Department of Precision and Microsystems Engineering Topology Optimisation Including Buckling Analysis S.J. van den Boom Report no Coach Professor Specialisation Type of report Date : : : : : : EM 2014.034 Dr. ir. M. Langelaar Prof. dr. ir. A. van Keulen Engineering Mechanics Graduation Thesis 5 December 2014 T OPOLOGY O PTIMISATION I NCLUDING B UCKLING A NALYSIS by S.J. van den Boom in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the Delft University of Technology, to be defended publicly on December 19t h at 13:00 Student number: Project duration: Supervisor: Thesis committee: 1354868 January 6, 2014 – December 19, 2014 Dr. ir. M. Langelaar Prof. dr. ir. A. van Keulen, TU Delft Dr. ir. P. Tiso TU Delft Dr. ir. G. J. van der Veen TU Delft An electronic version of this thesis is available at http://repository.tudelft.nl/. A BSTRACT Buckling is a failure mode of a structure caused by stiffness loss of compressed material. It arises primarily in slender structures, for which the bending stiffness is much lower than the axial stiffness. Slender, buckling sensitive structures occur especially in optimised designs, where an excellent strength-to-weight ratio is required. For this reason, buckling analysis of optimised designs is very important. However, buckling analysis is nowadays only performed in the post-processing phase, after the optimisation is completed. Inclusion of a buckling constraint in topology optimisation should lead to a design where failure by buckling is already excluded. In the following post-processing step, no major changes are needed on account of a buckling requirement, therefore allowing the final design to remain close to the optimal design. Ultimately this should lead to improved results. In literature, linear buckling analysis is included in topology optimisation on a couple of instances, albeit mostly in the role of an objective instead of as a constraint. In this report, an adjoint formulation for the sensitivities of the buckling load is found, resulting in much more efficient computation than other methods, such as finite differences. Furthermore, different practical aspects of inclusion of a buckling constraint are explored, with emphasis on the underlying physical problem. It is found that including a buckling constraint requires careful implementation, tailored to the specific optimisation problem at hand. An educated choice should be made on the admissibility of negative buckling loads. Allowing negative buckling loads leads to a non-convex design space, complicating the search for the globally optimal design. Furthermore, the switching of modes should be considered. While mode switching can introduce a number of issues, preventing this switching limits the design freedom the optimiser has to reach an optimal design. Even more importantly, the point is raised that a linear buckling analysis does not give any information on the post-buckling behaviour of the structure. The stability of the buckling load greatly influences the sensitivity of the structure to imperfections. For practical implementations, an optimal design that is extremely sensitive to imperfections is worthless. Therefore, ideally, an assessment is done on the stability of the structure, during optimisation, in order to enforce stable post buckling behaviour. However, current techniques for post-buckling analysis are elaborate and time-consuming in implementation and use. A method for rapid determination of the stability is required. Such a method for rapid estimation of the buckling load is found in the use of linear buckling analysis for structures that are perturbed with the mode shape of the perfect structure. This method is tested on very simple test structures and is found to be very promising for implementation in topology optimisation, because for a large part it can re-use code that is already available in the original formulation. iii S AMENVATTING Knik is een faalmechanisme dat wordt veroorzaakt door verlies van stijfheid in materialen die op druk worden belast. Het komt voornamelijk voor in slanke construties, waarbij de buigstijfheid veel kleiner is dan de axiale stijfheid. In geoptimaliseerde constructies, waar een optimale ratio tussen sterkte en gewicht gewenst is, komen slanke, knikgevoelige, constructies vaak voor. Het is daarom gebruikelijk om ontwerpen uit topologie-optimalisatie na te bewerken zodat ze voldoen aan de aanvullende eisen, zoals een eis aan de kniklast. Doormiddel van een knikconstraint in topologie-optimalisatie kan een optimaal ontwerp worden gevonden, waarbij al rekening is gehouden met de eisen aan de kniklast. In de nabewerking van het ontwerp hoeven dan geen grote veranderingen te worden doorgevoerd. Hierdoor ligt het uiteindelijke ontwerp dichter bij het optimum. In de literatuur zijn er een aantal papers te vinden waarin knikanalyse is toegevoegd aan topologie optimalisatie. In deze paper is de kniklast meestal als objective gebruikt. In dit rapport is ten eerste een adjoint formulering voor de gevoeligheden van de kniklast gevonden. Deze adjoint formulering leidt tot een efficiëntere manier van berekenen dan andere methodes. Verder zijn er verschillende praktische aspecten van het invoegen van een knikconstraint onderzocht, met nadruk op het onderliggende probleem Om een waardevol resultaat uit de optimalisatie te krijgen is het belangrijk een goede formulering voor het probleem te hebben. Wat precies een goede formulering is hangt af van het specifieke ontwerpprobleem. Een weloverwogen keuze moet worden gemaakt wat betreft de toelaatbaarheid van negatieve knikwaarden. Het toelaten van negatieve knikwaarden lijdt tot een niet-convexe design-space, wat het vinden van een globaal optimum bemoeilijkt. Ook moet er rekening worden gehouden met het wisselen van knikmodes. Dit wisselen van de modes brengt een aantal moeilijkheden met zich mee, maar wanneer dit wisselen wordt voorkomen wordt ook de vrijheid van het optimalisatie algorithme beperkt. Een ander belangrijk aspect is het feit dat lineaire knikanalyse geen enkele indicatie geeft voor het naknikgedrag van de structuur. De gevoeligheid van de structuur voor imperfecties hangt in grote mate af van de stabiliteit behorende bij de kniklast. Bij praktische toepassingen kan de aanwezigheid van imperfecties niet voorkomen worden, dus een structuur met een hoge imperfectiegevoeligheid is niet bruikbaar. Het is daarom belangrijk om tijdens optimalisatie niet alleen een constraint op de kniklast te hebben, maar ook op de stabiliteit hiervan. Voor deze toepassing is een zeer efficiënte methode gewenst om deze stabiliteit te bepalen. Het is gebleken dat de stabiliteit van een structuur efficiënt kan worden afgeschat doormiddel van lineaire knikanalyse van structuren met een kleine imperfectie uit te voeren. Deze methode is getest op zeer eenvoudige problemen, en is veelbelovend gebleken voor toepassing in topologie optimalisatie. v P REFACE AND A CKNOWLEDGEMENTS This report is the final result of my Master’s thesis. This Master’s thesis was completed in partial fulfillment of my Master’s studies at the department for Precision and Microsystems Engineering at the faculty of Mechanical Engineering at Delft University of Technology. I am very glad that my studies at the university finally led me to this department, after having started with a Bachelor’s degree in Industrial Design Engineering. It is certainly in this department that I have followed some of the most interesting courses. I am fortunate to have been able to combine subjects from my two favourite Master’s courses, "Engineering Optimisation: Concept & Application" and "Stability of Thin-Walled Structures", in this graduation project. I would like to thank my supervisors, Matthijs Langelaar and Fred van Keulen for the freedom they have given me during this project, for their constructive criticism and their sharp questioning. Furthermore, I would like to thank my family, friends and boyfriend for keeping me sane during this project. Their support during these last months has meant a lot to me. Finally, I would like to thank you, the reader, for your interest in my thesis. S.J. van den Boom Delft, December 2014 vii C ONTENTS Abstract iii Samenvatting v Preface and Acknowledgements vii List of Figures xi List of Tables xiii Nomenclature xv 1 Introduction 1 2 Literature review 2.1 Topology optimisation . . . . . . . . . . . . . . . . . 2.2 Buckling and optimisation . . . . . . . . . . . . . . . 2.2.1 Buckling in density based topology optimisation 2.3 Post-buckling behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Linear Buckling Analysis in Topology Optimisation 5 5 5 6 7 9 3 Linear buckling theory 11 3.1 Notation conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Eigenvalue buckling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Optimisation problem 15 4.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Method of Moving Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 Finite Element Model 5.1 Static deformations and stresses . . . . 5.2 Linear buckling analysis . . . . . . . . 5.3 Sensitivities . . . . . . . . . . . . . . . 5.3.1 Sensitivities of the buckling load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17 18 20 20 6 Numerical examples 6.1 Test case . . . . . . . . . . 6.2 Various buckling constraints 6.3 Varying number of elements 6.4 Various filter sizes . . . . . . 6.5 Various domain shapes . . . 6.6 Other test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23 24 25 26 27 28 7 Discussion 7.1 Negative buckling loads . . . . . . . . . . . . . . . . 7.2 Mode switching and the multiplicity of buckling loads 7.3 Buckling of void elements . . . . . . . . . . . . . . . 7.4 Stability and imperfection sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 32 33 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II Post-buckling Behaviour and Imperfect Systems 35 8 Stability assessment 37 8.1 Nonlinear buckling analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.2 Koiter initial post-buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.3 Imperfect structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ix x 9 Linear buckling analysis of imperfect structures 9.1 Linear pre-buckling assumptions . . . . . . . . . 9.2 Estimate post-buckling using imperfect structures 9.3 A simple structure. . . . . . . . . . . . . . . . . 9.4 Koiter-Roorda frame . . . . . . . . . . . . . . . C ONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 39 39 40 44 10 Outlook for application in topology optimisation 47 10.1 Stability constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 10.2 Other considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 11 Conclusions and recommendations 49 11.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 11.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Bibliography 51 A Detailed results 53 L IST OF F IGURES 1.1 An illustration of possible post-processing steps from topology optimisation to final design. Source: [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Simple example of the expected advantage of including buckling constraints in topology optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 2.1 Example from literature of topology optimisation with buckling constraints Source:[2] . . . . . . 2.2 Example from literature of buckling analysis in topology optimisation Source:[3] . . . . . . . . . 6 6 4.1 Visual representation of the linear density filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.1 Location of Gauss points in the elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Comparison of the eigenbuckling formulation with Euler buckling . . . . . . . . . . . . . . . . . . 5.3 Computation time comparison of the sensitivity calculation methods . . . . . . . . . . . . . . . . 