for 46 3.3- Isoparametric quadrilateral elements Figure 3.1: Example of superametric, isoparametric and subparametric elements constructing the solution for the entire domain from local approximations, the convergence can be attained if the functional itself converges as the size of the element diminishes. It is therefore to be noted that: 1. the local functions have to be constructed so that their discontinuities (in terms of their derivatives as well) should not make the functional itself undefined over the entire domain. In other words, not only the local functions but the derivatives of one order less than that occurring in the functional must be continuous; 2. the integrand of the functional must be single-valued and represent a constant as the element size approaches zero. Figure 3.2: Pascal’s triangle to generate various trial functions and corresponding elements Chapter 3 47 Figure 3.3: Two dimensional transformation between local or theory space (left) and the real or user space (right) of quadrilateral and triangular elements 3.3.2 Isoparametric quadrilateral elements Is useful to start from the most simple isoparametric quadrilateral finite element, and extend the result to other elements after some study. The shape functions are polynomial representations of behaviour inside each element. This behavior is written in terms of local coordinates. Consider 2D behaviour: 3D is a straightforward extension. Although there are diﬀerences in detail in the theoretical derivations between triangles and quadrilaterals, the end result are equivalent. The basic theory is developed in a square of side length 2 units, which can be termed the fundamental shape. This space is the theory space and has dimensionless coordinates (ξ 1 , ξ 2 ). The real, or user space, has the usual (x1 , x2 ) cartesian coordinates. These spaces are shown in Figure 3.3 for the quadrilateral and the triangular elements. If the degree of freedom at each node i are the displacement components ¡ h ¢ ū1,i , ūh2,i , as used in stress and dynamics analysis, then the shape functions can ¡ h h¢ express u1 , u2 at any point in the element based on these nodal values as: X X uh1 = Ui ūh1,i , u2 = Ui ūh2,i , (3.32) uh1 = U1 ūh1,1 + U2 ūh1,2 + . . . + Un ūh1,n (3.33) uh2 = U1 ūh2,1 + U2 ūh2,2 + . . . + Un ūh2,n (3.34) where the summations are over the nodes on the element. A requirement of each of these shape functions is that they equal unity at their reference node and zero at every other node. Also the Equation (3.29) have to be verified throughout the element. 48 3.3- Isoparametric quadrilateral elements Table 3.1: Shape functions of 4, 8 and 9 noded quadrilaterals elements include only if node i exists on the element coords shape function i=5 − 12 U5 −1 −1 U1 = 14 (1 − ξ 1 ) (1 − ξ 2 ) 1 1 U3 = 14 (1 + ξ 1 ) (1 + ξ 2 ) 1 −1 0 1 −1 1 −1 0 0 1 −1 0 0 0 U2 = 14 (1 + ξ 1 ) (1 − ξ 2 ) − 12 U5 U4 = 14 (1 − ξ 1 ) (1 + ξ 2 ) ¡ ¢ U5 = 14 1 − ξ 21 (1 − ξ 2 ) ¢ ¡ U6 = 14 (1 + ξ 1 ) 1 − ξ 22 ¡ ¢ U7 = 14 1 − ξ 21 (1 + ξ 2 ) ¢ ¡ U8 = 14 (1 − ξ 1 ) 1 − ξ 22 ¡ ¢¡ ¢ U9 = 14 1 − ξ 21 1 − ξ 22 i=6 − 12 U6 − 12 U6 i=7 − 12 U7 − 12 U7 i=8 i=9 − 12 U8 1 4 U9 1 4 U9 1 4 U9 − 12 U8 1 4 U9 − 12 U9 − 12 U9 − 12 U9 − 12 U9 The shape functions for the 4, 8, and 9 noded quadrilateral elements are given in Table 3.1. The performance of those elements will be evaluated in the next sections, for now it is interesting to pose the attention on Equation (2.24). After some manipulation, which in detail is not necessary here, it turns out that B can be written out in full as: ¯ ⎡ ⎤ g11 0 g12 0 ¯¯ 0 B = ⎣ 0 g21 0 g22 ¯¯ g2n ⎦ (3.35) g21 g11 g22 g12 ¯ g1n where the three rows correspond to the three components of strain, and the columns cover the n nodes. A typical term applying to node k is: ¶ ¶ ¸ ∙µ µ 1 ∂Ui ∂Uk ∂Ui ∂Uk x2,i − x2,i (3.36) g1k = ∂ξ 2 ∂ξ 1 ∂ξ 1 ∂ξ 2 det J ¶ ¶ ¸ ∙ µ µ 1 ∂Ui ∂Uk ∂Ui ∂Uk g2k = − x1,i + x1,i (3.37) ∂ξ 2 ∂ξ 1 ∂ξ 1 ∂ξ 2 det J An important factor here is the inverse det J term. The scalar det J is the Jacobian transformation, eﬀectively a numerical representation, or scaling, between the theory space and the real space of the element, over which the integration is performed. ⎡ ∂x ∂x2 ⎤ 1 ⎢ ∂ξ 1 ∂ξ 1 ⎥ ⎥ J= ⎢ (3.38) ⎣ ∂x1 ∂x2 ⎦ ∂ξ 2 ∂ξ 2 Chapter 3 49 ∂x1 ∂x2 ∂x1 ∂x2 − (3.39) ∂ξ 1 ∂ξ 2 ∂ξ 2 ∂ξ 1 A last note about the Jacobian lies on the fact that the most common finite element method is based on the principle of minimum potential energy, which means that when the structure is loaded and suitably fixed, the displacements and stresses are those which render the potential energy a minimum. Then the approximating equation system will yield a set of linear equations of the form det J = Kd + f = 0, (3.40) here, K is the global stiﬀness matrix and u assume the role of the total degree of freedom vector, and f is the corresponding vector of equivalent nodal loads. Each element contributes its own stiﬀness matrix Ke , added into the global stiﬀness matrix, plus any contributions to the load vector Kij = nelem X Keij e=1 fi = nelem X fie . The element stiﬀness matrix is given by Z 1Z 1 BT CB det Jdξdη Ke = −1 (3.41) e=1 −1 (3.42) in terms of the element’s 2D theory space. When det J is no longer constant, the stiﬀness matrix polynomial becomes of infinite order, and a strictly accurate numerical integration is not available. This can be seen from Equation (3.42), whose right-hand side contains det J in the integrand. The product which gives Ke is seen therefore to contain 1/ det J in the integrand, which is an infinite polynomial. We can term any element whose shape is such that det J is not constant over the element as distorted. Only undistorted elements can be accurately evaluated numerically. If singularities exist in the mapping, det J becomes zero, corresponding to certain types of gross distortion, discussed in the next section. Large numerical errors could result. Another requirement of the mapping is that it should not be both positive and negative in the same element. If negative throughout the element, the mapping sign can be reversed by software (e.g. by reversing the topology order). Over an element, if det J is not constant but a polynomial, then a useful check on the shape is to compare the maximum and minimum values of det J, or their ratio. Hence, checking det J is a valuable element shape check. To understand what are distortions it is useful to analyze the isoparametric maps of 4 and 8 node elements. Any change in shape in the user space from the original square is called a dis- 50 3.3- Four-node element in detail Figure 3.4: Reference coordinates for four-node and eight node element used for give physical meaning of coeﬃcients αi,j tortion. Several investigators have described the various shapes and their significance, see, for example, Burrows [Burrows & Enderby, 1993], Robinson [Robinson, 1987b]—[Robinson, 1988] and the references therein. 3.3.2.1 Four-node element in detail The isoparametric map x : [−1, 1] × [−1, 1] → B ⊂ R2 of a biunit on R × R onto a quadrilateral is defined for an 4-node element by ¾ X ¾ ½ ½ 4 x1 x1i = = x = Ui (ξ 1 , ξ 2 ) (3.43) x2 x1i i=1 ¾ ½ α1,0 + α1,1 ξ 1 + α1,2 ξ 2 + α1,3 ξ 1 ξ 2 = (3.44) α2,0 + α2,1 ξ 1 + α2,2 ξ 2 + α2,3 ξ 1 ξ 2 x1,i , x2,i are nodal coordinate of an element. Shape functions Ui (ξ 1 , ξ 2 ) for each node are given as follows: ¢¡ ¢ 1¡ for i = 1, 2, 3, 4 1 + ξ 1,i ξ 1 1 + ξ 2,i ξ 2 Ui = 4 the coeﬃcients αi,j , for i = 1, 2 and j = 0 − 3, in terms of the nodal coordinates x1,k , k = 1 − 4, are: 1 (x1,1 + x1,2 + x1,3 + x1,4 ) (3.45a) α1,0 = 4 1 (−x1,1 + x1,2 + x1,3 − x1,4 ) α1,1 = (3.45b) 4 1 (−x1,1 − x1,2 + x1,3 + x1,4 ) α1,2 = (3.45c) 4 1 (x1,1 − x1,2 + x1,3 − x1,4 ) α1,3 = (3.45d) 4 and similar α2,j , which are determined in terms of the coordinates x2,k . As show in [Robinson, 1987b] and whit respect to Equations (3.45) the physical significance of the αi,j coeﬃcients is shown in Figure 3.5. From Figure 3.5(a) it is clear that α1,0 and α2,0 define an origin (translation of axes), Figure 3.5(b) Chapter 3 51 Figure 3.5: Physical meaning of the coeﬃcients α1,i and α2,i for i = 1, 2, 3, 4 shows that α1,1 and α2,2 define the size of rectangle (aspect ratio), Figure 3.5(c) demonstrates that α1,2 and α2,1 give two rotations (skew and rotation of axes), and Figure 3.5(d), (e) shows that α1,3 and α2,3 give two tapers. If the local axes are defined as in Figure ?? the shape parameter are: max (α1,1 , α2,2 ) (3.46) aspect ratio = min (α1,1 , α2,2 ) α1,2 skew = (3.47) α2,2 α2,3 taper in the x − direction = (3.48) α2,2 α1,3 taper in the y − direction = (3.49) α1,2 3.3.2.2 Eight-node element in detail The isoparametric map x : [−1, 1] × [−1, 1] → B ⊂ R2 (3.50) of a biunit on R × R onto a quadrilateral is defined for an 8-node element by 52 3.3- Eight-node element in detail x = = ½ ½ x1 x2 ¾ = 8 X Ui (ξ 1 , ξ 2 ) i=1 ½ x1i x1i ¾ = α1,0 + α1,1 ξ 1 + α1,2 ξ 2 + . . . + α1,5 ξ 22 + α1,6 ξ 21 ξ 2 + α1,7 ξ 1 ξ 22 α2,0 + α2,1 ξ 1 + α2,2 ξ 2 + . . . + α2,6 ξ 22 + α2,6 ξ 21 ξ 2 + α2,7 ξ 1 ξ 22 (3.51) ¾ (3.52) x1,i , x2,i are nodal coordinate of an element. Shape functions Ui (ξ 1 , ξ 2 ) for each node are given as follows, for corner nodes: ¢¡ ¢¡ ¢ 1¡ 1 + ξ 1,i ξ 1 1 + ξ 2,i ξ 2 ξ 1,i ξ 1 + ξ 2,i ξ 2 − 1 Ui = for i = 1, 2, 3, 4 (3.53) 4 for boundary nodes: ¢¡ ¢ 1¡ for i = 5, 7 1 − ξ 21 1 + ξ 2,i ξ 2 (3.54) Ui = 2 ¢¡ ¢ 1¡ for i = 6, 8 1 + ξ 1,i ξ 1 1 − ξ 22 Ui = (3.55) 2 the coeﬃcients αi,j , for i = 1, 2 and j = 0 − 7, in terms of the nodal coordinates x1,k , k = 1 − 8, are: 1 1 (3.56) α1,0 = − (x1,1 + x1,2 + x1,3 + x1,4 ) + (x1,5 + x1,6 + x1,7 + x1,8 ) 4 2 1 (x1,6 + x1,8 ) 2 (3.57) 1 (−x1,5 + x1,7 ) 2 (3.58) 1 (x1,1 − x1,2 + x1,3 − x1,4 ) 2 (3.59) α1,4 = 1 1 (x1,1 + x1,2 + x1,3 + x1,4 ) − (−x1,5 + x7) 2 2 (3.60a) α1,5 = 1 1 (x1,1 + x1,2 + x1,3 + x1,4 ) − (−x1,6 + x1,8 ) 2 2 (3.61) α1,6 = 1 1 (−x1,1 − x1,2 + x1,3 + x1,4 ) + (x1,5 − x1,7 ) 2 2 (3.62) α1,1 = α1,2 = α1,3 = 1 1 (−x1,1 + x1,2 + x1,3 − x1,4 ) + (−x1,6 + x1,8 ) (3.63a) 2 2 and similarly α2,j , which are determined in terms of the coordinates x2,k . An interpretation of the eight coeﬃcients α1,i and α2,i for i = 1, 2, 3, 4, is shown in Figure 3.6. α1,7 = Chapter 3 Figure 3.6: Physical meaning of the coeﬃcients α1,i and α2,i for i = 4, 5, 6, 7 53 54 3.4- Numerical integration and optimal sampling points 3.4 Numerical integration and optimal sampling points The integration in Equation (3.42) cannot be performed analytically, because the equations are too complicated also for the simplest types of elements. Hence the famous numerical integration scheme called Gaussian quadrature is used. Figure 3.7: Optimal sampling points for the function (a) and its gradient (b) in one dimension (linear elements) The scheme integrates a polynomial curve in 1D, which eﬀectively calculates the area under the curve over the required interval. The rule order is the number of integration points, or Gauss points, needed in the interval to perform an accurate numerical integration. For each order, the Gauss points have to be situated at particular locations within the interval. For a rule with n such points, polynomials of up to and including order 2n − 1 are integrated accurately. In numerical analysis terms, Gaussian quadrature is very eﬃcient, requiring less points than other methods of integration. In the limit of a straight line, n = 1, the integration calculates the area of a simple quadrilateral. Polynomials higher than 2n − 1 cannot be integrated accurately, which in finite element terms results in instabilities such Chapter 3 55 as zero energy modes. If a given polynomial is integrated accurately by a rule of order k, fit is also integrated accurately by all orders higher than k. In our finite element calculation we often have a need for accurate estimates of the derivatives of the primary variable. For example, in a plane stress or plane strain analysis, the primary unknowns which we compute are the displacement components of the nodes. However, we often are equally concerned about the strains and stresses which are computed from derivatives of the displacements. Likewise, when we model an ideal fluid with a velocity potential, we actually have little or no interest in the computed potential, but we are very interested in the velocity components which are the derivatives of the potential. A logical question at this point is: what location in the element will give me the most accurate estimate of derivatives? Figure 3.8: Integration property of gauss points: (a) p = 1; (b) p = 2; (c) p = 3 which guarantees superconvergence At this stage, it is appropriate to restate the general problems discussed earlier. In the displacement method, the stresses are discontinuous between elements because of the nature of the assumed displacement variation. In analysis involving numerically integrated elements such an isoparametric elements experience has shown that the integration points are best stress sampling points. The nodes, which are the most useful output locations for stresses, appear to be worst sampling points. Reasons for this phenomenon are not immediately apparent even if they could be somehow explained basing on the arguments exposed in Section 3.1. However, it is well known that interpolation functions tend to behave badly near the extremities of the interpolation region. It is therefore reasonable to expect the shape function derivatives (and hence stresses) sampled in the interior of elements would be more accurate than those sampled on the element periphery. Indeed in one dimension at least we shall find that such points often exhibit the quality known as superconvergence (i.e., the values sampled at these points show an error which decreases more rapidly than elsewhere). Obviously, the user of fi- 56 3.4- Eight-node element in detail nite element analysis should be encouraged to employ such points but at the same time note that the errors overall may be much larger. To clarify ideas we shall start with a typical problem of second order in one dimension. Figure 3.9: Optimal sampling points for the function (a) and its gradient (b) in one dimension (quadratic elements) 3.4.1 A one-dimensional example Here we consider a problem of a second-order equation which may be typical of either one-dimensional heat conduction or the equilibrium of a thin membrane subjected to a transverse load b This equation can readily be written as µ ¶ du d k =b (3.64) dx dx with the boundary conditions either defining the values of the function u or of its gradients at the ends of the domain. Let us consider a typical problem shown in Figure 3.7. Here we show an exact solution for u and du/dx for a span of several elements and indicate the type of solution which will result from a finite element calculation using linear elements. We have already noted that on occasions we shall obtain exact solutions for u at nodes. This will happen when the shape functions contain the exact solution of the homogeneous diﬀerential equation, a situation which happens for Equation (3.64) Chapter 3 57 when b = 0 and polynomial shape functions are used. In all cases, even when b is non-zero and linear shape functions are used, the nodal values generally will be much more accurate than those elsewhere, Figure 3.7(a). For the gradients shown in Figure 3.7(b) we observe large discrepancies of the finite element solution from the exact solution but we note that somewhere within each element the results are nearly exact. It would be useful to locate such points and indeed we have already remarked in the context of two-dimensional analysis that values obtained within the elements tend to be more accurate for gradients (strains and stresses) than those values calculated at nodes. Clearly, for the problem illustrated in Figure 3.7(b) we should sample somewhere near the centre of each element. Figure 3.10: Cantilever beam with four 8-node elements. Stress sampling at cubic order (2x2) Gauss points with extrapolation to nodes Pursuing this problem further in a heuristic manner we shall note that, if higher order elements (e.g., quadratic elements) are used, the solution still remains exact or nearly exact at the end nodes of an element but may depart from exactness at the interior nodes, as shown in Figure 3.9(a). The stresses, or gradients, in this case will be optimal at points which correspond to the two Gauss quadrature points for each element as indicated in Figure 3.9(b). This fact was observed experimentally by Barlow in [Barlow, 1976]-[Barlow, 1977]-[Barlow, 1989], and such points are frequently referred to as Barlow points.2 We shall now state in an axiomatic manner that: 2 The waste bin is not the right place for our failures, when an intelligent response can generate ideas like the Barlow points. (by B. Irons from [Irons & Ahmadn, 1980]) 58 3.4- Optimal sampling points 1. the displacements are best sampled at the nodes of the element, whatever the order of the element is, 2. the best accuracy is obtainable for gradients or stresses at the Gauss points corresponding, in order, to the polynomial used in the solution. At such points the order of the convergence of the function or its gradients is one order higher than that which would be anticipated from the appropriate polynomial and thus such points are known as superconvergent. Figure 3.11: Optimal superconvergent sampling and minimum integration points for some elements 3.4.2 Optimal sampling points The single point at the centre of an element integrates precisely all linear functions passing through that point and, hence, if the stresses are exact to the linear form they will be exact at that point of integration. For any higher order polynomial of order p, the Gauss-Legendre points numbering p will provide points of superconvergent sampling. We see this from Figure 3.8 directly. Here we indicate one, two, and three point Gauss-Legendre quadrature showing why exact results are recovered there for gradients and stresses. For points based on rectangles and products of polynomial functions it is clear that the exact integration points will exist at the product points as shown in Figure 3.11 for various rectangular elements. In the same figure we show, however, some triangles and what appear to be good but not necessarily superconvergent sampling points. Though we find that superconvergent points do not exist in triangles, the points shown in Figure 3.11 are optimal. In Figure 3.11 we contrast these points with the minimum number of quadrature points necessary for obtaining an accurate (though not always stable) Chapter 3 59 Table 3.2: Basic element properties: the optimal points are also the reduced rule, althoug the latter may give mechanism for some of the element types Number of: number of intgn rules: stress var’n: el. type nodes dofs el. order complete optimal points complete pseudo 2D TRI3 3 6 linear 1 1 const - 2D TRI6 6 12 quadr 3×3 2×2 linear - 2D Q4 4 8 linear const linear 2D P8 8 16 quadr linear quadr 2D Q9 9 18 quadr linear quadr 2×2 3×3 3×3 1×1 2×2 2×2 stiﬀness representation and find these to be almost coincident at all times. 40 30 20 10 0 -10 (a) 0.5 0 -0.5 -1 (b) Figure 3.12: Bending of a slender beam using 3 × 3 integration rules points: (a) shear stress map, (b) longitudinal stress The Gaussian quadrature integration rules suitable for diﬀerent members of the isoparametric element families are shown in Table 3.2. In Figure 3.10, representing an analysis of a cantilever by four rectangular elements with 8 nodes, we see how well the stresses sampled at superconvergent points behave compared to the overall stress pattern computed in each element. It is from results like this that many suggestions have been made to obtain improved nodal values and one method proposed by Hinton and Campbell [Hinton & Campbell, 1974] has proved to be quite widely used. However, we shall discuss better recovery procedures later. The extension of the idea of superconvergent points from one-dimensional elements to two-dimensional rectangles was fairly obvious. However, the full superconvergence is lost when isoparametric distortion occurs. 60 3.5- Eﬀects of geometry distortion on the element performance 3.5 Eﬀects of geometry distortion on the element performance It is well known that many existing finite elements exhibit poor performance when the element geometry is distorted (see for example [Carey & Oden, 1983]— [Zienkiewicz & Taylor, 1989]—[Striklin et al. , 1977]—[Backlund, 1978]—[Giﬀord, 1979]—[Cannarozzi et al. , 1997]—[Macneal & Harder, 1992]. A set of basic distor- Figure 3.13: Classification of element distortions: (a) undistorted element; (b) ratio distortion; (c) parallelogram distortion; (d) skew distortion; (e) mid-side node distortion; (f) curved edge distortion. tion shapes is shown in Figure 3.13. In practice, most distortions are a combination of these. It has been a common knowledge that the interpolation precision of quadrangular finite elements deteriorates if the element geometry is not a parallelogram. This is because for more general element configurations, the global coordinates are no more linear functions of local coordinates in the geometry mapping. Accordingly, the local coordinates are no more polynomials of the global coordinates, but irrational functions. Thus, the interpolated field is badly dependent on the real shape of the element. Then, as hint before, there immediately occur two issues: the first is how to determinate the interpolation precision that can be exactly achieved by the shape functions for the element configuration, the second is whether we can find an approach to improve the element performance. A qualitative method is available to investigate the first issue: the polynomial we are interested in understanding when correctly captured (say x2 ) is expanded in terms of the local Chapter 3 61 Figure 3.14: Distortion groups for serendipity eigth node elements coordinates by using the geometry mapping and then we check if the shape functions contain every term needed to describe it [Lee & Bathe, 1993]. This method has been widely used, but due to the indirect analysis it gives no useful information to evaluate, control or predict the resulting error as well as no indication on how this trouble can be cured. To study element distortions, it could be convenient to divide the shapes into 5 groups. Groups 1 to 4 follow the convention of Barlow [Barlow, 1989] and are shown in Figure 3.14, while group 5 is covered by Figure 3.15. Each group has geometric distortions in the user space, which contain certain terms from the (ξ 1 , ξ 2 ) polynomial in the theory space. The higher the group, the more terms in this polynomial to represent the cartesian geometry. In ascending order of distortion severity, the groups are: • group 1: constant (ξ 1 , ξ 2 ) terms, a square shape, • group 3: quadratic (ξ 1 , ξ 2 ) terms, displaced midside nodes, tapered rectangles, simple curved sides, • group 4: quasi-cubic (ξ 1 , ξ 2 ) terms, more complicated distortion shapes, • • group 2: linear (ξ 1 , ξ 2 ) terms, a rectangle or a parallelepiped, group 5: extreme distortions, where singularities exist such that the Jacobian determinant, Equation (3.39) , is zero at one or more points. Most practical distortions will be combinations of the above, when the group- 62 3.5- Eﬀects of geometry distortion on the element performance Table 3.3: Total response errors from distributios for quadrilaterals 8-node serendipity elements, (see [Barlow, 1989]) 2 × 2 Rule 3 × 3 Rule Distortion groups: Order of strains 2 3 4 2 3 4 constant 0 0 0 0 0 0 linear 0 d2 d 0 d d quadratic 1 1 1 1 1 1 Table 3.4: Stress calculation errors from distortions for quadrilaterals 8-node serendipity elements, (see [Barlow, 1989]) General points Optimal 2 × 2 points Distortion groups: Order of strains 2 3 4 2 3 4 constant 0 0 0 0 0 0 2 linear 0 d d 0 d d quadratic 1 1 1 0 d d ing is decided by the most severe component. In figure 3.14, the amount of the distortion is indicated by a single measure of magnitude d (the α parameters of Figure 3.5 and 3.6). Note that as the length of the side is 1, then d < 1 and so d2 < d. The significance of these groupings is that errors due to the distortions depend on the group, the magnitude of d and the nature of the strain field experienced by the element. The higher the group number, the worse the error can be. The nature of the strain experienced by the element is an important consideration, since a highly distorted element may well model a certain component of strain accurately and another (such as a transverse strain) badly. Also, for elastic isotropic elements, strain is proportional to stress, so that we can consider either strain or stress in these arguments. The exact strain distribution across an element is not known a priori, so diﬀerent strain fields are considered in turn. The highest strain field that a single 8-node element can accommodate is quadratic (albeit incomplete), hence we consider the eﬀects of constant, linear and quadratic strain variations. In practice, they appear across the element as a combined eﬀect, but in this assessment they are considered in this separated form. Chapter 3 63 Distortion errors do not occur for group 1 shapes, since there is no distortion. For groups 2 to 4, Barlow [Barlow, 1989] has deduced for each distortion group the error variations occurring in each element due to the applied strain variations. Table 3.3 shows the order of magnitude of the errors for each distortion group, under diﬀerent orders of applied strain field. These errors are the combined errors of the calculated stresses (or strains) and the resulting stiﬀness matrix. Table 3.4 shows the corresponding orders of magnitude of the error in the calculated stresses (or strains) only. Here, general points are considered to be any locations in the element other than the optimal points. In these tables, the error of 1 denotes the same magnitude as the individual components of stiﬀness or stress, i.e. very large, whilst d is the order of the distortion, which in this analysis is assumed to be small, with d2 << d. In each entry, the error is for the worst case, so that some cases in each category may show much less error. Figure 3.15: Types of extreme distortions in four and eigth node elements In Table 3.3, it is seen that the total response errors due to any order of distortion are negligible under constant strain fields, but are of order of unity under quadratic strain fields, for both complete and reduced integration. The linear strain field errors are slightly more favourable for reduced integration, mainly of order d. Table 3.4 shows very small errors for stresses evaluated at the optimal points, never more than d even for quadratic strain fields. At other points, the errors are of order of unity under quadratic strain fields. 64 3.5- A simple distortion test for quadratic elements The above groups have distortions measured by the geometric quantity d for each particular shape. An alternative way to view element distortion is the Jacobian determinant, as discussed in previous section. For groups 1 and 2, det J is constant over the element. For the other groups, it is not constant and has been observed to vary: in general larger variations are observed for larger distortions. Hence, the ratio of the maximum to minimum values of det J over an element gives an a priori guide to distortion. Many points over the element need to be sampled, particularly on sides and faces; just sampling at the Gauss points is not suﬃcient. The ratio is thus easily calculated for the mesh at generation time and is available for diagnostic warnings in some commercial software. A value of typically 2 is often taken as a guide for the upper limit of this ratio, such that when exceeded the distortion may be too severe, and the user should check this part of the mesh. It should be emphasized that, in real life, some applied load cases may only give rise to insignificant distortion errors in elements where the det J ratio is high, and, conversely, significant errors where the ratio is low. Evaluating the ratio therefore has to be considered only as an indicative a priori guide to distortion. Distortion eﬀects can be assessed by calculating the individual distortion shapes of Figure 3.13 for each element. These can be checked, along with det J, by automatic software procedures, establishing which group is relevant to each element. Then, knowledge of the amount of distortion in each element, by diagnostic warnings of elements in certain of the above groups, or by the above det J ratios, enables suitable warnings to be highlighted. The eﬀects are, however, very dependent on the applied loading, so that particular caution would be required if multiple load cases were being run concurrently. 3.5.1 A simple distortion test for quadratic elements For elements that have complete polynomial basis functions of at least degree 2 (8 and 9-node elements), we use the constant-bending-moment problem shown in Figure 3.16 to demonstrate the eﬀects of element distortions [Lee & Bathe, 1993]. The Young’s modulus is E = 1.0 × 107 , Poisson’s ratio is ν = 0.3, thickness is t = 1.0 and we are in a plane stress condition. We solve this problem using 8node, 9-node isoparametric elements and the meshes of Figure 3.16 (b)-(c). The corresponding stress band plots are given in Figures 3.16 (d)-(e). The results show that the 9-node Lagrangian element is not aﬀected by angular distortions, no matter how severe they may be. The 8-node serendipity element, on the other hand, is badly aﬀected by angular distortions. We also see that the 8and 9-node elements are significantly aﬀected by the curved-edge distortions, and by the mid-side node distortion Figure 3.17. Chapter 3 65 Figure 3.16: The two element test: (a) the geometry and loading; (b) undistorted configuration; (c) progressive distortion due to δ parameter; (d) stress band plots of longitudinal stresses (σxx ) in constant-bending-moment problem using undistorted mesh; (e) stress band plots of longitudinal stresses (σxx ) using a distorted mesh These findings have several important practical implications, such as: 1. for a reliable analysis, the use of the serendipity elements is best restricted to cases where rectangular or parallelogram elements can be used; 2. Lagrangian elements can be freely used: one need not be concerned about awkward-looking meshes as long as the elements have straight sides and evenly spaced nodes; 3. elements with curved sides should be used to represent only curved boundaries since the performance of such elements is inferior to those that have straight sides; 4. Lagrangian elements are preferred in large-displacement, large-strain analyses since the elements in these analyses can quickly become distorted; 5. adaptive schemes in large deformation analysis should be based on the use of completely new meshes when mesh/interpolation refinement at a particular step is required. 66 3.6- A new procedure to evaluate element distortions Figure 3.17: Stress band plots of longitudinal stresses (σ xx ) in constant-bending-moment problem for a curved edge distortion and a mid-side node distortion Angular distorted element usually appear in the mesh generation of curved geometries and in transition regions from coarse to fine mesh, and also in the nonlinear problems involving large deformations. In some cases, extreme distortions actually model certain singularities correctly, such as the elastic and elasto-plastic strain fields around sharp crack tips. Automatic mesh generators would normally avoid generating such extreme distortions, although in some awkward geometrical shapes they could well arise. It is therefore sensible to utilize any pre-processor checks for zero det J. However, checking this quantity has to be made at many locations over each element, including on its boundaries, since det J has a polynomial variation which is of higher order the more the distortion, and so which requires a lot of sampling points to detect trends towards zero. This check is usefully made with the ratio of maximum to minimum det J over each element, to give some guidance to distortion eﬀects. Table 3.5 collect the polynomial displacement fields that can be solved exactly by the various serendipity and Lagrangian elements in their undistorted and distorted configurations, as shown by Lee and Bathe in [Lee & Bathe, 1993]. 3.6 A new procedure to evaluate element distortions In this section an analytical procedure to evaluate element distortions is presented [Castellazzi, 2001]-[Castellazzi & Ubertini, 2004]. 3.6- A new procedure to evaluate element distortions 62 x4 x3 x3 x3 x4 y x2 x2 x3 y x2 x2 x3 y x2 x3 y 1 xy 1 xy 1 xy x2 y 2 1 xy x2 y 2 x3 y 3 1 xy x2 y 2 y xy 2 y xy 2 y xy 2 y xy 2 x2 y 3 y xy 2 x2 y 3 x2 y 3 y2 y2 xy 3 xy 3 y2 xy 3 y2 y2 xy 3 Undistorted con guration ---------------Additional elds if also unrotated x x2 y x x2 y x x2 y x x2 y x3 y 2 x x2 y x3 y 2 x2 y 2 x3 y 3 y3 y3 y3 xy 4 y4 x3 x3 x2 x2 x2 x x x x x 1 1 1 1 y y y y y Quadratic curved-edge distortion y2 y3 Angular distortion y y y y xy 2 y xy 2 y3 1 1 1 xy 1 xy 1 xy y2 1 x x x x x2 y x x2 y y2 Table 3.5: Polynomial displacement …elds that can be solved exactly by the various serendipity and Lagrangian elements in their undistorted and distorted con…gurations Type of element 8-node serendipity element 12-node serendipity element 9-node Lagrangian element 16-node Lagrangian element 25-node Lagrangian element x3 y x3 y 2 68 3.6- A new procedure to evaluate element distortions Consider Figure 3.3 the coordinate transformation defined by T : x = x (ξ) , (3.65) over the master element B̂. The following condition on the determinant of the Jacobian matrix j(ξ) = det [∇ξ x] > 0, (3.66) ensures locally the existence and uniqueness of the inverse map: T −1 : ξ = ξ (x) . (3.67) Transformation T is called direct transformation, transformation T −1 is called inverse transformation. For convenience, in standard finite elements interpolation functions (e.g. shape functions) are defined on the master element, F̂ = F̂ (ξ). However, the interpolation quality should be evaluated over the physical domain. Obviously, to determine the actual dependence of interpolation functions on x the inverse map is needed: T −1 F (x) = F (ξ (x)) = F̂ (ξ) . Except for very simple element geometry, determining the inverse transformation is an hard task, due to nonlinearity of Equation 3.65. In the following, the inverse mapping is determined trough the Gröbner basis method [Cox et al. , 1992]. This means to compute a Gröbner basis of the ideal generated by polynomials x − x (ξ). The Gröbner basis is a new set of polynomials which has the same roots of the original one x − x (ξ) and, in addition has a triangular form. Thus, one of the new polynomials contains just one component of ξ and can be solved for in. Then by back substitution into the other polynomial the inverse mapping can be found. This procedure has no theoretical limitations, since Gröbner bases can always be computed in a finite number of steps [Buchberger, 2001]—[Tran, 2000], using for example a commercial software for algebraic manipulation such as MAPLE, although it may be a very time consuming. However, in practice the one component equation can be explicitly solved only up to the fourth degree. Moreover, the resulting expressions could be heavy to be handled. In the following some basic transformations are analyzed and discussed. The analysis is conducted basing on the fact that if the direct transformation is defined by polynomial P̂ : def x = P̂ (ξ) , (3.68) the inverse map, can be represented as: ξ (x) = P (x) + N P (x) , with P ⊆ P̂ , (3.69) where N P is the non-polynomial part. Indeed these equations have a completely Chapter 3 69 polynomial form only if T is linear. Based on the above remark and taking into account that the finite element generally crucially depends on the polynomial part of the interpolation functions, the analysis is focused on this part. In particular we consider vector Ξ̂ collecting monomials in ξ £ ¤ Ξ̂ = 1, ξ 1 , ξ 2 , ξ 1 ξ 2 , ξ 21 . . . , ξ n2 , (3.70) which can be viewed as the basis of the shape functions. Then through the computed inverse mapping we could determine the transformation basis Ξ, whose structure can be analyzed based on the following matrix: J0 = ∇Ξ, (3.71) where derivations carried out with respect to ¤ £ X = 1, x1 , x2 , x1 x2 , x21 , . . . , xm 2 . (3.72) It is to be intended as a formal derivation capable to extract from lengthy expressions the polynomial part of the transformation, as shown by the following example: Ξ̂ = [1, ξ 1 , ξ 2 , ξ 1 ξ 2 ] , £ ¤ X = 1, x1 , x2 , x1 x2 , x21 , . . . , (3.73) (3.74) a generic the inverse map, can be represented as: √ ξ 1 = x1 + x1 ξ2 (3.75) = x2 h √ i √ Ξ = 1, x1 + x1 , x2 , x1 x2 + x2 x1 (3.76) ⎡ ⎤ 1 0 0 0 ⎢ 0 1 0 0 ⎥ ⎥ J0 = ⎢ (3.77) ⎣ 0 0 1 0 ⎦ 0 0 0 1 By inspecting matrix J0 and, in particular, linear independence of rows and columns we can extract useful information on the polynomial part of the transformation. 3.6.1 Some basic distortions of plane elements For later convenience the transformation is written as ⎡ ⎡ 2 ⎤ ∙ ¸ ξ1 ξ1 ⎢ + C2×3 ⎣ ξ 1 ξ 2 ⎦ + D2×i ⎣ T : x = A2×1 + B2×2 ξ2 2 ξ2 ⎤ ξ 31 .. ⎥ + . . . . ⎦ ξ 32 (3.78) 70 3.6- A simple rotation Table 3.6: Relation between elements of X and elements of Ξ 1 ξ1 ξ2 ξ1ξ2 .. . 1 x1 x2 x1 x2 x21 l l l l l ↔ ↔ ↔ ¥ 0 ¥ 0 0 0 ↔ 0 0 ¥ ¥ 0 0 .. . 0 .. . 0 .. . ¥ .. . ¥ .. . 0 0 0 0 ··· ··· ··· ··· =⇒ J 0 ··· .. . Table 3.7: Trasformation and relation T : x = A2×1 → translation T : x = B2×2 · ξ02×1 → simple rotation if B2×2 = R T : x = A2×1 + B2×2 · ξ02×1 → generic linear transformation R ∈Orth Where A2×1 is 0th order matrix , B2×2 is the first order matrix and so on. Table 3.7 collects some basic transformations. In the following section we will denote by the letter Q quadrilateral Lagrangian elements plus a number to identify the degree of completeness, that, Q1 denotes four node Lagrangian elements, Q2 nine node Lagrangian element and Q3 sixteen node Lagrangian element. We will use also the letter P for serendipity elements : P2 denotes eigth-node serendipity element, P3 the twelve-node element. 3.6.2 A simple rotation The simpler linear transformation is a rotation. With respect to Figure 3.18(a) the transformation (3.78) reduces to T : x = R · ξ, with R−1 = RT , and in extended form ½ x1 = cos αξ 1 − sin αξ 2 , T : x2 = sin αξ 1 + cos αξ 2 . (3.79) (3.80) In this simple case, the inverse transformation can be easily computed and given by ½ ξ 1 = cos αx1 + sin αx2 , T −1 : (3.81) ξ 2 = − sin αx1 + cos αx2 . Chapter 3 71 Figure 3.18: Linear transformation: the local normal co-ordinate system for a simple rotation (a) and for a generic lianear transformation (b). Table 3.8: Relation between of the monomials 1 x1 x2 x1 x2 x21 x22 x31 x21 x2 x1 x22 x32 x41 x31 x2 x21 x22 x1 x32 x42 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ξ1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ξ 2 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 ξ1ξ2 0 0 0 ξ 21 ξ 22 ξ 21 ξ 2 ξ 1 ξ 22 ξ 21 ξ 22 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 Following the outlined procedure we can compute matrix J0 , given by Table 3.8. The analysis of this matrix can be summarized in the following three points. 1. The sub-matrix established by the first 4 rows and columns, corresponding to bilinear Lagrange element Q1, have independent rows, resulting in the following polynomial basis: [1 x2 2. − x1 x2 ]. (3.82) The sub-matrix corresponding to Serendipity element P2 has independent rows, resulting in the following polynomial basis: [1 x2 3. − x1 − x1 − x1 x2 2x22 2x21 − 2x1 x22 2x21 x2 ]. (3.83) The sub-matrix corresponding to Lagrangian element Q2 has independent 72 3.6- A generic linear transformation Table 3.9: Coordinate’s point, see Figure 3.18 P1 P2 P3 P4 x 0 l1 cos α1 l1 cos α1 + l2 cos α2 l2 cos α2 y 0 l1 sin α1 l1 sin α1 + l2 sin α2 l2 sin α2 rows, resulting in the following polynomial basis: [1 x2 − x1 − x1 x2 2x22 2x21 − 2x1 x22 2x21 x2 4x21 x22 ]. (3.84) This practical, and very simple, example show as rotation does not deteriorate element performance, as expected. 3.6.3 A generic linear transformation A generic linear transformation can be written as ¸ ∙ ¸ ∙ ∙ α1,0 α1,1 ξ1 = + T : x=A+B· ξ2 α2,0 α2,1 α1,2 α2,2 ¸∙ ξ1 ξ2 ¸ , (3.85) and represents a quadrilateral with parallel straight edges, eventually rotated and translated. For simplicity we set A = 0 (αi,0 = 0) that rules out the translation. Making reference to Figure 3.18(b) and Table 3.1, the direct and inverse transformations are ½ x1 = (l1 cos α1 )ξ 1 + (l2 cos α2 )ξ 2 , T : , (3.86) x2 = (l1 sin α1 )ξ 1 + (l2 sin α2 )ξ 2 , T where −1 : ½ ξ1 = ξ2 = sin α2 l1 d x1 sin α1 l2 d x1 − − cos α2 l1 d x2 , cos α1 l2 d x2 , , (3.87) d = cos α1 sin α2 − cos α2 sin α1 . (3.88) Ξ̂ = [1 ξ 1 ξ 2 ξ 1 ξ 2 ξ 1 2 ξ 2 2 ξ 1 2 ξ 2 ξ 1 ξ 2 2 ξ 1 2 ξ 2 2 ], (3.89) Setting: c1 = cos α1 , s1 = sin α1 , c2 = cos α2 , s2 = sin α2 , and making reference to Q2 element we have Xi = [1 x1 x2 x1 x2 x21 x22 x31 x21 x2 x1 x22 x32 x41 x31 x2 x21 x22 x1 x32 x42 ]. (3.90) Chapter 3 73 Table 3.10: Coeﬃcients of a generic linear transformation ³ e10 = c2 2 , l12 d2 e2 = − lc12d , e11 = s21 l22 d2 e3 = − ls21d , e12 = −2 sl21dc21 , e21 = − l1cd2 c31l2 2 , c1 l2 d , e13 = e22 = e5 = − l1sd2 s21l2 , ³ e6 = l1sd2 c21l2 + e14 = e1 = e4 = s2 l1 d , , e20 = c2 s1 l1 d2 l2 ´ , e15 s2 2 , l12 d2 c21 , l22 d2 ³ 2 s2 c1 l12 d3 l2 + ³ 2 = − l1c22 ds31l2 2 e16 = − ls22d3sl12 , e17 = 1 ⎢ ⎢ ⎢ ⎢ 0 J =⎢ ⎢ ⎢ ⎢ ⎣ 0 e1 e3 0 0 0 0 0 0 0 e2 e4 0 0 0 0 0 0 0 0 0 e6 e9 e12 0 0 0 0 0 0 e5 e8 e11 0 0 0 ´ s2 s1 , l1 d3 l2 2 c2 c1 , l12 d3 l2 0 0 0 0 0 0 e16 e20 0 ´ , 4s2 c2 s1 c1 +s2 2 c1 2 +c2 2 s1 2 , l1 2 d4 l2 2 , 1 2 0 0 0 e7 e10 e13 0 0 0 s2 c1 2 l1 d3 l2 2 2 c2 s1 2 l1 d3 l2 2 ´ 2 0 0 0 0 0 0 e14 e18 0 0 0 0 0 0 0 e15 e19 0 2s2 c2 s1 2 l1 2 d4 l2 2 2c2 2 s1 c1 l1 2 d4 l2 2 2 e25 = c2 c1 , l1 2 d4 l2 2 e26 = s2 2 s1 2 l1 2 d4 l2 2 ´ ´ , , , Then the functional matrix in Equation (3.71) takes the form: ⎡ 1 0 0 0 0 0 0 0 0 + ³ 2 2 s1 c1 − e23 = − 2s l1 2 d4 l2 2 ³ ´ 2 2 c2 c1 2 c2 c1 − 2s − , e24 = − 2s l1 2 d3 l2 l1 2 d4 l2 2 2s2 c2 s1 l21 d3 l2 ³ 2 s1 c1 e18 = − 2s − l1 d3 l2 2 e9 = −2 sl22dc22 , 2c2 s1 c1 l1 d3 l2 2 2 2 e7 = − l1cd2 c21l2 , e8 = e19 = 0 0 0 0 0 0 e17 e21 0 0 0 0 0 0 0 0 0 e26 0 0 0 0 0 0 0 0 e23 0 0 0 0 0 0 0 0 e22 0 0 0 0 0 0 0 0 e24 0 0 0 0 0 0 0 0 e25 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.91) where coeﬃcients ei are collected in Table 3.10. This matrix has full rank, the first six columns are linear independent, just two columns are linear independent from 7th to 10th and just one among the last five columns. Therefore the polynomial part admits the following representation £ ¤ 1 x1 x2 x1 x2 x21 x22 p1 p2 p3 , (3.92) where p1 and p2 are independent polynomials that are linear combinations of third degree monomials: p1 = p1 (x31 , x21 x2 , x1 x22 , x32 ), p2 = p2 (x31 , x21 x2 , x1 x22 , x32 ), whereas p3 is a linear combination of fourth degree monomials: p3 = p3 (x41 , x31 x2 , x21 x22 , x1 x32 , x42 ). For the eight-node serendipity element, Ξ̂ is the same except for ξ 1 ξ 2 and x takes the form ¤ £ X = 1 x1 x2 x1 x2 x21 x22 x31 x21 x2 x1 x22 x32 . (3.93) The corresponding functional matrix can be obtained by the first eight rows and ten columns of Equation (3.91). This new sub-matrix has full rank, and the most 74 3.6- A bilinear transformation Table 3.11: Eﬀects of a generic linear transformation degree 0◦ 1◦ ◦ Q1 1 x1 x2 P2 1 Q2 1 x1 x2 x1 x2 x21 P3 1 x1 x2 x22 x1 x2 x21 Q3 1 x1 x2 x22 x1 x2 x21 x1 x2 x22 x1 x2 x21 x22 2 p1 3◦ - p1 , p2 p1 , p2 ◦ 4 - - p3 p4 , p5 p6 , p7 , p8 5◦ - - - - p9 , p10 ◦ - - - - p11 6 x31 x21 x2 x22 x1 x32 x31 x21 x2 x22 x1 x32 general algebraic polynomials represented by F̂ belong now to whole described by ¤ £ 1 x1 x2 x1 x2 x21 x22 p1 p2 , (3.94) where p1 and p2 are already known. The functional matrix of Q1 element can be obtained by first four rows and first six columns of matrix contents in Equation (3.91). This new sub-matrix has full rank, and the most general algebraic polynomials is described by [1 x1 x2 p0 ] , (3.95a) where p0 is a linear combination of second degree monomial: p0 = p0 (x21 , x1 x2 , x22 ) The same approach can be repeated also for other types of elements. The results are summarized in Table 3.11. 