18 19 21 6.1 6.2 6.3 6.4 23 24 25 6.5 6.6 6.7 6.8 6.9 Domain and load case for the first optimisation tests . . . . . . . . . . . . . . . . . . . . . . . . . . Topology optimised designs for various values of the buckling constraint . . . . . . . . . . . . . . Topology optimised designs for a varying number of elements . . . . . . . . . . . . . . . . . . . . Topology optimised designs for various filter radii, for all cases the buckling load is constrained to 150 % of the unconstrained and unfiltered buckling load . . . . . . . . . . . . . . . . . . . . . . Topology optimised designs for various domain sizes . . . . . . . . . . . . . . . . . . . . . . . . . . Topology optimised designs for Test Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The objective value and first five positive buckling loads of Case A.19 during optimisation . . . . Topology optimised designs for Test Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First positive buckling load for Case A.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 27 28 29 30 30 7.1 7.2 7.3 7.4 7.5 Optimisations for different options for negative buckling loads . . . . . . . . . . . . . . . . . . . . First five buckling loads during optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Penalised Young’s modulus and buckling load for a single element for different values of q, p =3 First buckling of a structure with different values for q . . . . . . . . . . . . . . . . . . . . . . . . . Post buckling behaviour and imperfection sensitivity. Source: [4] . . . . . . . . . . . . . . . . . . . 31 32 33 34 34 9.1 9.2 9.3 9.4 9.5 Simple test case for linear buckling of imperfect structures Figure adapted from [4] Simple structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear buckling analysis of imperfect structures with different spring properties . . Finite element test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear buckling analysis of imperfect Koiter- Roorda frames . . . . . . . . . . . . . . A.1 Detailed results of test case nr. 1 . A.2 Detailed results of test case nr. 2 . A.3 Detailed results of test case nr. 3 . A.4 Detailed results of test case nr. 4 . A.5 Detailed results of test case nr. 5 . A.6 Detailed results of test case nr. 6 . A.7 Detailed results of test case nr. 7 . A.8 Detailed results of test case nr. 8 . A.9 Detailed results of test case nr. 9 . A.10 Detailed results of test case nr. 10 A.11 Detailed results of test case nr. 11 A.12 Detailed results of test case nr. 12 A.13 Detailed results of test case nr. 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 41 43 44 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 56 57 58 59 60 61 62 63 64 65 66 67 xii L IST OF F IGURES A.14 Detailed results of test case nr. 14 A.15 Detailed results of test case nr. 15 A.16 Detailed results of test case nr. 16 A.17 Detailed results of test case nr. 17 A.18 Detailed results of test case nr. 18 A.19 Detailed results of test case nr. 19 A.20 Detailed results of test case nr. 20 A.21 Detailed results of test case nr. 21 A.22 Detailed results of test case nr. 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 69 70 71 72 73 74 75 76 L IST OF TABLES 5.1 Comparison of the eigenbuckling formulation with Euler buckling . . . . . . . . . . . . . . . . . . 19 6.1 6.2 6.3 6.4 6.5 6.6 6.7 . . . . . . . 23 24 25 26 27 28 29 9.1 Imperfections used in Figure 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A.1 Parameters and results of the test case with a distributed load on the top edge . . . . . . . . . . . A.2 Parameters and results of the test case clamped on the upper and lower edge and a downward force inside the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Parameters and results of the test case clamped at the lower edge with a force to the right in the upper right corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Parameters for optimisation of the numerical test case . . . . Results for optimisations with various buckling constraints . Results for optimisations with a varying number of elements Results for optimisations with various filter sizes . . . . . . . Results for optimisations with various domain shapes . . . . Results for optimisations with Test Case 2 of Figure 6.6a . . . Results for optimisations with Test Case 3 of Figure 6.8a . . . xiii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 54 N OMENCLATURE µ First adjoint variable σ Stress vector λcr Critical loading factor for buckling λ∗cr Buckling constraint ν Poisson ratio ρe Element density, design variable for the optimisation problem ρ max Maximum value of the density ρ ρ mi n Minimum value of the density ρ B Element matrix containing the derivatives of the shape functions d Deformation vector E Constitutive matrix f Force vector G Element matrix containing the derivatives of the shape functions K Structural stiffness matrix K(e) Element stiffness matrix Kσ Structural geometric stiffness matrix K(e) σ Element geometric stiffness matrix L Localisation matrix S Element matrix containing stresses v Buckling mode w Second adjoint variable C Compliance of the structure E Youngs modulus Ni Shape functions of the element p Penalisation factor q Penalisation factor for calculation of stress R Filter radius V Occupied volume V∗ Volume constraint xv 1 I NTRODUCTION Throughout history engineers have been striving to achieve designs that are in some sense optimal, though what exactly may be defined as ’optimal’ varies. An optimal design can, for example, mean: the cheapest construction that still works; the stiffest structure that can be made with a certain quantity of material. The precise trade-off at the basis of the optimisations differs, but often it is essentially functionality versus costs of some sort. Especially in, for example, aircraft or formula one cars, where both perfect performance and minimal weight is of great importance, optimisation is a valuable aspect of the design process. The optimisation of designs has been done in a variety of ways: heuristic methods and experience or mathematical optimisation schemes. Recent developments in additive manufacturing, such as 3D-printing, enable the production of a much wider range of forms than was possible in the past. Simultaneous to the introduction of additive manufacturing techniques, an optimisation method has been developed that offers a similar degree of design freedom. This optimisation method is known as topology optimisation. Topology optimisation was first introduced by Michell [5] in 1904, but it was not until finite element analysis (FEA) came along in the 1980s that is was further developed. Since that time, different topology optimisation methods have been proposed, some of which will be briefly discussed in the literature review. Nowadays the Solid Isotropic Material with Penalisation (SIMP) method by Bendsoe [6] is the one used most often. Even though the concept of topology optimisation has been around for a few decades, it is still far from perfect and not yet widely used in commercial applications. While commercial use of topology optimisation is slowly emerging, a vast volume of ongoing research is being conducted to develop the method even further. Topology optimisation is mainly used as a design tool in the concept phase of the development of a product. Typically the stiffness is maximized while the amount of material is constrained. The optimised design is then used as a concept design for further development. Engineers and possibly also designers evaluate the optimal design and alter it to satisfy all the additional requirements concerning performance, manufacturability and looks. It is not until this post-processing phase that other constraints on the structure are included. Examples of additional constraints are the yield strength of the material, a minimum buckling load or a minimum eigenfrequency. These additional constraints can substantially change the concept, thus resulting in a very non-optimal final design. An example of post-processing a topology-optimised design is shown in Figure 1.1. In this case, the incorporation of these additional constraints is done by a separate shape optimisation step. While some post-processing will still be necessary for topology-optimised designs where these constraints were incorporated in the topology optimisation itself, the concept should be closer to the actual optimal design. One of the previously mentioned additional constraints is a constraint on the buckling load. Because of the nonlinear relationship between strain and deformation, the effective stiffness of a structure depends on the stress in the material. Material that is loaded in tension gains geometric stiffness, whilst material that is loaded in compression loses geometric stiffness. Due to such loss of stiffness, the effective stiffness in a certain direction can become zero for a certain compression load. In that case, the structure is free to 1 2 1. I NTRODUCTION Figure 1.1: An illustration of possible post-processing steps from topology optimisation to final design. Source: [1] deform in that direction without any change in the strain energy. Therefore, the stability of the structure in that particular direction is lost. This effect is called buckling, the load at which the effect occurs is known as the buckling load and the corresponding direction is the buckling mode. Buckling is a failure mode that occurs before the internal stresses reach the material limit. Buckling is often an elastic phenomenon, which means that the structure returns to its original shape once the load has been removed. Buckling is especially prominent in structures that are much stiffer in the axial direction than in bending, such as slender beams, plates and shells. Euler was the first scientist to derive an analytical equation for the buckling of columns. In Figure 1.2 the advantage of including buckling analysis in topology optimisation is illustrated. If optimisation without buckling analysis is applied to the design domain of Figure 1.2a, it is likely a structure as in Figure 1.2b is found. After post-processing this structure to meet the requirement on the buckling load, the structure will be that of Figure 1.2c. The structure of Figure 1.2d, where the buckling constraint is taken into account during optimisation, is more optimal. Eigenvalue buckling analysis has been implemented in a topology optimisation setting before. The chapter ‘literature review’ will present some of those papers and discuss them. In literature, this problem is solved mainly as a numerical exercise, with little to no regards to the physical problem at the basis of the optimisation. In this project, the physical meaning of inclusion of eigenvalue buckling analysis is investigated. One major drawback of eigenvalue buckling analysis is that no information is obtained on the post-buckling behaviour of the structure. It is important to have information on post-buckling behaviour to evaluate the imperfection sensitivity of the solution found. Depending on the post-buckling behaviour, the structure might be highly sensitive to imperfections in the material or the force direction. Furthermore, a cluster of close buckling loads can lead to interaction between the corresponding modes, which then causes an even higher degree of sensitivity to imperfections. As “A given form will be optimum if all failure modes which can possibly intersect occur simultaneously under the action of the load environment” [7] it is highly likely that a cluster of close buckling loads will arise during optimisation. It is for this reason that Thompson and Supple [8] 3 (a) Design domain (b) After optimisation (c) After post-processing (d) Desired design Figure 1.2: Simple example of the expected advantage of including buckling constraints in topology optimisation conclude that: “Any optimisation scheme which is to be applied to a structural system displaying multi-mode buckling must be so defined as to take into account the complete post-buckling characteristics including the effects of unavoidable manufacturing imperfections.” In this project, topology optimisation is enriched with eigenvalue buckling analysis. Different aspects that