3.6.4 A bilinear transformation A generic bilinear transformation is given by ⎡ ⎤ ¸ ∙ ξ1 α1,1 α1,2 α1,3 ⎣ ξ2 ⎦ . T : x= α2,1 α2,2 α2,3 ξ1ξ2 (3.96) It transforms B̂ into a generic quadrilateral element. For simplicity we can analyze three bilinear transformation considering to progressively distorted element configuration as described in Figure 3.19. Whit respect to Figure 3.19(a), transformation reduces to ½ x1 = α1,1 · ξ 1 + α1,3 · ξ 1 ξ 2 , (3.97) T : x2 = α2,2 · ξ 2 Chapter 3 75 Figure 3.19: Bilinear transformation: (a) taper distortion ), (b) taper distortion with skew, (c) generic bilinear distortion and the inverse mapping is T −1 ⎧ −x2 ⎪ ⎨ ξ1 = − α2,2 : α2,2 (−x1 + α1,1 ) , ⎪ ⎩ ξ2 = − α1,1 α2,2 + α1,3 x2 (3.98) The T −1 are therefore established from the following basis functions: [1 x2 r1 r2 ], (3.99) where r1 and r2 are rational functions. With respect to Figure 3.19(b) transformation is: ½ x1 = α1,1 · ξ 1 + α1,2 · ξ 2 + α1,3 · ξ 1 ξ 2 , (3.100) T : x2 = α2,2 · ξ 2 and the inverse takes the form ⎧ x1 α2,4 − α1,1 α2,4 − α1,4 x2 + α1,4 α2,1 ⎪ ⎨ ξ1 = α1,2 α2,4 , T −1 : α1,2 (x2 − α2,1 ) ⎪ ⎩ ξ2 = x1 α2,4 − α1,1 α2,4 − α1,4 x2 + α1,4 α2,1 Now the quadrilateral element is not a parallelogram, and results from previous case let to take oﬀ from (3.96) rotation terms, but without loosing generality. With respect to Figure 3.19(c) the transformation is: ½ x1 = α1,1 · ξ 1 + α1,2 · ξ 2 + α1,3 · ξ 1 ξ 2 . (3.101) T : x2 = α2,2 · ξ 2 + α2,3 · ξ 1 ξ 2 In this case the elimination process based in Gröbner basis leads to a second order equation which yields to two possible solutions for the inverse map, but just one is acceptable: p ½ ξ 1 = β 1,0 + β 1,1 x1 + β 1,2 x2 + β 1,3 pr (1, x1 , x2 , x1 x2 , x21 , x22 ) −1 , (3.102) T : ξ 2 = β 2,0 + β 2,1 x1 + β 2,2 x2 + β 2,3 r (1, x1 , x2 , x1 x2 , x21 , x22 ) ¢ ¡ where r 1, x1 , x2 , x1 x2 , x21 , x22 is a polynomial, and β i,j are coeﬃcients depending 76 3.6- Higher degree transformations: quadratic curved-edge Table 3.12: Bilinear transformation degree Q1 P2 Q2 P3 Q3 0◦ 1 1 1 1 1 ◦ 1 x1 x2 x1 x2 ◦ 2 - h1 (x1 x2 , . . .), h2 (x1 x2 , . . .) 3◦ - - x1 x2 x21 x1 x2 x1 x22 x1 x2 x2 x21 x1 x22 h3 (x31 , . . .) - x1 x2 x2 x21 x22 x31 x21 x2 x22 x1 x32 on element geometry: β 1,0 = (a1,2 a2,1 − a1,1 a2,2 ) , 2 (a1,1 a2,3 − a1,3 a2,1 ) β 1,2 = − β 2,0 = a1,3 , 2 (a1,1 a2,3 − a1,3 a2,1 ) a1,1 a2,2 − a1,2 a2,1 , 2 (a1,2 a2,3 − a1,3 a2,2 ) β 2,2 = − a1,3 , 2 (a1,2 a2,3 − a1,3 a2,2 ) β 1,1 = a2,3 , 2 (a1,1 a2,3 − a1,3 a2,1 ) β 1,3 = − 1 , 2a1,1 a2,3 − a1,3 a2,1 β 2,1 = a2,3 , 2 (a1,2 a2,3 − a1,3 a2,2 ) β 2,3 = 1 , 2 (α1,3 α2,2 − α2,3 α1,2 ) The T −1 is therefore based on the following: q [1 x1 x2 r (1, x1 , x2 , x1 x2 , x21 , x22 )]. (3.103) (3.104) Based on the above results on inverse mapping, the analysis of the polynomial part of interpolation functions can be carried out. The main results are summarized in Table 3.12, when hi are linear combination of n-degree monomial, being n the table row. 3.6.5 edge Higher degree transformations: quadratic curved- Here element geometry with quadratic curved edge is discussed. This most general quadratic transformation is given by ⎡ 2 ⎤ ∙ ¸∙ ¸ ∙ ¸ ξ α1,1 α1,2 ξ1 α1,3 α1,4 α1,5 ⎣ 1 ⎦ ξ1ξ2 . T : x= + (3.105) α2,1 α2,2 ξ2 α2,3 α2,4 α2,5 ξ 22 Among this class we focus the attention on two cases with an increasing level of distortion as shown in Figure 3.20. Consider the semirectangulars [Frey et al. , 1978] of Figure 3.20(a) and (b), whose transformation are given by ½ x1 = α1,1 ξ 1 + α1,4 ξ 1 ξ 2 + α1,7 ξ 1 ξ 22 , (3.106) T : x2 = α2,2 ξ 2 Chapter 3 77 Figure 3.20: Higher degree transformations and ½ x1 = α1,1 ξ 1 + α1,2 ξ 2 + α1,3 ξ 1 ξ 2 + α1,4 ξ 22 . (3.107) x2 = α2,2 ξ 2 + α2,3 ξ 1 ξ 2 + α2,4 ξ 22 Using the Gröbner basis method the inverse mappings of the first transformation can be computed in the form ( x1 α2,2 2 ξ 1 = α1,1 α2,2 2 +α1,3 −1 x2 α2,2 +α1,5 x2 2 . T : (3.108) 2 ξ 2 = αx2,2 T : In the second most severe case the Gröbner basis method leads to a fourth order triangular system. The only acceptable solution can be written as ⎧ t √ 3 (3) (2) p1 +γ 1,3 r1 ⎪ p ⎪ t 2 ξ = γ + γ x + γ x + + ⎨ 1 1,0 1,1 1 1,2 2 √ 3 (3) γ 1,4 γ 1,5 · p3 +γ 1,6 r1 t T −1 : ; √ 3 (1) (1) p4 +γ 2,1 r2 p5 ⎪ ⎪ t ξ = γ + + ⎩ 2 2,0 √ 3 (3) γ 2,2 γ 2,3 · p6 +γ 2,4 r2 (3.109) were r1 and r2 are polynomials of second degree, while γ i,j are coeﬃcients depend(k) ing on element geometry, while pj are polynomials, of degree k. Then, the T −1 has the following basis ∙ ¸ t t p2 p2 √ √ 3 3 1 , x1 , x2 , p1 + t1 r1 , s , (3.110) √ , p4 + t3 r2 , s √ 3 3 p3 + t2 r1 p6 + t4 r2 where ti are coeﬃcients depending on γ i,j . Now the analysis can be carried out and the main results are summarized in Table 3.13, where wi are polynomials whose degree is indicated in the first column. 3.7 Some numerical tests To numerically evaluate the eﬀect of coordinate transformations on the element interpolation capability and, so, verify the analytical results discussed in the previous section, single element test are used. The idea is the same of the Continuum Region Element (CRE) method of Robinson [Robinson, 1985]—[Robinson, 1987a] (Figure 3.21). A single element of any shape is constructed and displacement are 78 3.7- Bilinear transformations Table 3.13: Higher degree transformations: quadratic curved-edge degree 0◦ ◦ P2 Q2 P3 Q3 1 1 1 1 1 x1 x2 x1 x2 x1 x2 x1 x2 2◦ w1 (x1 x2 , . . .) w2 (x1 x2 , . . .) w3 (. . .), w4 (. . .) x21 x1 x2 x22 3◦ - - w5 (. . .) w6 (. . .), w7 (. . .) ◦ - - - w8 (. . .) 4 prescribed in all the nodes according to (a priori selected) analytical solution. Then the numerical solution is computed and checked against the analytical solution. In the following the basic element geometries discussed in the previous section are considered, together with linear, quadratic and cubic prescribed displacement fields. The results are presented by plotting the stress field and the finite element stress field, and comparing locally the two solutions overlapping the contours of the exact solution (red line) and the contour of the FEM solution (blue line) with the local error. This plots confirm also that unfortunately the Barlow points not always remain superconvergent points as already pointed out in [Barlow, 1989]. As, experienced, when linear displacement field are prescribed, no errors were recorded, independently of the element type used. Figure 3.21: The Continuum Region Element test: displacements u = (x) are prescribed 3.7.1 Bilinear transformations Consider the example shown in Figure 3.16 where a simple distortion test is performed on a cantilever beam. Here we reproduce this distortion case using a CRE approach. As shown by Table 3.12 all the elements represent correctly the linear Chapter 3 79 displacement field, and so we do not report them. Diﬀerently for a quadratic displacement field some diﬀerence occurs as shown by Table 3.12. Actually P2 and Q2 elements have some diﬀerence due quadratic terms. Consider first a quadratic displacement field. Figures 3.22 and 3.23 show that a P2 element is not capable to represent it correctly. Taking a cubic displacement field, Figures 3.24 and 3.25 shows the results for P2 elements. Figures 3.26 and 3.27 show that in this case also Q2 element is not capable to represent a quadratic displacement field correctly and suﬀer of the same error distributions of P2 element. Exact Vx Exact Vy Exact Wxy 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 FEM Vx FEM Vy FEM Wxy 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 FEM: Vx - MAX: 0.0079357 FEM: Vy - MAX: 0.010388 FEM: Wxy - MAX: 0.0065607 Figure 3.22: Element test for P8 elements: stress band plots stress for quadratic displacement field and error distributions with contour line step: 0.001 -0 0 .0 -0.007.00 0 5 -0.008 6 0.0 06 0 2 00 0. 1 00 0. 1 00 -0 . 06 0.000.0 5 - 0 .0 -0 . 00 -0 .00 .00 20 1 4 03 -0 .00 5 04 0.0 0. 00 3 0 02 0.0 001 0 0. -0 .0 02 03 -0 .0 0 4 .0 -0 1 00 0. 1 .00 -0 0 0.01 0 0.000 8 9 0.00.0.0 0. 0607 00 2 0.005 0.004 0.003 0 1 00 0 1 0. 0 .0 - 2 00 -0 .0 .00 34 - .00 -0 .00 5 -0-0.006 0.003 04 0. 0 5 00 0. FEM Wxy - Exact Wxy FEM V y - Exact V y FEM V x - Exact V x 0.0 0. 001 002 Figure 3.23: Element test for P8 elements: local error distributions for quadratic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.001) 80 3.7- Bilinear transformations Exact Vx Exact Vy Exact Wxy 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 FEM Vx FEM Vy FEM: Vx - MAX: 1.7697 FEM Wxy 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 FEM: Vy - MAX: 1.3771 FEM: Wxy - MAX: 1.5789 Figure 3.24: Element test for P8 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.1 FEM V x - Exact V x FEM V y - Exact V y FEM Wxy - Exact Wxy -0 .1 3 1. 2 1. 1.1 1 9 0. .8 0 .7 0 6 0. 5 0. 4 0. 3 0. 0.2 0.1 0 -0 .1 1 1. 1 9 0. 8 0. .7 0 .6 0 5 0. 4 0. 3 0. 0.2 0.1 4 1..3 1 .2 1 1.1 1 9 0. .8 0 .7 0 0.65 0. 4 0. 0. 3 0.20.1 0 0 Figure 3.25: Element test for P8 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.1) Chapter 3 81 Exact Vx Exact Vy Exact Wxy 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 FEM V x FEM V y FEM W xy 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 FEM: Vx - MAX: 1.7697 FEM: Vy - MAX: 1.3771 FEM: Wxy - MAX: 1.5789 Figure 3.26: Element test for Q9 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.1 FEM V x - Exact V x FEM Wxy - Exact Wxy FEM V y - Exact V y 1..21 1 1 9 0. 0.8.7 0 6 0. 0.1 5 0. .4 0 3 0. 2 0. 0 0.1 0 19 0. 0.8 7 0. .6 0 5 0. 0.4 .3 0 2 0. 0 3 1..2 1 .1 1 1 0.98 0. .7 0 .6 0 .5 0 0.4 0.3 0.2 0.1 Figure 3.27: Element test for Q9 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.1) 3.7.2 Higher order transformations Here we intend to reproduce distortions of Figure 3.17 using the CRE approach. As shown by Table 3.13 and Figure 3.17 all the "curved edge" elements represent correctly the linear displacement field, and so we do not report them. On the contrary for a bilinear displacement field some problems occur as shown by the same table. Figures 3.28 and 3.29 show the results corresponding to a bilinear 82 3.7- Higher order transformations displacement field, which confirm that a P2 element is not capable to correctly capture it. Moreover, Figures 3.30 and 3.31 confirm that Q2 element yields in this case a similar response. Summing up, the numerical results presented above confirm the results of the analysis presented in the previous section. Exact Vx Exact Vy Exact Wxy 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 FEM Vx 0.05 0.05 FEM Vy FEM: Vx - MAX: 0.14765 FEM Wxy 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 FEM: Vy - MAX: 0.63242 FEM: Wxy - MAX: 0.48026 Figure 3.28: Element test for P8 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.02 FEM V x - Exact V x -0.62 FEM Wxy - Exact Wxy 0. 3 0. 32 0.3 4 0.380.36 0.4 0.42 04 -0.12 FEM V y - Exact V y -0.62 -0.12 -0.14 4 0.4 0.46 0.4 -0.14 0.42 0.38 0.36 4 0.3 32 0.0. 3 Figure 3.29: Element test for P8 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.02) Chapter 3 83 Exact Vx Exact Vy Exact Wxy 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 FEM Vx 0.05 0.05 FEM Vy FEM: Vx - MAX: 0.14765 FEM Wxy 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 FEM: Vy - MAX: 0.63242 FEM: Wxy - MAX: 0.48026 Figure 3.30: Element test for Q9 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.02 FEM V x - Exact V x -0.62 FEM Wxy - Exact Wxy 0. 3 0. 32 0.3 4 0.380.36 0.4 0.42 04 -0.12 FEM V y - Exact V y -0.62 -0.12 -0.14 4 0.4 0.46 0.4 -0.14 0.42 0.38 0.36 4 0.3 32 0.0. 3 Figure 3.31: Element test for Q9 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.02) 3.8 Distortion measures One of the main concerns when generating a new mesh is the quality of elements produced. In particular, the eﬃciency of adaptive refinement analysis depends 84 3.8- Distortion measures for four node elements on the shape of the elements, and so estimating the quality of element shape is a requisite during the adaptive analysis being performed. Unfortunately, most posterior error estimates can not evaluate the shape error of an element, so that some diﬃculties remain in the application of an adaptive analysis. The use of element distortion factors (computable a priori) is, therefore, a necessary step in order to ensure that the elements shapes do not act as an extra cause of deterioration in finite element analysis accuracy. Element shape distortion can be defined in many forms, most notable of which are aspect ratios, skew, taper and warp. Robinson [Robinson, 1987b] used three of the latter properties to classify the distortion of four-noded quadrilaterals. A sensible choice of distortion measure would be the determinant of the element’s Jacobian matrix. Robinson expressed the shape parameters in terms of the Jacobian determinant for fournoded quadrilaterals. However, the diﬃculty arising in linking each element’s Jacobian determinant to such parameters (e.g., the aspect ratio or internal angles) for elements of displacement order higher than one with curved sides was pointed out. Eight further parameters in the form of the tangential and normal deviations for each of the four edges were later added [Robinson, 1988]. Although these 12 parameters for eight-noded curved quadrilaterals provide a high degree of accuracy, a higher computational workload is needed when calculating all twelve factors for each element. In [El-Hamalawi, 2000] a simple element distortion factor is proposed, based only on interior angles, and here will be used as a comparison factor. In [Yuan & Pian, 1994] a set of distortion measuring parameters for the quadrilaterals hybrid stress membrane element is presented, based on Robinson’s distortion parameters modified by geodesic coordinates, subsequently used in [Lautersztajn-S & Samuelsson, 1997b] and [Lautersztajn-S & Samuelsson, 1998b]. Thus, significant reduction of the number of these parameters, with respect to previous work, from 12 to 4, from is obtained. Figure 3.32: Simple distortione measure: (a) diagonal ratio c1 ; (b) middle point distance ratio c2 ; (c) middle point pojection ratio (on reference axes) c3 Chapter 3 85 Table 3.14: Test No.1: two element test. δ̂ 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 3.8.1 η -16.1778 -15.6361 -14.2400 -12.5666 -11.0137 -9.7113 -8.6383 -7.7203 -6.8674 -5.9574 e 0.0000 0.0335 0.1198 0.2232 0.3192 0.3997 0.4660 0.5228 0.5755 0.6318 c0 0.0000 0.2792 0.5381 0.7643 0.9542 1.1107 1.2389 1.3443 1.4315 1.5043 Distortion c1 1.0000 0.9050 0.8198 0.7442 0.6778 0.6202 0.5708 0.5293 0.4953 0.4681 measure c2 1.0000 0.9950 0.9806 0.9578 0.9285 0.8944 0.8575 0.8192 0.7809 0.7433 c3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Distortion measures for four node elements A quality measure for quadrilaterals can be depend as [El-Hamalawi, 2000] v u 4 uX c0 = t (δθi )2 (3.111) i=1 ¯ ¯ where δθi = ¯ π2 − θi ¯ is the deviation of interior angles θi from the optimum value of π2 , corresponding to the rectangular shape. In addition to the above indicator, three more measures can be included [Castellazzi & Ubertini, 2004]: (a) diagonal ratio c1 ; (b) middle point distance ratio c2 ; (c) middle point pojection ratio (on reference axes) c3 see Figure 3.32. Making reference to the notation defined in the previous section, the first parameter is given by q α21,1 + 2α1,1 α1,2 + α21,2 + α22,1 + 2α2,1 α2,2 + α22,2 c1 = q , (3.112) α21,1 − 2α1,1 α1,2 + α21,2 + α22,1 − 2α2,1 α2,2 + α22,2 the second is given by and the third parameter by q α21,2 + α22,2 , c2 = q α21,1 + α22,1 c3 = 3.8.2 |α1,1 | . |α2,2 | (3.113) (3.114) Test No.1 The test in Figure 3.16 is considered, with h = 1, L = 2, M = 1 and results for the tip deflection are given for the cases v = 0.3 and Young’s modulus equal to one. 86 3.8- Test No.2 (a) (b) (c) (d) Figure 3.33: Contour bands of stress for a four-node element: (a) undistorted configuration; (b) angular distortion with δ̂ = 0.3; (c) angular distortion with δ̂ = 0.6; (d) angular distortion with δ̂ = 0.9 In Table 3.14 the first column indicates the dimensionless distortion coeﬃcient δ̂, (which varies from 0 to 1 and defines the mesh) and the second column the tip deflection η, the third column the percentage relative deviation η̄ − η (3.115) e= η̄ where η̄ is the tip deflection when δ̂ = 0. In addition the distortion measures c0 , c1 , c2 , c3 are reported. In Figure 3.33 the stress bands plot of the σx stress component, are plotted for four diﬀerent value of δ̂. It is clear the eﬀect of progressive deterioration. 3.8.3 Test No.2 The cantilever beam of Figure 3.34 is analyzed with four node elements and different meshes. Young’s modulus is set to 1, Poisson’s coeﬃcient to 0.3 and F to 1. The results are collected in Tables 3.15—3.18. Chapter 3 87 Figure 3.34: A beam problem: (a) the geometry and load; (b) discretization for undistorted configuration, reference solution; (c) linear distortion; (d) first bilinear distortion; (e) second bilinear distortion; (f) taper distortion; (g) arbitrary bilinear distortion; Table 3.15: A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) δ̂ 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 η -3791.1856 -3786.8955 -3780.5241 -3772.0722 -3761.5454 -3748.9549 -3734.3160 -3717.6481 -3698.9729 -3678.3134 e 0.0000 0.0011 0.0028 0.0050 0.0078 0.0111 0.0150 0.0194 0.0243 0.0298 c0 0.0000 0.0926 0.1847 0.2760 0.3659 0.4541 0.5402 0.6240 0.7053 0.7838 Distortion c1 1.0000 0.9938 0.9878 0.9820 0.9762 0.9707 0.9653 0.9601 0.9550 0.9500 measure. c2 1.0000 0.9999 0.9997 0.9993 0.9988 0.9981 0.9972 0.9963 0.9951 0.9939 c3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 88 3.8- Test No.2 Table 3.16: A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) δ̂ 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 η -3791.1856 -3786.8033 -3781.9247 -3776.5385 -3770.6324 -3764.1936 -3757.2084 -3749.6631 -3741.5434 -3732.8347 e 0.0000 0.0012 0.0024 0.0039 0.0054 0.0071 0.0090 0.0110 0.0131 0.0154 c0 0.0000 0.0463 0.0926 0.1388 0.1847 0.2305 0.2760 0.3211 0.3659 0.4102 Distortion c1 1.0000 0.9784 0.9573 0.9367 0.9166 0.8971 0.8781 0.8596 0.8417 0.8243 measure. c2 1.0000 0.9936 0.9869 0.9799 0.9727 0.9650 0.9571 0.9489 0.9404 0.9315 c3 1.0000 0.9938 0.9879 0.9820 0.9764 0.9708 0.9654 0.9601 0.9549 0.9498 Table 3.17: A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) δ̂ 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 η -3791.1856 -3786.7882 -3780.9705 -3773.7398 -3765.1218 -3755.1588 -3743.9076 -3731.4362 -3717.8206 -3703.1413 e 0.0000 0.0012 0.0027 0.0046 0.0069 0.0095 0.0125 0.0158 0.0194 0.0232 c0 0.0000 0.0707 0.1410 0.2106 0.2792 0.3465 0.4122 0.4761 0.5381 0.5980 Distortion c1 1.0000 0.9753 0.9512 0.9278 0.9050 0.8828 0.8612 0.8402 0.8198 0.8001 measure. c2 1.0000 0.9997 0.9988 0.9972 0.9950 0.9923 0.9889 0.9850 0.9806 0.9756 c3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Table 3.18: A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) δ̂ 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 η -3791.1856 -3788.0883 -3782.5727 -3774.5421 -3763.8307 -3750.1957 -3733.3006 -3712.6732 -3687.5855 -3656.5848 e 0.0000 0.0008 0.0023 0.0044 0.0072 0.0108 0.0153 0.0207 0.0273 0.0355 c0 0.0000 0.1027 0.2071 0.3150 0.4279 0.5468 0.6724 0.8049 0.9437 1.0876 Distortion c1 1.0000 0.9861 0.9725 0.9591 0.9460 0.9332 0.9205 0.9081 0.8959 0.8839 measure. c2 1.0000 0.9861 0.9726 0.9595 0.9468 0.9345 0.9227 0.9113 0.9004 0.8900 c3 1.0000 0.9861 0.9725 0.9593 0.9463 0.9336 0.9212 0.9090 0.8970 0.8852 Chapter 4 A posteriori error estimation Nella prima parte di questo capitolo, vengono introdotti i concetti di base su cui si fonda la stima a posteriori dell’errore di discretizzazione in analisi agli elementi finiti. In seguito, l ’attenzione é incentrata sui cosiddetti "patch recovery methods", ovvero sui metodi che stimano l ’errore per confronto della soluzione agli elementi finiti con una più accurata ottenuta previa opportuna ricostruzione (recovery) del campo di sforzi operata localmente su piccoli insiemi di elementi contigui (patch). In questo contesto, sono illustrate alcune delle più note tecniche presenti in letteratura e viene sviluppata una nuova procedura di ricostruzione locale degli sforzi, denominata RCP (Recovery by Compatibility in Patches), ottenuta rilassando la compatibilità cinematica, attraverso la minimizzazione di un funzionale tipo energia complementare, definito sulla patch. Nell’ultima parte del capitolo, viene condotta un ’ampia sperimentazione numerica che mostra come la nuova procedura risulti versatile, robusta e superconvergente. 4.1 Aims and concepts of a posteriori error estimation The remarkable success of some a posteriori error estimators has opened a new chapter in computational mechanics. By eﬀectively estimating error, the possibility of controlling the entire computational process through new adaptive algo- 90 4.1- Aims and concepts of a posteriori error estimation rithms emerges. Fresh criteria for judging the performance of algorithms become apparent. Most importantly, the analyst can use a posteriori error estimates as an independent measure of the quality of the simulation under study. The a priori estimation of errors in numerical methods has long been an enterprise of numerical analysts. Such estimates give information on the convergence and stability of various solvers and can give rough information on the asymptotic behavior of errors in calculations as mesh parameters are appropriately varied. Traditionally, the practitioner using numerical simulations, while aware that errors exist, is rarely concerned with quantifying them. The quality of a simulation is generally assessed by physical or heuristic arguments based on the experience and judgment of the analyst. Frequently such arguments are later proved to be flawed. Some of the earliest a posteriori error estimates used in computational mechanics were developed for the solution of ordinary diﬀerential equations. These are typified by predictor-corrector algorithms in which the diﬀerence in solutions obtained by schemes with diﬀerent orders of truncation error is used as rough estimates of the error. This estimate can in turn be used to adjust the discretization step. It is notable that the original a posteriori error estimation schemes for elliptic problems had many features that resemble those for ordinary diﬀerential equations. Modern interest in a posteriori error estimation for finite element methods for two point elliptic boundary value problems began with the pioneering work of Babuška and Rheinboldt [Babuška & Rheinboldt, 1978]-[Babuška & Rheinboldt, 1979]. A posteriori error estimation techniques were developed by investigating the residuals occurring in a patch of elements or even in a single element what makes possible to estimate the errors in energy, or an energy norm, which arise locally. The original approaches for error estimation and indeed for correcting these by adaptive refinement started by simply looking at the elements with largest error and dividing these to achieve some acceptable accuracy. These formed the basis of adaptive meshing procedures designed to control and minimize the error. The use of complementary energy formulations for obtaining a posteriori error estimates was put forward by de Veubeke [de Veubeke, 1965]. However, the method failed to gain popularity being based an a global computation. The idea of solving element by element complementary problems together with the important concept of constructing equilibrated boundary data to obtain error estimates was advanced by Ladeveze and Leguillon [Ladeveze & Leguillon, 1983]. Related ideas are found in the work of Kelly [Kelly, 1984]. Chapter 4 91 During the early 1980s the search for eﬀective adaptive methods led to a wide variety of ad hoc error estimators. Many of these were based on a priori or interpolation estimates, that provided crude but eﬀective indications of features of error suﬃcient to drive adaptive processes. Since these early works many players entered the field and today the procedures available for error estimation are essentially reduced to two kinds. These are, first, the residual error estimators, continuing the original suggestions of Babuška but the emphasis is now on use of self-equilibrating patches. Here the work of Ainsworth and Oden [Ainsworth & Oden, 1992], [Ainsworth & Oden, 1993], and others is important. Such residual approaches have gained universal mathematical aplomb and for some years formed the basis of acceptable error estimators. The second kind of procedures was initially suggested by Zienkiewicz and Zhu in 1987 [Zienkiewicz & Zhu, 1987] and consist of using a recovery process to obtain a more accurate representation of the unknowns. In the original procedure, simple averaging and the so-called L2 projection recovery of Brauchli and Oden [Brauchli & Oden, 1971] were used to estimate errors. This is a simple error estimation technique that is eﬀective for some classes of problems and types of finite element approximations. Indeed, their method started the new category of recoverybased methods: gradients of solutions obtained on a certain domain partition are smoothed and then compared with the gradients of the original solution to assess error. This approach was later modified [Zienkiewicz & Zhu, 1992a]-[Zienkiewicz & Zhu, 1992b], leading to the well-known Superconvergent Patch Recovery (SPR) method, which provides very accurate recovered values that generally converge at a higher rate than the original finite element solution so giving a solid basis for error estimation. In the SPR, the standard process was to find, in a patch formed by the elements surrounding a node or a element, a set of points at which the accuracy of the finite element solution was higher, such as the superconvergence points, and then to pass a least squares approximation of the function of the stresses or gradients (as in the case of thermal problems) through such values. With the knowledge of these more accurate stresses or gradients of the solution, the energy norm could easily be estimated and, indeed, so could the local values of stresses and gradients [Lee et al. , 1997]—[Lo & Lee, 1998]. Today the methodology is widely used, industrially, following the same practice, but some variants of it have been introduced. In one of these variants the superconvergent values are taken not as the stresses or gradients of the solution, but on the values of the original functions themselves. It is known for instance that superconvergence will always occur in these values for triangular elements 92 4.1- Residual based approaches at the vertex nodes, whereas in the same triangular elements such superconvergence for the stresses or gradients does not exist. Therefore for triangular elements it is preferable to include in the patch superconvergence of the displacements or the basic function itself and then to find its gradients by establishing a suitable polynomial of higher order. This procedure of obtaining the recovered values is currently widely used and many papers by Wiberg have contributed very much to this recovery methodology [Wiberg & Abdulwahab, 1993]-[Wiberg, 1997]-[Wiberg & Abdulwahab, 1997a]-[Wiberg & Abdulwahab, 1997b]-[Wiberg & Li, 1999]. The subject of a posteriori error estimation for finite element approximation has now reached maturity. In fact, study of robustness of existing estimators and identifying limits on their performance has gained importance. Noteworthy in this respect is the work of Babuška et al. [Babuška et al. , 1994a]-[Babuška et al. , 1994b] who conducted an extensive study on the performance and robustness of the main error estimation techniques. This methodology is essentially a sort of patch test (Babuška patch test) in which arbitrary fields of finite elements, assumed to be repeatable in space, are subjected to the next higher order basic polynomial contained in the original shape functions. Obviously if the finite elements used include quadratics any solution which is quadratic terms will be exactly returned through the numerical computation and the main errors will be concentrated only in the cubic terms. Thus it is necessary, in a repeatable sense, to impose only cubic polynomials on the patch and test whether errors arise and, if so, their magnitude. By doing this it is easy to establish whether the procedures capture errors exactly or simply show the errors for inexact solutions. Here Babuška proposed to value each error estimator with a robustness index and after many tests concluded that the robustness index was definitively optimal for the recovery methods, with the residual methodology giving much poorer results. 4.1.1 Residual based approaches The residual type of error estimator was first introduced by Babuška and Rheinboldt [Babuška & Rheinboldt, 1978] and their research was later followed by many others [Ladeveze & Leguillon, 1983]—[Babuška & Miller, 1987]—[Ainsworth & Oden, 1992]—[Ainsworth & Oden, 1993]—[Ainsworth & Oden, 1997]. This type of error estimator is computed by using the residuals of the finite element solution implicitly or explicitly. The term explicit or implicit is used in a context similar to the characterization of explicit and implicit diﬀerence schemes for time-marching algorithms: explicit schemes involve a direct computation using available data while implicit schemes involve the solution of an algebraic system of equations. The implicit residual error estimators are determined by solving local boundary value Chapter 4 93 problems for the error, which use the residuals of the finite element solution as data. The explicit residual error estimators are, on the other hand, expressed directly by the residuals of the discretization solution. As might be expected, explicit schemes generally require less computational eﬀort than implicit schemes but involve compromises in robustness and in utility as means for accurate, quantitative error estimation. The accuracy of the implicit residual error estimator depends on how the local problem is solved. In particular, how the boundary conditions of the local problem are set. Several techniques have been developed to split the flux on the boundary of the element by satisfying some equilibrium conditions. The accuracy of the explicit residual error estimator depends on the accuracy of the constants contained in the error estimator. The robustness of this type of error estimator, as shown by Babuška and coworkers [Babuška et al. , 1994a]—[Babuška et al. , 1994b]—[Babuška et al. , 1995b]—[Babuška et al. , 1995a]—[Babuška et al. , 1997a]—[Babuška et al. , 1997c]—[Babuška et al. , 1998], depends on the regularity of the problem and the mesh used in the finite element analysis. 4.1.2 Recovery based approaches The finite element method produces the optimal approximation from the finite element subspace. However, it is frequently the case that the finite element analyst is more interested in the gradient of the finite element approximation, than in the approximation itself. For instance, in computational elasticity, the stresses and strains are the primary concern, rather than the displacements. Furthermore, the normal component of the gradient of the approximation is generally discontinuous across the element boundaries, meaning that the practitioner is presented with a discontinuous approximation to the main quantity of interest. For this reason, many finite element packages incorporate a post-processing procedure whereby the discontinuous approximation to the gradient is smoothed before being presented to the user. The reasons for performing such a post-processing are not purely cosmetic: under certain circumstances it is found that the accuracy of the smoothed gradient is superior to the approximation provided by the untreated gradient of the original finite element approximation. A rather natural approach to a posteriori error estimation is based on measuring the diﬀerence between the direct and post-processed approximations to the gradient. This ostensibly rather crude approach can result in astonishingly good estimates for the true error. In this approach, the error estimator is computed by first using some recovery techniques to achieve a more accurate, or even superconvergent solution from the finite element approximation. The recovered solution is then used in place of the exact solution to compute the 94 4.1- Recovery based approaches error: exact error estimated error = FEM solution - exact solution, (4.1) = FEM solution - recovered solution. (4.2) This type of error estimator is very easy to implement and is computationally very eﬃcient. Obviously the accuracy of the error estimator depends on the accuracy of the recovered solution. The appeal of such procedures to practitioners is easy to appreciate. The ready availability of a post-processed gradient in the finite element package means that the estimator may be easily implemented. This approach allows considerable leeway in the selection of the post-processed gradient, and there are as many diﬀerent estimators as there are post-processing techniques. One class of techniques that is particularly popular with the engineering community is averaging methods. The gradient of the finite element approximation provides a discontinuous approximation to the true gradient. This may be used to construct an approximation at each node by averaging contributions from each of the elements surrounding the node. These values may then be interpolated to obtain a continuous approximation over the whole domain. The specific steps used to construct the averaged gradient at the nodes distinguish the various estimators and have a major influence on the accuracy and robustness of the ensuing estimator. Nodal averaging of derivatives (or stresses) has been practised since the inception of the finite element method to present the user with more realistic results. Although the basis of the process was initially intuitive its success is closely linked with the existence of superconvergent points in many finite element solutions at which the normal rate of theoretical convergence is exceeded by one order. The existence of such optimal sampling points has been suggested quite early [Barlow, 1976]-[Barlow, 1977] and has been subject to much research since. Extrapolation from such superconvergent points and subsequent averaging at nodes (local projection) gives clearly a sound basis for improved nodal values obtainable for linear elements in which the superconvergent point is that of the element centroid for quadrilaterals and that of a mid side of triangles. A widely used technique of Hinton and Campbell [Hinton & Campbell, 1974] for quadrilateral quadratic elements follows a similar procedure using the 2 × 2 Gauss points sampling and bilinear extrapolation. Diﬀerent strategies were developed in recent years, namely, simple nodal averaging, L2 -projections (see [Brauchli & Oden, 1971]), and the so called Superconvergent Patch Recovery (SPR) technique, introduced by Zienkiewicz and Zhu Chapter 4 95 [Zienkiewicz & Zhu, 1992c], [Zienkiewicz & Zhu, 1992a], [Zienkiewicz & Zhu, 1992b]-[Zienkiewicz et al. , 1993]-[Zienkiewicz & Zhu, 1995]. Meanwhile, various enhancements of the SPR technique have been presented (see e.g. Wiberg and Abdulwahab [Wiberg & Abdulwahab, 1993]-[Wiberg, 1997]-[Wiberg & Abdulwahab, 1997a]-[Wiberg & Abdulwahab, 1997b], Blacker and Belytschko [Blacker & Belytschko, 1994], Kvamsdal and Okstad [Kvamsdal & Okstad, 1998], Duarte [Duarte & Dutra Do Carmo, 2000] and Gu [Gu et al. , 2004]). 4.2 Some recovery methods In this section, two of the most popular recovery based methods, that is the SPR method, the REP method are illustrated. 4.2.1 Superconvergent Patch Recovery: SPR The Superconvergent Patch Recovery procedure was introduced by Zienkiewicz and Zhu in 1992 [Zienkiewicz & Zhu, 1992c] and is characterized by single and continuous polynomial expansion of the function describing the derivatives used on an element patch surrounding the nodes at which recovery is desired. This expansion can be made to fit locally the superconvergent points in a least square manner or simply be an L2 projection of the consistent finite element derivatives. The first of these processes will be shown always to lead to superconvergent recovery of nodal derivatives or stresses. The second, although not generally superconvergent, will still always result in a considerable improvement of recovered nodal values. In the previous chapter we have shown that sampling of the gradients and stresses at some particular points is generally optimal and possesses a higher order accuracy when such points are superconvergent. However, we would also like to have similarly accurate quantities elsewhere within each element for general analysis purposes, and in particular we need such highly accurate gradients and stresses when the energy norm or other similar norms have to be evaluated in error estimates. We have already shown how with some elements very large errors exist beyond the superconvergent point and attempts have been made from the earliest days to obtain a complete picture of stresses which is more accurate overall. Here attempts are generally made to recover the nodal values of stresses and gradients from those sampled internally and then to assume that throughout the element the recovered stresses σ ∗ are obtained by interpolation in the same manner as the displacements by, first, recovering nodal values σ̄ ∗ and, then, interpolating these values by standard shape functions: σ ∗ = Uσ̄ ∗ , (4.3) 96 4.2- Implementation of the SPR procedure where U are the same basis functions as for the interpolation of displacements in Equation (2.22). Thus, the key step is to determine the nodal values σ̄ ∗ . This is a process used almost from the beginning of finite element calculations for triangular elements, where elements are sampled at the centroid (assuming linear shape functions have been used) and then the stresses are averaged at nodes. However this is not the best for triangles and for higher order elements such averaging is inadequate. 4.2.1.1 Implementation of the SPR procedure Local patches of elements are considered, assembled around a central node. If we accept the superconvergence of σ h at certain points s in each element, then it is a simple matter to compute σ ∗ which is superconvergent at all points within the element. The procedure is illustrated for two dimensions in Figure 4.1. At the superconvergent points the values of σ h are accurate to order p and not p − 1 as is true elsewhere (p is the order of the polynomials in the dispacement shape functions). However, we can easily obtain an approximation given by a polynomial of degree p, with identical order to these occurring in the shape function for displacement, which has superconvergent accuracy everywhere if this polynomial is made to fit the superconvergent points in a least square manner. Figure 4.1: Interior patches for quadrilateral elements: (a) 4-nodes elements; (b) 8-node elements; (c) 9-node elements Thus we proceed by introducing the following approximation for the recovered solution over the patch σ ∗p = P∗ a. ∗ (4.4) where P is a matrix of approximation functions, a is a vector of unknown parameters and subscript ·p emphasizes that the recovery is done over the patch. Chapter 4 97 For instance, in two dimensional elasticity P∗ and a are given by ⎤ ⎤ ⎡ ⎡ ⎤ ⎡ P̃ (x) σ 11 a1 ⎦ , a = ⎣ a2 ⎦ , σ ∗ = ⎣ σ 22 ⎦ , P∗ = ⎣ P̃ (x) τ 12 a3 P̃ (x) (4.5) where P̃ (x) is a 1×m vector that contains the polynomials terms in x coordinates, ai is a m × 1 vector of unknown coeﬃcients and m is the number of terms used to form the polynomial bases of order p i h x2 , . . . , x1 xp−1 , xp2 . (4.6) P̃ (x, y) = 1, x1 , x2 , . . . , xp1 , xp−1 1 2 To evaluate the vector of unknown parameters a, we minimize, the following functional n X £ h ¤2 σ (xi ) − σ ∗p (xi ) = F (a) = i=1 n X £ h ¤2 σ (xi ) − P∗ (xi )a , = (4.7) i=1 over an assigned patch Ωp Ωp = nep [ Ωj, (4.8) j=1 formed by nep elements, where xi is the i-th superconvergent point, and n = nep×k is the amount of sampling points being k the number of sampling points for element. Minimization of F (a) yields: n n X ∂F (a) X ∗T ∗ = P (xi )P (xi )a− P∗T (xi )σ h (xi ) = 0, (4.9) ∂a i=1 i=1 or in compact form Ha = g, where H= n X (4.10) P∗T (xi )P∗ (xi ), (4.11) P∗T (xi )σ h (xi ). (4.12) i=1 and g= n X i=1 Equation (4.10) can now be solved to compute the coeﬃcients a a = H−1 g. σ ∗p ∗ (4.13) The availability of allows the values of σ̄ to be determined at all nodes. As some nodes belong to more than one patch, average values of σ̄ ∗ are best obtained. Finally, the recovered solution σ ∗ throughout each element is achieved by Equation (4.3). 98 4.2- L2 recovery variant Figure 4.2: Recovery of exact σ of degree p by linear elements (p = 1) In Figure 4.2 and 4.3 we show, in a one-dimensional example, how the Superconvergent Patch Recovery works. The Superconvergent Patch Recovery can be extended to produce superconvergent displacements. The procedure for the displacements is quite simple if we assume the superconvergent points to be at nodes of the patch. However, as we have already observed it is always necessary to have more data than the number of coeﬃcients in the particular polynomial to be able to execute a least square minimization. Here of course we occasionally need a patch which extends further than before, particularly since the displacements will be given by a polynomial one order higher than that used for the shape functions. The recovered solution σ ∗ has on occasion been used in dynamic problems, because in dynamic problems the displacements themselves are often important. 4.2.1.2 L2 recovery variant An alternative procedure lies on the local L2 projection to obtain the polynomial solution locally on each patch. The functional F (a) to be minimized is given by Z Z (σ h − σ ∗p )2 dV = (σ h − P∗ a)2 dV, (4.14) F (a)= Ωp Ωp where, similar to (4.8), the patch domain is defined by nep [ Ωp = Ωj . (4.15) j=1 Minimizing F (a) yields Z Ωp and again P∗T P∗ dV a = Z P∗T σ h dV, (4.16) Ωp a = H−1 g, (4.17) Chapter 4 99 where H= Z Ωp ∗T P ∗ P dV and g= Z P∗T σ h dV. (4.18) Ωp It is interesting to note that if we use the numerical integration to compute the Equation (4.18), we obtain an analogous expression to Equation (4.11); in this case each terms of the sum will be multiplied by a weight coeﬃcient, and we obtain the same results obtained before if and only if the integration points are the same of the superconvergence points. The L2 recovery gives much inferior answers, showing superconvergence only for odd values of p and almost no improvement for even values of p. Figure 4.3: Recovery of exact σ of degree p by quadratic elements (p = 2) 4.2.2 Recovery by Equilibrium in Patches: REP Although SPR has proved to work well generally, the reason behind its capability of producing an accurate recovered solution even when superconvergent points do not in fact exist remains an open question. An alternative, known by the acronym REP (Recovery by Equilibrium in Patches), was first presented in [Boroomand & Zienkiewicz, 1997b] and later improved in [Boroomand & Zienkiewicz, 1997a][Boroomand & Zienkiewicz, 1999]-[Boroomand et al. , 1999]. To some extent the motivation is similar to that of Ladeveze et al. [Ladeveze & Leguillon, 1983] who sought to establish (for somewhat diﬀerent reasons) a fully equilibrating stress field which can replace that of the finite element approximation. The method operates in the some way of SPR. It is aimed at recovering a new solution σ ∗p over patches of elements. These solutions are used to compute the 100 4.2- Implementation of the REP procedure nodal values σ̄ ∗p which are formally interpolated by Equation (4.3) to determine the recovered stresses over the whole domain. 4.2.2.1 Implementation of the REP procedure Consider a patch, as shown in Figure 4.4. The finite element solution derives by imposing the discrete equilibrium of all elements. Thus also every isolated patch of elements Ωp will be in equilibrium. This can be shown by writing the equilibrium equations for such a patch as FW - W p Actions of remaining domain Figure 4.4: Example of a patch of elements Z Ωp T h B σ dV − Z T Ωp U bdV − Z UT t̄dV + FΩ−Ωp = 0, ∂Ωp t (4.19) where FΩ−Ωp represents the forces from the domain Ω−Ωp and its exterior boundary (see Figure 4.4) and B is given by Equation (2.24). Thus, we can write for any patch Z BT σ h dV = g, (4.20) Ωp with g = −FΩ−Ωp − FΩp , where FΩp are the equilibrant nodal forces: Z Z UT bdV − FΩp = − Ωp (4.21) UT t̄dS. (4.22) ∂Ωp t Now assuming the forces on the right-hand side of Equation (4.22) are known, the patch of elements can be considered as a separate system on which these forces are acting as nodal forces. Thus, again applying the virtual work principle, we can Chapter 4 101 write for any stress system σ ∗p the discrete equilibrium equation as Z BT σ ∗p dV = g, (4.23) Ωp and Z Z BT σ ∗p dV = Ωp and consequently Z BT σ h dV, (4.24) BT CBūh dV. (4.25) Ωp BT σ ∗p dV = Ωp Z Ωp At this stage we shall assume a continuous form of stresses over the patch. This can be done in several ways: 1. smoothing the stresses by defining these in a polynomial form: σ ∗p = P∗ a, resulting in ÃZ T ∗ B P dV Ωp 2. ! a= ÃZ (4.26) T B CBdV Ωp ! ūh ; smoothing the strains: ²∗p = P∗ a, resulting in ÃZ T ∗ B CP dV Ωp 3. (4.27) ! a= ÃZ (4.28) T B CBdV Ωp ! ūh ; (4.29) or smoothing the displacements: u∗p = P∗ a, resulting in ÃZ Ωp BT CDP∗ dV ! a= ÃZ Ωp (4.30) BT CBdV ! ūh . (4.31) In all the above cases the number of unknowns should be equal to or less than the number of equations implied in Equation (4.25). In the third case the rigidbody motion must be omitted from series of unknowns. One convenient way for this is to consider the displacements of all boundary nodes of the patch and minimize the square diﬀerence of recovered values and those of finite element method. As already mentioned, to achieve the recovery we proceed in an exactly analogous way to that used in the SPR procedure, first approximating the stress in each patch by a polynomial of appropriate order σ ∗p , second using this approximation to obtain nodal values of σ̄ ∗ and finally interpolating these values by standard 102 4.2- Implementation of the REP procedure shape functions. Therfore, in what follows we shall only use the first type of smoothing and assume that σ ∗p = P∗ a, and so ÃZ ∗ T B P dV Ωp ! a= ÃZ (4.32) T B CBdV Ωp ! ūh . (4.33) The stress is taken as a vector of appropriate components, and for equilibrium we shall attempt to ensure that the smoothed stress σ ∗p satisfies in the least square sense the same patch equilibrium conditions as the finite element solution. For instance, in plane elasticity the stress approximation is taken as ⎫ ⎤⎧ P̃ (x) 0 0 ⎨ a1 ⎬ a2 . σ ∗p = P∗ a = ⎣ 0 (4.34) P̃ (x) 0 ⎦ ⎩ ⎭ a3 0 0 P̃ (x) Obvious modifications are made for more or less components. It has been found in practice that the constraints provided in this way are not suﬃcient to always produce non-singular least square minimization. Accordingly, the equilibrium constraints are split into an alternative form in which each component of stress σ ∗pi is subjected to equilibrium requirements. This may be achieved by expressing the stress as X 1i σ ∗pi , (4.35) σ∗ = ⎡ i where 1i represents a zero vector having unit value at the position of i-th stress (i.e. 12 = [0, 1, 0]T for the second stress component), and imposing the set of constraints Z Z BT σ ∗pi dV = Ωp BT σ hpi dV. (4.36) Ωp The imposition of the above equation allows each set of coeﬃcients ai to be solved independently. The least square minimization is based on the following functional !T ÃZ ! ÃZ Z Z BT σ ∗pi dV − BT σ hi dV BT σ ∗pi dV − BT σ hi dV , F = Ωp Ωp Ωp Ωp (4.37) providing the number of parameter ai is smaller (or equal) than the number of equations implied. The above functional can be rewritten as T F (ai ) = (Hi ai − gi ) (Hi ai − gi ) , where Hi = Z BT 1i P̃dV, Ωp (4.38) (4.39) Chapter 4 103 and gi = Z T B σ hi dV Ωp = ÃZ T Ωp B 1i ⊗ 1T i CBdV ! ūh . (4.40) The minimization condition results in ¤−1 T £ ai = HT Hi gi . i Hi (4.41) Sometimes the application of this minimization to the patch alone is not suﬃcient to provide non-singular solution due to a large number of parameters a. Generally, this may be eliminated by modifying the patch requirement to the minimization of nep X F ∗ (ai ) = (Hi ai − gi )T (Hi ai − gi ) + α (Hei ai − gie )T (Hei ai − gie ) , (4.42) e where the added terms represent modification on individual elements and α is a parameter. Z e Hi = BT 1i P̃dV ai , (4.43) Ωp gie = Z BT σ hi dV. (4.44) Ωp Minimization now gives " ai = HT i Hi +α X e e HeT i Hi #−1 " HT i gi +α X e e HeT i gi # . (4.45) The REP procedure gives overall an approximation which does not require knowledge of any superconvergent points. The accuracy of REP and SPR is comparable. 4.2.2.2 Some remark on superconvergence Numerical tests show that the REP procedure exhibits superconvergence. The reason lies on the formulation that is based on using nodal forces. Referring to Equation (4.19) it can be seen that these forces are directly related to integrals of the displacement shape functions. Therefore, the rate of convergence of these forces is the same as that of displacements. This means that the rate of convergence of stresses in Equation (4.23) will, at least, be that of displacements if proper smoothing function is used. Alternatively, we can prove this by using the principle of minimum potential energy. We must show that the answer, after smoothing, will approach the exact solution with higher order of error than that of the standard finite element solution. For this we shall use the equivalence of minimization of error of potential energy and minimization of total potential energy. The equivalence has been proved by Herrmann in [Herrmann, 1972]. Starting from Equation (4.20) we can consider 104 4.2- Some remark on superconvergence this equation as an answer of minimization of total potential energy over a patch provided that the nodal forces of patch are known: Z ³ ´T ³ ´ Πp = (4.46) Būh C Būh dV − ūhT FΩ−Ωp . Ωp In fact, this energy is a part of total energy of system. If minimum supports for this sub domain are provided the answers for stresses will be the same as those from solution of the whole domain. But according to Herrmann’s theorem this is equivalent to minimization of error of energy with respect to ū. Thus, we minimize Z h ³ ´iT h ³ ´i ∆E = C D u − Uūh dV, (4.47) D u − Uūh Ωp where u is the exact displacement field. Now we have ∂∆E = 0, ∂ūh which gives Z Z BT CBūh dV. BT CDudV = Ωp (4.48) Ωp From Equation (4.25) we recall that the new approach is based on Z Z BT CDP∗ adV = BT CBūh dV. (4.49) Comparing (4.48) with (4.49), Z Z BT CDudV = (4.50) Ωp Ωp Ωp or Z Ωp BT CDP∗ adV, Ωp BT C (Du − DP∗ a) dV = 0. (4.51) The matrix BT C can be considered as a discontinuous weight function over the domain. So Equation (4.51) implies that Du ' DP∗ a. T Multiplying both sides of above equation by (DP∗ ) D and taking integral over the patch gives Z Z T T (DP∗ ) CDūh dV ' (DP∗ ) CDP∗ adV. (4.52) Ωp Ωp Now we can compare the above result with that of minimization of ∆E with respect to a: Z ∆E T = Ωp ∂∆E ∂a = 0, [D (u − P∗ a)] C [D (u − P∗ a)] dV, (4.53) (4.54) Chapter 4 which leads to 105 Z Ωp ∗ T h (DP ) CDū dV = Z T (DP∗ ) CDP∗ adV. (4.55) Ωp From equations (4.52) and (4.55) it can be stated that REP gives the answers of minimization of least square of the stress errors in energy norm. In fact, we have changed the finite element bases from shape functions with discontinuous derivatives to a continuous polynomials with higher order. Equality (4.52) is dependent on the form of weighting function BT C in Equation (4.51). So the form of elements used in the mesh can aﬀect the super-convergence and therefore it can be expected that for diﬀerent kinds of meshes the rate of convergence will diﬀer. It is clear that the order of polynomial used in finite element procedure determines the order of convergence of answers. So obviously if we are able to change the order of finite element bases, we can increase the rate of convergence by using a higher order polynomial. 4.3 Recovery by Compatibility in Patches: RCP In this section, a new stress recovery procedure is presented [Ubertini, 2002]— [Ubertini, 2004]—[Benedetti et al. , 2006]. The method proceeds in a way analogous to that used in both the SPR procedure and the REP procedure. The desired improved approximation for stresses over the whole domain, say σ ∗ , is constructed by, first, recovering nodal values σ̄ ∗ and, then, interpolating these values by standard shape functions (see Equation (2.24)). Thus, the key step is to determine the nodal values σ̄ ∗ . To accomplish this task, the problem domain is subdivided into local patches of elements. Each assembly node of the finite element discretization is associated to a patch. Such a patch is defined as a union of elements surrounding the particular assembly node considered. An illustration of typical patches associated to corner nodes internal to the domain is shown in Figure 4.5(a). Some additional details about patches associated to nodes in the interior or on the edge of an element as well as to nodes lying on the domain boundary are given in Section 4.3.4. 4.3.1 Formulation of the RCP procedure Let Ωp be the domain of the typical patch, ∂Ωp the boundary of the patch and σ ∗p the local stress field in Ωp which is to be determined aiming at improving the finite element solution. The current patch is considered as a separate system on which displacements are prescribed along the boundary ∂Ωp , as shown in Figure 4.5(b). 106 4.3- Formulation of the RCP procedure FEM displacements prescribed :p :p (a) (b) Figure 4.5: Example of pacth: (•) assembly node defining the patch The prescribed displacements are those resulting from the finite element analysis. Then, the local stress field σ ∗p is determined by minimizing the convex functional of complementary energy associated to this separate system: Z Z 1 ∗ ∗ T −1 ∗ Π(σ p ) = σ C σ p dV − uh T NT σ ∗p dS, (4.56) 2 Ωp p ∂Ωp over a set of stress fields fulfilling equilibrium equations within the patch: D∗ σ ∗p = b in Ωp . h (4.57) σ ∗p Since both u and are continuous over the patch, the above functional can be re-expressed in the equivalent form: ! Ã Z Z nep X 1 ∗ ∗ T −1 ∗ hT T ∗ Π(σ p ) = σ C σ p dV − u N σ p dS , (4.58) 2 Ωj p ∂Ωj j=1 where nep is the number of elements in the patch and Ωj and ∂Ωj are the domain and the boundary of the generic element of the patch, respectively. Notice that Equation (4.56) is the complementary energy of the separate patch system, which does not coincide in general with the exact complementary energy of the corresponding subdomain embedded into the original system, since uh does not coincide in general with the exact displacement u. This simple procedure is based on the idea of improving stresses by enhancing equilibrium while relaxing compatibility. In fact, using the Gauss-Green formula (see Equation (2.16)) together with the following uh = U (x) ūh , and with the Equation (4.57), functional (4.58) can be written as Z Z Z 1 h ∗ ∗ T −1 ∗ ∗T σ C σ p dV − σ p Bū dV + bT Uūh dV. Π(σ p ) = 2 Ωp p Ωp Ωp (4.59) (4.60) Denoting by ²h the strains resulting from the finite element solution and by ²∗p the strains associated with the recovered stresses: ²h = Būh , ²∗p = C−1 σ ∗p , (4.61) Chapter 4 107 the first variation of the above functional takes the form Z ¡ ¢ δΠ = δσ ∗p T ²∗p − ²h dV. (4.62) Ωp This expression shows that the present procedure leads to local compatibility conditions. In particular, it enforces strains coming from equilibrated stresses to be equal to compatible finite element strains, by projecting the error into the new stress space over each patch. Hence, the procedure is a Recovery of stresses by Compatibility in Patches (RCP). It is interesting to notice that the present procedure can be viewed as dual, in a certain sense, to the already mentioned procedure called Recovery by Equilibrium in Patches (REP). Indeed, dual should be intended only in the sense that the condition enforced by REP can be obtained from Equation (4.62) by exchanging the roles played by σ and ². This means to project the error between new and original stresses into the finite element strain space over the patch. 4.3.2 Implementation of the RCP procedure In this section, a simple and low-cost implementation of the RCP procedure is presented. The first step is to select an approximation for local stresses over the patch. In order to obtain stresses which satisfy interior equilibrium, the following additive decomposition is introduced σ ∗p = σ ∗ph + σ ∗pp , σ ∗ph (4.63) σ ∗pp where is the homogeneous solution of Equation (4.57) and is a particular ∗ ∗ solution of the same equation. More details about σ ph and σ pp are given in Section 4.3.4. Notice that only σ ∗ph is indeterminate at this stage. Now, an approximation for σ ∗ph over the patch is assumed: σ ∗ph = P∗ a, (4.64) where P∗ is the matrix of basis functions and a is the vector of unknown parameters. For the above assumptions, matrix P∗ should be selected such that D∗ P∗ = 0 in Ωp , (4.65) i.e., P∗ is a matrix of self-equilibrated stress modes. The minimization condition results in the following system of linear algebraic equations H a = g, with matrix H and vector g given by nep X H= Hj , j=1 g= nep X ¡ u ¢ gj − gjp , j=1 (4.66) (4.67) 108 4.3- Some remarks where Hj = gju = Z Z P∗T C−1 P∗ dV, Ωj P∗T Būh dV = Ωj gjp = Z ÃZ P∗T B dV Ωj P∗T C−1 σ ∗pp dV. (68) ! ūh , (69) (70) Ωj The outlined procedure can be easily implemented in existing finite element codes and possesses some features which make it simpler and cheaper than other patchbased recovery procedures. In fact, the number of unknown parameters a is equal to the number of equations of system (4.66) and the coeﬃcient matrix H is symmetric and positive definite. Thus, system (4.66) can be solved directly using standard procedures. Notice that this is not the case of other recovery techniques, such as SPR and REP, which need to resort to least-squares scheme. Unlike the SPR procedure, RCP does not require any knowledge of super-convergent points. Moreover, matrix H and vector g are simply obtained by summing contributions from each element forming the patch, as shown by Equation (4.67). Finally, numerical stability is always guaranteed, so that the RCP procedure can be applied also to small patches (such as boundary ones) without any special modification to add extra constraints. 4.3.3 Consistency of the RCP procedure A basic requirement for a recovery procedure, as well as for any numerical procedure, is that the exact solution should be reproduced under favorable circumstances. In the present context, such a condition can be formalized by requiring that if the finite element solution is exact, then also the local recovered stresses should be exact [Ainsworth & Oden, 2000]. It can be immediately realized that the RCP procedure satisfies this consistency condition if the local stress approximation is properly selected. In fact, let uh be a polynomial of degree p which coincides with the exact solution u. Then, Equation (4.56) actually represents the exact complementary energy for the subdomain corresponding to the current patch and the exact stresses are always recovered if the self-equilibrated field σ ∗ph belongs to the space of p-order polynomials which satisfy condition (4.65). Except for internally isostatic models such as rods or beams, this means that each component of σ ∗ph should be able to reproduce a complete polynomial of degree p. Thus, consistency is met if self-equilibrated stress approximations are selected among polynomials of degree not less than p, when finite element of order p are used. Chapter 4 4.3.4 109 Some remarks In this section, some issues about the proposed procedure are addressed. The first issue is the form of the typical patch. In principle, the number of elements forming the patch may be large at will. Also the whole domain could be considered as a patch. However, using large patches increases the computational cost and, as a consequence, the resultant procedure could become no more convenient in practice. Here, patches of conventional minimal form are used, as shown in Figure 4.5. In other words, each patch is formed using only one layer of elements surrounding the patch assembly node, so to minimize the computational cost. Figure 4.6: Computation of nodal values for four-, eight- and nine-node elements: (◦) nodal points, (•) nodal values determined by recovery procedure The second issue to be discussed is how to construct local stress approximations. Indeed, this can be easily done by employing the well-established techniques usually adopted to construct stress approximations for the original version of hybrid stress elements [Pian, 1973]. In particular, polynomial basis functions for self-equilibrated stresses σ ∗ph can be derived from assumed stress functions. It should be emphasized that these approximations should be derived having care to obtain complete polynomial expansions for each component of σ ∗ph , so to retain invariance of the procedure with respect to any coordinate change and, in case, consistency, as stated in the previous section. A scheme to construct such approximations for some structural models is illustrated in Appendix A. Notice that the present procedure poses no upper limit to the order of polynomial approximation or to the number of unknown parameters a with any form of the patch. This does not occur with the SPR and REP procedures because of the limited number of independent equations they involve. Thus, for finite elements of order p any approximation of degree not less than p could be used in principle, provided it satisfies Equation (4.65). The numerical results obtained on some bi-dimensional tests using approximations of degree p and p + 1 over patches of conventional form are compared and discussed in Section 4.6. Further increasing the order of approx- 110 4.3- Some remarks imation seems to be not recommendable since the computational cost increases without any appreciable improvement in accuracy. Another important issue regarding implementation of patch-based recovery procedures is how to compute smoothed stresses at nodes which are not located at the element vertices. In fact, when higher-order elements are used, improved stresses should be computed not only for corner nodes but also for edge nodes and/or nodes interior to the elements. Although the corresponding patches could be defined in diﬀerent ways, the rules recommended for the SPR and REP procedures are adopted here. Only patches associated to corner nodes are formed and each patch is used to recover stresses at all the nodes inside the patch itself, as shown in Figure 4.6. By doing so, it is clear that edge and interior nodes generally belong to several patches and diﬀerent recovered values are available from diﬀerent overlapping patches. In order to obtain a single value for each node, a simple averaging of these values is taken. It should be remarked that the reason to adopt the above strategy for the present recovery procedure is not dictated by stability requirements. It has been adopted because provides good accuracy and allows computational savings. Figure 4.7: Boundary nodal recovery: (a) and (b) interior patches, (c) and (d) boundary patches; (◦) nodal points, (•) nodal values determined by recovery procedure Another particular situation arises for nodes lying on the domain boundaries. Also in this case, there are many ways to define patches for boundary nodes and compute the corresponding stress values. Here, two diﬀerent techniques are Chapter 4 111 adopted. The first technique consists of forming and processing boundary patches in the same manner of interior patches. In particular, the smoothed stress field σ ∗p is taken as for interior patches. Notice that neither reducing the order of stress approximation nor adding any extra condition are needed for stability purpose. An illustration of boundary patches is reported in Figure 4.7. The second technique is the one recommended for the SPR procedure [Zienkiewicz & Zhu, 1995] and consists of evaluating stresses at boundary nodes by interior patches instead of boundary ones, as shown in Figure 4.7(a)-(b). Finally, it should be added that locally based coordinate origins are used to avoid ill-conditioning. 4.3.5 An alternative version of RCP recovery An alternative version of RCP recovery can be obtained by adopting a diﬀerent strategy to form the patch, while retaining the same core. In the original form, the patch is defined as the union of the elements surrounding an assembly node of the finite element discretization (node patch, see Figure 4.7), while, in the new form, the patch is defined as the union of the elements surrounding an element (element patch, see Figure 4.8). In other words, in the original RCP each assembly node is associated to a patch, while in the new version each element is associated to a patch. Indeed, the above description of the node patch configuration is not exhaustive, since it refers only to internal vertex nodes. In fact, some additional details about node patches associated to nodes in the interior or on the edge of an element as well as to nodes lying on the domain boundary have been given in the previous Section, so leading to the need of special procedures in order to treat these cases. On the contrary, the above description completely defines the element patch configuration. For better clarity, Figure 4.9 shows an element patch associated to an element with edges on the domain boundary. As it can be noted, the same criterion as for internal elementsis used to form the patch. Owing to the new patch configuration, parameters a can be directly used to recover stresses over the central element of the patch, without any additional interpolation and average process of values coming from adjacent patches. The new form of the RCP procedure is simpler than the original one and easier to implement into existing finite element codes. In fact, no special data structures are needed to form the patch or to distinguish between vertex nodes, interior nodes and nodes lying on the domain boundary or other physical interfaces. Another advantage oﬀered by the new form is that the larger support of the patch generally leads to higher accuracy without increasing the computational eﬀort, which is substantially dictated by the assumed stress approximation over the patch. Indeed, 112 4.4- Recovery on the boundary of the domain FEM displacements prescribed Ωp ∂Ωp elements of the patch (a) element defining the patch (b) Figure 4.8: Example of element pacth: assembly element defining the patch elements of the patch element defining the patch Figure 4.9: Boundary element recovery unlike the original version, discontinuous solutions across element boundaries are obtained in this way, although discontinuities could be eliminated by an averaging process. However, in the spirit of the RCP recovery, the equilibrated nature of improved stresses within each element is retained. Finally, notice that the new version of the RCP recovery preserves the attractive feature of numerical stability. 4.4 Recovery on the boundary of the domain It will be found that the accuracy of recovery at inner parts of the domain is greater than the accuracy on their boundary. There are many ways to compute boundary stress values. One is computing these stresses from the smoothed stresses from interior patches without adding any extra condition for any imposed tractions. In order to increase the accuracy at boundaries, we can enforce equilibrium with known traction boundary conditions, Figure 4.10(a) We can be sure that adding this form of constraint will not aﬀect the convergence of the solution because the exact solution satisfies this conditions and we do nothing except to force the solution toward the exact solution. This constraint has been used in SPR lead- Chapter 4 113 Figure 4.10: Typical boundary patches: (a) smooth boundary with traction; (b) geometrical singularity ing to improvement of results elsewhere [Zienkiewicz & Zhu, 1995]—[Boroomand & Zienkiewicz, 1997b]. Considering the number of equations and unknowns, REP allows us to use such boundary patches with some extra traction conditions. For this, in the least squares procedure we add some terms which represent the diﬀerence between exact boundary condition and assumed one. Then we can write: NP X ¡ T ∗ ¢T ¡ ¢ F (ai ) = (Hi ai − gi )T (Hi ai − gi ) + N σ p − t̄ i NT σ ∗p − t̄ i . (4.71) i=1 In the above equation, t̄ denotes the prescribed traction on the boundary, and N P is the number of integration points with prescribed tractions on the boundary of the patch. The experience shows that when a curved boundary exists and we are using low-order polynomial for smoothing, minimum number of integration points gives the best answer. For geometrical singularities, like that shown in Figure 4.10(b), there will be singularities in strains and stresses. To overcome this diﬃculty, instead of using polynomial forms for σ 11 , σ 22 and τ 12 , we could use polar expressions for smoothed values of σ r , σ θ and τ rθ . A combination of polynomial for in radial direction (i.e. 1, r−1 , r−2 , . . . ) and periodic functions in rotational direction (i.e. 1, cos(θ), sin(θ), ...) can be employed. This will help to have a discontinuous answer at r = 0. To evaluate H, an appropriate rotational matrix must be used: Z BT TP∗ dV. Ωp The above equation may need a larger number of integration points than conventionally used. Further it may be worthwhile to use large patches and add more unknown around singular points. To use higher order functions around these points, one can find all interior nodal stresses as a first step and use them, as the sec- 114 4.5- The quality of an error estimator ond step, for the nodes belonging to the boundary patch as additional constraints. This can also be done for other boundary nodes when the user wishes to increase the number of polynomial terms. 4.5 The quality of an error estimator The recovery procedures can be used to estimate the error resulting from a finite element analysis by replacing the exact solution with the recovered solution, as indicated by Equation (4.2). So, the estimated local error in the stresses is given by ∗ e∗ = σ h − σ ∗ , (4.72) where the symbol · is usual to distinguish the estimated error from the exact error (see Section 2.3 for more details on the error definitions). As for the exact error, if single-valued nodal errors are needed, because of interelement discontinuity the average of nodal errors can be computed in the same manner, element or global errors can be estimated. Adopting for instance the energy norm, the estimated element error is given by ¸ 12 ∙ Z ° ° ¡ h ¢ ¡ ¢ 1 ke∗ ke = °σ h − σ ∗ °e = σ − σ ∗ C−1 σ h − σ ∗ dV , (4.73) 2 Ωe where Ωe is the element domain under consideration. Finally, the estimated global error is ³P ´ 12 n−elem ∗ ke k e e=1 ke∗ k = , (4.74) E where E is the exact energy norm of the problem ∙ Z ¸ 12 1 T −1 E= σ C σdV . (4.75) 2 Ω Clearly the higher is the accuracy of the recovered solution, the higher is the quality of the error estimator. To measure the accuracy of the recovered solution, the error definitions recalled above can be used again. A local measure is simply given by the pointwise error in the recovered stress ees = σ ∗ − σ. (4.76) Notice that the superscript ·es is used to indicate this type of error, that is the exact error in the recovered solution, while symbol e is reserved for the exact error in the original finite element solution. In the case of interelement discontinuity the average of nodal errors can be considered also in this case. The (exact) element and global errors in the recovered solution, assuming the Chapter 4 115 energy norm, read as ∙ Z ¸ 12 1 ∗ −1 ∗ ke ke = kσ − σke = (σ − σ) C (σ − σ) dV , 2 Ωe es ∗ kees k = ³P n−elem e=1 kees k2e (4.77) ´ 12 . (4.78) E The eﬀectiveness of the error-estimators is measured by the eﬀectivity index , namely, the ratio between the estimated and the exact value of the error norm of interest: ke∗ k . θ= kek The error estimator is said to be asymptotically exact if θ approaches unity as the exact error kek tends to zero (as h → 0, or as p → ∞). This means that the error estimator will always converge to the exact error while this decreases. Obviously, the reliability of ke∗ k is dependent on the quality of σ ∗ , and it can be proved (Zienkiewicz and Zhu, [Zienkiewicz & Zhu, 1992b]) that kees k kees k ≤θ ≤1+ . kek kek In fact we can rewrite the error estimator ke∗ k as ° °¡ ° ¢° ¢ ¡ ke∗ k = °σ h − σ ∗ ° ≡ ° σ h − σ − σ ∗ − σ h ° = ke − ees k , 1− (4.79) (4.80) and using the triangular inequality, we have kek − kees k ≤ ke∗ k ≤ kek + kees k . (4.81) Dividing by kek, we obtain the (4.79). Based on these result we have that the error estimator is asymptotically exact if kees k →0 as kek → 0. (4.82) kek This is achieved if kees k converges at a higher rate than kek. It follows that if kees k is superconvergent then asymptotic exactness of the error estimator is assured. The energy norm of the error of a finite element solution, with a p-order shape functions can be represented as follow kek = O (hp ) ≤ Chp , (4.83) where p > 0 is the rate of convergence and C > 0 is a constant independent of the element dimension h, but dependent on the patch (see Section 3.2). For the 116 4.5- The benchmarking approach recovered solution we can write ¢ ¡ kees k = O hp+α , (4.84) with α > 0 , so the limits of the eﬀectivity index, by substituting the equations (4.83) and (4.84) into (4.79), are : 1 − O (hα ) ≤ θ ≤ 1 + O (hα ) . Naturally, one hopes that eﬀectivity indices close to unity can be obtained, but global eﬀectivity indices of 2.0÷3.0 or even higher are often regarded as acceptable in many engineering applications. 4.5.1 The benchmarking approach In engineering practice one often attempts to analyze the accuracy of the error estimators which are implemented in various finite element codes by checking the values of the eﬀectivity index in a few example problems which are often called benchmarks. The benchmarking approach is a well accepted procedure for checking finite element codes [Babuška et al. , 1997a]. Can we classify the error estimator as reliable or unreliable based on these results? To answer to this question the following remarks should be considered [Babuška et al. , 1997b]: 1. The error in an element can be partitioned into the local error (the part of the error due to the residuals in the element and its neighbors) and the pollution error (the part of the error due to the residuals in the rest of the mesh and especially those in the neighborhood of the singular points and sharp fillets). 2. All the error indicators which employ only local computations (i.e. they use the input-data and the finite element solution in the element and its neighbors) cannot account for the pollution error. 3. The pollution error is the most significant component of the error in the majority of the elements for the types of meshes and domains employed in engineering computations. Note that when a globally adaptive mesh (which almost equidistributes the element error indicators) is employed, then the pollution error is practically controlled (see Reference [Zienkiewicz & Zhu, 1987]). Therefore, it follow that 1. For uniform meshes (and singular problems), the global energy norm of the error practically coincides with the energy norm of the error over the patch of the elements which have a vertex at the singular point. Hence, for uniform meshes, the global eﬀectivity index reflects only the accuracy of the error estimator for the elements adjacent to the singularity. The pollution error in Chapter 4 117 these elements is practically negligible and the element error indicators are close to the exact element norms of the error. 2. There is significant pollution practically everywhere, except in the elements with the biggest errors (which are the elements with a vertex at the singular point), and the element error indicators underestimate the element energy norms of the error in all the elements, except the elements with the biggest errors. Hence, we conclude that if one is interested in the accuracy of the element error indicators in the elements adjacent to the singularity then a benchmark problem can be a proper choice. On the other hand, if one is interested in the accuracy of the element error indicators in the interior of the mesh or at the elements adjacent to a smooth boundary then benchmark tests could give no useful information unless we carefully avoid or properly control pollution. Let us now summarize the conclusions which should be drawn from the above discussion (see also References [Babuška et al. , 1994b]-[Babuška et al. , 1994a][Babuška et al. , 1995b]). 1. The error in a patch of elements has a local and a global component and any error estimator based on local computations can only estimate the local component. 2. The element error indicators estimate only the local part of the error. The local error is practically the same as the error in the finite element approximation of the local (p + 1) Taylor series expansion of the exact solution. 3. The element eﬀectivity indices depend on the local geometry of the mesh and should be reported for the worst (p + 1) degree Taylor series expansions corresponding to the particular mesh-geometry of interest. Based on the above conclusions, Babuska and coworkers [Babuška et al. , 1994a][Babuška et al. , 1997a] prepared a numerical methodology to test the robustness of an error estimator in the asymptotic limit (when the mesh size tends to zero) for smooth problem (that is to check the capability of computing the local error). The methodology described in the next section provides an eﬀective tool to evaluate and compare diﬀerent procedures in a systematic and objective manner. It can be viewed as a patch test procedure which fixes minimal criteria to be met and avoids drawing erroneous conclusions based only on a limited number of test problems. However, it should be emphasized it is an asymptotic test and certain eﬀects, such as those due to boundaries or singularities, have been isolated. In addiction no indications are given on the accuracy in the pre-asymptotic behaviour. Thus, this asymptotic testing should always be accompanied by some (carefully chosen) 118 4.5- The robustness test benchmarking computation which give valuable information on the pre asymptotic behaviour. 4.5.2 The robustness test The patch test idea was proposed by Bruce Irons in 1965 for to verify the quality of a model implemented in a finite element code, verifying, under regular condition, the capability of the element to reproduce, in a generic point, a constant state of strain when the dimension of the region surrounding that point tends to zero. So the elements surrounding this point must represent a constant strain state. As shown before (see Section 3.2) the convergency is assured by the consistency of the finite element method: when the characteristic dimension of the elements h tends to zero, the approximated equations represent exactly the diﬀerential problem, at least in weak form. Based on the above conditions, the elements, with p degree polynomials, can reproduce exactly every problem with an exact solution based on that polynomial. The eﬀectivity robustness test, introduced by Babuška and coworkers, is based on the same idea of the traditional patch test, and is useful to compute the robustness of the error estimators. The method start from the fact that a smooth problem have a solution where the terms with order (p + 1) are predominant, so, as before, we can study a simple patch and its (p + 1) exact solution, that will give the asymptotic behaviour, its dimension tend to zero. We can also use this method to compute, over a specific patch, the maximum and minimum asymptotic value of θ, that reach the maximum and minimum values for the family problems with exact solution of order (p + 1) . These maximum and minimum bounds of the eﬀectivity index are called θL and θU and for robustness in estimating of errors both values should be close to unity. Babuška and coworkers defined a single robustness index as: ¯ ¯ ¯¶ ¯ µ ¯ 1 ¯¯ ¯¯ 1 ¯¯ , R = max |1 − θL | + |1 − θU | , ¯¯1 − (4.85) + 1 − θL ¯ ¯ θU ¯ which obviously should be equal to zero in the ideal case when θL = θU = 1, (4.86) and which measures how far both values depart from unity. The particular form of the index is chosen to give equal weight to the value of θL (or θU ) below unity or above it, as clearly the coeﬃcients must always be positive. The test is based on the following assumptions: Assumption 1. Since we are dealing with an asymptotic problem, it immediately follows that we should assume the element sizes to be near to Chapter 4 119 Figure 4.11: Babuska patch test: (a) problem domain; (b) local (periodic) mesh; (c) master-cell zero so the dimensions of any patch of elements, with limited number of elements, will be close to zero as well. Assumption 2. We assume that the patch considered is far from singular points and that a smooth exact solution exists at the location of the patch. Assumption 3. We assume that the mesh at the location of the patch has a repeatable pattern in all directions (we usually consider a single one of such repeatable patterns as a patch). Summing up the test aims at assessing the asymptotic behaviour by considering that: (a) the mesh is locally uniform, (b) the exact solution is locally smooth and (c) the pollution error is negligible. The idea is sketched in Figure 4.11 . Focusing the attention on the local periodic mesh (Figure 4.11(b)) and considering a finite element solution with p order shape function in all elements, we know thus that all reliable elements can reproduce an exact field of order p. Therefore the terms of order p + 1 in the Taylor’s series expansion of exact solution u have a predominant eﬀect in the error (as h → 0): ° ° ° ° kek = °σh − σ ° ' °σ h − σ asy ° , (4.87) where σ asy = CDup+1 , (4.88) p+1 is the stress corresponding to the (p + 1)-order term u of the exact solution. Therefore for testing any recovery only such monomials need to be used as exact solutions: u ' up+1 . (4.89) The whole procedure of this test is based on estimating the asymptotic error and evaluating the eﬀectivity indices, considering each one of these monomials as an exact solution over a local periodic mesh (Figure 4.11(b)). Having obtained these eﬀectivity indices, the maximum and minimum bounds for the combination of the 120 4.5- Computing asymptotic finite element solution monomials can be found through the solution of an eigenvalue problem. The exact asymptotic error is given by ° ° keasy k = °σ hasy − σ asy ° , (4.90) where σ hasy is the asymptotic finite element solution on the periodic mesh. The estimated asymptotic error is ° ∗ ° ° h ° °easy ° = °σ asy − σ ∗asy ° , (4.91) where σ ∗asy is the asymptotic recovered solution. The procedure consists of three steps. 1. Select the master cell (mesh pattern and aspect ratio), Figure 4.11(c), and take the exact solution as a general (p + 1) degree polynomial in the form X X (p+1) u= αi ui , σ= αi σ i , (4.92) (p+1) where ui and are basis functions of degree p + 1, αi are unknown coeﬃcients (p+1) σ i = CDui . (4.93) For example, for scalar-value problems and linear elements we can take u = α1 x21 + α2 x1 x2 + α3 x22 . (4.94) (p+1) ui , 2. For each bases function determine the (asymptotic) finite element soh lution σ i and the (asymptotic) recovered solution σ ∗i over the periodic patch formed using the chosen master-cell. 3. Compute the extreme values of the (asymptotic) eﬀectivity index, for all possible values of αi , corresponding to the (asymptotic) solutions: X X σ hasy = αi σ hi , σ ∗asy = αi σ ∗i . (4.95) To apply the above procedure we need to explain how to compute each finite element solution σ hi . Indeed computing σ ∗i poses no particular diﬃculty as it can be done in the standard way using the recovery technique under investigation. In addition, we give some details on how to compute the extreme values of θ (step 3). The nature of this test is such that any recovery method which is truly superconvergent must produce the perfect answer of θL = θU = 1 or R = 0. 4.5.2.1 Computing asymptotic finite element solution (p+1) The relation between the finite element solution uhi and the exact solution ui can always be written as Z ³ ´ (p+1) BT CD uhi − ui (4.96) dV = 0. Ω Then, if ūhi are the nodal displacements of uhi and uhi = Uūhi , the above relation Chapter 4 121 can be put in the form Z BT CDuhi dV = Ω Z BT CDUūhi dV, (p+1) By interpolating the exact (p + 1)-degree solution ui degree shape functions we have (p+1) ûi (p+1) where ūi rewritten as (p+1) (p+1) = ûi using the standard p- (p+1) = Uūi (4.98) (p+1) . Then the exact solution can be are the nodal values of ui ūi (4.97) Ω ³ ´ (p+1) (p+1) + ūi − ûi and substituting in the Equation (4.96) yields Z Z ³ ´ (p+1) (p+1) BT CDêhi dV = BT CD ui − ûi dV, Ω (4.99) (4.100) Ω where êi is the diﬀerence between the finite element solution and the interpolant of the exact solution ³ ´ (p+1) (p+1) êi = uhi − ûi = U ūhi − ūi (4.101) = = Uēi . (4.102) Taking into account for this reason, Equation (4.97) can be rewritten as µZ ¶ Z ³ ´ (p+1) (p+1) T B CBdV ēi = BT CD ūi − ûi dV. Ω Ω ´ ³ (p+1) (p+1) − ûi is a periodic function with a period same It can be shown that ui as the patch length in all direction [Boroomand & Zienkiewicz, 1997a]. To prove this we consider the m-th monomial of the exact solution. With this important fact and with the knowledge of periodicity of the mesh we can write (4.97) for only the master-cell Ωmc : ¶ µZ Z ³ ´ (p+1) (p+1) T B CBdV ēi = BT CD ui − ûi (4.103) dV. Ωmc Ωmc This equation can be solved for ēi once the appropriate boundary conditions are specified. In particular êi should be periodic over the master cell, therefore periodic boundary conditions on ēi should be applied. It can be easily verified that the above formulation has a unique solution a part for a constant which can be computed by subjecting the domain to a uniform "body force" and requiring that the absorbed energy from using the finite element method solution and the exact one should be the same. The resulting additional condition is Z Z ³ ´ (p+1) (p+1) ēhi dV = − ûi (4.104) ui dV. Ωmc Ωmc Now, the set boundary conditions is complete and Equation (4.103) can be solved 122 4.5- Computing the eﬀectivity index for ēi and, subsequently, the finite element solution can be computed by ³ ´ (p+1) (p+1) uhi = ûi + êi = U ēi + ūi . 4.5.2.2 (4.105) Computing the eﬀectivity index For any procedure of error estimation we can independently determine the estimated error norm and, as the exact solution is known, also the exact error norm thus establishing the eﬀectivity index θ corresponding to each bases function (p+1) ui . It would appear that a search amongst the values of indices so obtained would give the bounds of θL and θU or the possible eﬀectivity. The determination of eﬀectivity for each monomial separately does not, however, guarantee that absolute lowest and highest bounds have been determined. Figure 4.12: Computing the eﬀectivity index: (a) master-cell; (b) periodic mesh We give below a process by which all combination of the individual terms can be examined for the case of recovery based estimators. As a starting point we (p+1) repeat the solution procedure for all ui so the general finite element solution takes the form: X X h αi uhi = αi Uūhi = UŪ α, (4.106) ūh = i i where Ūh collects the nodal values of all the finite element solutions ¯ ¯ ⎡ ⎤ ¯ ¯ ¯ ¯ Ūh = ⎣ ūh1 ¯¯ ūh2 ¯¯ . . . ⎦ , ¯ ¯ (4.107) and α collects the coeﬃcients αi . The general finite element solution in terms of stresses is given by h σ h = CBŪ α. (4.108) Chapter 4 123 Analogously, the recovered solution in compact form reads as X σ∗ = αi σ ∗i = S∗ α, (4.109) i where matrix S∗ collects the solutions σ ∗i obtained by applying the recovery procedure under investigation to the finite element solution uhi . Finally, also the exact solution can be put in a compact form as X (p+1) u= αi ui = U(p+1) α , i (p+1) where matrix U(p+1) collects the basis functions ui : ¯ ¯ ∙ ¸ ¯ ¯ ¯ ¯ Ues = . (p+1) ¯ (p+1) ¯ ... u2 u1 (4.110) Consequently the exact solution in terms of stresses can be written as σ = CBU(p+1) α. (4.111) Using the above expressions the exact asymptotic error over the master-cell (in energy norm) results in 2 keasy k Z ´T ´ ³ ³ h h CDU(p+1) α − CBŪ α C −1 CDU(p+1) α − CBŪ α dV, Ωmc ½Z ¾ ³ ´ ³ ´ h T T h (p+1) (p+1) T = α − BŪ C DU − BŪ dV α. (4.112) DU = 1 2 Ωmc Posing E= Z Ωmc yields ³ ´ h T DU(p+1) − BŪ ³ ´ h CT DU(p+1) − BŪ dV, 2 keasy k = αT Eα. (4.113) (4.114) On the other hand, the estimated asymptotic error over the master-cell can be expressed as ° ∗ °2 °easy ° Z ´T ´ ³ ³ h h S∗ α − CBŪ α C−1 S∗ α − CBŪ α dV = Ωmc ½Z ¾ ³ ´ ³ ´ h T −1 h = αT C S∗ − CBŪ S∗ − CBŪ dV α. (4.115) = 1 2 Ωmc So, posing E∗ = Z Ωmc we end with ³ ´ ³ ´ h T −1 h C S∗ − CBŪ S∗ − CBŪ dV, ° ∗ °2 °easy ° = αT E∗ α. (4.116) (4.117) 124 4.5- Summary of the procedure Now we can write the (asymptotic) eﬀectivity index as ° ∗ °2 °easy ° αT E∗ α 2 θasy = 2 = αT Eα . keasy k (4.118) The maximum and minimum values for θ2asy are the maximum and minimum eigenvalues of the following problem ¡ ∗ ¢ E − θ 2 E w = 0, (4.119) where θ 2 is the diagonal matrix of the eigenvalues of the problem. 4.5.2.3 Self equilibrated solutions In the test so far described we have used generic (p + 1) degree bases functions (p+1) . Indeed it is convenient to use only self equilibrated, that is only these ui solutions of the homogeneous form of the governing equations of degree (p + 1). Such solutions can be found from the general form by enforcing a proper restrictions on coeﬃcients α: α = Tα̃. (4.120) Then Equations (4.114) and (4.117) can be written as 2 keasy k = α̃T TT ETα̃ = α̃T Ẽα̃, (4.121) and ° ∗ °2 °easy ° = α̃T TT E∗ Tα̃ = α̃T Ẽ∗ α̃. (4.122) Therefore a modified procedure should be followed by finding the eigenvalues of ´ ³ 2 (4.123) Ẽ∗ − θ̃ Ẽ w̃ = 0. 4.5.2.4 Summary of the procedure In summary the test procedure is performed as described below: 1. select a mesh pattern for the master-cell; 2. select a proper basis function (monomial) one order higher than the order of shape-functions; 3. solve Equation (4.103) for ēi with periodic boundary conditions and minimum support for the master-cell; 4. use Equation (4.104) to find rigid-body motions; 5. find the finite element solution through Equation (4.105); 6. perform the recovery procedure using the finite element solution from previous step. This step itself consists of three smaller steps: (a) produce a periodic mesh from the patch; Chapter 4 125 (b) apply the recovery procedure on the periodic mesh; (c) consider the recovered answers only for one master-cell. (it should be noted that depending on the location of the patch the finite element solution may change but it can be shown that the errors will be periodic.) 