impede efficient implementation are explored. The physical meaning of eigenvalue buckling analysis of imperfect structures is investigated. This information could eventually be used as a indication for initial postbuckling behaviour, to ensure imperfection insensitivity to imperfections. To reduce computation time, only 2D domains will be used for optimisation. However, the formulation could be extended to 3D problems. Relatively simple optimisation problems will be used for numerical examples. The literature review of Chapter 2 will give an overview of the available literature on topology optimisation with buckling analysis. Papers on topology optimisation with other types of eigenvalue analysis will also be discussed, because of the similarity to the buckling problem. Furthermore, some literature on initial postbuckling behaviour is also briefly reviewed. After the literature review, the report is divided into two parts. The first part is on the implementation of eigenvalue buckling analysis in topology optimisation. The first chapter of this part, Chapter 3, is on buckling theory, there the theory of eigenvalue buckling analysis and initial post-buckling analysis is introduced and explained in detail. Chapter 4, entitled ’Optimisation problem’, will give an overview of the optimisation problem, with its design variables, objective functions, constraints, and filter. Chapter 5 is on the Finite Element model, and discusses the implementation of buckling analysis and the calculation of the sensitivities thereof. In Chapter 6, numerical examples of the optimisation are presented. In the discussion of Chapter 7, different aspects of the optimisation problem are described. The next part is on post-buckling and imperfect systems. In Chapter 8 the importance of such analysis is explained. Chapter 8 deals with linear buckling analysis of imperfect structures. An outlook for application in topology optimisation is given in Chapter 10. Lastly, conclusions are drawn and recommendations are given in Chapter 11. 2 L ITERATURE REVIEW 2.1. T OPOLOGY OPTIMISATION To give a complete description of the history of topology optimisation would be time-consuming and far beyond the scope of this project. Nevertheless, some works need to be mentioned here for their historical value or their relevance to the rest of this report. Topology optimisation is the name for any topology where no predefined topology is assumed. This has been done with ground structures of trusses, variable sheet thickness and density formulations. In all cases the truss size, sheet thickness or density should be allowed to approach zero and vanish from the structure. Nowadays the Solid Isotropic Material with Penalisation (SIMP) method, by Bendsoe and Sigmund [9], is used most often. In this method the Young’s modulus of the material is interpolated and penalised, and the densities of the elements are allowed to take any value between zero and one. Because of the penalisation, intermediate densities are unfavourable. Therefore a black and white design will be achieved. To use topology in practical applications a vast amount of research is being conducted. A big research area to enable the transition to practical application, is the area of filters. Filters in topology optimisation in general have two functions: to prevent checker-boarding and to prevent mesh dependency of the solution. Checker-boarding an effect that occurs during topology optimisation, where a checker-board pattern of densities has an artificially high stiffness to weight ratio. Because this artificially high stiffness has no physical meaning, this effect needs to be suppressed in the optimisation process. The other reason for a filter is the mesh dependency of the results. Because stiffness optimisation problems generally have no unique solution for an infinitely fine mesh, the optimiser will tend to make smaller structural features when the mesh is refined. Ideally, one would want to obtain a high resolution design by refining the mesh, while still retaining control over the minimal feature size. Another field of research for topology optimisation is the expansion of the method to other fields of physic and to multi-physics problems. The inclusion of additional constraints such as stress constraints and constraints from manufacturing are also researched. 2.2. B UCKLING AND OPTIMISATION On the subject of buckling in combination with optimisation some important papers and letters have been written by Thompson. The article that will be discussed is Thompson and Supple [8]. In this article, the writers argue that it is crucial to base the optimisation not only on the critical load, but also on the post buckling behaviour of the structure. It is argued that clusters of close buckling loads can lead to a nonlinear coupling between the corresponding modes. This coupling can result in instable equilibrium paths, even if the stability of each individual bifurcation point is stable. The paper illustrates this effect with a very simple example: a rigid bar that is connected with two rotational springs. The first buckling load is maximized by distributing a fixed amount of stiffness over the two springs. The optimal design is then, as is also intuitively 5 6 2. L ITERATURE REVIEW Figure 2.1: Example from literature of topology optimisation with buckling constraints Source:[2] Figure 2.2: Example from literature of buckling analysis in topology optimisation Source:[3] expected, to distribute this stiffness equally over the two springs. It is shown that this seemingly optimal solution is very sensitive to imperfections, and is thus in practice very non-optimal. Truss topology optimisation is at first sight extremely suitable for buckling constraints. All individual trusses are well defined and could be constrained on buckling through a simple Euler buckling constraint. However, Zhou [10] showed in his paper that using only local buckling constraints can easily lead to instable solutions. Removal of unnecessary nodes can solve the problem of instable solution, but with this formulation non optimal solutions are found. In an other paper, this truss topology optimisation with local buckling constraints is expanded to a truss topology optimisation with both local and global buckling constraints. This formulation however has a new set of challenges. ([11], [12], [13] ,[14]) One of these challenges is the occurrence of singular topologies, as described in Zhou [3].This means that different parts of the design domain are not well connected.This phenomenon is especially known in topology optimisation with local stress constraints, but also occurs for local buckling constraints. ([15]) 2.2.1. B UCKLING IN DENSITY BASED TOPOLOGY OPTIMISATION In the review paper of Zhou [3] also another problem of topology optimisation with buckling constraints is addressed: the problem of buckling in void areas. Because the void elements have a non-zero density they contribute to the geometric stiffness matrix. The first buckling modes will then arise in the void elements, which has no physical meaning. The occurrence of modes also is a problem for dynamic topology optimi- 2.3. P OST- BUCKLING BEHAVIOUR 7 sation problems with eigenfrequency analysis. Therefore, the literature review includes papers on dynamic topology optimisation as well. ([16]) There are three solution methods for the void buckling problem. The most intuitive method is to artificially set all stresses in element with a density below a certain threshold value to zero. This method is proposed by Neves et al. [17]. While this method suppresses the void buckling modes, it is not very suitable for the gradient based optimisation methods used in topology optimisation. The geometric stiffness matrix is in this formulation a discontinuous, non differentiable function of the element densities. A somewhat more refined method is to use a different penalisation on the physical stiffness matrix and the geometric stiffness matrix. This has the same effect: low density elements do not contribute to the geometric stiffness matrix. This method was first proposed by Bendsoe and Sigmund [9]. The main advantage of this method over the previous method is the fact that the geometric stiffness matrix is now a continuous, differentiable function of the element densities. This makes this method better suited for gradient based optimisation methods. A last approach to avoid buckling in void area is to completely remove the void elements. This gives the best resemblance with the physical situation. The removed elements should be able to ’grow back’. Therefore, not all void elements are removed, but a border of elements surrounding the structure is kept. If the buckling load is used as an objective, another problem arises. This problem is the switching of modes, and was first investigated by Ma et al. [18] in a eigenfrequency context. Switching of modes occurs if during optimisation, there is a switch in the mode with the lowest critical value. The resulting objective value is then a non-smooth function of the design variables. A solution for this problem is to include multiple buckling loads in the objective, for example using mean eigenvalues. 2.3. P OST- BUCKLING BEHAVIOUR Depending on the post-buckling behaviour of a structure, the structure might be highly sensitive to imperfections in the material or the force direction. Furthermore, a cluster of close buckling loads can lead to interaction between the corresponding modes, which causes an even higher degree of sensitivity to imperfections. These clusters arise especially when a structure is optimised. For these reasons, Thompson and Supple [8] advised to include the full post-buckling analysis into the optimisation. Path-following techniques can be used to describe the post-buckling behaviour, but they are very computationally expensive. Another disadvantage of path-following techniques is that they only describe one of the possible equilibrium paths at a time. Koiter’s asymptotic method provides an efficient way to describe the total post-buckling behaviour of the perfect structure. The influence of imperfections can be evaluated afterwards. Much has been written on Koiter post-buckling, most notably the paper by Budianski [4]. More recently, papers have been published on implementing Koiter’s analysis in a finite element setting. I L INEAR B UCKLING A NALYSIS IN T OPOLOGY O PTIMISATION 9 3 L INEAR BUCKLING THEORY 3.1. N OTATION CONVENTIONS In this chapter the formulation of eigenvalue buckling and initial post-buckling analysis will be derived. The derivation will introduce all assumptions that are made in this type of formulation. This will give a deeper understanding of the type of analysis that is conducted, and the nature of the problem for which this kind of analysis holds. The theory will be described in a very general finite element setting. As the method will be implemented in a finite element topology optimisation setting, an even more general continuum formulation is not necessary. Before we start with the derivation, it is important to establish some notation conventions. In this report, when indices are used, the Einstein summation convention will be used. In this notation, summation is implied over the repeating indices, to achieve notational conciseness. For the same reason of brevity, comma derivative notation will be used throughout this report. A superscript (e) will be used to denote element wise operations. Vectors will be displayed as bold face lower case letters, while matrices will be shown as upper case letters. These notation conventions are shown in Equation 3.1 and in Equation 3.2. 3 X ci x i = ci x i (3.1) i =1 . . a = . and A = . . . . . . . . . (3.2) 3.2. E IGENVALUE BUCKLING ANALYSIS First of all, the strains as a function of the nodal displacements need to be calculated. For this, the displacement as a function of the local coordinates need to be interpolated from the nodal displacements through the matrix N, which contains the shape functions, see Equation 3.3. For the relationship between the displacements and the strains a non-linear large-deformation strain definition is needed, such as the Green-Lagrange strain tensor displayed in Equation 3.4. The strains are now a non-linear function of the displacements, as opposed to the linear formulation that is often used in mechanics. u(ξ, η) = Nu (3.3) 1 ²i j = (u i , j + u j ,i + u k,i u k, j ) 2 (3.4) Now we are interested in how small variations in the displacements result in variations in the strains as in Equation 3.5. This results in a linear operation with the matrix B, shown in Equation 3.6, which is itself a function of the displacements d. This is different from the linear situation, where the matrix B would be a 11 12 3. L INEAR BUCKLING THEORY constant matrix. This relationship between variations in displacements and variations in strains imposes the first condition on the finite element solution: the condition of continuity. d² = Bdd (3.5) d²i dd j (3.6) Bi j = The principle of virtual work can now be applied in order to eventually arrive at the static equilibrium equation. The principle of virtual work states that a small, virtual, variation in external work should result in an equal variation in internal work. The external virtual work is defined as the external force f times a virtual displacement δd, as defined in Equation 3.5. The internal virtual work is defined as in Equation 3.7. The internal stresses are multiplied by the variations in strain, this is then integrated over the total volume. For finite elements this integration is done numerically on a certain amount of integration points. This results in generalized stresses and generalized deformations. dWi = σB(d)dd (3.7) Setting the internal virtual work equal to the external virtual work and inserting the previously derived equation for the small variations in strain gives δd in both sides of the equation. Dividing both sides of the equation by δd results in the second condition on the finite element solution: the constraint of equilibrium in Equation 3.8. In this report the force vector f is assumed to be independent of the displacements d. BT σ = f (3.8) The stress tensor σ can also be related to the strains through a material dependent E matrix, as in Equation 3.9. This matrix itself can be a function of the deformations ² for nonlinear material behaviour. In this report, however, we assume the material to be linear. This relation imposes the third (and last) condition on the finite element model: the condition of that introduces material parameters to the formulation. The final equilibrium equation thus looks like Equation 3.10. The matrix B in this equation depends linearly on the displacements d, the strain tensor ² depends quadratically on the d. The matrix E and the force vector f both are constants. σ = E²(d) (3.9) BT (d)E²(d) = f (3.10) Because we are dealing with a nonlinear function, we will look at the rate equations to obtain a local value for the stiffness. The problem is modelled in a quasi-static way, by setting the changes in time of the left hand side equal to the changes in time of the right hand side. This results in Equation 3.11. Ã ! d d X B pi E pq ²q = f i dt p.q dt (3.11) Expanding Equation 3.11 as in Equation 3.12 results in a tangent operator J that consists of two terms. The first part of the tangent operator consists of the material constitutive matrix E, pre-multiplied and postmultiplied with matrix B(d). This term is called the physical stiffness matrix, denoted K. The second term corresponds to the nonlinear part of the equation and contains the second derivatives of the strain tensor and the stresses. This term of the tangent operator is sometimes called the geometric stiffness matrix, denoted Kσ . (B pi E pq B q j + ²p,i j σp )u̇ j = f˙i (3.12) 3.2. E IGENVALUE BUCKLING ANALYSIS 13 This tangent operator describes the stiffness of the structure at a certain value of the displacements d. If there are no initial stresses, the tangent operator at d = 0 is the same as the physical stiffness matrix K in a linear formulation. Both K and Kσ are symmetric, sparse, and assembled in the same way. Kσ does not need to be positive definite. The load is now defined as a scalar load factor λ times a constant load vector f0 (Equation 3.13). The load factor at which the tangent operator becomes singular is called the buckling load λc r . At this stability point there will either be a bifurcation or a limit point of the equilibrium path. f = λf0 (3.13) If we now assume the pre-buckling solution to be linear, the physical stiffness matrix can be calculated at d = 0. This assumption of linearity holds for most buckling problems, yet not for all of them so caution is advised when using this formulation. We also assume the second derivatives of the strains to be constant, which means the geometric stiffness matrix will scale linearly with the buckling load λ. The stress in this geometric stiffness matrix will be calculated at the force f0 , in accordance with the assumed linear pre-buckling solution. With these assumptions we finally arrive at the general eigenvalue formulation for buckling analysis. This formulation is shown in Equation 3.14. The eigenvalues λcr stand for the buckling loads, and the eigenvectors v are the corresponding buckling mode. (K + λKσ )v = 0 (3.14) 4 O PTIMISATION PROBLEM 4.1. P ROBLEM FORMULATION Buckling analysis can be implemented in topology optimisation in one of two ways: either as an objective or as a constraint. When implemented as an objective, the optimiser strives for the highest possible buckling load, while constrained with a requirement on the stiffness and the occupied volume. If on the other hand the buckling load is implemented as a constraint, the stiffness is optimised, while adhering to the requirement on the buckling load and a volume constraint. As the same information is needed from the finite element model for both options, it is relatively straightforward to switch between the two. In this report the second option is used, unless otherwise specified. In most practical cases, this formulation corresponds best with how the design problem is defined. In this formulation the lowest buckling load can be controlled by a constraint: an expected worst-case loading scenario with a certain safety margin. Equation 4.1 shows the optimisation problem. min ρ s.t. C = dT Kd V∗− PN λ∗ cr λcr e=1 ρ e (4.1) ≤0 −1 ≤ 0 ρ mi n ≤ ρ ≤ ρ max The objective in this optimisation problem is to minimize the compliance C of the structure , in other words, to maximize the stiffness of the structure. This objective is generally used in topology optimisation, as it is often important that a structure does not deform much under the expected loading. This problem has two constraints. First of all there is a constraint on the volume the structure can occupy in the given design domain, V ∗ . This constraint is applied because usually, especially in high tech applications, the weight of the structures needs to be controlled. Secondly, there is a constraint on the lowest buckling load,λ∗cr . This will generally be the expected maximum load of the structure during operation, with a certain safety margin. The presence of material in each element are the design variables of the problem. There is no algorithm that can be used efficiently to optimise such a large number of discrete design variable. In order to make the design variables continuous the SIMP method is used, where the density values of the elements are used as design variables. To still obtain a black and white design, the element Young’s moduli are penalised with a penalisation factor p. The density for each element can vary between ρ mi n = 10−3 and ρ max = 1 15 16 4. O PTIMISATION PROBLEM A linear density filter is also implemented in the code, although it is not used in all of the numerical examples. The filter redistributes the density of an element over all elements within the filter radius R. This forces a minimum feature size of 2R and prevents checker-boarding. The procedure of the filter is visualized in Figure 6.1 and described in Equation 4.2 Figure 4.1: Visual representation of the linear density filter R − di wi = P R − di (4.2) 4.2. M ETHOD OF M OVING A SYMPTOTES As the optimisation algorithm, the method of moving asymptotes (MMA) by Svanberg [19] is used. This optimisation algorithm builds a convex asymptotic approximation of the problem and efficiently solves this convex subproblem in every iteration. Moving asymptotes are used to control the convergence of the process. MMA is generally used in topology optimisation because of its fast convergence and ability to deal with a large number of design variables efficiently. The method of moving asymptotes is a gradient-based algorithm: the convex sub-problems are generated using information on both the function values of the real problem at a certain design, but also the derivatives of those functions. Hence, the sensitivity information of the structure is required. In Chapter 5 the calculation of the sensitivities is described in detail. 5 F INITE E LEMENT M ODEL The objective and constraint values, and their sensitivities that are needed for the MMA optimiser are computed with a finite element model. The density of these finite elements serve as the design variables for optimisation. In this chapter, the details of the finite element model are described. 5.1. S TATIC DEFORMATIONS AND STRESSES The design domain is divided into square two-dimensional 4-node elements with 8 degrees of freedom per node. The shape functions for this type of element are shown in Equation 5.1. The derivatives of this shape function are used as entries in matrix B. With the use of matrix B the element physical stiffness matrix K(e) is calculated as in Equation 5.4. Note that this integration is usually done numerically. In this case, however, all elements are the same, so this integration only needs to be done once and not much computation time is gained by numerical integration. The constitutive matrix E that is used corresponds to a plane stress situation with isotropic material behaviour. This assumption holds for thin plates. 1 N1 = (1 − x)(1 − y) 4 1 N2 = (1 + x)(1 − y) 4 1 N3 = (1 + x)(1 + y) 4 1 N4 = (1 + x)(1 − y) 4 N1,x B= 0 N1,y 0 N1,y N1,x N2,x 0 N2,y N3,x 0 N3,y 0 N2,y N2,x ν 1 0 1 E ν E= (1 − ν2 ) 0 K(e) = Z 1 Z 1 (5.1) 0 N3,y N3,x N4,x 0 N4,y 0 N4,y N4,x (5.2) 0 0 (1−ν) 2 ; (5.3) BT EB (5.4) x=−1 y=−1 p The element stiffness matrices are then scaled with the penalised element density ρ e and assembled into the full physical stiffness matrix K f ul l as in Equation 5.5. The localisation matrix L is connects the degree of freedom of two neighbouring elements. After assembly, the degrees of freedom that are prescribed to be zero are removed from the full matrix to obtain the global stiffness matrix K. 17 18 5. F INITE E LEMENT M ODEL Figure 5.1: Location of Gauss points in the elements K f ul l = N X p ρ e L(e)T K(e) L(e) (5.5) e=1 The global stiffness matrix is then used to calculate the deformations d when a certain force vector f is applied to the structure (Equation 5.6). d = K−1 f (5.6) For normal topology optimisation the deformations of the structure are enough information, but for calculation of the buckling load the stresses in the structure are required too. The stresses are not integrated over the whole element, but are evaluated at a discrete set of integration points, or Gauss points. The location of these Gauss point in the element are shown in Figure 5.1. At these locations the stresses are calculated as in Equation 5.7, where Bi is evaluated at the Gauss points as well. q σ(e) = ρ e EBi d(e) i (5.7) Note that the Young’s modulus is again scaled with the element density values, this time penalised with another penalisation factor q. This penalisation factor q can be chosen to be different from the penalisation factor p. In Chapter 7, the effect of different values for q is investigated. For the numerical examples, p = q = 3 is chosen. 