7. repeat (2)-(6) for all possible basis functions (monomials); 8. evaluate E and E∗ for exact and recovered errors. Special care should be taken here for rectangular elements since these elements can exactly reproduce some higher-order solutions, e.g. x1 x2 in bilinear elements, and the FEM solution, itself, will be exact. Therefore the corresponding terms in E and E∗ , which are close to zero, must be eliminated otherwise roundoﬀ error will encounter the computation leading to some non-realistic answers. 9. find the eigenvalues of E∗ (or E) to determine the upper and lower bounds of the eﬀectivity index and robustness index. 4.6 Numerical tests In this section, the performance of the RCP recovery is validated numerically through a systematic testing based on the methodology proposed by Babuška et al. in [Babuška et al. , 1994a]-[Babuška et al. , 1997a], Section 4.6.1, and several benchmark tests proposed in the literature, Section 4.6.2. Comparisons are carried out with the improved version of the REP procedure [Boroomand & Zienkiewicz, 1997a], as quite natural considering the duality recognized in Section 4.3.1. For the sake of brevity, no further comparisons are included. However, the reader can refer to [Boroomand & Zienkiewicz, 1997b][Boroomand & Zienkiewicz, 1997a][Boroomand & Zienkiewicz, 1999] for a detailed comparison between the REP procedure and the widely used SPR procedure. As claimed by the authors of these papers, the two procedures yield very similar results. All the numerical results presented in the following are obtained using patches of conventional minimal form. In the REP procedure, complete polynomial expansions of order p are selected for each stress component, being p the interpolation order for displacements in the finite element solution (i.e. the order of shape functions U). In the RCP procedure, the influence of diﬀerent choices for the order of stress approximation is investigated. It is worth to emphasize that the present procedure allows to enrich the approximation for smoothed stresses without any need to enlarge the form of the patch. As recommended in [Boroomand & Zienkiewicz, 1999], the REP procedure is applied without using boundary patches or any correction on boundaries, so that nodal boundary values are computed from interior 126 4.6- Asymptotic behaviour and robustness Table 4.1: Versions of RCP recovery tested and related orders of the stress approximation Type order of the stress approxim ation form ation of the patch RCP1n p node patch; b oundary patch are used; RCP1nb p node patch; b oundary patch are not used; RCP2n p+1 node patch; b oundary patch are used; RCP2nb p+1 node patch; b oundary patch are not used; RCP1e p element patch; - RCP2e p+1 element patch; - patches. Since the present procedure allows to use boundary patches without introducing any additional condition or decreasing the order of approximation for smoothed stresses, this opportunity has been exploited. Thus, various versions of the RCP procedure have been implemented. Denoting by p theinterpolation order for displacements in the finite element solution, the various versions of the RCP procedure are summarized in Table 4.1. (a) (b) (c) (d) Figure 4.13: Boundary nodal recovery: (a) and (b) interior patches, (c) and (d) boundary patches; (◦) nodal points, (•) nodal values determined by recovery procedure 4.6.1 Asymptotic behaviour and robustness The analysis described in Section4.6.1 has been carried out for two diﬀerent types of problems: equilibrium of thin membranes of uniform thickness or heat transfer (Class I problems, see Appendix A.1) and plane strain elasticity (Class II problems, see Appendix A.2). The Young’s modulus and Poisson’s ratio are taken as 1000 and 0.3, respectively. Various mesh patterns have been selected for the basic cell, using Chapter 4 127 both triangular and quadrangular elements, as illustrated by Figures 4.14 and 4.15. In all these tests the performance of the RCPn-based error estimator is compared with both the RCPe form and the REP-based estimator, in the improved version presented in [Boroomand & Zienkiewicz, 1997a]. The interested reader can refer to [Boroomand & Zienkiewicz, 1997a]-[Boroomand et al. , 2004] for a comparison with the SPR-based estimator. As suggested in [Boroomand & Zienkiewicz, 1997a], the REP procedure is used with complete polynomial expansions of degree p for each stress component. Stress approximations are selected according to Equation (A.3) for the thin membrane case and according to Equation (A.18) for the plane strain case. (a) (b) (c) (d) (e) (f) (g) (h) Figure 4.14: Mesh patterns used in robustness tests for triangular elements 4.6.1.1 Class I problems Class I problems refer to thin membrane problems and are described in Appendix A.1. The results obtained for triangular and quadrangular elements in this case with the RCPn procedure are collected in Tables 4.2 and 4.3, respectively, while the corresponding results obtained with the RCPe procedure are collected in Tables 4.4 and 4.5. Moreover, Tables 4.6 and 4.7 collect some results obtained using regular mesh patterns of quadratic elements with various aspect ratios. It is interesting to observe that the RCP procedure exhibits a super-convergent behavior for two simple patterns (Figures 4.14(a) and (c)) of linear triangular 128 4.6- Class I problems (a) (b) (d) (c) (e) Figure 4.15: Mesh patterns used in robustness tests for quadrangular elements elements. Super-convergence is observed also for diﬀerent values of the aspect ratio of these mesh patterns. Indeed, super-convergence is lost by the RCPe on the criss-cross pattern (Figure 4.14(d)), where the RCPn and the REP are superconvergent, but the robustness index takes very small values: R = 0.0243 for RCP1e and R = 0.0057 for RCP2e (Table 4.2). On the other hand, full superconvergence is not achieved by none of the procedures considered for triangular elements of quadratic kind. Also in this case, however, the RCPe preserves small values of the robustness index, valuably smaller than those of the other procedures. In particular, R < 0.07 for RCP1e, R < 0.27 for RCP1n and R < 0.33 for REP (Table 4.4). This is confirmed also on regular meshes with diﬀerent aspect ratios (see Table 4.7). As regards quadrangular elements, the REP procedure is super-convergent for both four- and nine-node elements on regular meshes (Figure 4.15(a)), while the RCPn and the RCPe procedure only for four-node elements. However, although not exact, the RCP-results for nine-node elements are very good being R < 0.08 for RCPn (Table 4.6) and R < 0.007 for RCPe (Table 4.7). As for triangular elements, super-convergence is not observed for irregular patterns (Figures 4.15(b)-(e)), but excellent results are obtained. In general, the robustness index obtained with the RCPe procedure takes, values smaller than those obtained with the RCPn version The above results demonstrate the good quality of the RCPn and RCPe procedure. The RCP1 versions appear to be more robust than REP especially on irregular mesh patterns. No appreciable improvements of the asymptotic perfor- Chapter 4 129 Table 4.2: RCPn - Eﬀectivity bounds and robustness indices for triangular elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). REP Mesh θL RCP1n θU R RCP2n θL θU R θL θU R 1.0000 0.9201 1.0000 1.0000 0.9636 0.9405 0.7121 0.5975 1.0000 1.0039 1.0000 1.0000 1.0014 1.0035 1.0218 0.9806 0.0000 0.0907 0.0000 0.0000 0.0392 0.0667 0.4256 0.6935 1.0000 0.9298 1.0000 1.0000 0.9717 0.9479 0.7153 0.6311 1.0000 1.0052 1.0000 1.0000 1.0020 1.0051 1.0316 0.9763 0.0000 0.0807 0.0000 0.0000 0.0311 0.0600 0.4288 0.6089 0.9956 0.9839 0.9454 0.9454 0.9695 0.9929 0.8487 0.8033 1.0114 1.0157 1.0136 1.0136 1.0095 1.0083 0.9950 0.9764 0.0158 0.0318 0.0712 0.0712 0.0409 0.0154 0.1832 0.2691 0.9956 0.9815 0.9454 0.9454 0.9582 0.9681 0.7767 0.7401 1.0114 1.0070 1.0136 1.0136 1.0071 1.0058 1.0026 0.9613 0.0158 0.0258 0.0712 0.0712 0.0507 0.0387 0.2900 0.3914 Three-node elements a b c d e f g h 1.0000 1.0000 1.0000 1.0000 0.9715 0.9828 0.9352 0.9939 1.0000 1.0074 1.0000 1.0000 1.0190 1.0247 1.5400 1.8287 0.0000 0.0074 0.0000 0.0000 0.0480 0.0420 0.6047 0.8347 Six-node elements a b c d e f g h 0.9516 0.9950 0.9320 0.9259 0.9481 0.9816 0.8972 0.8868 1.0398 1.0222 0.9836 0.9594 1.0240 1.0045 1.1625 1.2142 0.0891 0.0272 0.0897 0.1223 0.0782 0.0232 0.2653 0.3274 Table 4.3: RCPn - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). REP Mesh θL RCP1n θU RCP2n R θL θU R θL θU R 0.0000 0.0187 0.0082 0.0882 0.0847 1.0000 0.9987 0.9978 0.9394 0.9449 1.0000 1.0174 1.0037 1.0118 1.0077 0.0000 0.0187 0.0059 0.0762 0.0660 1.0000 0.9841 0.9889 0.9363 0.9334 1.0000 1.0008 1.0047 1.0000 1.0031 0.0000 0.0169 0.0159 0.0680 0.0745 1.0107 0.9929 1.0016 0.9800 0.9730 1.0107 1.0083 1.0142 1.0018 1.0008 0.0214 0.0154 0.0158 0.0222 0.0285 1.0107 0.9681 0.9941 0.9467 0.9454 1.0107 1.0058 1.0074 0.9933 0.9934 0.0214 0.0387 0.0133 0.0630 0.0644 Four-node elements a b c d e 1.0000 0.9987 0.9979 0.9798 0.9833 1.0000 1.0174 1.0061 1.0680 1.0680 Nine-node elements a b c d e 1.0000 0.9816 0.9967 0.9911 0.9909 1.0000 1.0045 1.0131 1.0329 1.0334 0.0000 0.0232 0.0164 0.0418 0.0425 130 4.6- Class I problems Table 4.4: RCPe - Eﬀectivity bounds and robustness indices for triangular elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). REP Mesh θL RCP1e θU R RCP2e θL θU R θL θU R 1.0000 0.9705 1.0000 0.9765 0.9812 0.9853 0.8504 0.7493 1.0000 1.0006 1.0000 1.0002 1.0005 1.0023 1.0024 0.9942 0.0000 0.0310 0.0000 0.0243 0.0196 0.0173 0.1783 0.3404 1.0000 0.9677 1.0000 0.9946 0.9822 0.9888 0.8760 0.7701 1.0000 1.0008 1.0000 1.0003 1.0006 1.0019 1.0011 0.9848 0.0000 0.0342 0.0000 0.0057 0.0188 0.0133 0.1427 0.3140 0.9989 0.9986 0.9972 0.9972 0.9978 0.9978 0.9674 0.9465 1.0007 1.0023 1.0044 1.0076 1.0051 1.0038 1.0125 0.9961 0.0018 0.0038 0.0073 0.0104 0.0073 0.0060 0.0461 0.0605 0.9979 0.9980 0.9931 0.9932 0.9960 0.9942 0.9419 0.9138 1.0012 1.0105 1.0029 1.0015 1.0114 1.0054 1.0025 0.9879 0.0033 0.0125 0.0098 0.0084 0.0154 0.0111 0.0642 0.1066 Three-node elements a b c d e f g h 1.0000 1.0000 1.0000 1.0000 0.9715 0.9828 0.9352 0.9939 1.0000 1.0074 1.0000 1.0000 1.0190 1.0247 1.5400 1.8287 0.0000 0.0074 0.0000 0.0000 0.0480 0.0420 0.6047 0.8347 Six-node elements a b c d e f g h 0.9516 0.9950 0.9320 0.9259 0.9481 0.9816 0.8972 0.8868 1.0398 1.0222 0.9836 0.9594 1.0240 1.0045 1.1625 1.2142 0.0891 0.0272 0.0897 0.1223 0.0782 0.0232 0.2653 0.3274 Table 4.5: RCPe - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). REP Mesh θL RCP1e θU R RCP2e θL θU R θL θU R 1.0000 0.9999 0.9981 0.9364 0.9311 1.0000 1.0011 1.0016 1.0025 1.0032 0.0000 0.0011 0.0035 0.0704 0.0771 1.0000 0.9998 0.9926 0.9354 0.9280 1.0000 1.0079 0.9995 0.9924 0.9980 0.0000 0.0081 0.0079 0.0767 0.0797 0.9981 0.9971 0.9978 0.9951 0.9946 0.9981 1.0022 1.0002 1.0021 1.0021 0.0038 0.0050 0.0024 0.0071 0.0075 0.9981 0.9930 0.9983 0.9936 0.9920 0.9981 1.0025 1.0028 1.0026 1.0009 0.0038 0.0095 0.0045 0.0091 0.0090 Four-node elements a b c d e 1.0000 0.9987 0.9979 0.9798 0.9833 1.0000 1.0174 1.0061 1.0680 1.0680 0.0000 0.0187 0.0082 0.0882 0.0847 Nine-node elements a b c d e 1.0000 0.9816 0.9967 0.9911 0.9909 1.0000 1.0045 1.0131 1.0329 1.0334 0.0000 0.0232 0.0164 0.0418 0.0425 Chapter 4 131 Table 4.6: RCPn - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.15(a)). REP Aspect ratio θL RCP1n & RCP2n θU R θL θU R 0.9956 0.9968 0.9988 0.9997 0.9999 1.0000 1.0000 1.0114 1.0178 1.0250 1.0279 1.0288 1.0290 1.0290 0.0158 0.0210 0.0262 0.0283 0.0288 0.0290 0.0290 1.0107 1.0030 1.0008 1.0002 1.0000 1.0000 1.0000 1.0107 1.0297 1.0530 1.0659 1.0701 1.0713 1.0716 0.0214 0.0327 0.0538 0.0661 0.0702 0.0713 0.0716 Six-node triangular elements 1/1 1/2 1/4 1/8 1/16 1/32 1/64 0.9516 0.9516 0.9516 0.9516 0.9516 0.9516 0.9516 1.0398 1.0425 1.0450 1.0459 1.0462 1.0462 1.0463 0.0891 0.0916 0.0939 0.0948 0.0950 0.0950 0.0951 Nine-node quadrangular elements 1/1 1/2 1/4 1/8 1/16 1/32 1/64 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table 4.7: RCPe - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.15(a)). Aspect REP ratio θL RCP1e θU R RCP2e θL θU R θL θU R 0.9989 0.9984 0.9983 0.9983 1.0007 1.0021 1.0025 1.0025 0.0018 0.0037 0.0042 0.0042 0.9979 0.9981 0.9981 0.9981 1.0012 1.0049 1.0080 1.0082 0.0033 0.0068 0.0099 0.0102 0.9981 0.9998 1.0000 1.0000 0.0038 0.0060 0.0062 0.0062 0.9981 0.9942 0.9938 0.9938 0.9981 0.9998 1.0000 1.0000 0.0038 0.0060 0.0062 0.0062 Six-node triangular elements 1/1 1/4 1/16 1/64 0.9516 0.9516 0.9516 0.9516 1.0398 1.0450 1.0462 1.0463 0.0891 0.0939 0.0950 0.0951 Nine-node quadrangular elements 1/1 1/4 1/16 1/64 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.9981 0.9942 0.9938 0.9938 132 4.6- Homogeneous elestic bar mance are observed using the RCP2 versions. It should be remarked that REP is not stable for linear triangular elements on the mesh patterns in Figures 4.14(c) and (f). Thus, the corresponding results have been obtained using the stabilization technique suggested in [Boroomand & Zienkiewicz, 1997b]. 4.6.1.2 Class II problems Class II problems refer to plane elasticity problems and are described in Appendix A.2. In particular, plane strain conditions and elastic isotropic properties are considered. The values of Young’s modulus and Poisson’s ratio are assumed as 1000 and 0.3, respectively. Tables 4.8 and 4.9 collect the results obtained for triangular and quadrangular elements using the RCPn procedure and the mesh patterns illustrated in Figures 4.14 and 4.15, respectively. Tables 4.12 and 4.10 collect some results obtained with the RCPe procedure. Finally, Tables 4.12 and 4.13 shows the influence of the aspect ratio for regular meshes of quadratic elements. The above comments can be substantially repeated for triangular elements applied to the plane strain case. In particular, super-convergence is achieved for linear triangular elements on the simple patterns of Figures 4.14(a) and (c), also for aspect ratios diﬀerent from 1/1. This ideal behavior is not observed using quadratic triangular elements, but the results reported in Tables 4.6, 4.7, 4.12 and 4.13 indicate a good performance of the RCP procedure. In the case of quadrangular elements, both REP and RCP yield full superconvergence only for four-node elements on regular meshes. However, the robustness indices in Tables 4.9, 4.11, 4.12 and 4.13 reveal an excellent performance also for nine-node elements on regular meshes. The results obtained on distorted patterns are very good though not exact. These numerical tests confirm the good quality of the RCP procedure, already experienced on the previous class of problems. The asymptotic behaviour of the RCP recovery is similar to that of the REP recovery. Actually, the RCP1 version appears to be slightly superior to REP. Passing to the RCP2 versions no remarkable improvements of robustness are observed. 4.6.2 Benchmark tests In the following, the numerical validation of the RCP recovery is completed by some benchmark computations, which provide valuable information on the preasymptotic behaviour. Chapter 4 133 Table 4.8: RCPn - Eﬀectivity bounds and robustness indices for triangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). REP Mesh θL θU RCP1n R RCP2n θL θU R θL θU R 0.0000 0.0358 0.0000 0.0000 0.0548 0.0693 1.0163 0.9991 1.0000 0.8815 1.0000 1.0000 0.9447 0.8845 0.5533 0.4486 1.0000 1.0063 1.0000 1.0000 1.0030 1.0038 1.0124 1.0021 0.0000 0.1407 0.0000 0.0000 0.0615 0.1344 0.8196 1.2313 1.0000 0.9207 1.0000 1.0000 0.9630 0.8826 0.5753 0.5005 1.0000 1.0090 1.0000 1.0000 1.0062 1.0040 1.0308 0.9825 0.0000 0.0950 0.0000 0.0000 0.0446 0.1369 0.7683 1.0157 0.1461 0.1066 0.1653 0.1968 0.1403 0.1907 0.6794 0.6078 0.9923 0.9811 0.8743 0.8743 0.9468 0.8747 0.7188 0.6740 1.0361 1.0325 1.0216 1.0216 1.0216 1.0115 1.0034 0.9969 0.0438 0.0513 0.1650 0.1650 0.0773 0.1546 0.3946 0.4867 0.9923 0.9565 0.8743 0.8743 0.9184 0.8484 0.6617 0.6144 1.0361 1.0126 1.0216 1.0216 1.0093 1.0040 1.0129 0.9801 0.0438 0.0579 0.1650 0.1650 0.0980 0.1826 0.5240 0.6478 Three-node elements a b c d e f g h 1.0000 0.9763 1.0000 1.0000 0.9628 0.9590 0.8136 0.8270 1.0000 1.0116 1.0000 1.0000 1.0165 1.0273 1.8299 1.8261 Six-node elements a b c d e f g h 0.9558 0.9691 0.8587 0.8448 0.9174 0.8609 0.6944 0.7430 1.1018 1.0757 0.9993 0.9871 1.0529 1.0300 1.3145 1.3508 Table 4.9: RCPn - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). REP Mesh θL θU RCP1n R RCP2n θL θU R θL θU R 1.0000 0.9895 0.9921 0.8968 0.9064 1.0000 1.0249 1.0066 1.0237 1.0191 0.0000 0.0354 0.0145 0.1383 0.1220 1.0000 0.9655 0.9827 0.8925 0.9002 1.0000 1.0014 1.0100 1.0319 1.0317 0.0000 0.0371 0.0274 0.1514 0.1416 1.0000 0.9804 0.9974 0.9625 0.9562 1.0360 1.0203 1.0208 1.0028 1.0027 0.0360 0.0399 0.0234 0.0417 0.0485 1.0000 0.9420 0.9683 0.9049 0.9015 1.0360 1.0180 1.0127 0.9936 0.9937 0.0360 0.0792 0.0453 0.1116 0.1156 Four-node elements a b c d e 1.0000 0.9898 0.9924 0.9411 0.9517 1.0000 1.0231 1.0074 1.0975 1.0870 0.0000 0.0333 0.0150 0.1564 0.1353 Nine-node elements a b c d e 1.0025 0.9834 0.9919 0.9675 0.9664 1.0102 1.0219 1.0170 1.0246 1.0235 0.0127 0.0386 0.0251 0.0576 0.0577 134 4.6- Homogeneous elestic bar Table 4.10: RCPe - Eﬀectivity bounds and robustness indices for triangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). REP Mesh θL RCP1e θU R RCP2e θL θU R θL θU R 1.0000 0.9642 1.0000 0.9802 0.9761 0.9764 0.7612 0.6377 1.0000 1.0010 1.0000 1.0005 1.0012 1.0026 1.0020 1.0016 0.0000 0.0381 0.0000 0.0207 0.0257 0.0268 0.3157 0.5698 1.0000 0.9630 1.0000 0.9964 0.9798 0.9847 0.7371 0.6336 1.0000 1.0011 1.0000 1.0075 1.0014 1.0022 1.0039 1.0008 0.0000 0.0395 0.0000 0.0112 0.0220 0.0177 0.3605 0.5791 0.9971 0.9957 0.9971 0.9970 0.9953 0.9961 0.9156 0.8626 1.0018 1.0035 1.0345 1.0345 1.0149 1.0280 1.0157 1.0077 0.0046 0.0078 0.0374 0.0376 0.0197 0.0319 0.1077 0.1669 0.9973 0.9958 0.9894 0.9911 0.9937 0.9870 0.8676 0.8022 1.0044 1.0161 1.0349 1.0349 1.0300 1.0272 1.0106 1.0008 0.0070 0.0203 0.0455 0.0438 0.0363 0.0402 0.1631 0.2473 Three-node elements a b c d e f g h 1.0000 0.9763 1.0000 1.0000 0.9628 0.9590 0.8136 0.8270 1.0000 1.0116 1.0000 1.0000 1.0165 1.0273 1.8299 1.8261 0.0000 0.0358 0.0000 0.0000 0.0548 0.0693 1.0163 0.9991 Six-node elements a b c d e f g h 0.9558 0.9691 0.8587 0.8448 0.9174 0.8609 0.6944 0.7430 1.1018 1.0757 0.9993 0.9871 1.0529 1.0300 1.3145 1.3508 0.1461 0.1066 0.1653 0.1968 0.1403 0.1907 0.6794 0.6078 Table 4.11: RCPe - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). REP Mesh θL RCP1e θU R RCP2e θL θU R θL θU R 1.0000 0.9991 0.9935 0.9137 0.9219 1.0000 1.0018 1.0038 1.0031 1.0056 0.0000 0.0027 0.0103 0.0976 0.0903 1.0000 0.9967 0.9886 0.9096 0.9148 1.0000 1.0098 1.0010 0.9952 0.9990 0.0000 0.0131 0.0125 0.1041 0.0941 0.9965 0.9965 0.9976 0.9926 0.9931 1.0000 1.0079 1.0013 1.0041 1.0042 0.0035 0.0114 0.0037 0.0115 0.0111 0.9965 0.9918 0.9980 0.9903 0.9871 1.0000 1.0075 1.0079 1.0038 1.0028 0.0035 0.0157 0.0099 0.0136 0.0158 Four-node elements a b c d e 1.0000 0.9898 0.9924 0.9411 0.9517 1.0000 1.0231 1.0074 1.0975 1.0870 0.0000 0.0333 0.0150 0.1564 0.1353 Nine-node elements a b c d e 1.0025 0.9834 0.9919 0.9675 0.9664 1.0102 1.0219 1.0170 1.0246 1.0235 0.0127 0.0386 0.0251 0.0576 0.0577 Chapter 4 135 Table 4.12: RCPn - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.14(a)). REP Aspect ratio θL RCP1n & RCP2n θU R θL θU R 0.9923 0.9888 0.9845 0.9828 0.9824 0.9823 0.9822 1.036 1.0392 1.0437 1.0457 1.0462 1.0464 1.0464 0.0438 0.0505 0.0592 0.0628 0.0639 0.0641 0.0642 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0360 1.0524 1.0656 1.0700 1.0712 1.0715 1.0716 0.0360 0.0524 0.0656 0.0700 0.0712 0.0715 0.0716 Six-node triangular elements 1/1 1/2 1/4 1/8 1/16 1/32 1/64 0.9558 0.9548 0.9527 0.9519 0.9517 0.9516 0.9516 1.1018 1.0980 1.0928 1.0900 1.0889 1.0886 1.0885 0.1461 0.1432 0.1401 0.1381 0.1372 0.1370 0.1369 Nine-node quadrangular elements 1/1 1/2 1/4 1/8 1/16 1/32 1/64 1.0025 1.0025 1.0002 0.9997 0.9968 0.9950 0.9945 1.0102 1.0181 1.0136 1.0061 1.0061 1.0061 1.0061 0.0127 0.0205 0.0139 0.0064 0.0094 0.0111 0.0116 Table 4.13: RCPe - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.15(a)). Aspect REP ratio θL RCP1e θU R RCP2e θL θU R θL θU R 0.9971 0.9947 0.9944 0.9944 1.0018 1.0023 1.0027 1.0028 0.0046 0.0076 0.0083 0.0084 0.9973 0.9973 0.9973 0.9973 1.0044 1.0048 1.0067 1.0075 0.0070 0.0075 0.0094 0.0102 1.0000 1.0000 1.0000 1.0000 0.0035 0.0059 0.0062 0.0062 0.9965 0.9941 0.9938 0.9938 1.0000 1.0000 1.0000 1.0000 0.0035 0.0059 0.0062 0.0062 Six-node triangular elements 1/1 1/4 1/16 1/64 0.9558 0.9527 0.9517 0.9516 1.1018 1.0928 1.0889 1.0885 0.1461 0.1401 0.1372 0.1369 Nine-node quadrangular elements 1/1 1/4 1/16 1/64 1.0025 1.0002 0.9968 0.9945 1.0102 1.0136 1.0061 1.0061 0.0127 0.0139 0.0094 0.0116 0.9965 0.9941 0.9938 0.9938 136 4.6.2.1 4.6- Homogeneous elestic bar Homogeneous elestic bar Consider a one-dimensional sample problem governed by the following equations k d2 u = b, dx2 0 < x < L, u = 0, x = 0, L. This problem can be viewed as the equilibrium problem of a clamped one-dimensional homogeneous elastic bar, of length L and axial rigidity k, subjected to a distributed axial force b. The values of the parameters are assumed as L = 4 and k = 1. The loading function is taken as ³ πx ´ π b = − cos . L L The exact solution in terms of stress is ³ πx ´ du = sin σ=k . dx L This problem was used by Boroomand and Zienkiewicz [Boroomand & Zienkiewicz, 1997b] to illustrate the convergence properties of the REP procedure. The finite 0 stress -0.2 -0.4 Patch 1 2 3 4 5 L -0.6 -0.8 FEM REP RCP exact -1 node 3 node 4 node 5 Figure 4.16: Recovered stress solutions for linear elements element solution is obtained using uniform meshes of two-node (linear) elements. The undeterminate self-equilibrated stress in the RCP1n procedure is simply a constant term: σ ∗ph = β. Thus, the present procedure involves only one unknown parameter for each patch. The stress term in equilibrium with the prescribed load within the patch is taken as Z x σ∗pp = b dx. 0 In this problem, the stress fields recovered by the RCP1n procedure with and Chapter 4 137 stress error 0.08 (a) FEM (max .194) REP (max .049) RCP (max .019) 0.06 0.04 0.02 0 0 0.125 0.25 0.375 0.5 x /L stress error (energy norm) without boundary patches are almost coincident. Thus, no distinction is made between RCP1n and RCP1nb in what follows. In the REP procedure, a linear stress expansion is assumed within the patch, so that two unknown parameters for each patch are involved. 0.06 1 2 3 (b) 4 L 0.04 FEM REP RCP 0.02 0 1 2 3 4 elements Figure 4.17: Error distributions for linear elements: (a) local error (maximum values are reported in brackets), (b) element error in energy norm The local recovered solutions σ ∗p over a patch are shown in Figure 4.16 for the four-element mesh and compared with the finite element solution. Notice that the RCP-solution is indistinguishable from the exact one and much more accurate than the REP-solution. This is confirmed by the error distributions in the smoothed stresses σ ∗ reported in Figure 4.17 for the eight-element mesh. In particular, Figure 4.17(a) shows the local error and Figure 4.17(b) the element error. Due to symmetry, the distributions are plotted only on one half of the domain. The convergence properties of the recovery procedures are illustrated by Figure 4.18. Figure 4.18(a) shows the rate of convergence of the local error in the nodal stress value at point x = (3/4)L. Figure 4.18(b) shows the rate of convergence of the global error in energy norm. These graphs reveal that nodal stress values recovered by the RCP1n procedure are ultra-convergent, while those recovered by the REP procedure are super-convergent. Unfortunately, ultra-convergence is lost for the global error, because of the nodal interpolation by linear shape functions, and super-convergence is achieved by both the REP and RCP1n procedures. 4.6.2.2 Thin membrane subjected to transverse loading (Class I problem) The second benchmark test is a typical problem of the class described in Appendix 4.6- Thin membrane subjected to transverse loading (Class I problem) log(stress error) 0 Local error -2 log(stress error) 138 FEM -4 -8 1 REP 2 Global error FEM -4 1 REP -6 RCP -12 RCP 4 -16 -3 -8 (a) -2 -1 log(element size) -3 2 (b) -2 -1 log(element size) Figure 4.18: Rate of convergence for linear elements: (a) local error at point x/L = 3/4, (b) global error in energy norm (Problem No. 1) A.1. In particular, the equilibrium problem of a thin square membrane of uniform thickness with fixed boundary is considered. Both the sidelength of the domain and the parameter k are taken of unity value. Two diﬀerent load cases are considered. In load case No. 1, function b is selected such that the exact solution is given by u(x, y) = xy sin(πx) sin(πy). In load case No. 2, function b is selected such that the exact solution is given by u(x, y) = x(1 − x)y(1 − y)(1 + 2x + 7y). This problem has been extensively used in the literature to test recovery procedures on a bi-dimensional case. The finite element analysis is carried out on diﬀerent meshes using quadrangular four- and nine node elements. The error distributions for load case No. 1 on a regular mesh of four by four bilinear elements are shown in Figures 4.19 and 4.20, in order to asses the accuracy properties of the RCP procedure. In particular, the distributions of the local error are shown in Figure 4.19 and the distributions of the element error are shown in Figure 4.20, together with the values of the relative percentage error η (Equation (2.37)). The local error is defined as the absolute value of the pointwise diﬀerence between the recovered stress σ ∗ and the exact stress σ for RCP and REP, while between the finite element stress σ h and the exact stress σ for FEM. These results show a very good performance of the RCP procedure. In fact, RCP is much more accurate than the REP procedure almost everywhere, and the associated errors are more uniformly distributed over the domain. Moreover, RCP1e is, notwithstanding its simplicity, about as accurate as RCP1n. The figures reveal also that the RCPe seems to be slightly more sensitive than the RCPn to the increase of the stress Chapter 4 139 FEM: σx - MAX: 0.7147 REP: σx - MAX: 1.1671 RCP1n: σx - MAX: 0.3112 RCP2n: σx - MAX: 0.2878 RCP1e: σx - MAX: 0.3961 RCP2e: σx - MAX: 0.2016 Figure 4.19: Thin membrane subjected to transverse loading, load case No. 1 - Local error distributions on a regular mesh of four-node elements (contour interval = 0.1) approximation order. In particular, RCP2e remarkably improves accuracy all over the domain and the relative percentage error passes from 22% (RCP1e) to only 7%. Figure 4.21 shows the results of the convergence analysis of the RCP procedure, carried out on sequentially refined uniform meshes of bi-linear and bi-quadratic elements. In particular, the pointwise convergence of nodal stresses, evaluated at point (x, y) = (0.5, 0.5) where σ x = σ y , is shown in Figures 4.21(a) and (c). Because of inter-element discontinuity, here and in the following, the single-valued RCPe and the FEM values are computed as average of nodal errors. Moreover, note that, since uniform meshes are used, also the recovered stress components turn out to be equal, so that only one graph is reported for each type of elements. The rate of convergence of the relative percentage error for bi-linear and bi-quadratic elements is shown in Figures 4.21(b) and (d). These results confirm that the RCP procedure possesses the desirable feature of superconvergence. Moreover, it can be observed that RCP2 is more accurate than RCP1, although it exhibits a similar convergence behavior, as expected from robustness tests. The eﬀectiveness of the RCP procedure is also tested solving the present problem on an irregular unstructured mesh of bi-linear elements. The results are shown in Figures 4.22 and 4.23. In particular, Figure 4.22 shows the distributions of the local error in both the stress components, plotted as contours over the domain, and Figure 4.23 the distributions of the element error. The sensitivity of the RCP recovery to element geometry distortions is also investigated solving the some problem on a 140 4.6- Thin membrane subjected to transverse loading (Class I problem) FEM: η = 34.963 REP: η = 33.688 RCP1n: η = 23.852 0.08 0.06 0.04 0.02 0 RCP2n: η = 17.876 RCP1e: η = 22.363 RCP2e: η = 7.029 0.08 0.06 0.04 0.02 0 Figure 4.20: Thin membrane subjected to transverse loading, load case No. 1 - Distributions of element error in energy norm on a regular mesh of four-node elements (the relative percentage error is reported above each plot) sequence of progressively distorted meshes. The irregular pattern used in this test is illustrated in Figure 4.26. The distortion is measured by the parameter d/L, which ranges from −0.375 to 0.375. The local error in the nodal stress value at point x = 0.5 and y = 0.5 is plotted against the distortion parameter in Figure 4.27 for both four- and nine-node elements. Figures 4.27(a) and (d) show the results obtained for the load case No. 1, while Figures 4.27(b), (c), (e) and (f) show the results obtained for the load case No. 2. In the first load case, only the results for one stress component are reported owing to symmetry considerations. It is interesting to notice the very good performance of the RCP procedure also for irregular meshes. The nodal stress values recovered by the present procedure are weakly sensitive to element geometry distortions and very accurate also for patches of highly distorted elements. In particular, the RCP-results are much less sensitive to geometry distortions than the REP-results. This is especially evident for ninenode elements and confirms the good quality of RCP also in the pre-asymptotic regime. These experiments confirm what has been previously observed for uniform meshes, so demonstrating that RCPn is reliable also for arbitrary irregular meshes. This is a feature of importance in error estimation and adaptive analysis proce- Chapter 4 141 Local error at (0.5,0.5) 0 4-node elements FEM -2 REP 1 -4 log(stress error) log(stress error) 0 RCP1e 2 RCP2e -3 -2 -1.5 -1 -0.5 log(element size) -2.5 Relative % error 0 -3 2 REP RCP1e 9-node elements RCP1e REP -2 FEM 3.5 -4 -7 RCP1n -6 RCP1n & RCP2n (c) -2.5 RCP2n RCP2e 2 4 RCP2e -2 -1.5 -1 -0.5 log(element size) (b) -2 -1.5 -1 -0.5 log(element size) Local error at (0.5,0.5) 9-node elements FEM RCP1e RCP1n 2 log(stress error) log(stress error) REP (a) -8 -2.5 -5 FEM -1 RCP2n RCP2e RCP2n 3 -1 1 -2 RCP1n -6 Relative % error 4-node elements 3.7 -8 -2.5 (d) -2 -1.5 -1 -0.5 log(element size) Figure 4.21: Thin membrane subjected to transverse loading, load case No. 1 - Rate of convergence: (a) local error at point x = 0.5, y = 0.5 (σx = σy ) for four-node elements, (b) relative percentage error for four-node elements, (c) local error at point x = 0.5, y = 0.5 (σ x = σ y ) for nine-node elements, (d) relative percentage error for nine-node elements dures. In order to evaluate the eﬀect of using the boundary patch in the RCP procedure based on node patches, the above test have been conducted both with and without using boundary patches. The corresponding results, reported in Appendix B.1 show that using boundary patches actually enhances accuracy near boundaries. Finally, using the unstructured mesh, the error distribution estimated basing on the present RCP procedure, according to Equations (4.1) and (4.2), is compared with the exact FEM error by Figure 4.28. As it can be observed, an excellent behavior of the RCP error estimator is experienced, for both bi-linear and biquadratic elements. Similar results are obtained using triangular elements. These are not included for the sake of brevity. 142 4.6- Stretched plate with circular hole (Class II problem) FEM: σx - MAX: 0.7001 REP: σx - MAX: 0.2855 RCP1n: σx - MAX: 0.1769 RCP2n: σx - MAX: 0.1897 RCP1e: σx - MAX: 0.1500 RCP2e: σx - MAX: 0.0852 FEM: σy - MAX: 0.5998 REP: σy - MAX: 0.3313 RCP1n: σy - MAX: 0.1828 RCP2n: σy - MAX: 0.1965 RCP1e: σy - MAX: 0.1711 RCP2e: σy - MAX: 0.0622 Figure 4.22: Thin membrane subjected to transverse loading, load case No. 1 - Local error distributions on an unstructured mesh of four-node elements (contour interval = 0.05) Chapter 4 143 FEM: η = 18.061 REP: η = 8.836 RCP1n: η = 7.178 0.03 0.025 0.02 0.015 0.01 0.005 0 RCP2n: η = 6.200 RCP1e: η = 7.446 RCP2e: η = 2.628 0.03 0.025 0.02 0.015 0.01 0.005 0 Figure 4.23: Thin membrane subjected to transverse loading, load case No. 1 - Distributions of element error in energy norm on an unstructured mesh of four-node elements (the relative percentage error is reported above each plot) 4.6.2.3 Stretched plate with circular hole (Class II problem) The second class of problems considered refers to problem class described in the Appendix A.2. In particular, a portion of an infinite plate with a central circular hole of radius a subjected to a unidirectional tensile load σ 0 = 1 is considered. Plane stress conditions are assumed with elastic isotropic properties: Poisson’s ratio is taken as 0.3 and Young’s modulus as 1000. The geometry of the problem and the prescribed boundary conditions are shown in Figure 4.29. This is a popular benchmark test for which the exact solution reads as ¶ µ 3 a4 a2 3 σx = 1 − 2 cos 2θ + cos 4θ + cos 4θ, r 2 2 r4 ¶ µ 3 a4 a2 1 σy = − 2 cos 2θ − cos 4θ − cos 4θ, r 2 2 r4 ¶ µ 3 a4 a2 1 sin 2θ + sin 4θ + τ xy = − 2 sin 4θ. r 2 2 r4 where (r, θ) are the usual polar coordinates (see Figure 4.29). The radius of the hole is taken as a = 1. Three meshes are used in the finite element analysis and shown in Figure 4.29. The coarsest mesh consists of 72 elements. The subsequent meshes are created 144 4.6- Stretched plate with circular hole (Class II problem) FEM: Vx - MAX: 0.7001 REP: Vx - MAX: 0.2855 RCP1n: Vx - MAX: 0.1501 RCP1nb: Vx - MAX: 0.1769 RCP2n: Vx - MAX: 0.1585 RCP2nb: Vx - MAX: 0.1897 FEM: Vy - MAX: 0.5998 REP: Vy - MAX: 0.3313 RCP1n: Vy - MAX: 0.1164 RCP1nb: Vy - MAX: 0.1828 RCP2n: Vy - MAX: 0.1352 RCP2nb: Vy - MAX: 0.1965 Figure 4.24: Thin membrane subjected to transverse loading - Error distributions on an unstructured mesh of four-node elements (contour interval = 0.05,load case No. 1) Chapter 4 145 FEM: Vx - MAX: 0.07470 REP: Vx - MAX: 0.07123 RCP1n: Vx - MAX: 0.01113 RCP1nb: Vx - MAX: 0.04330 RCP2n: Vx - MAX: 0.00817 RCP2nb: Vx - MAX: 0.01966 FEM: Vy - MAX: 0.06859 REP: Vy - MAX: 0.07630 RCP1n: Vy - MAX: 0.01246 y RCP1bn: Vy - MAX: 0.03648 RCP2n: Vy - MAX: 0.00866 RCP2nb: Vy - MAX: 0.01703 Figure 4.25: Thin membrane subjected to transverse loading - Error distributions on an unstructured mesh of nine-node elements (contour interval = 0.005, load case No. 1) 146 4.6- Stretched plate with circular hole (Class II problem) y d/L d/L d/L x d/L Figure 4.26: Thin membrane subjected to transverse loading - Irregular pattern used in sensitivity test to mesh distortion by a uniform refining process: each element of the preceding mesh is divided into four new elements. The following quadrangular elements are employed: bi-linear four-node elements, quadratic serendipity eight-node elements and bi-quadratic nine-node elements. The stress approximations for the RCP and REP procedures are assumed as described in Section A.2.2. In particular, the same approximations are used for eight- and nine-node elements. The distributions of the local error for four-node elements are displayed in Figures 4.30 and 4.31. The distributions of the element error for the same elements are displayed in Figure 4.32. Inspecting these figures reveals that conclusions analogous to those of the preceding benchmark problem can be drawn, so confirming the excellent performance of the RCP procedure also for plane elasticity problems. The convergence properties of the procedure are illustrated in Figure 4.33: the pointwise convergence of the recovered stress components, evaluated at the vertex node (r = 1.5, θ = π/4) indicated on each mesh in Figure 4.29, is shown for four-node elements by Figures 4.33(a)-(c), the convergence of the relative percentage error is shown for four- and nine-node elements by Figures 4.33(d) and (f). As it can be noted, superconvergence of nodal stresses is experienced for the RCP procedure as well as for the REP procedure. On the contrary, the relative percentage error is not superconvergent with any procedure, as expected because the meshes are not fully regular. Anyway, the rate of convergence of the original finite element solution is remarkably improved. As in the previous test, the influence of using boundary patches in the RCPn procedure has also been investigated. The results, reported in Appendix B.2 demonstrate that boundary patches generally improve accuracy, especially on coarse meshes, but sometimes may deteriorate convergence. Finally, the convergence behavior of the RCP-based error estimator is shown in Figure 4.34(a) and (b) for four- and nine-node elements, respectively, and compared to the exact one. In particular, the errors estimated by both RCP1e and RCP2e are reported. The excellent properties of the RCP1e estimator are confirmed. Chapter 4 Load case 1 Load case 2 stress error (Vx) 4-node elements 0.4 FEM 0.3 0.2 REP 0.1 RCP1n Load case 2 4-node elements stress error (Vy) 0.5 stress error (Vx) 147 0.6 0.4 FEM 0.2 4-node elements 0.6 0.4 0.2 REP RCP2n RCP1n FEM REP RCP1n RCP2n RCP2n 0 0.2 0.4 0 0.2 0 -0.4 0.4 0.1 FEM 0.06 0.1 9-node elements 0.06 0.04 FEM REP 0.04 0.02 RCP1n 0.02 -0.2 0 0.2 0.4 (e) -0.2 0.4 Load case 2 0.08 0.06 FEM REP 0.02 RCP1n distortion parameter 0.2 0.04 RCP2n 0 -0.4 0 9-node elements REP RCP2n 0 -0.4 (d) -0.2 distortion parameter (c) Load case 2 Load case 1 9-node elements 0.08 -0.2 distortion parameter (b) stress error (Vx) 0.12 stress error (Vx) -0.2 distortion parameter (a) 0 -0.4 stress error (Vy) 0 -0.4 0 RCP1n RCP2n 0.2 0 -0.4 0.4 distortion parameter (f) -0.2 0 0.2 0.4 distortion parameter Figure 4.27: Thin membrane subjected to transverse loading - Sensitivity to mesh distortion of local error at point x = 0.5, y = 0.5: (a) four-node elements and load case No. 1, (b) and (c) four-node elements and load case No. 2, (d) nine-node elements and load case No. 1, (e) and (f) nine-node elements and load case No. 2 Moreover, it emerges that using the error estimator based on RCP2e could be convenient in conjunction with higher order elements. Similar results are obtained using triangular elements. 4.7 Recovery based error estimators can fail? Recovery-based estimators possess a number of attractive features that have led to their popularity. In particular, their ease of implementation, generality, and ability to produce quite accurate estimators have led to their widespread adoption, especially in the engineering community. However, the estimators also have drawbacks. 148 4.7- Recovery based error estimators can fail? Four-node elements exact error η = 18.0613 RCP1e estimator η∗ = 17.6521 θ = 0.9766 0.5 0.4 0.3 0.2 0.1 (RCP1n estimator: η∗ = 17.0330 θ = 0.9413) Nine-node elements exact error η = 1.2778 RCP1e estimator η∗ = 1.2904 θ = 1.0099 0.03 0.025 0.02 0.015 0.01 0.005 (RCP1n estimator: η∗ = 1.2453 θ = 0.9746) Figure 4.28: Thin membrane subjected to transverse loading, load case No. 1 - Exact and estimated distributions of the element error in energy norm on an unstructured mesh To illustrate the dangers in the indiscriminate use of such techniques, we present a simple example by [Ainsworth & Oden, 2000]. The setting is chosen to be as simple as possible so that there is no question that the source of the problems is due to extraneous eﬀects such as mesh distortions (the partition will be uniform) numerical quadrature (all integrals are evaluated exactly), nonsmooth coeﬃcients (the data are smooth) or nonsmooth solution (the true solution is smooth). Furthermore, the example is even taken to be one-dimensional to avoid any question regarding mesh topology. Nevertheless, even in such an idealistic setting, it is possible to construct an example whereby the recovery based estimators produce an estimated error of zero, while the actual error can be arbitrarily large. It would be quite wrong to dismiss such a situation as being purely academic and having no relevance to practical computation of much more complicated phenomena in multidimensions. The point is that if such anomalies can be present in such a simplified setting, then it is quite possible for similar eﬀects to occur, possibly only locally, in more complex cases. The danger would be that the estimated error in the neighborhood of the local feature would be zero, with the net result Chapter 4 149 y V0 r T a x a mesh 1 tractions according to exact solution y 3.0 V0 mesh 2 x 3.0 mesh 3 Figure 4.29: Stretched plate with circular hole that the adaptive refinement procedure would miss the local feature altogether and the error estimator would not flag any diﬃculties. Obviously, this is a potentially disastrous situation. The example consists of approximating the problem −u00 = f in Ω = (0, 1) , u (0) = u (1) = 0. (4.124) (4.125) The data f is chosen to be of the form f (x) = μ sin (2m πx) , (4.126) where m is a fixed integer and μ is an arbitrary constant. The approximation consists of using piecewise linear finite elements on a uniform partition consisting of 2n elements. Such a partition would arise beginning with a single element and performing n − 1 successive subdivisions of the partition, corresponding to a sequence of uniform refinements. The element nodes are located at the points k xk = n , k = 0, . . . , 2n . (4.127) 2 It is well known that the finite element approximation uh will actually be the piecewise linear interpolant to the exact solution u. Here, the exact solution is 150 4.7- Recovery based error estimators can fail? FEM: σx - MAX: 0.8137 FEM: σy - MAX: 0.5451 FEM: τxy - MAX: 0.4703 REP: σx - MAX: 0.6375 REP: σy - MAX: 0.4903 REP: τxy - MAX: 0.4876 RCP1n: σx - MAX: 0.6725 RCP1n: σy - MAX: 0.5100 RCP1n: τxy - MAX: 0.5015 RCP1e: σx - MAX: 0.6725 RCP1e: σy - MAX: 0.5100 RCP1e: τxy - MAX: 0.5236 Figure 4.30: Stretched plate with circular hole - Local error distributions for four-node elements on mesh No. 1: FEM solution, REP, RCP1n and RCP1e (contour interval = 0.05) Chapter 4 151 RCP2n: σx - MAX: 0.5013 RCP2n: σy - MAX: 0.3614 RCP2n: τxy - MAX: 0.3626 RCP2e: σx - MAX: 0.5581 RCP2e: σy - MAX: 0.4068 RCP2e: τxy - MAX: 0.3629 Figure 4.31: Stretched plate with circular hole - Local error distributions for four-node elements on mesh No. 1: RCP2n and RCP2e (contour interval = 0.05) 152 4.7- Recovery based error estimators can fail? FEM: η = 7.446 REP: η = 6.286 RCP1n: η = 6.692 x 10 -4 6 4 2 0 RCP2n: η = 5.087 RCP1e: η = 7.312 RCP2e: η = 4.612 x 10 -4 6 4 2 0 Figure 4.32: Stretched plate with circular hole - Distributions of element error in energy norm on mesh No. 1 of four-node elements (the relative percentage error is reported above each plot) Chapter 4 -2 4-node elements FEM -2 1 RCP1e -3 RCP1n RCP2e REP Local error (σy) 4-node elements FEM -3 1 -4 RCP1e -4 2 -3 log(stress error) Local error (σx) log(stress error) log(stress error) -1 153 -5 (a) 2 -6 -2.5 -2 -1.5 -1 -0.5 log(element size) 2 RCP1n RCP1e RCP1n 0.9 RCP2n (c) -4 -2 -1.5 -1 -0.5 log(element size) 9-node elements RCP1e RCP1n FEM -5 -6 1.9 REP -7 2.3 RCP2e REP -4.5 RCP2n 2 -9 -2.5 RCP2e -4 REP Relative % error log(stress error) log(stress error) -6 RCP1e -8 -2 -1.5 -1 -0.5 log(element size) 4-node elements RCP2n 1.3 -5 -2.5 RCP2e -5 (b) FEM -3.5 1 RCP2e Relative % error -3 FEM -4 RCP2n REP & RCP1n -2.5 4-node elements -7 RCP2n -5 -6 -2.5 Local error (τxy) (d) -2 -1.5 -1 -0.5 log(element size) -8 -2.5 2.8 (e) -2 -1.5 -1 -0.5 log(element size) Figure 4.33: Stretched plate with circular hole - Rate of convergence: (a) local error in σ x at point r = 1.5, θ = π/4 for four-node elements, (b) local error in σy at point r = 1.5, θ = π/4 for four-node elements, (c) local error in τ xy at point r = 1.5, θ = π/4 for four-node elements, (d) relative percentage error for four-node elements, (e) relative percentage error for nine-node elements Log(estimated relative % error) 4.7- Recovery based error estimators can fail? Log(estimated relative % error) 154 4-node elements 1 exact 0.5 RCP1e estimator 0.5 0 RCP2e estimator 1 -0.5 9-node elements 1 RCP1e estimator 0.5 exact 0 RCP2e estimator -0.5 1 -1 (a) 1 -1 2 3 Log(NDOFs) 1 2 3 Log(NDOFs) (b) 4 Figure 4.34: Stretched plate with circular hole - Convergence of the estimated relative percentage error: (a) four-node elements, (b) nine-node elements given by u (x) = 1 4m π2 μ sin (2m πx) (4.128) and if m ≥ n, then u (xk ) vanishes identically. Consequently, the finite element approximation uh is identically zero. Observe that, no matter how many times the partition is subdivided by increasing n, there always exists a set of data (corresponding to suﬃciently large m) such that uh will vanish identically. Obviously, the gradient of the finite element approximation also vanishes everywhere, and consequently the recovered gradient functions of uh will also vanish everywhere. The majority of practical a posteriori error estimates are based on either the construction of an improved solution which is extracted locally from the finite element solution by some local averaging (recovery) or on the solutions of local boundary value problems set over the elements (or small subdomains associated with the elements) with input-data obtained from the local residuals. This means that the estimated error will be zero. However, the actual error is proportional to |μ| and could therefore be arbitrarily large. Chapter 5 Adaptivity In questo capitolo viene illustrato e discusso il ruolo dello stimatore d’errore presentato nel precedente capitolo quale parametro guida per la ricostruzione della discretizzazione finalizzata alla minimizzazione dell’errore stesso, ovvero quale parametro di controllo di analisi di tipo adattativo. Dopo alcune considerazioni introduttive sulle varie strategie secondo cui si può essere condotta un’analisi adattativa, l’attenzione è focalizzata sull’impiego dello stimatore d’errore quale guida di analisi adattative condotte per completa rigenerazione della discretizzazione, mostrando l’eﬃcacia dello stimatore RCP anche in questo ambito. 5.1 Mesh generation Normal practice to solve engineering problems by means of the Finite Element Method involves increasing the number of discretization points in the computational domain and resolving the resulting system of equations to examine the relative change in the numerical solution. In general, this procedure is time consuming, it depends on the experience of the analyst, and it can be misleading if the solution has not entered an asymptotic range. Ideally, with a robust and reliable self-adaptive scheme, one would be able to specify an initial discrete model which is suﬃcient to describe the geometry/topology of the domain and the boundary conditions, and to specify a desired error tolerance, according to an appropriate criterion. Then, the system would 156 5.1- Mesh generation automatically refine the model until the error measure falls below the prescribed tolerance. The process should be fully automatic and without any user intervention. This approach increases the overall reliability of the analysis procedure since it does not depend on the experience, or inexperience, of the analyst. The need for developing better pre-processing techniques for the finite element method, for performing automated analysis, and for obtaining self-adaptive solutions (which is becoming a trend for commercial finite element software) have driven the development of automatic mesh generation algorithms, i.e. algorithms which are capable of discretizing an arbitrary geometry into a consistent finite element mesh without any user intervention. Figure 5.1: Simplified diagram of the iterative mesh design cycle: (1) self-adaptive tecnique; (1)+(2) self-adaptive tecnique with supervisor Such software is nowadays accompanied by interactive graphics, which eases the mesh generation task considerably. Standardized data interfaces such as IGES and STEP allow CAD-generated data to be transferred between diﬀerent software. The diﬃcult step then follows of assigning the topological element descriptions to fill the required volume. However, the generation of meshes, particularly for complicated 3D structures, can still be a daunting task. Mesh generation techniques fall into two broad classes, which can be termed top-down and bottom-up. Top-down takes for input an overall volume. This is defined to the program as a set of lines and curves in 2D, or equivalent surfaces in 3D. A mesh is automatically formed in the required volume. The problem here is that elements with badly distorted shapes can be generated to fill the space. When huge numbers of elements Chapter 5 157 are being generated in complicated 3D structures, such elements can be hard to spot by eye in the graphics. Some shape checks should be available in the software, and it is prudent to use this to check such distortions. Bottom-up is based on designing the mesh element by element, building up using sets of commands in sequence. Hence a more continuous control on each element is available. However, distortions can occur in the later stages of the process when final parts of the overall shape have to be filled. The process is also time consuming for the user. In both cases, the meshes produced should be checked for an accurate representation of the model, and that the elements are of suitable shape and size. Some elements will have to be somewhat distorted in the fitting process, particularly around detailed geometric features and curved boundaries [Zhu et al. , 1991b]. 5.2 Adaptive finite element techniques Adaptive procedures try to automatically refine, coarsen, or relocate a mesh and/or adjust the basis to achieve a solution having a specific accuracy in an optimal fashion. The computation typically begins with a trial solution generated on a coarse mesh with a low-order basis. The error of this solution is estimated. If it fails to satisfy the prescribed accuracy, adjustment are made with the goal of obtaining the desired solution with minimal eﬀort. While adaptive finite element methods have been studied for nearly twenty years, surprising little is known about optimal strategies. Common procedures studied to date include • local refinement and/or coarsening of a mesh • locally varying the polynomial degree of the basis • relocating or moving a mesh • combinations of the precedent We may guess that the relocating or moving a mesh method alone is generally not capable of finding a solution with a specified accuracy. If the mesh is too coarse, it might be impossible to achieve a high degree of precision without adding more elements or altering basis especially if geometry singularity occurs. This refinement is more useful with transient problems where elements move to follow an evolving phenomenon. By far the local refinement method is the most popular. It can increase the convergence rate, particularly when singularities are present. In some sense locally varying the polynomial degree as refinement is the most powerful. Exponential convergence rates are possible when solutions are smooth. When combined with a local refinement of the mesh size, these high rates are also 158 5.2- Adapt elements or functions? possible when singularities are present. The use of polynomial is most natural with a hierarchical basis, since portions of the stiﬀness and mass matrices and load vector will remain unchanged when increasing the polynomial degree of the basis. In the previous chapter, we have discussed at some length various methods of recovery by which the finite element solution results could be made more accurate and this led us to devise various procedures for error estimation. In this chapter, we shall be concerned with methods which can be used to reduce the errors generally once a finite element solution has been obtained. As the process depends on previous results at all stages it is called adaptive. Such adaptive methods were first introduced to finite element calculations by Babuska and Rheinboldt in the late 1970s [Babuška & Rheinboldt, 1978]-[Babuška & Rheinboldt, 1979]. Before proceeding further it is necessary to clarify the objectives of refinement and specify permissible error magnitudes, and here the engineer or user must have very clear aims. For instance, the naive requirement that all displacements or all stresses should be given within a specified tolerance is not always acceptable. The reasons for this are obvious as at singularities, for example, stresses will always be infinite and therefore no finite tolerance could be specified. The same diﬃculty is true for displacements if point or knife edge loads are considered. The most common criterion in general engineering use is that of prescribing a total limit of the error computed in the energy norm. Often this error is required not to exceed a specified percentage of the total energy norm of the solution and in the many examples presented later we shall use this criterion. However, using a recovery type of error estimator it is possible to adaptively refine the mesh so that the accuracy of a certain quantity of interest, such as the root mean square error in displacement and/or root mean square error in stress, satisfy some user-specified criterion. Zienkiewicz and Zhu in [Zienkiewicz & Zhu, 1987] have used root mean square error in stress in the adaptive procedure to obtain more accurate stress solutions. We should recognize that adaptive procedure based on reducing the root mean square error in displacement is in eﬀect reducing the average displacement error in each element; similarly adaptive procedure based on reducing the root mean square error in stress is the same as reducing the average stress error in each element. Here we could, for instance, specify directly the permissible error in stresses or displacements at any location. Others have used the requirement of constant energy norm density in the adaptive analysis, which is, in fact, equivalent to specifying a uniform distribution of root mean square error in stress in each element. We note that the recovery type of error estimators are particularly useful and convenient in designing adaptive analysis procedures for the quantities of interest. Chapter 5 5.2.1 159 Adapt elements or functions? As we have already remarked in the previous chapter we will at all times consider the error in the actual finite element solution rather than the error in the recovered solution. It may indeed be possible in special problems for the error in the recovered solution to be zero, even if the error in the finite element solution itself is quite substantial. (Consider here for instance a problem with a linear stress distribution being solved by linear elements which result in constant element stresses. Obviously the element error will be quite large. But if recovered stresses are used, exact results can be obtained and no errors will exist). The problem of which of the errors to consider still needs to be answered. At the present time we shall consider the question of recovery as that of providing a very substantial margin of safety in the definition of errors. Figure 5.2: Various procedures by h-refinement: (a) mesh enrichment by transition elements; (b) mesh enrichment; (c) complete mesh regeneration; (d) repositioning node Various procedures exist for the refinement of finite element solutions [Babuška & Miller, 1984]—[Carey, 2006]. Broadly these fall into two categories: • The h-refinement, in which the same class of elements continue to be used but are changed in size, in some locations made larger and in others made smaller, 160 5.2- Adapt elements or functions? to provide maximum economy in reaching the desired solution. • The p-refinement, in which we continue to use the same element size and simply increase, generally hierarchically, the order of the polynomial used in their definition. It is occasionally useful to divide the above categories into subclasses, as the h-refinement can be applied and thought of in diﬀerent ways. In Figure 5.2 we illustrate four typical methods of h-refinement. 1. The first of these h-refinement methods is element subdivision (enrichment) (Figure 5.2(a)). Here refinement can be conveniently implemented and existing elements, if they show too much error, are simply divided into smaller ones keeping the original element boundaries intact. Such a process is cumbersome as many hanging points are created where an element with mid-side nodes is joined to a linear element with no such nodes. On such occasions it is necessary to provide local constraints at the hanging points and the calculations become more involved. In addition, the implementation of de-refinement requires rather complex data management which may reduce the eﬃciency of the method. Nevertheless, the method of element subdivision is quite widely used. 2. The second method is the locally mesh enrichment by using transition zone (Figure 5.2(b)). Here, on the basis of a given solution, some element are subdivided by precise schemes and join together with other elements. Such a process needs sometime distorted element in the transition zone, and therefore is advise against use element sensitive to distortion. 3. The third method is that of a complete mesh regeneration or remeshing (Figure 5.2(c)). Here, on the basis of a given solution, a new element size is predicted in all the domain and a totally new mesh is generated. Thus a refinement and de-refinement are simultaneously allowed. This of course can be expensive, especially in three dimensions where mesh generation is diﬃcult for certain types of elements, and it also presents a problem of transferring data from one mesh to another. However, the results are generally much superior and this method will be used in most of the examples shown in this chapter. For many practical engineering problems, particularly of those for which the element shape will be severely distorted during the analysis, adaptive mesh regeneration is a natural choice. 4. The final method, sometimes known as r-refinement [Figure 5.2(d)], keeps the total number of nodes constant and adjusts their position to obtain an optimal approximation. While this procedure is theoretically of interest it is diﬃcult Chapter 5 161 to use in practice and there is little to recommend it. Further it is not a true refinement procedure as a prespecified accuracy cannot generally be reached. We shall see that with energy norms specified as the criterion, it is a fairly simple matter to predict the element size required for a given degree of approximation. Thus very few re-solutions are generally necessary to reach the objective. With p-refinement the situation is diﬀerent. Here two subclasses exist: 1. One in which the polynomial order is increased uniformly throughout the whole domain; 2. One in which the polynomial order is increased locally using hierarchical refinement. In neither of these cases a direct procedure, which allows the prediction of the best refinement to be used to obtain a given error, has been developed. Here the procedures generally require more resolutions and tend to be more costly. However, the convergence for a given number of variables is more rapid with p-refinement and it has much to recommend it. On occasion it is possible to combine eﬃciently the h- and p-refinements and call it the hp-refinement. In this procedure both the size of elements h and their degree of polynomial p are altered. Much work has been reported in the literature by Babuska, Oden and others and it has been proved that an hp-adaptive system gives the optimal convergence (maximum accuracy for a given number of equations). However, its programming is diﬃcult and requires careful planning of the data base structure [Babuška & Miller, 1984]—[Zienkiewicz et al. , 1989]—[Ainsworth & Oden, 1992]—[Ainsworth & Oden, 2000]. 5.2.2 Structured or unstructured mesh? The most basic form of mesh classification is based upon the connectivity of the mesh: structured or unstructured. Structured mesh is one in which the elements have the topology of a regular grid. Structured meshes are typically easier to compute with (saving a constant factor in runtime) but may require more elements or worse-shaped elements. Unstructured mesh is characterized by irregular connectivity. Unstructured meshes are often computed using quadtrees, or by Delaunay triangulation of point sets; however there are quite varied approaches for selecting the points to be triangulated. Compared to structured meshes, the storage requirements for an unstructured mesh can be substantially larger since the neighborhood connectivity must be explicitly stored. A hybrid mesh is a mesh that contains structured portions and unstructured 162 5.2- Rules: quadtree balancing portions. Note that this definition requires knowledge of how the mesh is stored (and used). There is disagreement as to the correct application of the terms "hybrid" and "mixed". The term "mixed" is usually applied to meshes that contain elements associated with structured meshes and elements associated with unstructured meshes (presumably stored in an unstructured fashion). 5.2.3 h-refinement on structured mesh The quadrilateral type elements generally produce better solutions than the triangulartype, and have been frequently used in the adaptive h−refinement algorithm. However, when an area of complicated geometry is refined locally, the use of four node elements often only leads either to the meshes with distorted shapes or to the meshes with too many degrees of freedom not to result in an ineﬃcient solution like those of Figure 5.3. The adaptive mesh refinements using variable-node elements have been reported to be eﬀective [Gupta, 1978]—[Lo et al. , 2006]. Figure 5.3: Examples of refinement without changing element type The subdivided elements called daughter elements can maintain the same shapes as that of the original element, or mother element, when the variable-node elements are used for mesh refinement, as shown in Figure 5.4. If the well-composed initial mesh is used and the variable-node element is utilized in the transition zone, the mesh distortion will not occur or will be minimized. In evaluating the element mass and stiﬀness matrices of a variable-node element, a normal numerical integration may not be applied directly over the entire element domain; this is because the slope discontinuity of shape function derivative assumed in the element may cause a singular integral. Mesh refinement strategies for elliptic (steady) problems need not consider coarsening. We can refine an initially coarse mesh until the requested accuracy is obtained. This strategy might not be optimal and won’t be, for example, if the coarse mesh is too fine in some regions. Chapter 5 5.2.3.1 163 Rules: quadtree balancing The recursive binary refinement algorithm is simple. Each test point is located with respect to the current tree, as shown in Figure 5.5. The cell where the point is located is recursively divided by two until the size of the resulting cell is less than the characteristic size of the element associated with the test point. This process is repeated for each element edge segment along the curve. Each segment mid point is located in the tree resulting from the refinement due to the previous point. Figure 5.4: Quadtree balancing: (a) split of neighbors; (b) initial element split with new nodes and daughter-elements In any spatial data structure, the domain is enclosed by unit squares (root) that are sub-divided into four equal elements (cells) which are the children of the root. This process can be repeated several times on each of the children until a stopping criteria is met. Two cells are adjacent if they have a common edge. Each child of the cell represents an element. A cell is called a leaf if it does not have any children. The level of a cell is the number of refinements needed to obtain that cell; the root is at level zero. Figure 5.5: Quadtree connection: refinement by subsequent subdivision level 164 5.2.4 5.3- Predicting the required element size in h-adaptivity h-refinement on unstructured mesh It is diﬃcult to make general statements about unstructured mesh generation algorithms because the most prominent methods are very diﬀerent in nature. The most popular family of algorithms are those based upon Delaunay triangulation, but other methods, such as quadtree/octree approaches are also used. Many of the commonly used unstructured mesh generation techniques are based upon the properties of the Delaunay triangulation and its dual, the Voronoi diagram. Given a set of points in a plane, a Delaunay triangulation of these points is the set of triangles such that no point is inside the circumcircle of a triangle. The triangulation is unique if no three points are on the same line and no four points are on the same circle. A similar definition holds for higher dimensions, with tetrahedra replacing triangles in 3D. 5.3 Predicting the required element size in h-adaptivity The error estimators discussed in the previous chapter allow the global energy (or similar) norm of the error to be determined and the errors occurring locally (at the element level) are usually also well represented. If these errors are within the limits prescribed by the analyst then clearly the work is completed. More frequently these limits are exceeded and refinement is necessary. The question which this section addresses is how best to eﬀect this refinement [Zienkiewicz & Zhu, 1987]—[Zienkiewicz & Zhu, 1991]—[Zhu et al. , 1991a]. Here obviously many strategies are possible and much depends on the objectives to be achieved. In the simplest case we shall seek, for instance, to make the relative energy norm percentage error η less than some specified value (say 5% in many engineering applications). Thus η ≤ η̄, (5.1) is to be achieved. In an optimal mesh it is desirable that the distribution of energy norm error (i.e., kek) should be equal for all elements. Thus if the total permissible error is determined (assuming that it is given by the result of the approximate analysis) as ³° °2 ´1/2 Permissible error ≡ η̄ kuk ≈ η̄ °uh ° + kek2 , (5.2) here we have used ° °2 kek2 = kuk2 − °uh ° . (5.3) Chapter 5 165 We could pose a requirement that the error in any element ( ·e ) should be Ã ° °2 !1/2 °uh ° + kek2 keke ≤ η̄ = ēne , ne (5.4) where ne is the number of elements involved. Elements in which the above is not satisfied are obvious candidates for refinement. Thus if we define the ratio keke = ξe, (5.5) ēne we shall refine whenever ξ e > 1, (5.6) ξ e can be approximated, of course, by replacing the true error in Equations (5.4) and (5.5) with the error estimates. The refinement could be carried out progressively by refining only a certain number of elements in which ξ is higher than a certain limit and at each time of refining halve the size of such elements. This type of element subdivision process is also known as mesh enrichment. This process of refinement though ultimately leading to a satisfactory solution being obtained with a relatively small number of total degrees of freedom, is in general not economical as the total number of trial solutions may be excessive. It is more eﬃcient to try to design a completely new mesh which satisfies the requirement that ξ e ≤ 1. (5.7) keke ∝ hpe , (5.8) hnew = ξ −1/p he . e (5.9) One possibility here is to invoke the asymptotic convergence rate criteria at the element level (although we have seen that these are not realistic in the presence of singularities) and to predict the element size distribution. For instance, if we assume hpe where is the current element size and p the polynomial order of approximation, then to satisfy the requirement of Equation (5.4) the new generated element size should be no larger than The reason for the success of the mesh regeneration based on the simple assumption of asymptotic convergence rate implied in Equation (5.8) is the fact that with refinement the mesh tends to be optimal and the localized singularity influence no longer aﬀects the overall convergence. Of course the eﬀects of singularity will still remain present in the elements adjacent to it and improved mesh subdivision can be obtained if in such elements 166 5.3- The h−refinement strategy with RCP we use the appropriate convergence and write, if in Equations (5.8) and (5.9) p is replaced by λ, he , hnew = ξ −1/λ e (5.10) in which λ is the singularity strength. A convenient number to use here is λ = 0.5 as most singularity parameters lie in the range 0.5 ÷ 1.0. With this procedure, added to the refinement strategy, we frequently achieve accuracies better than the prescribed limit in one remeshing. 5.3.1 The h−refinement strategy with RCP The refinement strategy depends on the nature of the accuracy criteria to be satisfied. A very common requirement is to specify the achievement of a certain value of the percentage error. Resorting to a recovery procedure, the percentage error in Equation (5.1) can be estimated as ke∗ k η∗ = ³ ´1/2 . 2 kσ h k + ke∗ k2 Aiming at an equal distribution of error between all elements, requirement (5.1) can be reformulated by placing a limit on the error in each element. Following the strategy proposed in the previous section and denoting by ne the total number of elements in the current analysis, the following requirement on the permissible error in each element ( ·e ) is posed: Ã ° °2 !1/2 °σ h ° + kek2 keke ≤ η̄ = ēm . (5.11) ne Thus, the ratio keke , (5.12) ēm defines the elements to be refined (ξ e > 1) as well as the areas where a coarser subdivision is permissible (ξ e < 1). In practice, an approximated value of ξ e can be computed for each element by replacing the true error kek in Equations (5.11) and (5.12) with an error estimator ke∗ k. Then, the predicted values of ξ e provide guidance for the refinement process. Here, the processes based on both mesh enrichment and complete mesh regeneration are considered. In the latter case the values ξ e are used to predict the element size distribution. Invoking the asymptotic convergence rate criteria, the new generated element size should be no larger than ξe = hnew = he ξ −1/p , e e (5.13) where he is the current element size and p the assumed convergence rate of the error in the area covered by the element. This value is generally taken as equal Chapter 5 167 to the polynomial order of approximation, though of course it is not realistic near singularities. The hp−refinement strategy 5.3.2 In an hp-adaptive solution one needs to pick which item to change first. Since p changes are relatively expensive and must be limited to integers it may be best to select pnew , first and to restrict the change in degree, say n to 0 or ±1 Then, due to the integer choice on p some of the estimated refinement (or de-refinement) still needs to occur by also selecting a new mesh size. We can envision the refinement indicator as having two contributions, ξ = ξ p ·ξ h . If the new integer degree, (p + n), was based on the current element size then the now known numerical value hp ξ p = (p+n) , (5.14) h can be used to get the needed remaining spatial refinement indicator, ξ h . Note that the product relation is hp = hp · h(p+n) = ξp · ξh, (p+n) h(p+n) h(p+n) hnew new which with ξ and ξ h known simplifies to ¶(p+n) µ ξ h ξh = = , ξp hnew ξ= (5.15) (5.16) or finally hnew = h . (5.17) ξh Even with these rough estimates of desired changes one may need other rules to assure that the mesh size and local degree do not change rapidly from one solution iteration to the next, or oscillate between large and small values. 5.4 1 (p+n) Numerical examples To illustrate the eﬃciency of the RCP error estimator in adaptive analyses we present now some examples of adaptive refinement. The refinement strategies shown in the previous section that will be used are: h-refinement with transition element and h-refinement with full mesh regeneration. 5.4.1 Class I problems The first class of problems considered refers to Class I problems (equilibrium problem of a thin membrane of uniform thickness, subjected to a distributed transverse force) described in Appendix A.1. The parameter k is set equal to 1, and the load 168 5.4- Square domain: load case No.1 term b is.specified in the following. 5.4.1.1 Square domain: load case No.1 This example [Zienkiewicz & Taylor, 1989] is fairly straightforward and starts from a simple square domain in which the loading term b is selected such that the exact solution is given by u(x, y) = x(1 − x)y(1 − y) arctan [α(ρ − ρ0 )] , (5.18) √ where ρ = (x + y)/ 2, ρ0 = 0.8 and α = 20. Figure 5.6 shows the exact solution in terms of stresses. As it can be observed, the exact stresses exhibit a relatively sharp concentration. In Figure 5.7 we show the first subdivision of this domain into regular linear and quadratic elements and the subsequent refinements. Base contour value = -0.365100 Maximum contour value = 0.852200 Contour interval = 0.060865 Base contour value = -0.365100 Maximum contour value = 0.852200 Contour interval = 0.060865 (a) (b) Figure 5.6: Exact solution: (a) contours of σx , (b) contours of σ y The elements are of both triangular and quadrilateral shape. For the linear elements the target error is 10% of total energy, while for quadratic elements the target error is 1% of total energy. In all cases, three refinements suﬃce to reach a very accurate solution satisfying the requirements despite the fact that the original mesh cannot capture in any way the high intensity region illustrated in the previous figure. Notice that the prescribed accuracy has been achieved in all cases. Moreover, Figures 5.8-5.11 illustrate a comparison between the distributions of the exact and the estimated error in energy norm, showing a very good agreement. Finally, convergence rates of the exact and of the estimated error on the adaptive mesh refinement are shown in Figures. 5.12 and 5.13. The excellent behaviour of the RCP error estimator can be observed. Chapter 5 169 Mesh 1 – 32 elements (9 DOFs) K = 78.71 % K* = 65.21 % T = 0.778 Mesh 2 – 828 elements (370 DOFs) K = 16.85 % K* = 17.02 % T = 0.010 Mesh 1 – 1881 elements (911 DOFs) K = 9.20 % K* = 9.45 % T = 1.027 Mesh 1 – 32 elements (9 DOFs) K = 57.48 % K* = 55.04 % T = 0.938 Mesh 2 – 580 elements (563 DOFs) K = 13.52 % K* = 13.61 % T = 1.007 Mesh 3 – 1294 elements (1277 DOFs) K = 7.98 % K* = 8.11 % T = 1.017 Mesh 1 – 32 elements (49 DOFs) K = 41.11 % K* = 43.53 % T = 1.124 Mesh 2 – 690 elements (1319 DOFs) K = 2.03 % K* = 2.02 % T = 0.939 Mesh 3 – 1064 elements (2077 DOFs) K = 0.84 % K* = 0.85 % T = 1.018 Mesh 1 – 16 elements (49 DOFs) K = 28.18 % K* = 24.95 % T = 0.874 Mesh 2 – 382 elements (1501 DOFs) K = 1.82 % K* = 1.89 % T = 1.037 Mesh 3 – 876 elements (3477 DOFs) K = 0.72 % K* = 0.78 % T = 1.078 Figure 5.7: Adaptive solutions for: (a) and (b) linear elements with target error set to 10 %, (c) and (d) quadratic elements with target error set to 1 % 170 5.4- Square domain: load case No.1 Figure 5.8: Distributions of element error in energy norm for linear triangular elements: (a) and (c) exact error, (b) and (d) error estimate Figure 5.9: Distributions of element error in energy norm for quadratic triangular elements: (a) and (c) exact error, (b) and (d) error estimate Chapter 5 171 Figure 5.10: Distributions of element error in energy norm for linear quadrilateral elements: (a) and (c) exact error, (b) and (d) error estimate Figure 5.11: Distributions of element error in energy norm for quadratic quadrilateral elements: (a) and (c) exact error, (b) and (d) error estimate 172 5.4- Square domain: load case No.1 Log(relative % error) 2 3-node 1.5 1 0.5 1 6-node 0.5 1 0 -0.5 1 RCP exact 1 1.5 2 2.5 3 3.5 Log(NDOFs) Figure 5.12: Convergence of adaptive refinement for triangular elements Log(relative % error) 2 4-node 1.5 1 0.5 1 9-node 0.5 0 -0.5 1 RCP exact 1 1.5 1 2 2.5 3 3.5 Log(NDOFs) Figure 5.13: Convergence of adaptive refinement for quadrilateral elements Chapter 5 5.4.1.2 173 Square domain: load case No.2 A square domain is considered also in this example [Ainsworth & Oden, 2000], but the source term b is chosen so that the exact solution of the problem is Base contour value = -62273.7487 Maximum contour value = 18906.9979 Contour interval = 4272.6709 (a) Base contour value = -62273.7487 Maximum contour value = 18906.9979 Contour interval = 4272.6709 (b) Figure 5.14: Exact solution: (a) contours of σ x , (b) contours of σy ´ ´ ³ ³ 2 2 u(x, y) = 5x2 (1 − x)2 e10x − 1 y 2 (1 − y)2 e10y − 1 (5.19) The solution is smooth, but possesses step gradients in the neighborhood of the boundaries as shown in Figure 5.14(a)-(b). For this reason, in this second load case the target error is higher than in the previous one: 15% of total energy for linear elements and, 5% of total energy for quadratic elements. In Figure 5.15 we show the first subdivision of this domain into regular linear and quadratic elements and the subsequent refinements Also in this example three refinements suﬃce to reach a very accurate solution satisfying the requirements, so confirming the good performance of the RCP estimator. Moreover, Figures 5.16—5.19 compare the distribution of the exact error with that ao the estimated one, obtained both with the RCPn and the RCPe estimator. 174 5.4- Square domain: load case No.2 Figure 5.15: Adaptive solutions for: (a) and (b) linear elements with target error set to 15 %, (c) and (d) quadratic elements with target error set to 5 % Chapter 5 175 RCP: K* = 75.6859 , T = 0.31252 K = 93.0156 RCPE: K* = 96.3847 , T = 0.9762 3.5 3.5 3.5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 RCP: K* = 32.9078 , T = 1.2526 K = 26.8024 RCPE: K* = 25.6085 , T = 0.95221 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 RCP: K* = 8.3185 , T = 0.98685 K = 8.4286 RCPE: K* = 8.0767 , T = 0.95797 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 Figure 5.16: Distributions of element error in energy norm for linear triangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate 176 5.4- Square domain: load case No.2 RCP: K* = 76.0242 , T = 0.26009 K = 97.5864 RCPE: K* = 97.6168 , T = 0.99972 3.5 3.5 3.5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 Mesh 1 - 16 elements (9 DOFs) RCP: K* = 23.3559 , T = 1.0583 K = 22.1334 RCPE: K* = 21.7384 , T = 0.98126 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 Mesh 2 - 768 elements (804 DOFs) RCP: K* = 14.5341 , T = 0.98944 K = 14.6858 RCPE: K* = 14.3001 , T = 0.97318 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 Mesh 3 - 1492 elements (1566 DOFs) Figure 5.17: Distributions of element error in energy norm for linear quadrangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate Chapter 5 177 RCP: K* = 75.2798 , T = 0.97361 K = 64.8979 RCPE: K* = 75.5045 , T = 0.98036 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 Mesh 1 - 32 elements (72 DOFs) RCP: K* = 3.4159 , T = 0.99604 K = 3.4294 RCPE: K* = 3.4456 , T = 1.0047 0.2 0.2 0.2 0.18 0.18 0.18 0.16 0.16 0.16 0.14 0.14 0.14 0.12 0.12 0.12 0.1 0.1 0.1 0.08 0.08 0.08 0.06 0.06 0.06 0.04 0.04 0.04 0.02 0.02 0.02 0 0 0 Mesh 2 - 710 elements (1452 DOFs) RCP: K* = 0.80648 , T = 0.98497 K = 0.81878 RCPE: K* = 0.81881 , T = 1 0.045 0.045 0.045 0.04 0.04 0.04 0.035 0.035 0.035 0.03 0.03 0.03 0.025 0.025 0.025 0.02 0.02 0.02 0.015 0.015 0.015 0.01 0.01 0.01 0.005 0.005 0.005 0 0 0 Mesh 3 - 1669 elements (3423 DOFs) Figure 5.18: Distributions of element error in energy norm for quadratic triangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate 178 5.4- Square domain: load case No.2 RCP: K* = 71.6472 , T = 1.8682 K = 47.863 RCPE: K* = 48.1312 , T = 0.99885 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 Mesh 1 - 32 elements (72 DOFs) RCP: K* = 6.8549 , T = 1.0009 K = 6.849 RCPE: K* = 6.833 , T = 0.99765 0.4 0.4 0.4 0.35 0.35 0.35 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 Mesh 2 - 556 elements (2284 DOFs) RCP: K* = 3.3175 , T = 0.98545 K = 3.3664 RCPE: K* = 3.3738 , T = 1.0022 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 Mesh 3 - 1669 elements (3423 DOFs) Figure 5.19: Distributions of element error in energy norm for quadratic quadrangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate Chapter 5 5.4.2 179 Class II problems In this section some adaptivity examples concerning plane stress problems (Class II problems) are presented. Young modulus is set equal to 1 and Poissons ratio to 0.3. 5.4.2.1 Short cantilever with uniformly distributed load A short cantilever subjected to uniformly distributed load is considered, Figure 5.20 [Zienkiewicz & Taylor, 1989]—[Zienkiewicz & Zhu, 1991]. Figure 5.20: Short cantilever beam The allowable error η is set equal to 5%. There are two singular stress points at the two corners of the edge clamped on the wall. The reference solution is obtained with a fine mesh of 64 × 64 elements. Adaptive analysis is performed employing mesh refinement with transition element. The details about the transition elements used in this example are given in Appendix D. As expected, in the refinement process, elements are significantly refined around the singular stress points. 5.4.2.2 L-shaped domain This example considers a finite plate subjected to tensile loading along the orizontal direction, one quarter of which is shown in Figure 5.22. In the plate centre, there is a square cut out that introduces four singular points at the four corners of the cut out. As discussed in the previous section (see Equation (5.10)), the eﬀects of the singularity will still be present in the elements adjacent to it and improved mesh subdivision can be obtained if, in such elements we use the appropriate singularity strength coeﬃcient λ. As shown before most singularity parameters lie in the range 0.5 ÷ 1.0. A convenient number to be use here is λ = 0.5 A complete 180 5.4- Machine part Mesh 1 Mesh 2 K = 41.4224 K = 26.4290 K = 23.2132 T = 0.87832 K = 36.3530 T = 0.87762 16 elements Mesh 4 4 elements Mesh 3 K = 15.9863 K = 9.4736 K = 14.3631 K = 8.5933 T = 0.90707 T = 0.89846 241 elements Mesh 6 64 elements Mesh 5 K = 6.1050 K = 4.2994 K = 4.0700 T = 0.94663 K = 5.6330 T = 0.92269 445 elements 690 elements Figure 5.21: Short cantilever beam solved by mesh enrichment: linear quadrilateral elements mesh regeneration with quadratic quadrilateral elements is used. The allowable error η is set equal to 1%. Figure 5.22 shows the progressive steps of adaptive refinement and Figure 5.23 shows the corresponding distributions of the estimated error with both RCPn and RCPe estimator. The convergence rate is given in Figure 5.24 5.4.2.3 Machine part The machine part shown in Figure 5.25 is considered [Zienkiewicz & Taylor, 1989]— [Zienkiewicz & Zhu, 1991]. The allowable error η is set equal to 5%. Adaptive analysis is performed using complete mesh regeneration strategy. Figure 5.26 and 5.27 show the RCP-based adaptive mesh refinement with quadrangular and triangular elements, respectively. As it can be noted only two steps suﬃce to reach the desired accuracy. Chapter 5 181 50 50 50 1 100 Mesh 1 – 27 elements (252 DOFs) η = 8.26 % η* = 10.03 % θ = 1.214 Mesh 2 – 169 elements (1424 DOFs) η = 2.67 % η* = 2.83 % θ = 1.058 Mesh 3 – 358 elements (2992 DOFs) η = 0.88 % η* = 0.90 % θ = 1.014 Figure 5.22: Adaptive refinement of an L-shaped domain in plane stress with prescribed error of 1%: nine-node elements 182 5.4- Machine part K = 8.274 - RCP:* K = 10.0332 , T = 1.2126 K = 8.274 - RCPE: K* = 15.8408 , T = 1.9145 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 Mesh 1 - 27 elements (119 DOFs) K = 2.6739 - RCP: K = 2.8292 , T = 1.0581 Mesh 2 - 169 elements (705 DOFs) K = 2.6739 - RCPE: K = 4.0278 , T = 1.5063 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 * K = 0.88329 - RCP: K = 0.89598 , T = 1.0144 * K = 0.88329 - RCPE: K = 1.3807 , T = 1.5631 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 Mesh 3 358 elements (1487 DOFs) Figure 5.23: Adaptive solutions for quadratic elements with target error set equal to 2 % Chapter 5 183 Log(relative % error) 1.2 1 RCP 0.8 0.6 exact 0.4 0.2 1 1 0 -0.2 2.4 2.6 2.8 3 3.2 3.4 Log(NDOFs) Figure 5.24: Convergence for L-shaped domain Figure 5.25: Machine part domain 184 5.4- Machine part Figure 5.26: Adaptive refinement of machine part using linear quadrilateral elements. Chapter 5 Figure 5.27: Adaptive refinement of machine part using linear triangular elements. 185 Chapter 6 Some advanced applications in structural mechanics La formulazione dell’elemento piastra è un caso speciale della formulazione generale dell’elemento guscio ed è basata sulla teoria delle piastre con deformazioni trasversali taglianti incluse. Questi teoria, dovuta a E. Reissner e R. D. Mindlin, si basa sul presupposto che le particelle della piastra originalmente su una linea retta, normale alla superficie indeformata centrale rimangano su una linea retta durante il procedimento di deformazione, Tale linea non sarà necessariamente normale alla superficie centrale deformata. Le strutture sottili come piastre e gusci costituiscono una categoria importante fra le analisi per elementi finiti a causa del loro grande campo di applicazione. Il presente capitolo punta l’attenzione sulla presentazione di un’estensione dello stimatore d’errore basato sulla procedura RCP all’analisi per elementi finiti piastre alla Reissner-Mindlin. Come ben noto tali modelli strutturali soﬀrono di problemi legati alla loro formulazione quando il rapporto tra larghezza della piastra e spessore della piastra aumenta, manifestandosi in un abnorme aumento di rigidezza della piastra. Al fine di evitare tali problemi si ricorre abitualmente ad elementi di tipo misto agli sforzi, ove le variabili spostamento e tensione vengono coinvolti separatamente. Nel presente capitolo vengono inoltre presentate alcune applicazioni avanzate, quali l’impiego della tecnica proposta con elementi finiti di tipo misto agli sforzi, l’applicazione alle piastre e l’estensione alla ricostruzione del profilo delle tensioni tangenziali lungo 188 6.1- Recovery for assumed stress finite elements lo spessore di compositi laminati. In this chapter we address some advanced problems in structural mechanics, such as structures having a "thin" dimension in one or more directions and constrained elastic problems. In particular we focus the attention on shear deformable plates structures, which are an important class of structures because of their large application fields. Indeed this application is one of the first to which finite elements were directed, but it still is a subject of continuing research [Alfano et al. , 2001]—[Auricchio et al. , 2006]. The reason lies on the well known pathology called shear locking, that is an abnormal stiﬀening exhibited by standard finite elements as the thickness/length ratio decreases. This locking performance is typical of constrained elastic problems such as nearly incompressible elasticity, and is more pronounced for lower order elements and in the presence of geometry distortion. A successful approach to cure this trouble is to develop non standard finite element based on mixed formulations. In the following the RCP based error estimation is extended to tackle with such advanced applications. First the recovery procedure is extended to non standard finite element formulation????, such as assumed stress formulations. Then the recovery procedure is applied to plate structures based on Reissner-Mindlin theory [Castellazzi et al. , 2006a]—[Castellazzi et al. , 2006c]—[Castellazzi et al. , 2006b]. In particular, the RCP procedure is applied to a hybrid stress (locking-free) element recently proposed in [de Miranda & Ubertini, 2006] besides standard assumed displacement elements. Finally, laminated composite plates are considered, with the formulation of First order Shear Deformation Theory (FSDT), and the RCP procedure is applied to capture the transverse shear stresses along the thickness [Daghia et al. , 2007], which is crucial in order to predict critical phenomena such as delamination. Indeed, this last application can be also viewed as an attempt to pass from a "gross" model (FSDT plate model) to a "finer" model (3D elasticity) and, in this sense, it could be viewed as a first step towards model validation. 6.1 Recovery for assumed stress finite elements Here, the procedures presented in the previous Chapter are applied to recover stresses from mixed and hybrid stress finite elements, based on the formulation early proposed by Pian [Yuan et al. , 1993]. Reference is made to the model problem discussed in Section 2.2 and the associated mixed and hybrid functionals (see Equations (2.20)—(2.21)). Both displacements and stresses are involved as independent variables and are Chapter 6 189 independently approximated as uh σ h = Uūh (6.1) = Pβ (6.2) In particular a standard interpretation is assumed for displacements, based on standard shape functions U and nodal values uh . Identifying nodal values at common element boundaries by the usual assemblage procedure ensures the required displacement continuity. On the other hand, stress are approximated independently in each element, through a matrix of stress modes P and a vector of unknown parameters β which is local to each element. In the hybrid version, stresses should satisfy the equilibrium equations pointwise within each element. This is obtained by the stress into two parts: σ h = σ hp + σ hh . (6.3) The first is a known function equilibrating the external load within the element D∗ σ hp = b. (6.4) The second is approximated according to Equation (6.2) by properly selecting self-equilibrating stress modes: D∗ P = 0. (6.5) Introducing the assumed displacements and stresses into mixed or hybrid functional referred to the single element and making it stationary yield the following discrete element equations: ¸ ∙ ¸ ∙ ¸ ¸∙ ∙ gσ 0 β −H G − = , (6.6) h ūh gu GT 0 where H= Z PT C−1 PdV, GT = Ωe Z T (DU) PdV, (6.7) Ωe h are the element nodal forces, gσ and gu are terms due to the prescribed loads. In the hybrid version matrix G reduces to Z T G = UT NT PdV. (6.8) Ωe At this level, inner parameters β can be condensed out and the following elemental equations involving only nodal displacements are obtained h = Kūh − f , (6.9) where K is the element stiﬀness matrix and f the equivalent nodal loads. Notice that the above equations are in the standard format of assumed displacement finite elements. All the recovery procedures presented in Section 4.3 can be applied to finite 190 6.2- Plate structures element solutions obtained by mixed and hybrid stress elements. However, these elements oﬀer new possibilities which deserve to be discussed. In particular, the finite element solution in terms of stresses is directly given by σ h (see Equation (6.2)) or it can be computed by uh as σ hu = CDuh = CBūh . (6.10) As a consequence the recovery procedure can operate either on σ h or on σ hu . Considering the RCP procedure and the compatibility condition it enforces (see Equation (4.62)), the following two alternatives can be derived ⎧ Z ⎨ (1) h = Būh ¡ ¢ ∗T ∗ h dV = 0 δσ , (6.11) p− ⎩ Ωp (2) h = C−1 σ h depending on if we evaluate the finite element strains from displacement uh or from stresses σ h . Both the choices can be used in principle. The first is coherent with the RCP formulation as described n Section 4.3, while the second can not be traced back to the minimization of functional (4.56), but should be formulated directly in the form of Equation (6.11), that is based on arguments related to the principle of virtual works. This does not appear as a limitation. Indeed this second choice is expected to perform better because stresses σ h are more accurate than those computed from displacements. It is expected to be especially true in constrained problems. These two RCP versions are compared in Section 6.3 through some numerical tests on plate problems. Based on the same arguments two versions of the other recovery procedures can be devised. In particular, the REP procedures can be implemented as a least square of the following conditions ⎧ Z ⎨ (1) σ h ¡ ¢ T ∗ ∗ B σ pi − σ i dV = 0 . (6.12) ⎩ h h Ωp (2) σ = CBū These two possibilities are both tested in Section 6.3. 