5.2. L INEAR BUCKLING ANALYSIS The stresses at the Gauss points are first used to create the element geometric stiffness matrices K(e) σ . This is done by adding the contribution of each Gauss point as in Equation 5.8 K(e) σ = N1,x N1,y G= 0 0 0 0 N1,x N1,y N2,x N2,y 0 0 4 X i =1 0 0 N2,x N2,y GTi Si Gi N3,x N3,y 0 0 (5.8) 0 0 N3,x N3,y N4,x N4,y 0 0 0 0 N4,x N4,y (5.9) 5.2. L INEAR BUCKLING ANALYSIS (a) Width of 1 element 19 (b) Width of 2 elements (c) Width of 4 elements (d) Width of 10 elements Figure 5.2: Comparison of the eigenbuckling formulation with Euler buckling σxx σx y S= 0 0 σx y σy y 0 0 0 0 σxx σx y 0 0 σy y σy y (5.10) Then the element contributions are assembled in a similar manner as for the physical stiffness matrix. f ul l However, for the geometric stiffness matrix Kσ the penalisation is already included in the calculation of the element matrices, so it is not done during the assembly. (Equation 5.11) f ul l Kσ = N X e=1 (e) L(e)T K(e) σ L (5.11) Lastly, the fixed degrees of freedom are removed in the same fashion as for the physical stiffness matrix to arrive at the geometric stiffness matrix Kσ . Now, the buckling loads are calculated by solving the eigenvalue problem of Equation 5.12. (K + λcr Kσ ) v = 0 (5.12) This approach is tested by comparing a clamped-free beam to an Euler buckling calculation of the same beam (Equation 5.13). In Figure 5.2 the results are shown. F cr = π2 E I = 0.002056 (2L)2 (5.13) Table 5.1: Comparison of the eigenbuckling formulation with Euler buckling Width of 1 element Width of 2 elements Width of 4 elements Width of 10 elements Buckling factor λ Buckling load f cr [N] Percentage difference [%] 3.0328 2.2937 2.1055 2.052 0.0030328 0.0022937 0.0021055 0.002052 47.5 11.6 2.4 -0.2 20 5. F INITE E LEMENT M ODEL From the comparison of Table 5.1 it becomes clear that the buckling load, is overestimated when only one or two elements are used in width. A width of at least four elements is necessary for a reasonable estimation of the buckling load. When using this formulation in topology optimisation for practical purposes, it is important to use a density filter to ensure all structures have a width of at least four elements. 5.3. S ENSITIVITIES As we are using a gradient based optimiser, sensitivity information of the objective and the constraints with respect to the design variables is needed. The sensitivities of the objective -the compliance- and the volume constraint are easily calculated as in Equation 5.14 and Equation 5.15 respectively. Both of these sensitivities can be calculated element-wise and are therefore very efficient. ∂C = −pρ (p−1) dT Kd ∂ρ (5.14) ∂V =1 ∂ρ (5.15) 5.3.1. S ENSITIVITIES OF THE BUCKLING LOAD Calculating the sensitivities of the buckling load is rather tedious. Initially, it seems that the similarity to eigenfrequency optimisation can be used, as in Equation 5.19. However, in contrast to the mass matrix sensitivity for eigenfrequency analysis, the geometric stiffness sensitivity is not readily available. The geometric stiffness sensitivity K’σ cannot be evaluated element-wise, and also depends on the deformation sensitivities of the structure. The derivation of Equation 5.19 is given in Equations 5.16, 5.17 and 5.18, where the first term of Equation 5.17 vanishes because (K + λKσ ) v = 0 vT (K + λKσ ) v = 0 (5.16) ¡ ¢ T 2v0 (K + λKσ ) v + vT K0 + λK0 σ + λ0 Kσ v = 0 (5.17) ¡ ¢ vT K0 + λK0 σ + λ0 Kσ v = 0 (5.18) vT (K’ − λcr K’σ ) v ∂λcr =− ∂ρ i vT Kσ v (5.19) Where: K’ = ∂K ∂ρ i and K’σ = ∂Kσ ∂ρ i (5.20) In the literature found on topology optimisation with buckling analysis either the geometric stiffness sensitivity K’σ is calculated using finite differences, or a non-gradient-based optimiser is used. Both the use of finite differences and a non-gradient-based optimiser are not suitable for topology optimisation, as the computation times increase exponentially with the number of design variables. A better way to compute the sensitivities of the buckling load is by adding adjoint variables and constraint functions. By choosing the adjoint variables correctly, it is possible to replace the complicated parts of the sensitivity equation by expressions that are easier to calculate. The starting point of the derivation of adjoint sensitivities for a buckling load is given in Equation 5.21, which consists of equation , augmented with two constraints, multiplied by adjoint variables µT and wT . The constraints link the stresses to the deformations, and the deformations to the force vector. The derivative of this equation is given in Equation 5.22, where, similarly to in Equation 5.17, the first term equals zero and can be dropped to arrive at Equation 5.23 ¡ ¢ vT (K + λKσ ) v + µT (σ − EBd) + wT BT EBd − f = 0 µ ¶ ¡ ¢ ¡ ¢ ∂Kσ 0 T 2v0 (K + λKσ ) v + vT K0 + λ σ + λ0 Kσ v + µT σ0 − E0 Bd − EBd0 + wT BT E0 Bd + BT EBd0 = 0 ∂σ (5.21) (5.22) 5.3. S ENSITIVITIES 21 Figure 5.3: Computation time comparison of the sensitivity calculation methods µ ¶ ¡ ¢ ¡ ¢ ∂Kσ 0 vT K0 + λ σ + λ0 Kσ v + µT σ0 − E0 Bd − EBd0 + wT BT E0 Bd + BT EBd0 = 0 ∂σ (5.23) Now, the first adjoint variable µT is chosen in such a way that σ0 drops out of the equation. This is done by setting the left-hand side of Equation 5.24 equal to zero. After some careful rearranging, the adjoint variable µ is computed as in Equation 5.25. ¶ µ ¡ ¢ ∂Kσ 0 T σ v + µT σ0 = 0 (5.24) v λ ∂σ µ j = −λv l ∂K σl k vk ∂σ (5.25) Substituting this value for µ into Equation 5.23 leaves Equation 5.26. From here, adjoint variable wT is computed such that d0 vanishes (Equation 5.27). wT is computed in Equation 5.28. ¡ ¢ ¡ ¢ ¡ ¢ vT K0 + λ0 Kσ v + µT −E0 Bd − EBd0 + wT BT E0 Bd + BT EBd0 = 0 (5.26) − µT EBd0 + wT BT EBd0 = 0 (5.27) ¡ ¢−1 ¡ T ¢ wT = BT E0 B µ EB (5.28) Finally, this leaves the formulation for the buckling sensitivity from Equation 5.29. In Figure 5.3 the computation time of these adjoint sensitivity is compared to the computation time of the finite difference computation that is used in the existing literature. It is clear that the adjoint formulation reduces computation time. ¡ ¢ ¡ ¢ vT K’v + µT −E0 Bd + wT BT E0 Bd ∂λcr =− (5.29) ∂ρ i vT Kσ v 22 5. F INITE E LEMENT M ODEL NOTE! When dealing with the equations symbolically, it is easy to make a mistake in the computation of wT in Equation 5.27. Instead of the correct method of Equation 5.28 it is tempting to use the incorrect Equation 5.30. This system is inconsistent and does not have a solution. wT BT = µT (5.30) Applying this adjoint variable vanishes all the adjoint terms, as shown in Equation 5.33. This, obviously, cannot be correct, as it would mean that a change in geometric stiffness does not influence the buckling load. ¡ ¢ ¡ ¢ ¡ ¢ vT K0 + λ0 Kσ v + µT −E0 Bd + wT BT E0 Bd = 0 (5.31) ¡ ¢ vT K0 + λ0 Kσ v = 0 (5.32) λ0 = − vT K0 v vT Kσ v (5.33) 6 N UMERICAL EXAMPLES The optimisation is run on a number of test cases to provide an insight in the effect of a buckling constraint on the first positive buckling load on the final designs. More detailed results of the designs shown in this chapter can be found in Appendix A. There, along with the design, the deformation and the first five positive modes of the structure are shown. Furthermore, the objective value and the value of the first five positive modes during optimisation are graphed. 6.1. T EST CASE The optimisation model described in the previous chapters is now implemented on a test case. The test case is shown in Figure 6.1 and consists of a domain with an equally distributed load on the top edge, and clamping conditions on the bottom edge. Figure 6.1: Domain and load case for the first optimisation tests The parameters from Table 6.7 are used in all numerical examples, unless otherwise specified. Table 6.1: Parameters for optimisation of the numerical test case Width [m] Height [m] # of elements E-modulus [Pa] Poisson ratio Volume constraint 10 10 30 x30 1 0.3 0.5 23 24 6. N UMERICAL EXAMPLES 6.2. VARIOUS BUCKLING CONSTRAINTS First, the optimisation is run for an increasing buckling constraint. This tests the effectiveness of the inclusion of a buckling constraint. In Case A.1, no buckling constraint is used. From here, cases are tested with a buckling constraint of 120%, 150% and 200% of the original buckling load of Case A.1 Table 6.2: Results for optimisations with various buckling constraints Case A.1 Case A.2 Case A.3 Case A.4 Buckling constraint % increase of λcr Objective Buckling factor 0 0.4864 0.6079 0.8106 0 20 50 100 0.23883 0.23814 0.22292 0.22349 0.40532 0.4864 0.6079 0.8106 From the results in Table 6.2 it is observed that the optimiser finds a structure with a first buckling load that is exactly on the constraint. It does this mainly by stiffening the structure, as is clear from the decreased compliance of Case A.1, Case A.3 and Case A.4 with respect to the compliance of Case A.1. Upon closer inspection of the designs in Figure 6.2 it appears that this improved compliance is obtained by checker-boarding of the material. Furthermore, in Figure 6.2c and Figure 6.2d the members of the structure are made wider to extent to the outside of the domain, so that a higher bending stiffness is achieved. Lastly, Figure 6.2d shows that the design for this constraint is no longer symmetric. This might indicate that the optimiser had some trouble finding a design that fulfils the constraint. (a) Optimised design, unconstrained buckling load (b) Optimised design, buckling load constrained to 120% of the original buckling load (c) Optimised design, buckling load constrained to 150% of the original buckling load (d) Optimised design, buckling load constrained to 200% of the original buckling load Figure 6.2: Topology optimised designs for various values of the buckling constraint 6.3. VARYING NUMBER OF ELEMENTS 25 6.3. VARYING NUMBER OF ELEMENTS The optimisation is run for an increasing number of elements, while keeping the domain size constant. As described in Chapter 5, linear buckling analysis for these elements only gives a reasonable estimation for structural element widths of at least four elements. Therefore, the optimisation should in practice be done with a fine mesh and a minimum feature size controlling filter. The checker-boarding that appeared in Section 6.2 also indicates the necessity of a filter. In these test cases, however, no filter is used. Table 6.3: Results for optimisations with a varying number of elements Case A.5 Case A.6 Case A.7 Case A.8 Number of elements Buckling constraint Objective Buckling factor 10 x 10 10 x 10 10 x 10 10 x 10 0 2.7861 0 0.4247 0.0293 0.0556 0.2831 0.5960 1.8574 2.7861 0.6573 0.4247 As can be seen in Table 6.3, the buckling loads for the 10 x 10 element systems are higher than those of the 50 x 50 element systems, because the bending stiffness of these structures is overestimated. Furthermore, it is found from Figure 6.3that the optimiser prefers grey elements in the 10 x 10 case, while it prefers checkerboarding in the 50 x 50 case. Therefore, the compliance is improved for the 50 x 50 case, while it is worsened in the 10x10 case. (a) Optimised design, 10 x 10 elements, unconstrained buckling load (b) Optimised design, 10 x10 elements, buckling load constrained to 150 % of the unconstrained buckling load (c) Optimised design, 50 x 50 elements, unconstrained buckling load (d) Optimised design, 50 x50 elements, buckling load constrained to 150 % of the unconstrained buckling load Figure 6.3: Topology optimised designs for a varying number of elements 26 6. N UMERICAL EXAMPLES 6.4. VARIOUS FILTER SIZES The optimisation is also run for different filter sizes of the linear filter. This filter should serve two purposes: it prevents checker-boarding and ensures a minimum feature size. For all cases, the buckling constraint is set at 150 % buckling load of the unconstrained and unfiltered Case A.1 Table 6.4: Results for optimisations with various filter sizes Case A.9 Case A.10 Case A.11 Case A.12 Filter size Buckling constraint Objective Buckling factor 40 cm 70 cm 90 cm 110 cm 0.6079 0.6079 0.6079 0.6079 0.2798 0.8161 0.3431 0.3592 0.6077 0.6079 0.6077 0.6077 In terms of design, in this case a filter with a radius R of 40 cm, as in Figure 6.4a gives the best solution. For larger filter sizes, the designs consist mostly of intermediate values. This filter radius of 40 cm is only slightly larger than the element size. Therefore, the minimum feature width of 4 elements is not met. For structure with a higher resolution, it is possible to increase the filter to meet this requirement. The design of Figure 6.4a nicely shows a new configuration with a buckling load op 150 % of the unconstrained one. The members are made wider with connecting members in between, to provide a higher bending stiffness. (a) Optimised design, R = 40 cm (b) Optimised design, R = 70 cm (c) Optimised design, R = 90 cm (d) Optimised design, R = 110 cm Figure 6.4: Topology optimised designs for various filter radii, for all cases the buckling load is constrained to 150 % of the unconstrained and unfiltered buckling load 6.5. VARIOUS DOMAIN SHAPES 27 6.5. VARIOUS DOMAIN SHAPES To investigate the effect of different buckling sensitivities, the shape of the domain is varied to be more rectangular. The buckling load of the unconstrained structure is found, and is used to constrain the buckling load of the next optimisation at 150 %. Table 6.5: Results for optimisations with various domain shapes Case A.13 Case A.14 Case A.15 Case A.16 Domain size Buckling constraint Objective Buckling factor 10 m x 20 m 10 m x 20 m 20 m x 10 m 20 m x 10 m 0 0.3048 0 0.5499 0.4408 0.4222 0.4627 0.4349 0.2032 0.3048 0.3666 0.5819 Obviously, the vertical domain shape has a lower buckling load than the horizontal domain, while having a comparable compliance. Therefore, the optimiser has much more trouble finding a solution for the constrained vertical problem of Figure 6.5b than for the horizontal one of Figure 6.5d. This is visible from the fact that in Figure 6.5b, there is a lot of checker-boarding present and material is moved to the outer edge of the domain, while in Figure 6.5d only minor changes are made compared to Figure 6.5c. The possible improvement of the buckling load of a structure thus cannot be easily predicted. (a) Optimised design for a domain of 10 x 20 m, unconstrained buckling load (b) Optimised design for a domain of 10 x 20 m, buckling load constrained to 150 % of the unconstrained buckling load (c) Optimised design for a domain of 20 x 10 m, unconstrained buckling load (d) Optimised design for a domain of 20 x 10 m, buckling load constrained to 150 % of the unconstrained buckling load Figure 6.5: Topology optimised designs for various domain sizes 28 6. N UMERICAL EXAMPLES 6.6. OTHER TEST CASES Because of the nature of the test case, in the previous sections the occurrence of negative buckling loads was not an issue. However, if the optimiser is to find the solution from Figure 1.2d, negative buckling loads also enter the optimisation problem. Figure 6.6a shows the new test case. For this test case, a volume constraint of 5% is used. Figure 6.6b, Figure 6.6c and Figure 6.6d show the results for different buckling constraints. At first sight, these results look promising, as the desired configuration is indeed obtained. However, an extremely high buckling constraint is applied in order to find the result of 6.6d. Upon a closer look into the optimisation process, it becomes clear that it depends entirely on the switching of buckling modes. In Figure 6.7 the objective value and first five positive buckling loads during optimisation are visualised. At the twelfth iteration, the objective has a peak and all buckling loads approach zero. This suggests that the optimiser made a dramatic change in the configuration near this step. After this step the positive buckling loads improve drastically, which indicate they belong to non-physical modes in the void regions. The buckling loads corresponding to physical buckling loads are now negative. Table 6.6: Results for optimisations with Test Case 2 of Figure 6.6a Case A.17 Case A.18 Case A.19 Buckling constraint Objective Buckling factor 0 0.021 0.21 0.10822 0.12828 0.13822 0.014023 0.023437 0.5796 (a) Domain and load case for Test Case 2 (b) Optimised design for Test Case 2, unconstrained buckling load (c) Optimised design for Test Case 2, buckling load constrained to 150 % of the unconstrained buckling load (d) Optimised design for Test Case 2, buckling load constrained to 1500 % of the unconstrained buckling load Figure 6.6: Topology optimised designs for Test Case 2 6.6. OTHER TEST CASES 29 Objective value during optimization 1.1 1 0.9 Compliance 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 15 20 25 Iteration number 30 35 40 45 40 45 (a) The objective of Case A.19 during optimisation First five buckling loads during optimization 1.4 1.2 Buckling load 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Iteration number 30 35 (b) The first five positive buckling loads of Case A.19 during optimisation Figure 6.7: The objective value and first five positive buckling loads of Case A.19 during optimisation The formulation of constraining the first positive buckling load is also tested on Test Case 3, shown in Figure 6.8a. When a buckling constraint of 400 % of the original buckling load is applied, the optimiser uses checkerboarding to artificially increase the stiffness of the structure. Only when an extremely high buckling is applied, as in Figure 6.8d, the lay-out of the structure changes. Again, the positive buckling loads of this structure correspond to non-physical modes of the void elements, as is shown in Figure 6.9. The members of the structure that are subjected to tension are reduced in width, while the member that experiences compression is widened. Additionally, the member to the right, that is the first member to buckle for the unconstrained case, is shortened and widened. Table 6.7: Results for optimisations with Test Case 3 of Figure 6.8a Case A.17 Case A.18 Case A.19 Buckling constraint Objective Buckling factor 0 2.4613 24.613 0.20466 0.20197 0.33377 0.61535 2.4614 3.9361 30 6. N UMERICAL EXAMPLES (a) Domain and load case for Test Case 3 (b) Optimised design for Test Case 3, unconstrained buckling load (c) Optimised design for Test Case 3, buckling load constrained to 400 % of the unconstrained buckling load (d) Optimised design for Test Case 3, buckling load constrained to 4000 % of the unconstrained buckling load Figure 6.8: Topology optimised designs for Test Case 3 Buckling mode1 Buckling load = 3.9861 Compliance = 0.33377 Figure 6.9: First positive buckling load for Case A.22 7 D ISCUSSION Numerous issues arise when including eigenvalue buckling analysis in a topology optimisation context. Some of these issues were already predicted in literature, while others were not. These issues can generally be approached in different ways, depending on the design problem at hand. For every individual case, the formulation should be tuned to obtain an optimisation problem that runs smoothly and gives the desired results. In this chapter a number of these issues are discussed, together with possible solutions. 7.1. N EGATIVE BUCKLING LOADS In contrast to another common eigenvalue problem -eigenfrequency optimisation- linear buckling analysis can lead to both positive and negative eigenvalues. In the case of buckling, a negative eigenvalue corresponds to buckling load in the opposite direction from the applied load. How to deal with these negative buckling loads depends on the design problem. In some cases, loads in the opposite direction might also occur, while in other cases the loading conditions are fixed. A constraint on both the positive and the negative buckling load can be used to deal with this problem of negative buckling loads. These constraints can be different, depending on the design problem. For problems where no loading in the opposite direction is expected, the negative buckling constraint can be set to zero. A big disadvantage of the occurrence of negative buckling loads is the fact that it makes the feasible design space non-convex. This introduces local optima, which complicates finding the global optimum. In many cases, for example for the case in Figure 6.6d, it is necessary for the optimiser to overcome this non-convexity. Starting the optimisation with different initial designs might help with finding a good optimum. In Figure 7.1 the results are shown of different optimisations on a structure where the lowest buckling load can be either positive or negative. The first design is the optimum design when no buckling constraint is applied, in the second design only the first positive buckling load is constrained, in the last design a constraint is applied to the absolute value of the first buckling load. Figure 7.1: Optimisations for different options for negative buckling loads 31 32 (a) Constraint on lowest buckling load 7. D ISCUSSION (b) Constraint on mean buckling load (c) Constraint on weighted mean buckling load Figure 7.2: First five buckling loads during optimisation 7.2. M ODE SWITCHING AND THE MULTIPLICITY OF BUCKLING LOADS When a constraint or objective is applied only to the first buckling load, the optimiser makes use of the fact that the higher buckling loads are not constrained. It will shift the lowest buckling load to a higher value, while not changing -or even lowering- the higher buckling loads. This leads to a cluster of very close buckling loads, as was first shown by Thompson and Supple [8]. This clustering of buckling modes is problematic for a number of reasons. The first problem with close eigenvalues is the calculation of the buckling load sensitivities. When there is a multiplicity in buckling load, there is more than one corresponding buckling mode for that buckling load. Therefore, the sensitivities can be calculated with all combinations of eigenmodes, and they become nonunique. At the point in the optimisation where switching occurs, the constraint function is non-differentiable. The other problem that occurs with a multiplicity of buckling load is of a physical nature. When there are close buckling loads, a coupling effect between the modes can arise, which can be highly sensitive to imperfections. Even the coupling between higher buckling loads could drastically knock down the snapping load of an imperfect structure. Therefore, multiplicity in buckling loads should be avoided in any optimisation. However, in order to find the optimal solution, the optimiser requires the freedom of mode switching. An impromptu solution to the problem of multiplicity can be to separately constrain the first few buckling loads with a certain spacing, instead of using a constraint on the lowest buckling load only. A more advanced solution is to constrain the mean buckling load instead of the only the lowest buckling load, similarly to the method of Ma et al. [18] for vibrating structures. In Figure 7.2 the first five buckling loads during optimisation with three formulations are shown. Figure 7.4a is optimised with a constraint on the lowest buckling load only, in Figure 7.4b the mean of those first five buckling loads is constrained and in Figure 7.4c a weighted mean is used. While using the mean buckling load as a constraint does indeed prevent clusters of buckling loads, the control of the lowest buckling load is lost in this formulation. 