6.2 Plate structures In this section the RCP recovery procedure is generalized to plate structures. In particular, shear deformable plates are considered. The corresponding generalization of the SPR and REP procedures can be found in [Boroomand et al. , 2004]. The governing equations for shear deformable plate structures are given in Appendix A.3. As well known standard finite elements for this class of problems suﬀers from shear locking in the thin plate element. In this work a four-node locking-free finite element based on hybrid stress formulation is employed for finite Chapter 6 191 element analysis [de Miranda & Ubertini, 2006]. The element is denoted by 9βQ4 and is briefly described in Appendix C. For completeness, the standard assumed displacement four node element (C4) is also included in the analysis. The RCP recovery procedure proceeds as illustrated in Chapter 4. The two ways to form the patch are shown in Figure 6.1. FEM displacements prescribed FEM displacements prescribed :p :p w:p w:p element patch node patch Figure 6.1: Example of node and element patch: (•) assembly node definig the patch, the dark element is the assembly element definig the patch. The condition to be enforced on each patch specializes as ¢ ¡ ∗ ¢ ¡ ∗ ¢¤ R £ ∗T ¡ ∗ δN μp − μh + δM∗T χp − χh + δS∗T γ p − γ h dV = 0, p p Ωp ∀ δN∗ , δM∗ , δS∗ , (6.13) where (δN∗ , δM∗ , δS∗ ) are the recovered stress resultants (over the patch) that ¢ ¡ should satisfy a priori the plate equilibrium equations, μ∗p , χ∗p , γ ∗p are the strain ¡ h h h¢ components corresponding to the recovered stress resultants and μ , χ , γ are the finite element strain components. The recovered stress resultantes are expressed as ⎡ ∗ ⎤ ⎡ ∗ ⎤ ⎡ ∗ ⎤ Nph Np Npp ⎣ M∗p ⎦ = ⎣ M∗pp ⎦ + ⎣ M∗ph ⎦ , (6.14) S∗p S∗pp S∗ph where the first term is chosen to be in equilibrium with the first term chosen to be in equilibrium with the applied loads over the patch (i.e. as a particular solution as the plate equilibrium equations over the patch), whereas the second term is a self-equilibrated term, approximated in the form ⎡ ∗ ⎤ Nph ⎣ M∗ph ⎦ = P∗ a, (6.15) S∗ph such that ⎡ ⎣ D∗p D∗p ⎤ I ⎦ P∗ = 0. D∗p (6.16) 192 6.3- Some numerical tests on bending response The equilibrium operators in the above condition are defined in Appendix A.3. The expressions adopted in this work for P∗ as well as for the first term in Equation (6.14) are given in Appendix C. ¢ ¡ The recovered strain components μ∗p , χ∗p , γ ∗p in Equation (6.13) are computed from the recovered stress resultants via the plate constitutive equations. Then, to apply the RCP condition (6.13) we need to specify how to compute the finite element strain components. When the recovery is used together with assumed stress elements, we have two possibilities as outlined in the previous section: they can be computed from the finite element kinematics via compatibility equations or from the finite element statics, that is from finite element stress resultants via the plate constitutive equations. Indeed, the hybrid stress element used in this work has a non standard displacement interpolation, or called linked interpolation. Thus the first version based on finite element kinematics can be implemented by ¡ ¢ computing μh , χh , γ h either through a standard interpolation or through the true linked interpolation. All these possibilities are tested in the following. However in the thin plate limit, the version based on finite element statics is expected to be superior to the other since finite element stress resultants (that are directly interpolated) do not suﬀer from the appearance of internal constraints. In the next section, the performance of RCP recovery is numerically verified. Since the membrane response has been already tested the attention is focused here on the bending response by considering homogeneous isotropic linearly elastic plates. In Section 6.4 the coupled membrane bending response is investigated by considering laminated composite plates. 6.3 Some numerical tests on bending response As shown in the previous chapters, two main approaches may be employed to investigate the performance of recovery methods. The first approach uses benchmark problems with exact solutions and attempts to generalize the conclusions to wider range of problems though no proof is given. Majority of studies on error estimators fall within this form of approach. In the second approach, the asymptotic behaviour is investigated. Although such a study does not play the role of a proof, the mathematical basis is much stronger and the range of application is much wider than the first approach. Unfortunately the second approach has not been yet generalized to plate bending problems, and such an extension is beyond the scope of this chapter. Hence the first approach is used in the following. Two homogeneous isotropic linearly elastic square plates, are considered for Chapter 6 w = 0 & Tx = 0 (SS2) w = 0 & Ty = 0 (SS2) L/2 y 193 L/2 x (a) (b) Figure 6.2: (a) Test No.1- SS2 square plate under uniform load; (b) test No.2 - square plate with prescribed displacement various aspect ratios h/L from thick to thin cases (Figure 6.2). The first square plate, of side length L = 8, is subjected to uniformly distributed transverse load q = 1, with simply supported boundary conditions of hard (SS2) type. This test case is referred to as Test No.1. Four meshes, created from a coarse mesh of 4×4 elements by a uniform refining process, are used in the finite element analysis, which is carried out for two values of the aspect ratio (h/L = 0.2 and h/L = 0.002) and assuming: E = 10.92 and ν = 0.3. Conventional 4-node elements and hybrid stress 9βQ4 elements (see Appendix C) are employed. For this test some results on unstructured meshes are also discussed. The second square plate, of side length L = 4, is subjected to prescribed transverse displacements along the boundary according to the following solution w = sin (x) e(−y) y, (6.17) This test case is referred to as Test No.2. Four meshes, created from a coarse mesh of 4×4 elements by a uniform refining process, are used in the finite element analysis, which is carried out for an intermediate value of the aspect ratio (h/L = 0.025) and assuming: E = 10.92 and ν = 0.3. Because of locking, only hybrid stress elements are employed for this test. Depending on the thickness of the plate, the exact solution for the first benchmark test is given with assumption of thick or thin plate formulation. Fourier series can be used for both cases expressing the plate deflection as follows ∞ X ∞ X nπy mπx w= sin , Cmn sin (6.18) a b m=1 n=1 with a and b being the plate dimensions and Cmn representing a series of coeﬃcients to be determined from the solution process. For thin plate solution, the deflection 194 6.3- Some numerical tests on bending response Table 6.1: Versions of RCP recovery tested and related approximations Q4 & 9βQ4 REP RCP l i n e a r RCP q u a d r a t i c linear stress expansion na = 3 + 3 quadratic stress expansion na = 9 cubic stress expansion na = 17 assumed above is directly substituted in to the governing diﬀerential equation derived from Kirchhoﬀ assumptions. The solution is straightforward. For thick plate solution, however, some similar expressions for θx and θy are needed ∞ X ∞ X nπy mπx sin , θx = Amn cos (6.19) a b m=1 n=1 θy = ∞ ∞ X X m=1 n=1 Bmn sin nπy mπx cos . a b (6.20) The solution may be found in Reference [Dobyns, 1981]. In the following the RCP recovery is tested by using diﬀerent versions as summarized in Table 6.1 and compared with REP recovery. The comparisons are carried out based on the following norms Total energy norm: ³ ´1/2 2 2 kukt = kukb + kuks , (6.21) Energy norm of bending stresses ∙ Z ¸1/2 1 kukb = M T C−1 M dΩ , b 2 Ω Energy norm of shear stresses ∙ Z ¸1/2 1 kuks = ST C−1 S dΩ , s 2 Ω The associated error definitions read as (6.22) (6.23) ³ ´1/2 , ke∗ kt = ke∗ k2b + ke∗ k2s (6.24) ¸1/2 ¢ ¡ h ¢ 1R¡ h ∗ T −1 ∗ M −M ke kb = Cb M − M dΩ , 2Ω ∙ ∗ ¸1/2 ¢ ¡ h ¢ 1R¡ h ∗ T −1 ∗ S −S ke ks = Cs S − S dΩ , 2Ω ∗ η ∗t = ke∗ kt × 100, kukt ∙ η ∗b = ke∗ kb × 100, kukb η ∗s = ke∗ ks × 100. kuks (6.25) (6.26) (6.27) Chapter 6 195 The results are presented in terms of convergence of both global error and pointTotal energy error % Bending energy error % Shear energy error % 0 C4 -1 1 0 log(stress error) log(stress error) log(stress error) C4 0 C4 1 -1 RCPE -1 1 RCPE REP RCP -2 REP RCP -2 REP 1.4 RCP, RCPE -2 1.5 -1 1.9 -0.5 0 0.5 log(element size) 1 -1 -0.5 0 0.5 log(element size) -3 -1 1 -0.5 0 0.5 log(element size) 1 Figure 6.3: Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on linear stress approximations Total energy error % Bending energy error % 1 0 log(stress error) C4 -1 Shear energy error % 0 log(stress error) log(stress error) 0 C4 -1 C4 -1 1 1 REP RCPE RCPE RCP -2 1.9 -1 -0.5 0 0.5 log(element size) RCP, RCPE RCP -2 2.0 1 -1 -0.5 0 0.5 log(element size) REP -2 REP 1.8 1 -3 -1 -0.5 0 0.5 log(element size) 1 Figure 6.4: Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on quadratic stress approximations wise error at point A located at the centre of the plate, from thick to thin cases. Convergence of the global errors in energy norm, for the thick case (h/L = 0.2), is shown in Figures 6.3 for the linear stress approximation and in Figure 6.4 for the quadratic stress approximation. These figures demonstrate the performance of RCP and RCPE when compatible C4 elements are used. As expected, the average rate of convergence of FEM solution is close to unity while those of the recovered solutions are higher than one, so evidencing a superconvergent behaviour similar 196 6.3- Some numerical tests on bending response to that observed for two-dimensional elastic problems. Notice that the convergence rates for the various recovery methods are similar and tend to two when quadratic stress approximation is adopted. The same case has been solved using hybrid stress elements instead of conventional C4 elements to obtain the finite element solution, and the corresponding results are shown in Figures 6.19—6.6 for linear and quadratic tress approximations, respectively. These results are similar to those observed in the previous analysis and no appreciable diﬀerences are noted among the various versions of the recovery procedures. In particular, RCP, RCPC and RCPL all give similar responses. Bending energy error % Total energy error % 9EQ4 -1 1 0 REP log(stress error) log(stress error) log(stress error) Shear energy error % 0 0 9EQ4 -1 1 REPL REPC -1 RCP RCPE 9EQ4 RCPE RCP -2 RCPL, RCPc REPL -2 RCPE -0.5 0 0.5 log(element size) 1.3 1 -2 REP, REPc 1.4 -1 1.9 RCP 1 -1 -0.5 0 0.5 log(element size) REP 1 -3 -1 -0.5 0 0.5 log(element size) 1 Figure 6.5: Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on Linear stress approximations Passing to a thinner plate (h/L = 0.025) some curious responses are observed. The results obtained (using hybrid stress elements) in terms of both global and local errors are shown by Figures 6.7—6.10, for linear and quadratic stress approximations. The values of percentage point-wise errors in stress resultants in the plate centre are collected in Tables 6.2—6.6. As it can be noted, the responses of recovery procedures based on FEM displacements (RCP, RCPEC , RCPEL and RCPL ) drastically deteriorates due to the poor performance on shear stresses. This becomes even more pronounced by increasing the order of stress approximation and can be interpreted as a sort of locking which destroyes the superconvergent properties of the recovery procedure in the thin plate limit. The reason lies on the fact that the recovery procedure is applied to the kinematic part of the finite element solution, which suﬀers from the internal constraint of thin plate limit. In fact the RCP procedures based on FEM stresses Chapter 6 197 Total energy error % Shear energy error % Bending energy error % 0 0 0 REPL RCPE -1 1 REP 9EQ4 -1 RCPE 1 RCPEL RCPE -1 9EQ4 RCP 1 -2 RCP RCPEL RCPL, RCPEC -2 log(stress error) 9EQ4 REPC REPL log(stress error) log(stress error) REP, REPc -2 RCPEC REP RCP RCPL, RCPC 1.7 -3 -1 -3 -0.5 0 0.5 log(element size) 1 1.7 1.8 -1 -0.5 0 0.5 log(element size) -3 -1 1 -0.5 0 0.5 log(element size) 1 Figure 6.6: Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on quadratic stress approximations (RCP and RCPE) do not exhibit such an analogous behaviour and maintain the superconvergent properties. This behaviour is confirmed by the predictions for the very thin plate, shown in Figure 6.11. As it can be observed, superconvergence is experienced using both RCP and RCPE. Total energy error % Shear energy error % Bending energy error % 1 1 0 REPC 9EQ4 9EQ4 log(stress error) 0 REPL log(stress error) log(stress error) REPL -1 1 REPC 0 9EQ4 -1 RCP -1 1 REP -2 1 RCPL, RCPC RCPL 1.6 -1 -0.5 0 log(element size) -2 REP RCP -2 -3 -1.5 RCPC RCP RCPc REP 1.7 0.5 -3 -1.5 -1 -0.5 0 log(element size) -3 0.5 -1.5 1.5 RCPL -1 -0.5 0 log(element size) 0.5 Figure 6.7: Test No.1 (h/L = 0.025) - convergence of global errors for recovery procedures based on linear stress approximations For the very thin plate, that has been recognized as the most sever test problem, the local error distributions on the 8 × 8 uniform mesh are given in Figures ??— ??. In particular, Figure ?? refers to the element error in energy norm whereas Figures ??—?? to the pointwise errors in Mx , Mxy and Sx . As expected, the recovery procedures sensibility reduce the error also locally within the domain, while appear less eﬀective near boundaries where small patches are used. Moreover, 198 6.3- Some numerical tests on bending response Bending energy error % Total energy error % Shear energy error % 0 1 1 REPL REPC 9EQ4 0 RCPC log(stress error) 9EQ4 REP log(stress error) log(stress error) REPL RCPC -1 1 -1 REPC 0 RCPC -1 9EQ4 RCPL 1 RCPL -2 REP 1.5 1.6 -1 -0.5 0 log(element size) -3 -1.5 0.5 RCPL 1 RCP RCP -3 -1.5 -2 REP -2 -3 -1 -0.5 0 log(element size) 0.5 -1.5 1.8 RCP -1 -0.5 0 log(element size) 0.5 Figure 6.8: Test No.1 (h/L = 0.025) - convergence of global errors for recovery procedures based on quadratic stress approximations no significant diﬀerences in accuracy emerge among the various procedures, so confirming the very good performance of the RCP procedure expecially in the element version. Finally, the sensitivity to mesh distortion has been tested. For the sake of brevity, only the results obtained for the very thin plate (Test No.1) are reported using an unstructured mesh. In particular, Figures 6.16—6.18 show the pointwise errors in the stress resultants. These results confirm those experienced on regular geometry and demonstrates that the RCP procedure is reliable also for arbitrary irregulars meshes. Local error - Mx Local error - My 1 RCPE -2 1 -1 -2 RCPE -1.5 -1 -0.5 0 log(element size) -3 0.5 -1.5 -1 RCPE 9EQ4, REP RCP -2 RCP 2 0 REP RCP, REP -3 log(stress error) 9EQ4 -1 9EQ4 0 log(stress error) log(stress error) 0 Local error - Mxy 2 2 -1 -0.5 0 log(element size) 0.5 -3 -1.5 -1 -0.5 0 log(element size) 0.5 Figure 6.9: Test No.1 (h/L = 0.025) - convergence of local error at the plate centre for recovery procedures based on linear stress approximations Summing up, the numerical experimentation evidences that the RCP procedure Chapter 6 199 Local error - Qx Local error - Qy 2 2 REPC REPL 1 log(stress error) log(stress error) REPC REPL 9EQ4 0 RCPC RCPEC -1 1 0 RCPC RCPEC 9EQ4 -1 0.9 -2 RCPL RCPEL REP -2 RCPEL 0.9 RCPE RCP REP RCPL RCP -3 2 -4 -1.5 -3 RCPE -1 -0.5 0 log(element size) 2 0.5 -4 -1.5 -1 -0.5 0 log(element size) 0.5 Figure 6.10: Test No.1 (h/L = 0.025) - convergence of local error at the plate centre for recovery procedures based on quadratic stress approximations Table 6.2: Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Mx at point A ,9βQ4. mesh FEM REP RCP RCPE REP C REP L RCP C RCP L RCPE C RCPE L 4 × 4 102.2980 63.6449 65.8533 65.8533 65.4537 63.6449 65.8533 65.8533 65.8533 65.8533 8×8 65.9772 17.1101 16.9315 23.1429 17.8086 17.1101 16.9315 16.9315 21.6229 21.6269 16 × 16 32.2545 4.8034 4.8050 6.1959 5.0017 4.8034 4.8050 4.8050 6.0956 6.0984 32 × 32 15.7180 1.2325 1.2331 1.5951 1.2840 1.2325 1.2331 1.2331 1.5864 1.5889 64 × 64 7.7118 0.3106 0.3107 0.4036 0.3236 0.3106 0.3107 0.3107 0.4012 0.4032 is reliable and eﬀective with either node or element patches, and that the linear stress approximation oﬀers the best trade-oﬀ between accuracy and computational cost, especially for thin plates. Using quadratic stress approximations tends to increase accuracy in any case but improves also the rate os convergence only for thick plates. Table 6.3: Test No. 2 (h/L=0.025): Point-wise errors (per cent) of My at point A, 9βQ4. mesh FEM REP RCP RCPE REP C REP L RCP C RCP L 4×4 107.6726 45.5386 20.0198 20.0198 55.0367 45.5386 20.0198 20.0198 8×8 145.2619 15.8789 14.6587 54.7693 17.9765 15.8789 14.6587 14.6587 16 × 16 82.9392 4.5577 4.4799 10.5333 5.2186 4.5577 4.4799 4.4799 32 × 32 44.0788 1.2092 1.2031 2.4882 1.3800 1.2092 1.2031 1.2031 64 × 64 22.7161 0.3067 0.3061 0.6182 0.3497 0.3067 0.3061 0.3061 RCPE C 20.0198 58.9709 10.8004 2.4986 0.6130 RCPE L 20.0198 58.9732 10.8072 2.5061 0.6194 200 6.3- Some numerical tests on bending response Table 6.4: Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Mxy at point A, 9βQ4. mesh FEM REP RCP RCPE REP C REP L RCP C RCP L RCPE C RCPE L 4 × 4 130.9319 128.3576 130.9319 130.9319 130.9319 128.3576 130.9319 130.9319 130.9319 130.9319 8×8 36.9052 35.6116 36.9052 64.2015 36.9052 35.6116 36.9052 36.9052 68.5926 61.9399 16 × 16 10.5748 10.2368 10.5748 15.3272 10.5748 10.2368 10.5748 10.5748 16.4785 14.6672 32 × 32 2.7141 2.6275 2.7141 3.8286 2.7141 2.6275 2.7141 2.7141 4.1263 3.6687 64 × 64 0.6830 0.6612 0.6830 0.9557 0.6830 0.6612 0.6830 0.6830 1.0330 0.9230 Table 6.5: Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Qx at point A, 9βQ4. mesh FEM REP RCP RCPE REP C REP L RCP C RCP L RCPE C RCPE L 4 × 4 42.6506 1032.3483 28.5764 28.5764 23.9424 7585.5042 23.0386 13.5889 23.0386 13.5889 8 × 8 18.8570 1412.1877 6.2487 9.5095 19.0370 2225.5636 8.7652 1.0312 7.8487 3.2804 16 × 16 14.6230 623.2831 5.3557 1.3339 3.3935 692.7196 1.5386 6.8012 0.8631 3.1556 32 × 32 8.9459 174.4279 6.6641 3.0204 0.7210 179.6036 0.3516 7.0531 0.1598 3.4948 64 × 64 5.4583 44.8515 4.8443 2.7780 0.1520 45.2808 0.0880 4.9525 0.0447 2.8996 Table 6.6: Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Qy at point A, 9βQ4. mesh FEM REP RCP RCPE REP C REP L RCP C RCP L RCPE C RCPE L 4 × 4 90.5749 5867.8725 72.6527 72.6527 43.4051 6692.6620 77.3863 73.7673 77.3863 73.7673 8 × 8 20.6936 971.0851 20.3957 31.6741 10.8521 2265.1508 17.7092 7.8062 34.1705 30.8678 16 × 16 9.5253 253.7822 8.8973 8.3888 2.8788 605.2156 4.9511 5.8076 7.5439 2.8718 32 × 32 4.8065 63.5138 5.2463 3.5083 0.7642 153.5090 1.2319 8.5986 1.7873 2.9443 64 × 64 2.7021 15.8915 3.1256 2.0107 0.2068 38.5098 0.2943 6.4601 0.4108 3.5097 Chapter 6 201 Total energy error % Bending energy error % Shear energy error % 9EQ4 REP -1 -2.5 log(stress error) 0 log(stress error) log(stress error) 0 9EQ4 REP -1 -3 RCPE -3.5 1 1.4 1 1 RCP 9EQ4 RCPE -2 RCPE RCP -2 1.9 -4 REP RCP 1.9 -4.5 -1 -0.5 0 0.5 log(element size) 1 -1 -0.5 0 0.5 log(element size) 1 -1 -0.5 0 0.5 log(element size) 1 Figure 6.11: Test No.1 (h/L = 0.002) - convergence of global errors for recovery procedures based on linear stress approximations FEM: 0.11124 RCP: 0.065623 RCPE: 0.071684 120 120 120 100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 REP: 0.066021 RCP2: 0.045475 RCPE2: 0.034645 120 120 120 100 100 100 80 80 80 60 60 60 40 40 40 20 20 20 Figure 6.12: Test No.1 (h/L = 0.002) Distributions of element error in energy norm on uniform mesh wuth 9βQ4 elements (the relative percentage error is reported above each plot) 202 6.3- Some numerical tests on bending response FEM: Mx − MAX: 0.36974 RCP: Mx − MAX: 0.70326 RCPE: Mx − MAX: 0.70326 REP: Mx − MAX: 0.44237 RCP2: Mx − MAX: 0.25783 RCPE2: Mx − MAX: 0.25783 Figure 6.13: Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mx on uniform mesh wuth 9βQ4 elements (contour interval =0.1) FEM: Mxy − MAX: 0.38113 RCP: Mxy − MAX: 0.14197 RCPE: Mxy − MAX: 0.17655 REP: Mxy − MAX: 0.094585 RCP2: Mxy − MAX: 0.12912 RCPE2: Mxy − MAX: 0.12911 Figure 6.14: Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mxy on uniform mesh wuth 9βQ4 elements (contour interval =0.1) Chapter 6 203 FEM: Sx - MAX: 0.6491 RCP: Sx - MAX: 1.3391 RCPE: Sx - MAX: 1.3391 REP: Sx - MAX: 0.73389 RCP2: Sx - MAX: 0.82623 RCPE2: Sx - MAX: 0.82623 Figure 6.15: Test No.1 (h/L = 0.002) Distributions of poinwise errors in Sx on uniform mesh wuth 9βQ4 elements (contour interval =0.1) FEM: Mx − MAX: 6.4319 RCP: Mx − MAX: 0.98721 RCPE: Mx − MAX: 0.98721 REP: Mx − MAX: 0.86907 RCP2: Mx − MAX: 0.71924 RCPE2: Mx − MAX: 0.57734 Figure 6.16: Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mx on uniform mesh wuth 9βQ4 elements (contour interval =0.2) 204 6.3- Some numerical tests on bending response FEM: Mxy − MAX: 1.9895 RCP: Mxy − MAX: 0.33842 RCPE: Mxy − MAX: 0.33842 REP: Mxy − MAX: 0.27379 RCP2: Mxy − MAX: 0.34464 RCPE2: Mxy − MAX: 0.32093 Figure 6.17: Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mxy on uniform mesh wuth 9βQ4 elements (contour interval =0.2) FEM: Sx - MAX: 14.6828 RCP: Sx - MAX: 3.1392 RCPE: Sx - MAX: 1.542 REP: Sx - MAX: 2.2198 RCP2: Sx - MAX: 3.7329 RCPE2: Sx - MAX: 1.0312 Figure 6.18: Test No.1 (h/L = 0.002) Distributions of poinwise errors in Sx on uniform mesh wuth 9βQ4 elements (contour interval =0.2) Chapter 6 205 FEM: 0.20245 RCP: 0.1075 RCPE: 0.11792 500 500 500 400 400 400 300 300 300 200 200 200 100 100 100 REP: 0.098613 RCP2: 0.10556 RCPE2: 0.085749 500 500 500 400 400 400 300 300 300 200 200 200 100 100 100 Figure 6.19: Test No.1 (h/L = 0.002) Distributions of element error in energy norm on uniform mesh wuth 9βQ4 elements (the relative percentage error is reported above each plot) 206 6.4 6.4- Laminated composite plates Laminated composite plates This section deals with laminated composite plates based on First order Shear Deformation Theory, Figure 6.20. Figure 6.20: First order Shear Deformation Theory (FSDT) scheme for a laminated composite plate The governing equations are briefly recalled in Appendix A, which can be derived from 3D elasticity as show in Appendix A.3.2. Using the same notation, the stress-strain relation for the kth layer can be written as τ (k) = κ ¯ C(k) s γ, S(k) = C(k) p e, (k) Cp (6.28) (k) Cs where is the reduced in-plane elasticity matrix and is the transverse shear elastic matrix. Thus, following Appendix A.3.2, the plate constitutive equations can be computed by integrating the local constitutive equations (6.28) through the thickness: N = Cm μ + Cmb χ, (6.29) M = Cmb μ + Cb χ, (6.30) S = Cs γ, (6.31) where the laminate stiﬀness matrices are given by n.layers X Cm = k Cmb = Cb = 1 2 1 3 (zk − zk−1 ) C(k) p , n.layers X k n.layers X k ¡ 2 ¢ (k) 2 zk − zk−1 Cp , ¡ 3 ¢ (k) 3 zk − zk−1 Cp , (6.32) (6.33) (6.34) Chapter 6 207 Cs = κ ¯ n.layers X k (zk − zk−1 ) C(k) s . (6.35) The two vertical abscissas zk−1 and zk define the bottom and top surfaces of the kth layer. Notice that bending and extension are generally coupled. In this work, laminated composite plates are analyzed using the hybrid stress element called 9βQ4 (Appendix C) [de Miranda & Ubertini, 2006]. Figure 6.21: Variations of strains and stresses through layer and laminated thicknesses. (a) Variation of typical in-plane strain. (b) Variation of corresponding stress The RCP recovery discussed in the previous section can be successfully applied to this class of problems, as shown by the numerical tests. Indeed, with this framework, the RCP recovery is found to be an eﬀective tool in order to face a key issue for laminates: the evaluation of transverse shear stresses along the thickness of the laminate. The kinematic hypothesis introduced in the first order shear deformation theory yields, obviously, inaccurate piecewise constant shear stresses which are discontinuous between the laminae. However, the specialistic literature suggests that accurate transverse shear stress profiles can be evaluated in the post-processing through three dimensional equilibrium equations (see Appendix A.3.2) D∗p s(k) + ∇z τ (k) + bx = 0. (6.36) From the FEM internal moments Mh and membrane forces Nh it is possible to evaluate the curvatures χh and the membrane strains μh through the laminate constitutive equation and, so, the in plane strains eh (see Appendix A.3.2). The constitutive equations of each lamina allow, finally, to determine the in plane stresses s(k) , which can be introduced in the three dimensional equilibrium equations to 208 6.5- Some numerical tests evaluate the transverse shear stresses: Z zk ³ ´ (k) (k) τz = τ0 − D∗p s(k) + bx dz, (6.37) zk−1 (k) where τ 0 represents the value of the shear stress vector at z = zk−1 i.e. (k) τ 0 = τ (k−1) (zk−1 ) , (6.38) with (1) τ 0 = −p(−) x (6.39) The shear stresses so obtained should satisfy the equilibrium boundary conditions on the top surface of the laminate: (n) τ 0 (zn ) = p(+) x , (6.40) being n the number of layers. Moreover, in order to have consistency between the recovered shear stresses and the FSDT model, the shear stress resultants along the plate thickness should be equal to the transverse shear forces evaluated via finite element procedure: Z Sk = τ dz = 0 (6.41) f It can be shown that both conditions are automatically satisfied if the internal moments and membrane forces satisfy the plate equilibrium equations. Indeed, accuracy of the stress profiles depends upon the derivatives of the internal moments and membrane forces, therefore a good approximation not only of the internal forces but also of their derivatives is important. According to this, a two steps procedure is proposed to recover transverse shear stresses. First, RCP procedure is applied in order to improve the rate of convergence of the FEM internal forces. Then, using the recovered internal forces, the transverse shear stress profiles are determined as described above. In particular, RCP is applied using element patches so to preserve local equilibrium. In this way, transverse shear stresses can be actually recovered by simply appling Equation (6.37): conditions (6.40) and (6.41) were be automatically satisfied. In the next section the eﬀectiveness of RCP and the transverse shear stress recovery has been verified on various test problems. 6.5 Some numerical tests In this section, some numerical tests are carried out in order to investigate the RCP performance and the transverse shear stress recovery procedure. A simply supported square plate of side L (Figure 6.22) is considered under both uniform and sinusoidal load with maximum intensity q. The side to thickness ratio Chapter 6 209 of the laminate is η = L/h = 10. Two diﬀerent stacking sequences are analyzed: a 0/90/0 symmetric laminate and a 0/90 antisymmetric one. Due to the double symmetry, only one quarter of the plate is studied. Some regular and distorted meshes used in the analyses are shown in Figure 6.23. The lamina mechanical properties are the following: E1 = 25E2 , ν 21 = 0.25, G13 = G12 = 0.5E2 , G23 = 0.2E2 . (6.42) The shear correction factors are evaluated according to cylindrical bending as shown in [Vlachoutsis, 1992]—[Laitinen et al. , 1995]. The reference solution used for the convergence analysis and in the tables is obtained according to Reddy [Reddy, 1997]. y u=w=fx =0 L/2 D v=w=f y=0 C A L/2 B x Figure 6.22: Simply supported square plate Figure 6.24 shows the convergence in energy norm in the case of sinusoidal load for the two diﬀerent stacking sequences. Figures 6.25 and 6.26 show the convergence of the FEM stress resultants at (L/4, L/4) (point C, see Figure 6.22) under uniform load for the symmetric and antisymmetric case, respectively. The values obtained are compared with the MITC laminated plate element implemented in the commercial software ADINA. The next set of graphs highlight the importance of the RCP procedure in this context. 67 Figure 6.27 shows the convergence in energy norm of the shear stress profiles obtained using the internal moments and membrane forces before and after recovery, respectively. The error is defined as µ Z ¶1/2 1 e= (τ̂ − τ )T C−1 (τ̂ − τ ) dV , (6.43) 2 Ω where τ̂ is computed using directly thhe finite element stress resultants, in the 210 6.5- Some numerical tests Figure 6.23: Some examples of regular and distorted meshes first case, or the recovered stress resultants, in the second case. Notice the 9βQ4 element yelds stress resultants that satisfying the plate equilibrium equations and, thus, they can be used for shear stress recovery without validating conditions (6.40) and (6.41). Both stacking sequences are reported, the load is sinusoidal. As can be clearly seen, the error does not decrease for the shear stresses obtained before the RCP, whereas for those obtained after the RCP the convergence rate is approximately 1. This result can be explained by pointing out that the shear stress profiles obtained via three dimensional equilibrium depend upon the derivatives of the in plane stress components, thus the derivatives of the internal moments and membrane forces are involved. A Recovery by Compatibility in Patches procedure, while improving the convergence rate of the generalized stresses and thus of their derivatives, maintains the equilibrium within the element. Some results concerning the eﬀectiveness of the RCP procedure to improve the convergence rate are presented in Figures 6.28 and 6.29. Here, the convergence of the internal moments at point D are shown before and after the RCP procedure, respectively. Both stacking sequences are considered and the load is sinusoidal. Point D coincides with a node in all the considered regular meshes. Since the RCP procedure yields generalized stresses which are discontinuous across elements, in this case the value considered are those for the elements which have point D as the Chapter 6 211 0 -1 -1.5 -2 1 -2.5 -1 0/90 - sinusoidal load -0.5 Log(error %) Log(error %) -0.5 -3 0 9bQ4 regular 0/90/0 - sinusoidal load 9bQ4 distorted -1 -1.5 -2 -2.5 -0.5 0 Log(L/N) 0.5 -3 1 -1 -0.5 0 Log(L/N) 0.5 Figure 6.24: Convergence in energy norm (sinusoidal load and diﬀerent staking sequences) top right-hand node. The stress values obtained at nodal points such as point D are usually more aﬀected by error than the internal values. The improvement in convergence caused by the RCP is significant (the convergence rate changes from around 1 to around 2). Figures 6.30 to 6.33 show the dimensionless transverse shear stress profiles at point C for sinusoidal load and the two diﬀerent stacking sequences: 100 τ̄ zs = τ zs (6.44) qη In the first two figures, the profiles obtained after the Rep are shown for diﬀerent meshes. The recovered profiles converge to the reference solution [Reddy, 1997]. In Figures 6.32 and 6.33, on the other hand, the profiles obtained before and after recovery for a 9 × 9 mesh are shown. Observing for example the result obtained for τ zy using the distorted mesh in the 0/90/0 laminate (black dots in Figure 6.32), it is evident that the shape of the profile is very diﬀerent from the reference solution, while the resultant force (represented by the area under the graph) is very similar. This means that, while the value of the generalized shear stress is near to the exact one, the profile does not converge unless a recovery procedure is implemented first. The importance of the RCP recovery is emphasized even more by observing Figures 6.34 and 6.35. They show the error density before and after recovery for two diﬀerent meshes and the two stacking sequences. As can be noted, before recovery the order of magnitude of the error density does not change with mesh refinement, whereas a significant improvement can be seen in the error maps after recovery. 212 6.5- Some numerical tests Figure 6.25: Convergence of stress resultant at point C (stacking sequence 0/90/0, uniform load) Chapter 6 213 Figure 6.26: Convergence of stress resultant at point C (stacking sequence 0/90, uniform load) 9bQ4 regular 0/90/0 - sinusoidal load 9bQ4 regular, recovery 9bQ4 distorted 9bQ4 distorted, recovery Log(error %) -0.5 -1 -1.5 -2 0 0/90 - sinusoidal load -0.5 Log(error %) 0 -1 -1.5 -2 1 -2.5 -2.5 1 -3 -1 -0.5 0 Log(L/N) 0.5 -3 -1 -0.5 0 Log(L/N) 0.5 Figure 6.27: Convergence in energy norm of the shear stress profiles (sinusoidal load, diﬀerent stacking sequences) 214 6.5- Some numerical tests 0 9bQ4 regular 9bQ4 regular, recovery 0 Mx -1 Log(error %) Log(error %) -1 My -2 1 -3 -2 1 2 -3 2 -4 -1 -0.5 0 Log(L/N) -4 0.5 -1 -0.5 0 Log(L/N) 0.5 0 Mxy Log(error %) -1 1 -2 -3 2 -4 -1 -0.5 0 Log(L/N) 0.5 Figure 6.28: Convergence of the moments at point D before and after the recovery (stacking sequence 0/90/0, sinusoidal load) 0 9bQ4 regular 9bQ4 regular, recovery 0 Mx and My -2 -1 Log(error %) Log(error %) -1 1 -3 -4 Mxy 1 -2 -3 2 2 -1 -0.5 0 Log(L/N) 0.5 -4 -1 -0.5 0 Log(L/N) 0.5 Figure 6.29: Convergenge of the moments at point D before and after the recovery (stacking sequence 0/90, sinusoidal load) Chapter 6 215 0.5 0.5 0.25 0.25 Ref. solution (Reddy) Mesh 3x3 Mesh 9x9 Mesh 27x27 z/h 0 z/h 0 -0.25 -0.5 -20 -0.25 -15 -10 tzx -5 0 -0.5 -8 -6 -4 tzy -2 0 Figure 6.30: Shear stress profiles at point C after recovery for diﬀerent regular meshes (stacking sequence 0/90/0, sinusoidal load) 0.5 0.25 0.5 Ref. solution (Reddy) Mesh 3x3 Mesh 9x9 Mesh 27x27 0.25 z/h 0 z/h 0 -0.25 -0.25 -0.5 -20 -15 -10 tzx -5 0 -0.5 -20 -15 -10 tzy -5 0 Figure 6.31: Shear stress profiles at point C after recovery for diﬀerent regular meshes (stacking sequence 0/90, sinusoidal load) 216 6.5- Some numerical tests 0.5 0.5 0.25 0.25 Ref. solution (Reddy) Regular mesh Regular mesh, recovery Distorted mesh Distorted mesh, recovery z/h 0 z/h 0 -0.25 -0.5 -20 -0.25 -15 -10 tzx -5 0 -0.5 -8 -6 -4 tzy -2 0 Figure 6.32: Shear stress profiles at point C after recovery for the mesh 9×9, regualr and distorted (stacking sequence 0/90/0, sinusoidal load) 0.5 0.25 0.5 Ref. solution (Reddy) Regular mesh Regular mesh, recovery Distorted mesh Distorted mesh, recovery 0.25 z/h 0 z/h 0 -0.25 -0.25 -0.5 -20 -15 -10 tzx -5 0 -0.5 -20 -15 -10 tzy -5 0 Figure 6.33: Shear stress profiles at point C after recovery for the mesh 9×9, regualr and distorted (stacking sequence 0/90, sinusoidal load) Chapter 6 217 Figure 6.34: error density of the shear profiles for diﬀerent meshes before and after recovery (stacking sequence 0/90/0, sinusoidal load) Figure 6.35: error density of the shear profiles for diﬀerent meshes before and after recovery (stacking sequence 0/90, sinusoidal load) Chapter 7 Concluding remarks: looking toward Validation In questo capitolo, conclusivo, si intende richiamare alla mente l’intero processo di Verifica e Validazione per collocare il presente lavoro di tesi all’interno di questa ideale catena logica, anche alla luce della recente pubblicazione da parte dell ’ASME della Guida alla Verifica e Validazione nella Meccanica Computazionale dei Solidi. Di seguito sono riportati alcuni contenuti del documento, essenziali dunque alla suddetta collocazione, e vengono tratteggiati i possibili sviluppi del presente lavoro verso la validazione. A tal proposito mi piace ricordare quanto aﬀermava il Professor A.A. Cannarozzi nel lontano 1991: "Gli aspetti di ideazione e di innovazione dei procedimenti di calcolo competono in linea di principio al Ricercatore. Il Produttore del codice, interagendo col Ricercatore (col quale talvolta si identifica) e col Produttore dell’apparecchiatura di calcolo, cura lo sviluppo, commercializza, aggiorna il codice e ne segue l’impiego presso l’utente. Quest’ultimo si avvale dei mezzi di calcolo come risorse per lo svolgimento della propria attività intellettuale, nell’ambito della quale i codici, per le opzioni finalizzate che oﬀrono, diventano strumento assoluto di realizzazione di una consistente parte del processo logico. Infatti il ruolo attivo nell’impiego di un codice consiste essenzialmente nella comprensione e nella scelta della procedura logica di analisi, nella preparazione e nella consegna dei dati, nell’interpretazione e nel controllo dei risultati, mentre il coinvolgimento nell’impostazione concettuale e nelle modalità di esecuzione è per lo più impossibile. Il pericolo di impiego "a scatola 220 7.1- Assessing the numerical error in the finite element solution chiusa", da molti lamentato, appare quindi eﬀettivo. D’altra parte, la responsabilità dell’utente del codice per le conseguenze della propria attività, nei confronti del Committente o più in generale della collettività all’interno della quale egli opera, è assai spesso totale. In considerazione di tutto questo appare comprensibile l’importanza di definire adeguate forme di regolamentazione e di controllo dell’operato delle figure interessate a vario titolo all’elaborazione automatica nella progettazione strutturale, dal Ricercatore all’utente" [Cannarozzi, 1991]. 7.1 Assessing the numerical error in the finite element solution During the last two decades, the National Agency for Finite Element Methods and Standards (NAFEMS, www.nafems.org) has developed some of the most widely known V&V benchmarks. Roughly 30 verification benchmarks have been constructed by NAFEMS. The majority of these benchmarks have targeted solid mechanics simulations, though some of the more recent benchmarks have been in fluid dynamics. Most of the NAFEMS verification benchmarks consist of an analytical solution or an accurate numerical solution to a simplified physical process described by a partial diﬀerential equation. The NAFEMS benchmark set is carefully defined, numerically demanding, and well documented. However, these benchmarks are currently very restricted in their coverage of various mathematical and/or numerical diﬃculties and in their coverage of physical phenomena. Further, the performance of a given code on the benchmark is subject to interpretation by the user of the code. It is also likely that the performance of a code on the benchmark is dependent on the experience and skill of the user. Several large commercial code companies specializing in solid mechanics have developed an extensive set of well-documented verification benchmarks that can be exercised by licensed users of their codes. Such benchmarks are intended to be applied only to a particular code, and they describe how that code performed on the benchmark problems. The performance results of a code tested on the benchmark problems by a commercial company can be clearly compared with the results obtained by a user who tests the code with the same benchmark problems. These company and user-testing activities give the user a better understanding of the minimal performance that can be expected from a code. It should be noted here that information about a code’s performance on a set of benchmark problems Chapter 7 221 prior to purchase of the code is often diﬃcult to obtain, as this information is proprietary. Moreover the most important step in any finite element analysis is clearly choosing the appropriate mathematical model to solve. This model must contain all the important ingredients regarding the geometry, loading, boundary conditions, and material data. The mathematical model must be reliable and eﬀective and is solved using a finite element program with a mesh defined by the analyst. The finite element solution of the mathematical model will contain some numerical error which the analyst clearly wants to be suﬃciently small. This will of course be the case if a very fine mesh is used with appropriate finite elements. But, in practice, when the mesh used is not very fine, the assessment of the numerical error between the "exact" solution and the finite element solution can be important. Since the exact solution is unknown, lower and upper bounds, proven to bracket closely the exact solution, and obtainable with an acceptable computational eﬀort, would be very valuable. Unfortunately, these bounds are not available for general analysis, not yet even in academic research. Hence for general analyses, the currently available error assessments are based on estimates without these bounds. Since this error estimation is an important issue, and presents an excellent research challenge, a very large research eﬀort by engineers and mathematicians has focused on the development of error estimators. The current state of art and hopefully some improvements on error estimation has been discussed in this work. But which is the role played by error estimation in the whole V&V process shown in Figure 7.1? Recently the American Society of Mechanical Engineers (ASME) Standard Committee on Verification and Validation in Computational Solids Mechanics (PTC 60 / V&V 10) approved their first document (July 2006). This Guide has been submitted to ASME publications (http://catalog.asme.org/) and to American National Standards Institute (ANSI) for public review. In the following section some parts of this Guide are presented, underlying the role of error estimators. 7.2 Validate the model, not the codes The notion of benchmarking a code refers to comparisons of results of the code to some standard "benchmark", which may be an analytical solution or, as many argue, a single physical experiment. Various uses of the notion of benchmarking are found in the literature. Among these are: 1. Code-to-Code Comparisons of Code Performance. Here execution speeds or 222 7.2- Validate the model, not the codes Figure 7.1: The scenario numerical results, for example, of diﬀerent codes solving one or more benchmark problems are compared on the same computer or on diﬀerent computers running the same code. 2. Code-to-Code Comparisons Versus Verified Code. Here one assumes that a fully-verified code is in hand, and one compares results of diﬀerent codes with those produced by the master code on a suite of benchmark problems. 3. Comparison of Code Predictions to Results of Physical Experiments. Here results of one or more codes are compared with results obtained by carefully executed physical experiments. 4. Comparison of Code Results to Analytical or Highly Accurate Solutions. Here a variety of features of basic algorithms in a code are tested by comparison of computed results with known analytic solutions or with "overkill" solutions obtained with very fine meshes using verified codes. All of these approaches suﬀer from defects, some more serious than others. Use 1 (Code-to-Code for Performance) has nothing to do with verification. Use 2 (Code-to-Verified Code Comparisons) is not entirely useless, but is an exercise fraught with pitfalls. If the "verified" code is not as accurate as the tested code, how can useful conclusions be made by simply comparing results? Which is correct? Such comparisons may be meaningless, without reference to other information. Use 3 (Comparison with Experiments), while a common benchmark scenario, is the worst of all. If discrepancies between computed and observed responses are observed, what is the source of error? Is the model bad or is the code inadequate or does the code have bugs? There is no systematic way to distinguish between Chapter 7 223 modeling error and approximation error by a simple comparison of computed predictions of a single experiment (which itself is subject to gross error). One must first "verify the code". Use 4 (Comparison with Accurate Solutions) is somewhat better, but also imperfect. Exactly what should be compared? Here we come to the central question of benchmarking: what exactly are the comparisons with results of benchmark problems supposed to test? Going to the core of the issue: what do we want to know about a code to determine if it is "solving the model correctly"? The issue is often much more complicated than this: if we are given results of a "black box" commercial code where the precise model used to predict response is unknown (as it is propriety information, belonging to the code developer) what information can a suite of benchmark calculations provide? We argue that: 1) benchmarking is primarily a verification exercise, not one of validation. Comparisons of computed predictions with experiments is the essence of validation, thus the subject code to be tested against experiments must have been already subject to verification test using, among other tools, benchmarking tests; 2) one must clearly set out tests designed to establish how well the subject code can solve the model used to depict natural phenomena or the behavior of engineering systems; 3) on the other hand, benchmarking need not necessarily be tied exclusively to the model used to develop the code, as the user has right to know where and when the model becomes invalid or inaccurate; 4) benchmarking, or the selection of a benchmark suite, is a delicate and complex proposition: tests on performance of a code on distorted meshes, on problems with nice smooth solutions, or on specific quantities of interest in a calculation must be made with a clear understanding of the capabilities of the model on which the code is based. Some principles usually recommended for designing benchmark problems are: 1. Failure - Benchmark problems should be designed to establish the limits of the code under consideration. Limits means failure: when does the code fail to produce results in agreement with a suite of carefully-selected analytical solutions? Design the benchmarks so that bad results, not always glorious agreement, are obtained, by altering time steps, load steps, mesh size, approximation order, material mismatch, boundary-layers, point or line-loads, etc. Surely, good agreement on simple problems is necessary, but not suﬃcient to determine the real limitations of interest to the user. 2. Convergence - If a code produces pretty results but doesn’t converge or doesn’t converge at acceptable rates, then its use as a simulation tool is suspect at best. Also, the real meaning of the limit to which the code converges to is important. Benchmark problems should be designed to determine via 224 7.2- Validate the model, not the codes numerical experiments the rates of convergence of a model as the mesh size, the order of the approximation, time step, load step, etc. are varied, convergence being measured in a space of meaningful norms. Whenever possible these should be compared with a priori estimates if they are available for the problem at hand. Then a sequence of benchmarks can be designed to establish the rates-of-convergence for various regularities. Thus, if a solution of a problem is known, the error can be calculated for each choice of mesh size and order. 3. Local Behavior, Diﬀusion, Propagation - Users interested in simulating physical phenomena are concerned with how well the code in hand can depict so-called local phenomena. What are stresses at a point? How does the code handle convection of a signal propagating through a domain? Are bifurcations, localizations mesh dependent? Or do the predicted critical parameters agree with those depicted in appropriate analytical solutions? These issues must be considered in designing benchmark problems. 4. Model Limitations - Given a code (often a black box), design benchmark problems to fool or verify or, at least, determine the limitations of the model on which the code is based, this is not always easy. As a general rule, the design of eﬀective benchmark problems (as a verification tool) should be based on models which supersede in complexity and sophistication those on which the code is based. Limitations of the subject code can be determined in a broader context. Then the limits of the model on which the code is based can also be roughly evaluated. The conclusion: the results of benchmark problems must be viewed and evaluated within the context of what is predictable by the theory upon which the underlying model is based. Once again, the goal of verification is to determine if the model is solved correctly. Thus, model verification depends upon what can and should be deliverable by the model. Such examples suggest that when the base model used in a code is unknown, the benchmarks should be based on models presumed to be of the highest sophistication and breadth appropriate for the class of problems under consideration. For this and other reason in 1999 an ad hoc verification & validation specialty committee was formed under the auspices of the United States Association for Computational Mechanics (USACM). The purpose of this committee was to pursue the formation of a verification & validation standards committee under a professional engineering society approved to produce standards under the rules of the American National Standards Institute (ANSI). This goal was achieved in 2001 when the then Board on Performance Test Codes (PTC) of the American Society Chapter 7 225 of Mechanical Engineers (ASME) approved the committee’s charter: ”To develop standards for assessing the correctness and credibility of modelling and simulation in computational solid mechanics.” and the committee was assigned the title and designation of the ASME Committee for Verification & Validation in Computational Solid Mechanics (PTC 601). The membership is diverse with three major groups being industry, Government, and academia. The industry members include representatives from auto and aerospace industries and the Government members are primarily from the Departments of Defense and Energy. Particularly well represented are members from the three national laboratories under the National Nuclear Security Administration. This latter membership group is key to the committee as much of the recent progress in verification & validation has come from these laboratories and their efforts under the Advanced Simulation and Computing (ASC) Program, started in 1995. 7.3 An overview of the PTC 60 / V&V 10 In this section, we give an overview of the "Performance Test Code #60" (PTC60) recently appeared on the IACM bulletin (No. 20, January 2007) on the verification and validation topic. This article depicts the main steps done by the committee related to the necessity of standards and rules in the computational Solid Mechanics field, and starts with the following funny example: question: "Are the sometimes lengthy and costly processes of verification & validation really necessary?"; answer: "Consider the following scenario that perhaps you can relate to first hand. A project review meeting is taking place and the project manager needs to make a critical decision to accept or reject a proposed design change. A relatively new employee, freshly minted from the nearby engineering university, makes an impressive presentation full of colorful slides of deformed meshes and skillfully crafted line plots indicating the results of many CPU and labour hours of non-linear numerical analyses, ending with a recommendation to accept the design change. Hopefully, an astute project manager, aware of the vagaries of nonlinear numerical analyses, will not accept the analysis and its conclusion at face value, especially given the inexperience of the analyst. Rather, the project manager should seek some assurance that not only are the results reasonable, but a sound procedure was followed in developing the model and documenting the numerous physical and numerical parameters required for a typical analysis. The degree of assurance sought by the project manager is directly related to the criticality of the decision to 226 7.3- The Introduction Section be made". The processes of verification & validation are how evidence is collected, and documented, that help establish confidence in the results of complex numerical simulations. The intended audience for this Guide is not the occasional computational mechanics user, e.g. a modern-day draftsman using an automated CAD/FEA package, rather it is computational analysts, experimentalists, code developers, and physics model developers, and their managers, who are prepared to read a technical document with a mixture of discussion concerning mathematics, numerics, experimentation, and engineering analysis processes. 7.3.1 The guide and the present work The guidelines are based on the following key principles: • Verification must precede validation. • The need for validation experiments and the associated accuracy requirements for computational model predictions are based on the intended use of the model and should be established as part of V&V activities. • Validation of a complex system should be pursued in a hierarchical fashion from the component level to the system level. • Validation is specific to a particular computational model for a particular intended use. • Validation must assess the predictive capability of the model in the physical realm of interest, and it must address uncertainties that arise from both simulation results and experimental data. The Guide contains four major sections: 1. Introduction - the general concepts of verification and validation are introduced and the important role of a V&V Plan is described. 2. Model Development - from conceptual model, to mathematical model, and finally the computational model are the keys stages of model development. 3. Verification - is subdivided into two major components: code verification seeking to remove programming and logic errors in the computer program, and calculation verification - to estimate the numerical errors due to discretization approximations. 4. Validation - experiments performed expressly for the purpose of model validation are the key to validation, but comparison of these results with model results depends on uncertainty quantification and accuracy assessment of the results. Chapter 7 227 In the next sections the contents of this four points will be sketched. 7.3.1.1 The Introduction Section The processes of verification & validation start, and end, with modelling and models, and so we seek to verify & validate a computational model, for making predictions within the domain of intended use of the model. Three types of models, from the general to the specific, are described. The logic flow from the most general Conceptual, to Mathematical, to the most specific Computational Model, is illustrated in Figure 7.2. Figure 7.2: The path from Conceptual and Computational Model Neither part of verification addresses the question of the adequacy of the selected Conceptual and Mathematical models for representing the reality of interest. Answering this question is the domain of validation, i.e. are the mechanics (physics) included in the Conceptual and Mathematical models suﬃcient for answering the questions in the problem statement. The manner in which the Verification and Validation interact is illustrated in the flow chart shown in Figure 7.3. After the selection of the Conceptual model, the V&V process has two branches: the left branch contains the modelling elements and the right branch the experimental elements. This figure is intentionally designed to illustrate the paramount importance of physical testing in the V&V process, as ultimately, it is only through experimentation that assessments about the adequacy of the selected Conceptual and 228 7.3- The Model Development Section Mathematical models for representing the reality of interest can be made. Close cooperation among modelers and experimentalist is required during all stages of the V&V process, until the experimental outcomes are obtained. Close cooperation is required because the two groups will have quite diﬀerent views of the Conceptual model, i.e. the mathematical and physical model will be diﬀerent. As an example consider the last example shown in the previous chapter about composite laminated plate with simply supported boundary conditions. Mathematically this boundary condition is quite easy to specify, but in the laboratory there is no such thing as a simply supported or ’clamped’ boundary. In general, some parts of the Conceptual model will be relatively easy to include in either the mathematical or physical model, and others more diﬃcult. A dialogue between the modelers and experimentalist is critical to resolve these diﬀerences. To aid in this dialogue, the ’cross-talk’ activity labelled as "Preliminary Calculations" in Figure 7.3 is intended to emphasize the goal that both numerical modelers and experimentalist attempt to model the same Conceptual model. Of equal importance is the idea that the experimental outcomes should not be revealed to the modelers until they have completed the simulation outcomes. The chief reason for segregation of the outcomes is to enhance the confidence in the model’s predictive capability. When experimental outcomes are made available to modelers prior to establishing their simulation outcomes, the human tendency is to ’tune’ the model to the experimental outcomes to produce a favorable comparison. This tendency decreases the level of confidence in the model’s ability to predict, and moves the focus to the model’s ability to mimic the provided experimental outcomes. 7.3.1.2 The Model Development Section Before modelling begins, a reality of interest is identified, i.e. what is the physical system to be modelled. The reality of interest is typically described in the problem statement presented to the analyst, e.g. "We need to know the behaviour of a certain slab under a distributed load of x Newtons/meter". In this case the reality of interest is the composite laminated plate. The most general form of the model addressed in the Guide is the Conceptual Model - "the collection of assumptions and descriptions of physical processes representing the solid mechanics behaviour of the reality of interest from which the mathematical model and validation experiments can be constructed". Continuing with the slab example, the conceptual model could be a simply supported plate of a laminated composite material, loaded uniformly. With the Conceptual Model defined, the analyst next defines the Mathematical Model - "The mathematical equations, boundary values, initial conditions, and Chapter 7 229 Figure 7.3: Verification and Validation activities and outcomes modelling data needed to describe the conceptual model". Referring again to the slab example the mathematical model could be established based on the First order Shear Deformation Theory. The final model in the sequence is the Computational Model - "The numerical implementation of the mathematical model, usually in the form of numerical discretization, solution algorithm, and convergence criteria". In our slab example, the computational model is derived within with the finite element framework using, for example, the 9βQ4 hybrid stress element. At this point the computational model can be exercised (run) and the results compared to available experimental data for validation of the model. It is frequently the case that the results do not compare as favourably as requested in the original problem statement. Assuming a high degree of confidence in the experimental data, the analyst has two basic choices for revising the model: changing the model form or calibrating model parameters. Changing the model form can apply to either the Mathematical or Conceptual 230 7.3- Calculation Verification model. As an example, the first order shear theory assumption reveals to be too restrictive and needs to be replaced by a higher-order plate theory or the plate model reveals to be inadequate and should be replaced by a plate-like body. 7.3.1.3 The Verification Section The Guide emphasizes that Verification must precede Validation. The logic is that attempting to validate a model using a code that may still contain (serious) errors can lead to a false conclusion about the validity of the model. Two types of verification are generally recognized and defined in computational simulation: 1. Code Verification - establish confidence, through the collection of evidence, that the mathematical model and solution algorithms are working correctly. 2. Calculation Verification - establish confidence, through the collection of evidence, that the discrete solution of the mathematical model is accurate. Code Verification In general, Code Verification is the domain of software developers who hopefully use modem Software Quality Assurance techniques along with testing of each released version of the software. Users of software also share the responsibility for code verification, even though they typically do not have access to the software source. The large number of software users, typical of most commercial codes, provides a powerful potential code verification capability, if it is used wisely by the code developers. Among the code verification techniques, the most popular method is to compare code outputs with analytical solutions; this type of comparison is the mainstay of regression testing. Unfortunately, the complexity of most available analytical solutions pales compared to even rather routine applications of most commercial software. One code verification method with the potential to greatly expand the number and complexity of analytical solutions is what is termed in the V&V literature as manufactured solutions. The basic concept of a manufactured solution is deceptively simple. Given a partial diﬀerential equation (PDE), and a code that provides general solutions of that PDE, an arbitrary solution to the PDE is manufactured, i.e. made up, then substituted into the PDE along with associated boundary and initial condition, also manufactured. The result is a forcing function (right-hand side) that is the exact forcing function to reproduce the originally selected (manufactured) solution. The code is then subjected to this forcing function and the numerical results compared with the manufactured solution. If the code is error free the two solutions should agree. Chapter 7 231 Calculation Verification The above illustration of a manufactured solution used as part of code verification is only half of the verification eﬀort. The other half is what is termed calculation verification, or estimating the errors in the numerical solution due to discretization. Calculation verification, of necessity, is performed after code verification, so that the two error types are not confounded. Any comparison of the numerical and analytical results will contain some error, as the discrete solution, by definition, is only an approximation of the analytical solution. So the goal of calculation verification is to estimate the amount of error in the comparison that can be attributed to the discretization. The discretization error is most often estimated by comparing numerical solutions at two more discretizations (meshes) with increasing mesh resolution, i.e. decreasing element size. The objective of this mesh-to-mesh comparison is to determine the rate of convergence of the solution. The main responsibility for Calculation Verification rests with the analyst, or user of the software. While it is clearly the responsibility of the software developers to assure their algorithms are implemented correctly, they cannot provide any assurance that a user-developed mesh is adequate to obtain the available algorithmic accuracy, i.e. large solution errors due to use of a coarse (unresolved) mesh are attributable to the software user. 7.3.1.4 The Validation Section The validation process has the goal of assessing the predictive capability of the model. This assessment is made by comparing the predictive results of the model with validation experiments. If these comparisons are satisfactory, the model is deemed validated for its intended use, as stated in the V&V Plan. There is perhaps a subtle point here to be emphasized. The original reason for developing a model was to make predictions for applications of the model where no experimental data could, or would, be obtained. However, in the V&V Plan it was agreed that if the model could adequately predict some related, and typically simpler, instances of the intended use, where experimental data would be obtained, then the model would be validated to make predictions beyond the experimental data for the intended use. Simply put, if the model passes the tests in the V&V Plan, then it can be used to make the desired predictions with confidence. The V&V Plan is of paramount importance to the V&V process. When it is said that the model is validated for the intended use, it is not just the Computational model, which likely will have to change for the predictions of interest, but the Mathematical and Conceptual models upon which the Computation model was built that have been validated. It is through the validation of the Conceptual model that confidence is gained that the correct physics (mechanics) 232 7.3- The Validation Section were included in the model development. The key components of the validation process are the: • Validation Experiments - experiments performed expressly for the purpose of validating the model. • Accuracy Assessment - quantifying how well the experimental and simulation outcomes compare. The goal of a validation experiment is to be a physical realization of an initial boundary value problem, since an initial boundary value problem is what the computational model was developed to solve. Most existing experiments do not meet the requirements of a validation experiment, as they were typically performed for purposes other than validation. Certainly appropriate existing experimental data should be used in the validation process, but the resulting confidence in the model’s ability to make predictions, based on these experimental results, is diminished, relative to validation experiments. The reduced confidence arises from the necessity of an analyst needing to select physical and numerical parameters required for the model that were left undefined in the experiment. The important qualities of a validation experiment include: • Redundancy of the Data - repeat experiments to establish experimental variation. • Supporting Measurements - not only are measurements of the important system response quantities of interest recorded, but other supporting measurements are recorded. An example would be to record the curvature of a beam to support a strain gauge measurement. • Uncertainty Quantification - errors are usually classified as being either random error (precision) or systematic error (bias). Once the experimental and simulation outcomes are obtained, the accuracy assessment phase of the validation process can begin. If possible, the comparison of the experimental and simulation outcomes should be made by an interested third party, as this helps to remove a bias that favors either the experimental or the simulation results. In addition to deciding what response quantities should be compared, the V&V Plan should state how the quantities are to be compared. Validation metric is the term used to describe the comparison of validation experiment and simulation outcomes. These metrics can range from simple binary metrics, e.g. was the material’s yield strength exceeded, to more complex comparisons involving magnitude and phase diﬀerence in wave forms, e.g. deceleration history in a vehicle crash. Whatever the form of the validation metric, the result should be a quantitative assessment of the agreement between the experiment and Chapter 7 233 simulation. Hopefully, this quantification will also include an estimate of the variability in the agreement and a confidence statement about the variability, e.g. the relative error between the experiment and simulations was 18% plus or minus 6% with a 85% confidence level. This three-part comparative statement is provided to the decision maker, along with all the supporting V&V documentation, to aide in their decision making process about the validity of the model for the intended use. 7.3.1.5 The Conclusion Section Some of the remaining important V&V activities requiring guidance from the community: • Verification - this ’poor’ sister of validation needs more attention from the V&V research community. Reliance on regression testing for code verification provides minimal confidence when using today’s complex multi-physics and multi-scale software. Methods, and their implementation as tools, for verification of increasing software complexity are needed. • Quantification of the Value of V&V - if program managers are asked to spend resources on V&V, they needed some measure of the value they are receiving for the resources expended. • Incomplete V&V - if the V&V process is terminated before a successful conclusion, what is the best path forward for decision maker? • Validation Experimentation - most experiments consume large amounts of resources, the value of these experiments to the V&V process needs to be quantified to enable decision makers to appropriately allocate resources for this important activity. • Uncertainty Quantification - meaningful comparisons of simulations with experiments requires an estimate of the uncertainty in both sets of results, and a comparative assessment of these two uncertain outcomes. • Predictive Confidence - when validated models are applied beyond the limited range of validation experiments, how can the confidence in these results be quantified? 7.4 Concluding remarks As a conclusion of the present work I would like to re interpret the last example shown in the previous chapter: the laminated composite plate, where our error verification tool (the RCP recovery procedure) is used to reconstruct the transverse shear stress along the laminate thickness. This is an attempt to improve simulation results by reviewing the underlying model, a sort of an ideal hierarchy of model 234 7.4- Concluding remarks (from plates to plate-like bodies). Thus this could be viewed as an attempt to go beyond verification, looking at validation. Looking back in the past, I would like to end this work with some advises coming from the far 1980 by the brilliant mind of Bruce Irons [Irons & Ahmadn, 1980] (that it still up-to-date), and with the encouraging willpower of J. Tinsley Oden. • Finite elements are no substitute for commonsense. • Trouble-shooting will continue to involve engineers in the sort of calculations they have always done but have seldom been taught to do. • A good engineer tries to understand every failure and learn from it. • An element may succeed in one problem and fail in another. Finite elements are not infallible, but their strength lies in their versatility. This is the kind of strength valued by any engineer, who by choice would never buy a tool that does only one job. • The matched solution involves thinking about computer results in an ordinary, engineering way and not merely accepting them as magic. • When things go wrong, we want to try other approaches quickly, and when something quite new is demanded we do not expect to be kept waiting six months for a spare part. • In seeking a suitable mesh, either by trial and error, or guessing from an intuitive impression of what the answers will be like, it is useful to regard the elements as interpolation formulae. If an element gives linear stresses, we judge the size the elements should be on the basis that they are representing the field by linear segments. The curvatures indicate, very roughly, what errors we should expect. Barlow points improve the estimates, and also make prediction more hazardous. • Perhaps ultimately the most useful thing to emerge from the matched solution is the consciousness of, and a technique for identifying, the unwanted sideeﬀects in the performance of an element. This will be important when we develop ’hybrid’ elements to do a better job. For we can then choose the responses we want, in terms of stress fields, and exclude those we do not. As a last point of this list, Irons inserts a curious picture, Figure 7.4, that probably represent the little snares hidden behind the computer manipulations. Chapter 7 Figure 7.4: Cimex Fortranarius, magnification 109 :1 (See 235 [Irons & Ahmadn, 1980]) Professor J. Tinsley Oden in 1996, speaking about Computational mechanics: "Modeling and simulation represent catalysts capable of synthesizing formal theory and experimental results, breaking artificial barriers between these two main approaches to scientific discovery and engineering advances. Recently, computational simulation has become a third approach - along with theory and laboratory simulation - to studying and solving scientific and engineering problems. Computational simulation is based on the use of high-performance computers to model and simulate complex systems. In this approach, a computer equipped with problem-solving software tools may represent a virtual laboratory in which researchers can build a model for a given problem and run it under varying conditions. These increasingly complex computational methodologies require sophisticated models and numerical algorithms, and vice versa" (Excerpts from the Timoshenko Medal Acceptance Speech delivered by Professor J. Tinsley Oden in 1996. "The Revolution in Applied Mechanics from Timoshenko to Computation"). Appendix A Structural models A.1 Class I problems The first class of problems considered refers to problems governed by the following diﬀerential equation µ 2 ¶ ∂ u ∂2u k in Ω. + 2 = b, (A.1) ∂x2 ∂y As usual, stresses are defined by ∂u ∂u σy = k . (A.2) σx = k , ∂x ∂y Within the context of linear elasticity, this problem may be viewed, for example, as the equilibrium problem of a thin membrane of uniform thickness, subjected to a distributed transverse force b. In the numerical tests, the parameter k is set to k = 1. Figure A.1: Class I problems A.1.1 The RCP stress approximation The stress approximation for the RCP procedure can be derived from a stress function φ(x, y) as follows ¤ £ ∂φ ∂φ σ ∗h x = , σ ∗h y = − . (A.3) σ ∗phT = σ ∗h x σ ∗h y , ∂y ∂x 238 The particular solution σ ∗pp is taken as Z ¤ £ ∗ 1 x ∗T ∗ ∗ σ pp = σ p x σ p y , σp x = b dx, 2 0 7.0- Class II problems σ ∗p y 1 = 2 Z y b dy. (A.4) 0 Thus, selecting the stress approximation simply reduces to select the stress function. Here, φ is assumed as a complete polynomial expansion of degree (q + 1), so that each stress component is approximated by a complete polynomial of degree q. For finite elements of order p, two diﬀerent values for q are tested: q = p and q = p + 1. The corresponding implementations of the present procedure are labelled by RCP1 and RCP2, respectively. Thus, local stresses are approximated by linear (RCP1) and quadratic (RCP2) polynomials, in the case of three- and four-node elements, and by quadratic (RCP1) and cubic (RCP2) polynomials, in the case of six- and nine-node elements. The expressions of these local approximations are reported in the following. The number of unknown parameters involved by linear, quadratic and cubic approximations are 5, 9 and 14, respectively. In the REP procedure, complete linear polynomials are used for three- and four-node elements (3 + 3 unknown parameters) and complete quadratic polynomials for sixand nine-node elements (6 + 6 unknown parameters). The local stress approximations used in the RCP procedure for Class I problems are: (a) linear 5-parameters approximation used in the RCP1 version for 4-node elements ⎤ ⎡ ∙ ¸ a1 1 0 y 0 x ⎢ .. ⎥ σ ∗ph = (A.5) , 0 1 0 x −y ⎣ . ⎦ a5 (b) quadratic 9-parameters approximation used in the RCP2 version for 4-node elements and in the RCP1 version for 9-node elements ⎤ ⎡ ¸ a1 ∙ 2 2 1 0 y 0 x y 0 x −2xy ⎢ . ⎥ σ ∗ph = (A.6) ⎣ .. ⎦ , y2 0 1 0 x −y 0 x2 −2xy a9 (c) cubic 14-parameters approximation used in the RCP2 version ments ∙ 1 0 y 0 x y2 0 x2 2xy y 3 ∗ σ ph = 0 1 0 x −y 0 x2 −2xy −y 2 0 ⎡ a1 ¸ 3 2 2 x −3xy 2x y ⎢ .. y3 −2xy 2 ⎣ . −3x2 y a14 for 9-node ele0 x3 ⎤ ⎥ ⎦. (A.7) Chapter 7 A.2 239 Class II problems The second class of problems is plane elasticity. A.2.1 Basic equations Figure A.2: Class II problems Consider the elastic plane problem, shown in Figure A.2, governed by the following equations: D∗ σ = b in Ω, (A.8) = C² in Ω, (A.9) ² = Du in Ω, (A.10) σ where Ω is the domain, together with the following boundary conditions on ∂Ω u = ū in ∂Ωu , T N σ = t̄ in ∂Ωt , (A.11) (A.12) where matrix N collects the components of the outward unit normal to ∂Ω ¸ ∙ nx 0 ny . (A.13) N= 0 ny nx According to the figure, displacement u, strains ² and stress σ are given by ⎡ ⎤ ⎡ ⎤ ¸ ∙ σx x ux , ² (x) = ⎣ y ⎦ , σ (x) = ⎣ σ y ⎦ . (A.14) u (x) = uy γ xy τ xy 240 7.0- The RCP stress approximation Compatibility and equilibrium operators D and D∗ take the form ⎡ ⎤ ∙ ¸ ∂/∂x 0 −∂/∂x 0 −∂/∂y ∗ ⎣ ⎦ 0 ∂/∂y . D= , D = 0 −∂/∂y −0∂/∂x ∂/∂y 0∂/∂x (A.15) b is a prescribed source term, ¸ bx . (A.16) b (x) = by In the case of isotropic material in plane stress conditions, the elasticity matrix C takes the form ⎡ ⎤ 1 ν 0 Et ⎣ ν 1 0 ⎦, C= (A.17) 1 − ν2 1−ν 0 0 2 where E is the Young’s modulus, ν is the Poisson’s ratio and t is the thickness. ∙ A.2.2 The RCP stress approximation The stress approximation can be easily constructed employing the well-established techniques usually adopted for hybrid stress elements. In particular, for the equilibrium problem of plane elasticity, σ ∗ph and σ ∗pp can be selected according to the following relations: ¤ £ σ ∗phT = σ ∗h x σ ∗h y τ ∗h xy , (A.18) σ ∗h x σ ∗ppT σ ∗p x ∂2φ ∂2φ ∂2φ , , σ ∗h y = , τ ∗h xy = − 2 2 ∂y ∂x ∂x∂y ¤ £ = σ ∗p x σ ∗p y τ ∗p xy , Z x Z y = − bx dx, σ ∗p y = − by dy, τ ∗p xy = 0. = 0 (A.19) (A.20) (A.21) 0 The particular solutions given in Equation (A.20) are those commonly adopted for hybrid finite elements. Indeed, other choices are possible which would lead, obviously, to diﬀerent final solutions. However, the various choices become all equivalent in practice when the mesh is refined, since all the recovered solutions have been proved to converge to the exact one [Ubertini, 2004]. Based on the above assumptions, choosing the stress approximation simply reduces to choose the stress function φ. Regarding this, it should be observed that the consistency of the procedure requires the self-equilibrated stress approximation to be selected among polynomials of degree not less than p, being p the interpolation order for displacements in the finite element solution. Moreover, the invariance of the procedure with respect to any coordinate change, requires that all the components of σ ∗ph are represented by complete polynomial expansions of the same degree. Therefore, the stress function should be chosen as a complete polynomial Chapter 7 241 of degree not less than (p + 2) in plane elasticity problems (see Equation (A.18)). RCP1 and RCP2 denote the RCP procedure with p- and (p + 1)-degree stress approximation, respectively. In particular RCP1 and RCP2 refer to the linear 7-parameters and quadratic 12-parameters approximations, when linear elements are used, while to the quadratic 12-parameters and cubic 18-parameters approximations, when quadratic elements are used. In the REP procedure, complete linear polynomials are used for linear elements (3 + 3 + 3 unknown parameters) and complete quadratic polynomials for quadratic elements (6 + 6 + 6 unknown parameters). The local stress approximations used in the RCP procedure for Class II problems are: (a) linear 7-parameters approximation ments ⎡ 1 0 0 y σ ∗ph = ⎣ 0 1 0 0 0 0 1 0 used in the RCP1 version for 4-node ele⎤⎡ 0 ⎢ y ⎦⎣ −x ⎤ a1 .. ⎥ , (A.22) . ⎦ a7 (b) quadratic 12-parameters approximation used in the RCP2 version for 4-node elements and in the RCP1 version for 8- and 9-node elements ⎡ ⎤⎡ a ⎤ 1 1 0 0 y 0 x 0 y2 0 x2 2xy 0 ⎢ ⎥ y2 0 2xy ⎦ ⎣ ... ⎦ , y 0 x2 σ ∗ph = ⎣ 0 1 0 0 x 0 2 2 0 0 1 0 0 −y −x 0 0 −2xy −y −x a12 0 x x 0 0 −y (A.23) (c) cubic 18-parameters approximation used in the RCP2 version for 8- and 9node elements ⎡ 1 0 0 y 0 x 0 y2 0 x2 2xy 0 y3 0 ∗ 2 2 ⎣ y 0 2xy 0 x3 0 1 0 0 x 0 y 0 x σ ph = 2 0 0 1 0 0 −y −x 0 0 −2xy −y −x2 0 0 ⎤⎡ a ⎤ 1 x3 3xy 2 0 3x2 y ⎢ ⎥ 0 3x2 y y 3 ⎦ ⎣ ... ⎦ . (A.24) 3xy 2 2 3 3 2 −3x y −y −x −3xy a18 A.3 Class III problems The third class of problems is shear deformable plates. A.3.1 Basic equations Consider a plate referred to a Cartesian reference frame (O, x, y, z) with the origin O on the mid-surface Ω and the z-axis in the thickness direction, −h/2 ≤ z ≤ h/2 242 7.0- Basic equations where h is the thickness. Figure A.3: Class III problems Let ∂Ω be the boundary of Ω. The Reissner-Mindlin theory is employed. The compatibility equations are given by μ = Dp u, χ = Dp θ, γ = Ds w + θ, (A.25) where uT = [ux , uy ] is the in plane displacement, w is the transverse displacement, ¤ ¤ £ £ γ xy is the vector θ T = θx θy is the vector of rotations, μT = x y ¤ £ of in-plane strains, χT = χx χy χxy is the vector of curvatures, γ T = ¤ £ γ x γ y is the vector of transverse shear strains and ⎡ ⎤ ∙ ¸ ∂/∂x 0 ∂/∂x ⎣ ⎦ 0 ∂/∂y , Ds = Db = . (A.26) ∂/∂y ∂/∂y ∂/∂x The equilibrium equations can be obtained via the principle of virtual work in the form (A.27) D∗p N = qx , D∗p M + S = c, D∗s S = qz , ¤ ¤ £ £ T T where vectors N = Nx Ny Nxy , M = Mx My Mxy and ST = ¤ £ Sx Sy collect the resultant stresses, D∗p and D∗s are diﬀerential operators adjoint to Dp and Ds , respectively, and qz , qx , and c are the prescribed generalized loads. The constitutive equations can be written in the following form typical of monoclinic laminated plates: N = Cm μ + Cmb χ, (A.28) M = Cmb μ + Cb χ, (A.29) S = Cs γ. (A.30) Chapter 7 243 where Cm , Cmb , and Cb are respectively the membrane, membrane-bending coupling and bending elasticity matrices, whereas Cs represents the resultant elastic shear matrix. For an isotropic linearly elastic material, the constitutive equations reduces to N = Cm μ, M = Cb χ, S = Cs γ, (A.31) ⎡ ⎤ 1 ν 0 Eh ⎣ ⎦, ν 1 0 Cm = (A.32) (1 − ν 2 ) 0 0 (1 − ν)/2 ⎡ ⎤ 1 ν 0 3 Eh ⎣ ν 1 ⎦, 0 Cb = (A.33) 12(1 − ν 2 ) 0 0 (1 − ν)/2 ∙ ¸ κEh 1 0 , Cs = (A.34) 2(1 + ν) 0 1 being E the Young’s modulus and ν the Poisson’s ratio and h is the plate thickness and κ the shear correction factor, (κ = 56 if not otherwise specified). Finally, the plate equilibrium problem can be stated once the boundary conditions on ∂Ωu on ∂Ω are specified. A.3.2 A plate-like body Figure A.4: A 3D plate-like body Consider the 3D plate-like body in Figure A.4, where volume forces are indicated by b and surface tractions on top and bottom surfaces by p(+) and p(−) . In accordance with the Reissners-Mindlin plate theory, the displacement field 244 7.0- A plate-like body d is represented as a linear function of the thickness coordinate z in the form ∙ ¸ u (x) + zθ (x) (A.35) d (x,z) = , w (x) where the notation introduced in the previous section is used to indicate quantities defined on the plate mid-plane Ω. The strain tensor ² can be decomposed as follows ¸ ∙ 1 e γ ε = ∇(s) d = 1 T 2 (A.36) 0 2γ where e is the in-plane strain tensor and γ the transverse shear vector. Using the matrix notation, they can be written as: ⎤ ⎡ e =⎣ x ⎦ = μ+zχ, γ = Ds w + θ. y γ xy (A.37) Analogously, the stress tensor σ can be separated into the in-plane stress tensor S and the transversal shear stress vector τ . As usual the out of plane normal stress is assumed to be null, trough it could be regarded as the "reactive" stress corresponding to the internal constant z = 0 [Lembo & Podio-Guidugli, 2007]. The in-plane equilibrium equations are given by ¸ ∙ bx , (A.38) ∇x s + ∇z τ + bx = 0, bx = by or, in matrix notation, by ⎡ ⎤ σx D∗p S + ∇z τ + bx = 0, S = ⎣ σ y ⎦ . τ xy Introducing the stress resultants Z Z N = sdz, M = zSdz, f S− f Z τ dz, (A.39) (A.40) f the plate equilibrium equations can be derived in the form of Equation (A.31) where Z ³ ´ (−) bx dz + p(+) + p qx = , x x f qz = Z Z f ³ ´ bz dz + p(+) + p(−) , z z ³ ´ (−) h . zbx dz + p(+) x − px 2 f Introducing the constitutive equations and taking into account for the above assumptions, the stress-strain relations can be written as c= s = Cp e, τ = κ ¯ Cs γ, (A.41) where Cp is the reduced in plane elastic matrix, Cs is the transverse shear elastic Chapter 7 245 matrix, matrix κ collects the shear correction factors ¸ ∙ κ11 κ12 , κ= κ12 κ22 (A.42) N = Cm μ + Cmb χ, (A.43) M = Cmb μ + Cb χ, (A.44) = Cs γ. (A.45) assumed to be constant along the plate thickness, symbol ¯ denotes a product component by component. In the case of homogeneous plates: κ11 = κ22 = 56 and κ12 = 0. By interpolating the stress-strain relations through the thickness, the plate constitutive equations are defined S A.3.3 The RCP stress approximation The RCP stress approximation is selected as follows. The part which equilibrates external loads within the element is selected as ¤ Rx ⎤ ⎡ Rx£ cx + 12 0 qz dx dx 0 ⎥ ⎢ Rx£ ¤ R 1 x ⎥ (A.46) M∗pp = ⎢ ⎣ 0 cy + 2 0 qz dy dy ⎦ 0 # " Rx q dx z 1 0 S∗pp = − (A.47) Ry 2 qz dy 0 Two local approximations are tested for the sef-equilibrated part: a linear approximation ruled by 9 parameters ∙ M∗ph S∗ph ¸ ⎡ ⎢ ⎢ =⎢ ⎢ ⎣ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 y 0 0 0 0 0 x 0 0 0 x 0 0 −1 0 0 y 0 0 −1 0 0 y −1 0 0 0 x 0 −1 ⎤ ⎡ ⎥ ⎥⎢ ⎥⎣ ⎥ ⎦ ⎤ a1 .. ⎥ , . ⎦ a9 (A.48) 246 7.0- The RCP stress approximation and a quadratic approximation ruled ⎡ 1 0 0 y 0 ⎢ 0 1 0 0 x ¸ ∙ ∗ ⎢ Mph = ⎢ ⎢ 0 0 1 0 0 S∗ph ⎣ 0 0 0 0 0 0 0 0 0 0 by 17 parameters x 0 0 −1 0 x2 0 −xy −x y 0 y 0 0 −1 0 y2 −xy x −y 0 0 y −1 0 0 0 x 0 −1 0 0 x2 0 −2 x xy 0 0 −y 0 0 0 y2 −2 y 0 ⎤ 0 xy 0 0 −x y2 0 0 0 0 0 x2 0 0 0 ⎤ ⎡ a1 ⎥ ⎥⎢ . ⎥ ⎥ ⎣ . ⎦ . (A.49) . ⎥ ⎦ a17 Membrane forces can be approximated as described in Section A.2.2. Chapter 7 247 Appendix B More benchmark tests B.1 Class I problems FEM: Vx - MAX: 0.7147 REP: Vx - MAX: 1.1671 RCP1n: Vx - MAX: 0.3112 RCP1nb: Vx - MAX: 0.3112 RCP2n: Vx - MAX: 0.2752 RCP2nb: Vx - MAX: 0.2878 Figure B.1: Thin membrane subjected to transverse loading - Error distributions on a regular mesh of four-node elements (contour interval = 0.1, with load case No. 1) 248 FEM (global error: 0.350) 7.0- Class I problems REP (global error: 0.337) RCP1n (global error: 0.221) 0.08 0.06 0.04 0.02 min 1.02E-02, max 8.02E-02 min 1.24E-02, max 7.98E-02 min 0.56E-02, max 4.43E-02 0 RCP1nb (global error: 0.238) RCP2n (global error: 0.160) RCP2nb (global error: 0.179) 0.08 0.06 0.04 0.02 min 1.00E-02, max 4.43E-02 min 0.42E-02, max 3.14E-02 min 0.67E-02, max 3.68E-02 0 Figure B.2: Thin membrane subjected to transverse loading - Distributions of element error in energy norm on a regular mesh of four-node elements, load case No. 1 (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) Chapter 7 249 FEM: Vx - MAX: 0.0953 REP: Vx - MAX: 0.4557 RCP1n: Vx - MAX: 0.0336 RCP1nb: Vx - MAX: 0.2083 RCP2n: Vx - MAX: 0.0186 RCP2nb: Vx - MAX: 0.0209 Figure B.3: Thin membrane subjected to transverse loading - Error distributions on a regular mesh of nine-node elements (contour interval = 0.015, load case No. 1) 250 7.0- Class I problems FEM (global error: 0.036) REP (global error: 0.116) RCP1n (global error: 0.018) 0.02 0.01 min 0.07E-02, max 0.77E-02 min 0.59E-02, max 2.82E-02 min 0.10E-02, max 0.30E-02 RCP1nb (global error: 0.037) RCP2n (global error: 0.009) RCP2nb (global error: 0.011) 0 0.02 0.01 min 0.18E-02, max 0.93E-02 min 0.05E-02, max 0.16E-02 min 0.10E-02, max 0.22E-02 0 Figure B.4: Thin membrane subjected to transverse loading - Distributions of element error in energy norm on a regular mesh of nine-node elements, load case No. 1 (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) Chapter 7 251 Local error at (0.5,0.5) 0 4-node elements log(stress error) log(stress error) 0 FEM -2 1 REP -4 RCP1n 2 Global error 4-node elements 1 FEM -1 REP RCP1nb RCP1n -2 RCP2n -6 RCP2nb RCP2n 3 -3 -8 -2.5 -2 -1.5 -1 -0.5 log(element size) (a) -2.5 Local error at (0.5,0.5) 0 9-node elements FEM -3 2 REP -5 -2 -1.5 -1 -0.5 log(element size) (b) log(stress error) log(stress error) -1 2 Global error 9-node elements RCP1nb REP -2 FEM 3.5 -4 RCP2bn RCP1n 2 4 RCP2n -6 -7 -2.5 (c) RCP1n & RCP2n -2 -1.5 -1 -0.5 log(element size) 3.5 -8 -2.5 (d) -2 -1.5 -1 -0.5 log(element size) Figure B.5: Thin membrane subjected to transverse loading - Rate of convergence for load case No. 1: (a) local error at point x = 0.5, y = 0.5 (σ x = σ y ) for four-node elements, (b) global error in energy norm for four-node elements, (c) local error at point x = 0.5, y = 0.5 (σx = σy ) for nine-node elements, (d) global error in energy norm for nine-node elements 252 7.0- Class I problems Local error at (0.25,0.75) 0 4-node elements log(stress error) log(stress error) -1 1 FEM REP -3 2 RCP1n 1 FEM -2 REP RCP1n -4 2 RCP2n -5 Local error at (0.25,0.75) 4-node elements RCP2n -6 2 -7 -2.5 (a) -2 -1.5 log(element size) -2.5 (b) -2 -1.5 log(element size) Global error 0 4-node elements log(stress error) log(stress error) 0 FEM -1 REP 1 -2 Global error 9-node elements RCP1nb -2 FEM REP -4 3.5 RCP1nb RCP1n RCP2nb -6 RCP2nb RCP2n -3 (c) -2 -1.5 -1 -0.5 