7.3. B UCKLING OF VOID ELEMENTS 33 7.3. B UCKLING OF VOID ELEMENTS Another problem with eigenvalue analysis in topology optimisation that is predicted in literature is the occurrence of modes in the void areas. This happens because void areas are never completely free from material, so instabilities can originate in the void areas as well as in the material areas. In literature, this problem comes directly from the similarity with eigenfrequency analysis, where eigenmodes can occur in the void areas. The instabilities are caused by the penalised values for the physical stiffness, when the geometrical stiffness is insufficiently penalised. Literature gives a number of solutions for this problem. First of all, the geometric stiffness can be reduced to zero below a certain threshold value for the density, while the physical stiffness remains unchanged. This method, however, introduces a discrete change at which the sensitivities cannot be computed. Another option is to use a slightly different penalisation for the geometric stiffness, to remove the non-differentiable change. Lastly, the void elements can be removed completely, except the void elements at the boundary of the structure. These boundary void elements are needed to give the optimiser to grow a structure into the void. However, the problem could not be reproduced with consistent penalisation of the E-modulus. The Emodulus is the value that is usually scaled with the design parameter, and thus also the parameter that is penalised. Consistent penalisation of the E-modulus results in a higher penalisation for the geometric stiffness matrix, as the E-modulus is used both in calculating the deformations and in calculating the corresponding stresses. Figure 7.3 shows the effect of different values of penalisation factor q on the buckling load of a single element. The dashed lines represent the E-modulus and the solid lines represent the buckling loads. For q =3, these lines overlap, as the penalisation is done consistently. Buckling of void elements is prevented when p ≤ q. This is shown in Figure 7.4. Penalized response for p = 3 & q = 0 Penalized response for p = 3 & q = 1 1 1 E−modulus Buckling load λ 0.9 0.8 0.7 0.7 0.7 0.6 0.6 0.6 Response 0.8 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 0 0.1 0.2 0.3 0.4 0.5 0.6 Density ρ 0.7 0.8 0.9 0 1 (a) Penalisation q = 0 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Density ρ 0.7 0.8 0.9 Penalized response for p = 3 & q = 3 0 1 (b) Penalisation q = 1 0.6 0.6 Response 0.7 0.6 Response 0.8 0.7 0.5 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0.4 0.5 0.6 Density ρ (d) Penalisation q = 3 0.7 0.8 0.9 1 0.5 0.6 Density ρ 0.7 0.8 0.9 1 0.8 0.9 1 0.5 0.4 0.3 0.4 Penalized response for p = 3 & q = 5 0.8 0.5 0.3 0 0.1 0.2 0.3 E−modulus Buckling load λ 0.9 0.7 0.2 0.2 1 E−modulus Buckling load λ 0.9 0.8 0.1 0.1 (c) Penalisation q = 2 1 E−modulus Buckling load λ 0 0 Penalized response for p = 3 & q = 4 1 0.9 E−modulus Buckling load λ 0.9 0.8 Response Response 0.9 Response Penalized response for p = 3 & q = 2 1 E−modulus Buckling load λ 0.4 0.5 0.6 Density ρ (e) Penalisation q = 4 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Density ρ (f) Penalisation q = 5 Figure 7.3: Penalised Young’s modulus and buckling load for a single element for different values of q, p =3 0.7 34 7. D ISCUSSION Buckling mode1 Buckling load = 9.2917 (a) First buckling mode for p = 3, q = 2 Buckling mode1 Buckling load = 104.5898 (b) First buckling mode for p = 3, q = 3 Buckling mode1 Buckling load = 114.925 (c) First buckling mode for p = 3, q = 4 Figure 7.4: First buckling of a structure with different values for q 7.4. S TABILITY AND IMPERFECTION SENSITIVITY This last issue is not observed while running the optimisation, but rather when the optimised design is to be translated into a real product. With linear buckling analysis only the bifurcation point of the structure is found, but what happens after buckling remains unknown. In perfect cases the post-buckling behaviour would not be an issue in itself, as buckling of the structure is to be avoided. It does however have a great influence of the imperfection sensitivity of the structure. Imperfection sensitivity is indeed of high importance as in reality there will always be imperfections present, whether in the structure itself or in the loading of the structure. Therefore, the task of the optimiser should not only be to find a structure with a certain bifurcation point, but the buckling response of the structure should also not be sensitive to imperfections. Post-buckling behaviour can be simplified to fall in to one of three categories, shown in Figure 7.5. Only one of those categories is symmetric and stable, and therefore insensitive to imperfections. An ideal structure would be one with a sufficiently high linear buckling load, that also falls into this category. To establish stable post-buckling behaviour, post-buckling analysis of the structure should be conducted. In an optimisation setting, this post-buckling analysis has to be very computationally inexpensive. In Part II, such an efficient method is investigated. Figure 7.5: Post buckling behaviour and imperfection sensitivity. Source: [4] II P OST- BUCKLING B EHAVIOUR AND I MPERFECT S YSTEMS 35 8 S TABILITY ASSESSMENT As mentioned in Chapter 7, a structure’s sensitivity to imperfections depends on its post-buckling behaviour. In real-life structures, imperfections in load direction and shape are inevitable. Consequently, the post-buckling behaviour of the structure should be analysed during optimisation, in order to achieve a meaningful design. This can be done by including a criterion that ensures that the post-buckling behaviour of the structures falls into the third, stable, category of Figure 7.5. The post-buckling behaviour of a structure can be determined in different ways. 8.1. N ONLINEAR BUCKLING ANALYSIS One option to determine post-buckling behaviour is with the use of a total nonlinear buckling analysis. In this iterative analysis the tangent operator J = K + Kσ is computed at every step. With a path-following technique such as the arc-length method the equilibrium path of the structure is tracked. The buckling load of the structure is then found as a limit load in the equilibrium path. The biggest advantage of this full nonlinear analysis is that less assumptions are made about the structure. This procedure works for any structure with any imperfection. With this method the complete post-buckling characteristics of the structure are evaluated, as advised by Thompson and Supple [8]. However, this advise was written in 1973, before the rise of topology optimisation. Whereas a nonlinear analysis might seem feasible for size or shape optimisation, where the number of design variables is relatively low, a full nonlinear analysis is not suited for topology optimisation. Because many computationally expensive function evaluations are needed, and sensitivities of limit loads are difficult to evaluate, nonlinear buckling analysis is not a valid option for this purpose. 8.2. KOITER INITIAL POST- BUCKLING Another option is Koiter initial post-buckling analysis. This method is an extension of eigenvalue buckling analysis, where a Taylor expansion of the equilibrium path is made. The value of the coefficients λ1 and λ2 is calculated. A nonzero value for λ1 corresponds to an asymmetric equilibrium path. A value of λ1 = 0 and λ2 < 0 means an instable symmetric behaviour, and λ1 = 0 and λ2 > 0 indicates a stable equilibrium path, the desired situation. The advantage of Koiter buckling to establish a stability criterion is that it supplements the eigenvalue buckling analysis. This makes Koiter initial post-buckling analysis is much more computationally efficient than full nonlinear buckling. A disadvantage of Koiter buckling analysis is that higher order derivatives of the potential energy are needed. While it is possible to apply this to finite element methods, it is not straightforward. 37 38 8. S TABILITY ASSESSMENT 8.3. I MPERFECT STRUCTURES The idea of using of imperfect structures to determine the post-buckling behaviour of a structure is based on the strong correlation between the post-buckling behaviour and the imperfection sensitivity of a structure. Normally this connection is used to estimate the buckling sensitivity from the known post-buckling response of the structure. In this case, however, the correlation is reversed to evaluate the stability of the structure by looking at the linearized behaviour of imperfect systems. It should be mentioned that this is not yet an established method for post-buckling analysis, but it is solely investigated on the basis of the similarity to other optimisation schemes where imperfections are considered. If proven accurate, this method does provide some great advantages over the other two methods, making further investigation into it worthwhile . The main advantages of this method are the ease with which the method can be implemented and the low computation time. Most of the infrastructure necessary for performing this analysis is already present in the current formulation of the topology optimisation. While it will be necessary to perform an eigenvalue buckling analysis a total of three times, it is still believed to be more computationally efficient than the other methods. A disadvantage of this procedure is the fact that it is not an established method and as such, the performance is not guaranteed. A number of implicit assumptions are made with the use of linear eigenvalue buckling. Therefore this method will be further investigated in Chapter 9. 9 L INEAR BUCKLING ANALYSIS OF IMPERFECT STRUCTURES In previous chapters, linear buckling analysis is used without a detailed look at the assumptions for this analysis. When using an eigenvalue buckling analysis for imperfect structures, these assumptions might be violated. In this chapter these implicit assumptions are investigated in more detail. The validity of eigenbuckling analysis on imperfect structures is examined. 9.1. L INEAR PRE - BUCKLING ASSUMPTIONS In linear buckling analysis, it is assumed that the force f is conservative and it scales linearly with a parameter λ. The deformations d, the stresses σ and the geometric stiffness matrix Kσ are also assumed to scale linearly with the parameter λ. (Equation 9.1) f = λf0 d = λd0 σ = λσ0 Kσ = λK0σ (9.1) These assumptions imply linear pre-buckling behaviour. They hold as long as no large deformations or rotations are present in the structure before the buckling load is reached. Imperfect structures already show bending in the pre-buckling solution, for which the structure is more compliant. Therefore, this assumption of linear pre-buckling behaviour does not hold for imperfect structures. With these assumptions it can be derived that the buckling of linear systems corresponds to a bifurcation point, where the buckling mode is orthogonal to the load vector. (Equation 9.2) vT f = 0 (9.2) 9.2. E STIMATE POST- BUCKLING USING IMPERFECT STRUCTURES In order to estimate the post-buckling behaviour of the structure one needs to know whether the initial post-buckling deformation has a stabilizing or a destabilizing effect on the structure. Therefore, the geometric stiffness Kσ is evaluated using the stresses calculated at the linear deformations, augmented with a deformation with the shape of the mode σ(d + αv). To evaluate the effect of this post-buckling geometric stiffness, the eigenvalue problem of Equation 9.3 is solved. This procedure converts the post-buckling geometric stiffness matrix into a meaningful scalar. In case 39 40 9. L INEAR BUCKLING ANALYSIS OF IMPERFECT STRUCTURES (a) Simple structure (b) Corresponding imperfect structure Figure 9.1: Simple test case for linear buckling of imperfect structures Figure adapted from [4] this scalar is higher than the buckling load for both α and −α, the post-buckling behaviour of the structure is initially stable. post −bucki ng (K + λKσ )w = 0 (9.3) The aforementioned scheme estimates the post-buckling behaviour of a system, but does not make use of an imperfect system. The use of imperfect system is based on the fact that a very small imperfection causes a small change in the linear stiffness matrix K of the system, but introduces large deformations that approxpost −buckl i ng imate d + αv. Therefore, a small change in K causes a big change in Kσ that approximates Kσ of the perfect system. One advantage of using this imperfect system is the fact that it can make use of the eigenvalue buckling analysis of existing finite element packages without the need of alteration of those packages. Another reason to investigate imperfect structures in more detail is that they might give an insight on the validity of eigenvalue buckling on arbitrary structures. 