log(element size) RCP1n RCP2n -8 2 -4 -2.5 2 -2.5 (d) 3.5 -2 -1.5 -1 -0.5 log(element size) Figure B.6: Thin membrane subjected to transverse loading - Rate of convergence for load case No. 2: (a) local error in σx at point x = 0.25, y = 0.75 for four-node elements, (b) local error in σ y at point x = 0.25, y = 0.75 for four-node elements, (c) global error in energy norm for four-node elements, (d) global error in energy norm for nine-node elements Chapter 7 B.2 253 Class II problems RCP2n: Vx - MAX: 0.5884 RCP2n: Vy - MAX: 0.4166 RCP2n: Wxy - MAX: 0.3927 RCP2nb: Vx - MAX: 0.5013 RCP2nb: Vy - MAX: 0.3614 RCP2nb: Wxy - MAX: 0.3626 Figure B.7: Stretched plate with circular hole - Error distributions for four-node elements on mesh No. 1: RCP2 and RCP2b (contour interval = 0.05) 254 7.0- Class II problems FEM: Vx - MAX: 0.8137 FEM: Vy - MAX: 0.5451 FEM: Wxy - MAX: 0.4703 REP: Vx - MAX: 0.6375 REP: Vy - MAX: 0.4903 REP: Wxy - MAX: 0.4876 RCP1n: Vx - MAX: 0.5833 RCP1n: Vy - MAX: 0.4332 RCP1n: Wxy - MAX: 0.4176 RCP1nb: Vx - MAX: 0.6725 RCP1nb: Vy - MAX: 0.5100 RCP1nb: Wxy - MAX: 0.5015 Figure B.8: Stretched plate with circular hole - Error distributions for four-node elements on mesh No. 1: FE solution, REP, RCP1 and RCP1b (contour interval = 0.05) Chapter 7 255 RCP2: Vx - MAX: 0.1244 RCP2: Vy - MAX: 0.1147 RCP2: Wxy - MAX: 0.1077 RCP2b: Vx - MAX: 0.1258 RCP2b: Vy - MAX: 0.1186 RCP2b: Wxy - MAX: 0.1438 Figure B.9: Stretched plate with circular hole - Error distributions for nine-node elements on mesh No. 1: RCP2 and RCP2b (contour interval = 0.05) 256 7.0- Class II problems FEM: Vx - MAX: 0.2137 FEM: Vy - MAX: 0.2542 FEM: Wxy - MAX: 0.1398 REP: Vx - MAX: 0.3629 REP: Vy - MAX: 0.2801 REP: Wxy - MAX: 0.2368 RCP1n: Vx - MAX: 0.1448 RCP1n: Vy - MAX: 0.1355 RCP1n: Wxy - MAX: 0.1122 RCP1nb: Vx - MAX: 0.2890 RCP1nb: Vy - MAX: 0.2312 RCP1nb: Wxy - MAX: 0.2180 Figure B.10: Stretched plate with circular hole - Error distributions for nine-node elements on mesh No. 1: FE solution, REP, RCP1 and RCP1b (contour interval = 0.05) Chapter 7 FEM (global error: 0.07446) 257 REP (global error: 0.06286) RCP1n (global error: 0.06795) x 10-4 6 4 2 min 0.36E-04, max 6.95E-04 min 0.16E-04, max 5.09E-04 min 0.12E-04, max 5.42E-04 RCP1nb (global error: 0.06692) RCP2n (global error: 0.05783) RCP2nb (global error: 0.05087) 0 x 10-4 6 4 2 min 0.12E-04, max 5.47E-04 min 0.16E-04, max 5.28E-04 min 0.16E-04, max 4.37E-04 0 Figure B.11: Stretched plate with circular hole - Distributions of element error in energy norm on mesh No. 1: four-node elements (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) 258 7.0- Class II problems FEM (global error: 0.01803) REP (global error: 0.02336) RCP1n (global error: 0.01286) x 10-4 2.5 2 1.5 1 0.5 min 0.18E-05, max 1.67E-04 min 0.14E-05, max 2.56E-04 min 0.08E-05, max 1.19E-04 RCP1nb (global error: 0.02210) RCP2n (global error: 0.01017) RCP2nb (global error: 0.01044) 0 x 10-4 2.5 2 1.5 1 0.5 min 0.12E-05, max 2.42E-04 min 0.05E-05, max 1.02E-04 min 0.05E-05, max 1.19E-04 0 Figure B.12: Stretched plate with circular hole - Distributions of element error in energy norm on mesh No. 1: nine-node elements (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) Chapter 7 -2 4-node elements FEM -2 1 RCP1n -3 REP Local error (Vy) 4-node elements FEM -3 1 -4 -4 RCP2n RCP2n 2 4-node elements FEM RCP1n RCP1nb -3.5 0.9 RCP2n -6 -4 REP & RCP1nb -6 1.9 RCP2n 2 -2 -1.5 -1 -0.5 log(element size) Global error 8-node elements FEM -5 REP 2 -4 9-node elements RCP1nb FEM -5 REP -6 1.9 RCP1n RCP2nb -7 RCP2n REP -4.5 1.3 (d) RCP1n -9 -2 -1.5 -1 -0.5 -2.5 (c) log(element size) -4 -5 -2.5 FEM -5 Global error log(stress error) log(stress error) -6 -2 -1.5 -1 -0.5 -2.5 (b) log(element size) Global error -3 1 -8 log(stress error) (a) 4-node elements -4 -7 REP & RCP1n -6 -2.5 Local error (Wxy) 2 -5 -5 -2.5 -3 log(stress error) Local error (Vx) log(stress error) log(stress error) -1 259 RCP1n -7 RCP2n RCP2nb -8 -2.5 -2 -1.5 -1 -0.5 (e) log(element size) 2.6 -2 -1.5 -1 -0.5 (f) log(element size) RCP2nb -8 -2.5 2.6 -2 -1.5 -1 -0.5 log(element size) Figure B.13: Stretched plate with circular hole - Rate of convergence: (a) local error in σx at point r = 1.5, θ = π/4 for four-node elements, (b) local error in σ y at point r = 1.5, θ = π/4 for four-node elements, (c) local error in τ xy at point r = 1.5, θ = π/4 for four-node elements, (d) global error in energy norm for four-node elements, (e) global error in energy norm for eight-node elements, (f) global error in energy norm for nine-node elements Appendix C The hybrid stress element for plate structures (9β Q4) Recently, a new quadrilateral 4-node element for the analysis of shear deformable plates has been presented in [de Miranda & Ubertini, 2006]—[Daghia et al. , 2007] and labelled 9βQ4. The formulation is of hybrid stress type, involving equilibrating stress resultants within each element and compatible displacements along the interelement boundary. The assumed stress approximation, expressed in terms of skew coordinates, has the minimum number of stress modes (9 + 5 modes) and is coordinate invariant. Displacements are modelled using a linked interpolation ruled by the standard nodal degrees of freedom (3 + 2 dofs per node). The hybrid stress approach, where displacement acts on element boundary only, together with the linked interpolation allow to derive an element which is locking-free and passes all the patch tests. Moreover, the resulting element has been tested to be stable, accurate and relatively insensitive to geometry distortions. The main features of the element are: (1) it has 4-nodes (12 + 8 dofs) and involves only compatible displacement functions, (2) it is locking-free, (3) it passes all the patch tests, (4) it is readily implementable into existing finite element codes. The key ingredients of the element formulation are: • a mixed variational approach involving stresses and displacements as independent variables, which allow to meet requirements (1) and (4); • a linked interpolation for the transverse displacement field, which goes towards avoiding locking eﬀects but it does not suﬃce to remove them; • the stress resultants are assumed to satisfy a priori the plate equilibrium equations within each element. It can be proved that the last ingredient together with the previous ones allows to meet also requirement (2) and (3). The assumed approximation for stress resultants is ⎡ h ⎤ ⎡ h ⎤ ⎡ hh ⎤ Np N N ⎣ Mhh ⎦ = ⎣ Mhp ⎦ + ⎣ Mhh ⎦ (C.1) Shh Shh Shp where the first term is particular solution of the plate equilibrium equations, while the second term is the self equilibrated component which is approximated as follows 262 7.0- The hybrid stress element for plate structures (9βQ4) η 4 η 4 3 3 y ξ x ξ 1 2 2 1 Figure C.1: Reference coordinate system and their coordinate axes £ ∙ Mhh Shh ¸ ⎡ 1 0 0 ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ 0 ⎣ 0 ⎡ 1 0 0 a21 η̄ ¤ h N = ⎣ 0 1 0 b21 η̄ 0 0 1 a1 b1 η̄ 0 1 0 0 0 1 0 0 Coeﬃcients ai and bi ⎡ a1 ⎣ a2 a3 a21 η̄ a23 ξ̄ b21 η̄ b23 ξ̄ a1 b1 η̄ 0 0 a23 ξ̄ b23 ξ̄ a3 b3 ξ̄ −a1 ξ̄ − a3 η̄ ⎤ ⎦ βN , 0 (C.2) −a21 ξ̄η̄ −b21 ξ̄η̄ − b1 ξ̄ − b3 η̄ a3 b3 ξ̄ 0 0 0 −a1 b1 ξ̄η̄ 0 0 1 0 a1 η̄ 0 0 0 1 b1 η̄ −a23 ξ̄η̄ ⎤ ⎥ ⎥ ⎥ −a3 b3 ξ̄η̄ ⎥ ⎥ βMS , ⎥ ⎥ a3 ξ̄ ⎦ −b23 ξ̄η̄ b3 ξ̄ (C.3) in the above expressions depend on the element geometry: ⎡ ⎤ ⎡ ⎤ x y1 ⎤ −1 1 1 −1 ⎢ 1 b1 1 x2 y2 ⎥ ⎥. b2 ⎦ = ⎣ 1 −1 1 −1 ⎦ ⎢ (C.4) ⎣ x3 y3 ⎦ 4 b3 −1 −1 1 1 x4 y4 Appendix D Formulation of a transition element for plane problems D.1 Shape functions The typical configuration of a variable-node element is shown in Figure D.1. The midside nodes need to be generated when the refined elements are attached to the side of the unrefined element. In particular a quadrilateral element with a bilinear displacement field is considered but, the method is general and can be easily applied to other elements. uC A Du uB C uA I C II III B A (-1) C (0) B (+1) (a) (b) Figure D.1: The transition quadrilateral element and displacement field In Figure D.1 edge AB of element I has discontinuity in the displacement slope at node C. The variation of any displacement u along edge AB can be expressed as 1 1 1 u = (1 − ξ) uA + (1 + ξ) uB + (1 − |ξ|) ∆uC , (D.1) 2 2 2 A B where u and u are displacements at nodes A and B, respectively, and ∆uC is the departure of displacement at C from the straight variation defined by the displacements at A and B ; ξ is the local coordinate along AB, ξ = −1 at A, ξ = +1 at B, ξ = 0 at C; |ξ| denotes the absolute value of ξ. Equation D.1 gives uC = uA + uB + ∆uC , 2 (D.2) uA + uB . 2 (D.3) or ∆uC = uC − 264 7.0- Shape functions Substituting Equation (D.3) into Equation (D.1) one gets 1 1 1 u = (|ξ| − ξ) uA + (|ξ| + ξ) uB + (1 − |ξ|) uC , 2 2 2 or u = N A uA + N B uB + N C uC , (D.4) where 1 (|ξ| − ξ) , (D.5) 2 1 (|ξ| + ξ) , NB = (D.6) 2 1 (1 − |ξ|) . NC = (D.7) 2 Equations (D.5-D.7) give the shape functions along an edge with a transition node. The procedure to obtain these shape function, as shown in [Gupta, 1978] can be summarized as follows: NA = • write the shape functions for the corner nodes as if there were no intermediate transition node; • write the shape function for the intermediate node treating the displacement at the intermediate node as a departure ∆uC from the linear displacement variation defined by the displacements at corner nodes; • modify the shape function at the corner nodes such that the displacement at the intermediate node becomes the total displacement at the node, uC . Note that the shape function at the intermediate node does not change; h (-1,1) 4 (0,1) x = -1 5 (-1,0) 7 1 (-1,-1) 3 (1,1) 6 h = +1 (1,0) x x = +1 h = -1 8 (0,-1) 2 (1,-1) Figure D.2: The transition quadrilateral element The above procedure was established for an edge, but the same procedure can be applied to the complete element. Figure D.2 shows a quadrilateral element with Chapter 7 265 corner nodes 1−2−3−4 and intermediate nodes 5−6−7−8. It is assumed that any intermediate node in this figure will introduce discontinuity in the displacement slope. However, it is not necessary that any or all of these intermediate nodes be present. Indeed, an element can be formulated with corner nodes only, or with one, two, three or four intermediate nodes. The procedure summarized above is applied as follows: • • • write the shape functions for the corner nodes as if there were no intermediate nodes 1 1 (1 − ξ) (1 − η) , N = (D.8) 4 1 2 (1 − ξ) (1 + η) , N = (D.9) 4 1 3 (1 + ξ) (1 − η) , N = (D.10) 4 1 4 (1 + ξ) (1 + η) , N = (D.11) 4 write the shape function for the intermediate node treating the displacement at the intermediate node as a departure from the linear displacement variation defined by the displacements at corner nodes 1 (1 − |ξ|) (1 − η) , N5 = (D.12) 4 1 (1 − |ξ|) (1 + η) , N6 = (D.13) 4 1 (1 + ξ) (1 − |η|) , N7 = (D.14) 4 1 (1 + ξ) (1 + |η|) , N8 = (D.15) 4 if any intermediate node is not present, simply set its shape function equal to zero. For instance, if intermediate nodes 6 and 7 are not present, set N 6 = 0 and N 7 = 0. Modify the shape function at the corner nodes such that the displacement at the intermediate node becomes the total displacement at the respective nodes ¢ 1¡ 7 1 N + N8 , N1 = N − (D.16) 2 ¢ 1¡ 8 2 N + N5 , N2 = N − (D.17) 2 ¢ 1¡ 5 3 N + N6 , N3 = N − (D.18) 2 ¢ 1¡ 6 4 N + N7 , N4 = N − (D.19) 2 Note, if nodes 6 and 7 were not present, their shape functions would be set to zero as explained above, which would automatically reflect in no change in 266 7.0- Shape functions the shape function for node 2. The shape functions for 5-node and 6-node elements are shown by Figures D.3 and D.4. Notice that some care should be posed on numerical integration when finite element equations are compute because of discontinuity of the shape functions. In particular a modified Gaussian quadrature rule is adopted. The patch test has been used to check the capability of the element above to represent constant strain states. All the elements pass the patch test. N1 N2 N3 1 1 1 0.5 0.5 0.5 0 −1 1 0 0 −1 0 ξ 1 −1 1 0 η 0 ξ N4 1 0.5 0.5 0 −1 1 0 ξ 1 −1 1 0 η 0 ξ 1 −1 N5 1 0 1 −1 0 −1 η 0 −1 1 0 0 ξ 1 −1 η Figure D.3: The shape function for 5 mode element η Chapter 7 267 N1 N3 N2 1 1 0.5 0.5 1 0.5 0 0 −1 1 0 0 −1 0 ξ 1 −1 1 0 η −0.5 −1 0 ξ N4 1 −1 η 0 ξ N5 1 1 0.5 0.5 0.5 0 −1 1 0 0 1 −1 η 0 −1 1 0 0 ξ 1 −1 1 −1 η N6 1 ξ 1 0 η 0 −1 1 0 0 ξ 1 −1 Figure D.4: The shape function for 6 mode element η Subject index taper, 50, 74 a posteriori Error estimation, 19, 20, 35, 79, 80, 82, 83, 130 a priori Error estimation, 20, 34, 41, 80, 175 Adaptive analysis, 74 h-refinement, 137 mesh regenerarion, 136 procedure, 134 process, 26 refinement, 74, 80, 129 scheme, 60 Ainsworth, M., 19, 81 Argyris, J.H., 15, 16 aspect ratio, 108, 115, 118, 159 Babuška, I., 82 Babuška, I., 18, 80—83, 105, 106, 113, 133 Barlow, J., 17, 54 points, 17, 54, 72, 185 Bathe, K.J., 17, 18, 43 Boroomand, B., 19 Bubnov, I.G, 14 Castellazzi, G., 21 Clough, R., 15—17 Completeness completeness requirements, 43— 45 Convergence order of convergence, 27, 41, 94 Courant, R., 15, 16 de Miranda, S., 21 Distortion parameter aspect ratio, 50, 75 skew, 50, 74 Eﬀectivity index, 103—105 robustness test, 106 Error Pollution, 104 Felippa, C., 15 Fix, G., 17, 38 Galerkin, B.G., 14, 15, 23, 30, 39— 41 Bubnov-Galerkin, 14, 30, 39 Petrov-Galerkin, 30 Gauss, C.F., 54, 84 Gauss-Green, 31 Gauss-Legendre, 55 Gröbner basis, 61 Hellinger-Reissner functional, 32 Hughes, T.J., 17, 18 Hybrid stress functional, 32 Index eﬀectivity, 103—105 Irons, B., 17, 106, 185 Jacobian, 48 determinant, 57, 59, 74, 75 matrix, 61, 74 transformation, 48 Locking, 44, 201 locking-free, 201 shear locking, 154 Martin, H.C., 15 Master-cell, 108—112 matrice funzionale, 67 Melosh, R.J, 16 mesh periodic, 107, 112 270 7.0- Subject index Model problem, 31 Oden, J.T., 18, 19, 81 Patch Element, 85, 99, 100 Patch test, 44, 105, 106 Babuska patch test, 82 Rayleigh, Lord (J.W. Strutt), 14, 24, 30, 41 Recovery-based estimator, 19, 81, 83, 85, 128, 129 Rheinboldt, W.C., 18, 80, 82, 133 Ritz, W., 14—16, 23, 24, 30, 40, 41 Rayleigh-Ritz, 23, 24 Strang, G., 17, 38 Taylor, R.L., 16, 17 test eﬀectivity robustness, 106 Topp, L.J., 15 Turner, M.J., 15, 16 Ubertini, F., 19, 21 Verification & Validation, 6 V&V, 6, 7, 172, 173, 178, 179, 181, 183, 184 Verification & Validation V&V, 182, 184 Wilson, E.L., 15—17 Zhu, J.Z., 19, 81, 84, 85, 103 Zienkiewicz, O.C., 16, 17, 19, 43, 81, 84, 85, 103 Chapter 7 271 List of Tables 2.1 Classification of various physical problems . . . . . . . . . . . . . . . . . . . . . 28 3.1 Shape functions of 4, 8 and 9 noded quadrilaterals elements . . . . . . . . 48 3.2 Basic element properties: the optimal points are also the reduced rule, althoug the latter may give mechanism for some of the element types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3 Total response errors from distributios for quadrilaterals 8-node serendipity elements, (see [Barlow, 1989]) . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Stress calculation errors from distortions for quadrilaterals 8-node serendipity elements, (see [Barlow, 1989]) . . . . . . . . . . . . . . . . . . . . . . 62 3.5 Polynomial displacement fields that can be solved exactly by the various serendipity and Lagrangian elements in their undistorted and distorted configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6 Relation between elements of X and elements of Ξ . . . . . . . . . . . . . . . 70 3.7 Trasformation and relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.8 Relation between of the monomials . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.9 Coordinate’s point, see Figure 3.18 . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.10 Coeﬃcients of a generic linear transformation . . . . . . . . . . . . . . . . . . 73 3.11 Eﬀects of a generic linear transformation . . . . . . . . . . . . . . . . . . . . . . 74 3.12 Bilinear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.13 Higher degree transformations: quadratic curved-edge . . . . . . . . . . . . 78 3.14 Test No.1: two element test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.15 A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.16 A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.17 A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.18 A beam problem: clamped beam along one end and vertically loaded at the free end with distorted element as shown in Figure (3.34)(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.1 Versions of RCP recovery tested and related orders of the stress approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.2 RCPn - Eﬀectivity bounds and robustness indices for triangular 272 7.0- List of Tables elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 129 4.3 RCPn - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 129 4.4 RCPe - Eﬀectivity bounds and robustness indices for triangular elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 130 4.5 RCPe - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 130 4.6 RCPn - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.15(a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.7 RCPe - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class I problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.15(a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.8 RCPn - Eﬀectivity bounds and robustness indices for triangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 133 4.9 RCPn - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 133 4.10 RCPe - Eﬀectivity bounds and robustness indices for triangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.14, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 134 4.11 RCPe - Eﬀectivity bounds and robustness indices for quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figure 4.15, the aspect ratio is 1/1). . . . . . . . . . . . . . . . 134 4.12 RCPn - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.14(a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.13 RCPe - Eﬀectivity bounds and robustness indices for regular meshes of triangular and quadrangular elements applied to Class II problems (the mesh patterns are illustrated in Figures 4.14(a) and 4.15(a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.1 Versions of RCP recovery tested and related approximations . . . . . . . . 194 6.2 Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Mx at point Chapter 7 273 A ,9βQ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.3 Test No. 2 (h/L=0.025): Point-wise errors (per cent) of My at point A, 9βQ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.4 Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Mxy at point A, 9βQ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.5 Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Qx at point A, 9βQ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.6 Test No. 2 (h/L=0.025): Point-wise errors (per cent) of Qy at point A, 9βQ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Chapter 7 275 List of Figures 1.1 A famous image about error: train wreck at Montparnasse, Paris, France, 1895, (photgrapher unknown). From Wikipedia, the free encyclopedia, www.wikipedia.org . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Verification and Validation processes . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Model of a Ford Crown Victoria for performing crashworthiness simulations: (a) physical system; (b) discretization process; (c) discrete model; (d) subdomain subdivision (http://www.arasvo.com/crown_ victoria/crown_ vic.htm) . . . . . . . . 12 1.4 Divide and conquer (divide et impera) strategy: (a) physical system; (b) idealization process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Discretization of a domain Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Problem domain Ω and boundary ∂Ω . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Problem domain Ω and boundary ∂Ωt and ∂Ωu . . . . . . . . . . . . . . . . . 32 3.1 Example of superametric, isoparametric and subparametric elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Pascal’s triangle to generate various trial functions and corresponding elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Two dimensional transformation between local or theory space (left) and the real or user space (right) of quadrilateral and triangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Reference coordinates for four-node and eight node element used for give physical meaning of coeﬃcients αi,j . . . . . . . . . . . . . . . . . . . . . . 50 3.5 Physical meaning of the coeﬃcients α1,i and α2,i for i = 1, 2, 3, 4 . . . . . 51 3.6 Physical meaning of the coeﬃcients α1,i and α2,i for i = 4, 5, 6, 7 . . . . . 53 3.7 Optimal sampling points for the function (a) and its gradient (b) in one dimension (linear elements) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.8 Integration property of gauss points: (a) p = 1; (b) p = 2; (c) p = 3 which guarantees superconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.9 Optimal sampling points for the function (a) and its gradient (b) in one dimension (quadratic elements) . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.10 Cantilever beam with four 8-node elements. Stress sampling at cubic order (2x2) Gauss points with extrapolation to nodes . . . . . . . . . . . . . 57 3.11 Optimal superconvergent sampling and minimum integration points for some elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.12 Bending of a slender beam using 3 × 3 integration rules points: (a) shear stress map, (b) longitudinal stress . . . . . . . . . . . . . . . . . . . . . . . 59 276 7.0- List of Figures 3.13 Classification of element distortions: (a) undistorted element; (b) ratio distortion; (c) parallelogram distortion; (d) skew distortion; (e) mid-side node distortion; (f) curved edge distortion. . . . . . . . . . . . . . . 60 3.14 Distortion groups for serendipity eigth node elements . . . . . . . . . . . . . 61 3.15 Types of extreme distortions in four and eigth node elements . . . . . . . 63 3.16 The two element test: (a) the geometry and loading; (b) undistorted configuration; (c) progressive distortion due to δ parameter; (d) stress band plots of longitudinal stresses (σ xx ) in constant-bending-moment problem using undistorted mesh; (e) stress band plots of longitudinal stresses (σ xx ) using a distorted mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.17 Stress band plots of longitudinal stresses (σ xx ) in constant-bendingmoment problem for a curved edge distortion and a mid-side node distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.18 Linear transformation: the local normal co-ordinate system for a simple rotation (a) and for a generic lianear transformation (b). . . . . . 71 3.19 Bilinear transformation: (a) taper distortion ), (b) taper distortion with skew, (c) generic bilinear distortion . . . . . . . . . . . . . . . . . . . . . . 75 3.20 Higher degree transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.21 The Continuum Region Element test: displacements u = (x) are prescribed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.22 Element test for P8 elements: stress band plots stress for quadratic displacement field and error distributions with contour line step: 0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.23 Element test for P8 elements: local error distributions for quadratic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.001) . . . . . . . . 79 3.24 Element test for P8 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.25 Element test for P8 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.1) . . . . . . . . . . 80 3.26 Element test for Q9 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.27 Element test for Q9 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show Chapter 7 277 the positions of integrations points (contour interval: 0.1) . . . . . . . . . . 81 3.28 Element test for P8 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.29 Element test for P8 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.02) . . . . . . . . . 82 3.30 Element test for Q9 elements: stress band plots stress for cubic displacement field and error distributions with contour line step: 0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.31 Element test for Q9 elements: local error distributions for cubic displacement field plotted on stress contour plots for exact solution (red line) and FEM solution (blu line). The red star markers show the positions of integrations points (contour interval: 0.02) . . . . . . . . . 83 3.32 Simple distortione measure: (a) diagonal ratio c1 ; (b) middle point distance ratio c2 ; (c) middle point pojection ratio (on reference axes) c3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.33 Contour bands of stress for a four-node element: (a) undistorted configuration; (b) angular distortion with δ̂ = 0.3; (c) angular distortion with δ̂ = 0.6; (d) angular distortion with δ̂ = 0.9 . . . . . . . . . 86 3.34 A beam problem: (a) the geometry and load; (b) discretization for undistorted configuration, reference solution; (c) linear distortion; (d) first bilinear distortion; (e) second bilinear distortion; (f) taper distortion; (g) arbitrary bilinear distortion; . . . . . . . . . . . . . . . . . . . . 87 4.1 Interior patches for quadrilateral elements: (a) 4-nodes elements; (b) 8-node elements; (c) 9-node elements . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2 Recovery of exact σ of degree p by linear elements (p = 1) . . . . . . . . . 98 4.3 Recovery of exact σ of degree p by quadratic elements (p = 2) . . . . . . 99 4.4 Example of a patch of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5 Example of pacth: (•) assembly node defining the patch . . . . . . . . . . . 106 4.6 Computation of nodal values for four-, eight- and nine-node elements: (◦) nodal points, (•) nodal values determined by recovery procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.7 Boundary nodal recovery: (a) and (b) interior patches, (c) and (d) boundary patches; (◦) nodal points, (•) nodal values determined by recovery procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.8 Example of element pacth: assembly element defining the patch . . . . . . 112 4.9 Boundary element recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.10 Typical boundary patches: (a) smooth boundary with traction; (b) 278 7.0- List of Figures geometrical singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.11 Babuska patch test: (a) problem domain; (b) local (periodic) mesh; (c) master-cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.12 Computing the eﬀectivity index: (a) master-cell; (b) periodic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.13 Boundary nodal recovery: (a) and (b) interior patches, (c) and (d) boundary patches; (◦) nodal points, (•) nodal values determined by recovery procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.14 Mesh patterns used in robustness tests for triangular elements . . . . . . 127 4.15 Mesh patterns used in robustness tests for quadrangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.16 Recovered stress solutions for linear elements . . . . . . . . . . . . . . . . . . . 136 4.17 Error distributions for linear elements: (a) local error (maximum values are reported in brackets), (b) element error in energy norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.18 Rate of convergence for linear elements: (a) local error at point x/L = 3/4, (b) global error in energy norm (Problem No. 1) . . . . . . . . 138 4.19 Thin membrane subjected to transverse loading, load case No. 1 Local error distributions on a regular mesh of four-node elements (contour interval = 0.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.20 Thin membrane subjected to transverse loading, load case No. 1 Distributions of element error in energy norm on a regular mesh of four-node elements (the relative percentage error is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.21 Thin membrane subjected to transverse loading, load case No. 1 - Rate of convergence: (a) local error at point x = 0.5, y = 0.5 (σ x = σ y ) for four-node elements, (b) relative percentage error for four-node elements, (c) local error at point x = 0.5, y = 0.5 (σ x = σ y ) for nine-node elements, (d) relative percentage error for nine-node elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.22 Thin membrane subjected to transverse loading, load case No. 1 - Local error distributions on an unstructured mesh of four-node elements (contour interval = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.23 Thin membrane subjected to transverse loading, load case No. 1 Distributions of element error in energy norm on an unstructured mesh of four-node elements (the relative percentage error is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.24 Thin membrane subjected to transverse loading - Error distributions on an unstructured mesh of four-node elements (contour interval = 0.05,load case No. 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.25 Thin membrane subjected to transverse loading - Error distributions Chapter 7 279 on an unstructured mesh of nine-node elements (contour interval = 0.005, load case No. 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.26 Thin membrane subjected to transverse loading - Irregular pattern used in sensitivity test to mesh distortion . . . . . . . . . . . . . . . . . . . . . 146 4.27 Thin membrane subjected to transverse loading - Sensitivity to mesh distortion of local error at point x = 0.5, y = 0.5: (a) four-node elements and load case No. 1, (b) and (c) four-node elements and load case No. 2, (d) nine-node elements and load case No. 1, (e) and (f) nine-node elements and load case No. 2 . . . . . . . . . . . . . . . . . . . . 147 4.28 Thin membrane subjected to transverse loading, load case No. 1 Exact and estimated distributions of the element error in energy norm on an unstructured mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.29 Stretched plate with circular hole . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.30 Stretched plate with circular hole - Local error distributions for four-node elements on mesh No. 1: FEM solution, REP, RCP1n and RCP1e (contour interval = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.31 Stretched plate with circular hole - Local error distributions for four-node elements on mesh No. 1: RCP2n and RCP2e (contour interval = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.32 Stretched plate with circular hole - Distributions of element error in energy norm on mesh No. 1 of four-node elements (the relative percentage error is reported above each plot) . . . . . . . . . . . . . . . . . . . 152 4.33 Stretched plate with circular hole - Rate of convergence: (a) local error in σ x at point r = 1.5, θ = π/4 for four-node elements, (b) local error in σ y at point r = 1.5, θ = π/4 for four-node elements, (c) local error in τ xy at point r = 1.5, θ = π/4 for four-node elements, (d) relative percentage error for four-node elements, (e) relative percentage error for nine-node elements . . . . . . . . . . . . . . . . . . . . . . . 153 4.34 Stretched plate with circular hole - Convergence of the estimated relative percentage error: (a) four-node elements, (b) nine-node elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.1 Simplified diagram of the iterative mesh design cycle: (1) self-adaptive tecnique; (1)+(2) self-adaptive tecnique with supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.2 Various procedures by h-refinement: (a) mesh enrichment by transition elements; (b) mesh enrichment; (c) complete mesh regeneration; (d) repositioning node . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.3 Examples of refinement without changing element type . . . . . . . . . . . . 162 5.4 Quadtree balancing: (a) split of neighbors; (b) initial element split with new nodes and daughter-elements . . . . . . . . . . . . . . . . . . . . . . . 163 5.5 Quadtree connection: refinement by subsequent subdivision 280 7.0- List of Figures level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.6 Exact solution: (a) contours of σ x , (b) contours of σ y . . . . . . . . . . . . . 168 5.7 Adaptive solutions for: (a) and (b) linear elements with target error set to 10 %, (c) and (d) quadratic elements with target error set to 1 % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.8 Distributions of element error in energy norm for linear triangular elements: (a) and (c) exact error, (b) and (d) error estimate . . . . . . . . 170 5.9 Distributions of element error in energy norm for quadratic triangular elements: (a) and (c) exact error, (b) and (d) error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.10 Distributions of element error in energy norm for linear quadrilateral elements: (a) and (c) exact error, (b) and (d) error estimate . . . . . . . . 171 5.11 Distributions of element error in energy norm for quadratic quadrilateral elements: (a) and (c) exact error, (b) and (d) error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.12 Convergence of adaptive refinement for triangular elements . . . . . . . . . 172 5.13 Convergence of adaptive refinement for quadrilateral elements . . . . . . . 172 5.14 Exact solution: (a) contours of σ x , (b) contours of σ y . . . . . . . . . . . . . 173 5.15 Adaptive solutions for: (a) and (b) linear elements with target error set to 15 %, (c) and (d) quadratic elements with target error set to 5 % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.16 Distributions of element error in energy norm for linear triangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.17 Distributions of element error in energy norm for linear quadrangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.18 Distributions of element error in energy norm for quadratic triangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.19 Distributions of element error in energy norm for quadratic quadrangular elements: (a), (c) and (e) exact error, (b), (d) and (f) error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.20 Short cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.21 Short cantilever beam solved by mesh enrichment: linear quadrilateral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.22 Adaptive refinement of an L-shaped domain in plane stress with prescribed error of 1%: nine-node elements . . . . . . . . . . . . . . . . . . . . 181 5.23 Adaptive solutions for quadratic elements with target error set equal to 2 % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Chapter 7 281 5.24 Convergence for L-shaped domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.25 Machine part domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.26 Adaptive refinement of machine part using linear quadrilateral elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.27 Adaptive refinement of machine part using linear triangular elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.1 Example of node and element patch: (•) assembly node definig the patch, the dark element is the assembly element definig the patch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.2 (a) Test No.1- SS2 square plate under uniform load; (b) test No.2 square plate with prescribed displacement . . . . . . . . . . . . . . . . . . . . . 193 6.3 Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on linear stress approximations . . . . . . . . . . . . . . . . 195 6.4 Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on quadratic stress approximations . . . . . . . . . . . . . 195 6.5 Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on Linear stress approximations . . . . . . . . . . . . . . . 196 6.6 Test No.1 (h/L = 0.2) - convergence of global errors for recovery procedures based on quadratic stress approximations . . . . . . . . . . . . . 197 6.7 Test No.1 (h/L = 0.025) - convergence of global errors for recovery procedures based on linear stress approximations . . . . . . . . . . . . . . . . 197 6.8 Test No.1 (h/L = 0.025) - convergence of global errors for recovery procedures based on quadratic stress approximations . . . . . . . . . . . . . 198 6.9 Test No.1 (h/L = 0.025) - convergence of local error at the plate centre for recovery procedures based on linear stress approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.10 Test No.1 (h/L = 0.025) - convergence of local error at the plate centre for recovery procedures based on quadratic stress approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.11 Test No.1 (h/L = 0.002) - convergence of global errors for recovery procedures based on linear stress approximations . . . . . . . . . . . . . . . . 201 6.12 Test No.1 (h/L = 0.002) Distributions of element error in energy norm on uniform mesh wuth 9βQ4 elements (the relative percentage error is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.13 Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mx on uniform mesh wuth 9βQ4 elements (contour interval =0.1) . . . . . . . . . 202 6.14 Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mxy on uniform mesh wuth 9βQ4 elements (contour interval =0.1) . . . . . . . . . 202 6.15 Test No.1 (h/L = 0.002) Distributions of poinwise errors in Sx on uniform mesh wuth 9βQ4 elements (contour interval =0.1) . . . . . . . . . 203 282 7.0- List of Figures 6.16 Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mx on uniform mesh wuth 9βQ4 elements (contour interval =0.2) . . . . . . . . . 203 6.17 Test No.1 (h/L = 0.002) Distributions of poinwise errors in Mxy on uniform mesh wuth 9βQ4 elements (contour interval =0.2) . . . . . . . . . 204 6.18 Test No.1 (h/L = 0.002) Distributions of poinwise errors in Sx on uniform mesh wuth 9βQ4 elements (contour interval =0.2) . . . . . . . . . 204 6.19 Test No.1 (h/L = 0.002) Distributions of element error in energy norm on uniform mesh wuth 9βQ4 elements (the relative percentage error is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.20 First order Shear Deformation Theory (FSDT) scheme for a laminated composite plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.21 Variations of strains and stresses through layer and laminated thicknesses. (a) Variation of typical in-plane strain. (b) Variation of corresponding stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.22 Simply supported square plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.23 Some examples of regular and distorted meshes . . . . . . . . . . . . . . . . . 210 6.24 Convergence in energy norm (sinusoidal load and diﬀerent staking sequences) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.25 Convergence of stress resultant at point C (stacking sequence 0/90/0, uniform load) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.26 Convergence of stress resultant at point C (stacking sequence 0/90, uniform load) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.27 Convergence in energy norm of the shear stress profiles (sinusoidal load, diﬀerent stacking sequences) . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.28 Convergence of the moments at point D before and after the recovery (stacking sequence 0/90/0, sinusoidal load) . . . . . . . . . . . . . . . . . . . . 214 6.29 Convergenge of the moments at point D before and after the recovery (stacking sequence 0/90, sinusoidal load) . . . . . . . . . . . . . . . 214 6.30 Shear stress profiles at point C after recovery for diﬀerent regular meshes (stacking sequence 0/90/0, sinusoidal load) . . . . . . . . . . . . . . . 215 6.31 Shear stress profiles at point C after recovery for diﬀerent regular meshes (stacking sequence 0/90, sinusoidal load) . . . . . . . . . . . . . . . . 215 6.32 Shear stress profiles at point C after recovery for the mesh 9 × 9, regualr and distorted (stacking sequence 0/90/0, sinusoidal load) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.33 Shear stress profiles at point C after recovery for the mesh 9 × 9, regualr and distorted (stacking sequence 0/90, sinusoidal load) . . . . . . 216 6.34 error density of the shear profiles for diﬀerent meshes before and after recovery (stacking sequence 0/90/0, sinusoidal load) . . . . . . . . . . 217 Chapter 7 283 6.35 error density of the shear profiles for diﬀerent meshes before and after recovery (stacking sequence 0/90, sinusoidal load) . . . . . . . . . . . 217 7.1 The scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.2 The path from Conceptual and Computational Model . . . . . . . . . . . . 227 7.3 Verification and Validation activities and outcomes . . . . . . . . . . . . . . 229 7.4 Cimex Fortranarius, magnification 109 :1 (See [Irons & Ahmadn, 1980]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A.1 Class I problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 A.2 Class II problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 A.3 Class III problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 A.4 A 3D plate-like body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 B.1 Thin membrane subjected to transverse loading - Error distributions on a regular mesh of four-node elements (contour interval = 0.1, with load case No. 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 B.2 Thin membrane subjected to transverse loading - Distributions of element error in energy norm on a regular mesh of four-node elements, load case No. 1 (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 B.3 Thin membrane subjected to transverse loading - Error distributions on a regular mesh of nine-node elements (contour interval = 0.015, load case No. 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 B.4 Thin membrane subjected to transverse loading - Distributions of element error in energy norm on a regular mesh of nine-node elements, load case No. 1 (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 B.5 Thin membrane subjected to transverse loading - Rate of convergence for load case No. 1: (a) local error at point x = 0.5, y = 0.5 (σ x = σ y ) for four-node elements, (b) global error in energy norm for four-node elements, (c) local error at point x = 0.5, y = 0.5 (σ x = σy ) for nine-node elements, (d) global error in energy norm for nine-node elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 B.6 Thin membrane subjected to transverse loading - Rate of convergence for load case No. 2: (a) local error in σ x at point x = 0.25, y = 0.75 for four-node elements, (b) local error in σ y at point x = 0.25, y = 0.75 for four-node elements, (c) global error in energy norm for four-node elements, (d) global error in energy norm for nine-node elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 B.7 Stretched plate with circular hole - Error distributions for four-node elements on mesh No. 1: RCP2 and RCP2b (contour interval = 284 7.0- List of Figures 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 B.8 Stretched plate with circular hole - Error distributions for four-node elements on mesh No. 1: FE solution, REP, RCP1 and RCP1b (contour interval = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 B.9 Stretched plate with circular hole - Error distributions for nine-node elements on mesh No. 1: RCP2 and RCP2b (contour interval = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 B.10 Stretched plate with circular hole - Error distributions for nine-node elements on mesh No. 1: FE solution, REP, RCP1 and RCP1b (contour interval = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 B.11 Stretched plate with circular hole - Distributions of element error in energy norm on mesh No. 1: four-node elements (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . 257 B.12 Stretched plate with circular hole - Distributions of element error in energy norm on mesh No. 1: nine-node elements (minimum and maximum values are reported below each plot, the global error in energy norm is reported above each plot) . . . . . . . . . . . . . . . . . . . . . . 258 B.13 Stretched plate with circular hole - 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Finite Elements in Analysis and Design, 19, 11—23. Zienkiewicz, O.C., Zhu, J.Z., & Gong, N.G. 1989. Eﬀective and Pratical h−p-Version Adaptive Analysis Procedures for the Finite Element Method. Int. J. Numer. Methods Engrg., 28, 879—891. Zienkiewicz, O.C., Zhu, J.Z., & Wu, J. 1993. Superconvergent Patch Recovery Techniques - some Further Tests. Commun. Numer. Meth. Engng., 9, 251—258. Curriculum Giovanni Castellazzi received the Laurea degree in Civil Engineering from the University of Bologna, Italy, in summer 2001. After receiving his Laurea degree, he spent time as Support Tech Specialist at Laboratory of Computational Mechanics (DISTART) Universityy of Bologna. From January 2004 to January 2007 he attended at Ph.D. program in Structural Mechanics - University of Bologna. This thesis is available in PDF format on LAMC web site: www.lamc.ing.unibo.it Contacts Laboratory of Computational Mechanics DISTART - Structural Department Faculty of Engineering ALMA MATER STUDIORUM - University of Bologna Viale del Risorgimento 2, 40136 Bologna - Italy Phone: +39 051 2093503 Fax: +39 051 2093496 E-mail: [email protected] Web: www.lamc.ing.unibo.it

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