9.3. A SIMPLE STRUCTURE To evaluate the linear buckling analysis of imperfect structures, the simplest possible structure is used as a starting position. This structure consists of a vertical rod, connected to the ground by a rotational spring, and is shown in Figure 9.1a while the corresponding imperfect structure is shown in Figure 9.1b. The spring, however, can be non-linear, as in Equation 9.4. The buckling load of the perfect system is given in Equation 9.5 M (ξ) = k 1 ξ + k 2 ξ2 + k 3 ξ3 + ... f cr = λcr f 0 = k1 L (9.4) (9.5) This structure is chosen, because the Koiter initial post-buckling solution for the perfect structure is readily available for comparison (see Equation 9.6). Additionally, the structure only has one degree of freedom, which allows investigation of the buckling load only, without having a different buckling mode for the imperfect 9.3. A SIMPLE STRUCTURE 41 structure. Furthermore, the different stability situations of Figure 7.5 can easily be induced by changing the spring constants k 1 k 2 and k 3 . Figure 9.2: Simple structure λ= ¶ µ k1 k2 k1 k3 2 ξ + ... + ξ+ + L L 6L L (9.6) Now, the material stiffness (Equation 9.7) and geometric stiffness (Equation 9.8) for this structure is determined. Similarly to in the finite element model, the geometric stiffness can only be evaluated once the linear solution is known. Note here that the geometric stiffness depends non-linearly on the rotation ξ, and therefore does not scale linearly with λ. Therefore, the choice of f 0 influences the answer for imperfect structures. As mentioned previously, the buckling load for the corresponding perfect structure should be used as f 0 . K= Kσ = − f cos(ξ̄) + k1 L (9.7) k2 k3 ξ + ξ2 + ... L L (9.8) The linear solution is given in Equation 9.9 and is substituted in the geometric stiffness in Equation 9.10. ξ= f 0 L sin(ξ̄) k1 µ ¶2 k 2 f 0 L sin(ξ̄) k 3 f 0 L sin(ξ̄) Kσ = − f cos(ξ̄) + + + ... L k1 L k1 (9.9) (9.10) 42 9. L INEAR BUCKLING ANALYSIS OF IMPERFECT STRUCTURES By solving Equation 9.11 the buckling load of the imperfect structure can be calculated as in Equation 9.12. It should be noted that for an imperfection ξ̄ of zero, the buckling factor λcr is the same as for the perfect system in Equation 9.5. (K + λcr K σ ) = 0 λcr = (9.11) k1 µ L − f cos(ξ̄) + kL2 f 0 L sin(ξ̄) k1 + kL3 ³ f 0 L sin(ξ̄) k1 ´2 (9.12) ¶ + ... Table 9.1: Imperfections used in Figure 9.3 Imperfection angles [rad] 0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.4 In Figure 9.2 this process is visualised for an imperfect structure with a linear spring. The black dotted line shows the real equilibrium path of the imperfect system. The red dashed line shows the linear solution, with the linear buckling load visualised as a red circle. The Koiter post-buckling solution of the perfect system is plotted with a blue line. It is clear that for this system, the linear buckling analysis of the imperfect system gives a poor estimation of the behaviour of the imperfect system itself, but it gives an indication of the postbuckling of the corresponding perfect system. In Figure 9.3 the results for different spring constants k 1 k 2 and k 3 and different imperfection angles are shown. 9.3. A SIMPLE STRUCTURE 43 (a) Linear spring (b) Asymmetric nonlinear spring, negative (c) Asymmetric nonlinear spring, positive (d) Symmetric nonlinear spring, negative (e) Symmetric nonlinear spring, positive Figure 9.3: Linear buckling analysis of imperfect structures with different spring properties 44 9. L INEAR BUCKLING ANALYSIS OF IMPERFECT STRUCTURES (a) Koiter-Roorda frame (b) Imperfect Koiter-Roorda frame Figure 9.4: Finite element test case 9.4. KOITER-R OORDA FRAME To test the procedure on a finite element model, the Koiter-Roorda frame is used. This frame, presented in Figure 9.4a, is well known for its asymmetrical post-buckling behaviour. This frame was modelled in ANSYS using 8 beam elements. First, the buckling load and buckling mode of the perfect system were calculated. Then the mode was used to introduce a very small imperfection to the system. A path-following technique was used to follow the nonlinear equilibrium path of the imperfect system. In Figure 9.5, the non-linear equilibrium paths are graphed as black dotted lines. Then, the linear buckling loads for different imperfection factors are computed and shown as red circles in Figure 9.5. The asymmetrical behaviour is clearly visible from linear buckling analysis of the imperfect structures, however, they do not really follow the expected post buckling of the perfect system. This procedure correctly predicts the post-buckling category. 9.4. KOITER-R OORDA FRAME 45 Koiter−Roorda frame 140 120 100 load f [N] 80 60 40 20 0 −0.05 0 displacement u 0.05 x Figure 9.5: Linear buckling analysis of imperfect Koiter- Roorda frames 10 O UTLOOK FOR APPLICATION IN TOPOLOGY OPTIMISATION 10.1. S TABILITY CONSTRAINT So how could this method of determining the stability of the structure be used in topology optimisation? First of all, the linear buckling analysis of the perfect structure had to be performed. The buckling mode v of this perfect structure can than be used to pertubate the system in both directions. Two additional linear buckling analyses are performed on the pertubated systems, so three distinct buckling loads are obtained. Two new constraints can be added to the problem formulation, that enforce that the buckling loads for the perturbed systems should always be larger than the buckling load for the perfect system. This means that the system is constrained to be stable. While the function values of these constraints are easy to obtain, the sensitivities are more cumbersome. The geometric stiffness matrix of the pertubated system does not only depend on the linear deformations d, but also on the mode shape of the perfect system v. Therefore, the mode sensitivities are required. 10.2. OTHER CONSIDERATIONS In previous chapters it has been shown that linear buckling analysis of imperfect structures contains information about the stability of the corresponding perfect structure. However, it is also very clear that linear buckling analysis of the imperfect structures gives an extremely poor estimation of the buckling load of the imperfect structure itself. For arbitrary structures, it is not guaranteed that they are ’perfect’ in a sense that they show linear prebuckling behaviour. Linear buckling analysis of an imperfect structure does not have any meaning when the imperfect structure cannot be related to a corresponding perfect structure. With the choice of eigenvalue buckling analysis in optimisation, an assumption has been made on the linearity of the structure, even before the actual structure is known. Especially in topology optimisation, where slender structures with organic shapes are to be expected, it is highly likely to find structures that do not adhere to this assumption of linearity. The optimiser could misuse this assumption of linearity to find a structure that has a high eigenbuckling load, but is in fact highly nonlinear. Therefore, the use of eigenvalue buckling analysis in topology optimisation, where the linearity of the structure cannot be known on forehand, is questionable. Nonetheless, a first evaluation of the linearity condition that the mode v should be orthogonal to the force vector f for the test cases from Part does not indicate non-linear pre-buckling behaviour for the first mode of each structure, except for Cases A.20, Case A.21 and Case A.22. This is not surprising, as the domain and load direction for those cases introduce bending to the domain. 47 11 C ONCLUSIONS AND RECOMMENDATIONS 11.1. C ONCLUSIONS The first part of this report considered the practical implementation of a buckling constraint in a topology optimisation setting. Furthermore, this part discusses implications of such a constraint. From the first part of this project, a number of conclusions can be drawn: • With the use of adjoint sensitivities, a buckling constraint can efficiently be included in a topology optimisation problem. An adjoint formulation dramatically reduces the computation time. Such a formulation requires two adjoint variables. • The problem of buckling in void elements, which was predicted in literature on occasions, does not occur when the Young’s modulus is penalised consistently. • The calculated eigenvalues can take negative values as well as positive ones. How to handle those negative values is choice that should be based on each individual design problem. For structures that are subjected to loading in both directions, a constraint can be put on the absolute value of the lowest buckling load. Other possibilities are to put separate constraints on the positive and negative buckling loads, which can differ in value, or to only constrain the positive buckling loads. • During optimisation the buckling loads can switch, resulting in a non-differentiable point in the constraint function. A constraint on the mean buckling load can prevent the switching of modes, but it does not provide a direct control over the lowest buckling load. Furthermore, it restricts the design freedom of the algorithm. • While the inclusion of linear buckling analysis can indeed be used to attain a certain buckling load for the perfect structure, no information is obtained on the stability. Imperfection sensitivity is highly dependent on the stability. • Once again it has been shown that constraining the buckling load while optimising leads to clusters of close buckling loads. A multiplicity of buckling loads often leads to an interaction of the corresponding modes, leaving the structure highly sensitive to imperfections. In the second part of the report investigates post-buckling behaviour and the effect of imperfections. The use of linear buckling analysis for imperfect structures was investigated. This part of the report leads to the following conclusions: • The post-buckling behaviour of a structure that exhibits linear pre-buckling behaviour can efficiently be estimated by eigenvalue buckling analysis of the corresponding imperfect structures. • Eigenvalue buckling analysis of arbitrary structures does not necessarily give any indication of the actual snapping load of said structure. This complicates the use in topology optimisation, where structures with linear pre-buckling behaviour are not guaranteed. Everything considered, including buckling analysis in topology optimisation in a way that is physically meaningful, remains a challenge. 49 50 11. C ONCLUSIONS AND RECOMMENDATIONS 11.2. R ECOMMENDATIONS Additional research is necessary into various aspects of this report to ensure that including buckling constraints in topology optimisation has a beneficial effect on the final design. Recommendations for additional research are: • A more realistic test case with realistic load cases should be found, in order to make better choices in the implementation of the negative buckling loads. • Including the buckling load as an objective instead of as a constraint could lead to better results. • Verification of a number of optimised designs should be done with a full non-linear analysis. Most importantly, the assumption of linear pre-buckling analysis should be verified for more complicated test cases. • Based on the findings of the above-mentioned recommendation, the implementation of a linear buckling analysis in topology optimisation might need reconsideration. A non-linear analysis, although extremely computationally expensive, might prove to be necessary for some design problems • Furthermore, the procedure of estimating post-buckling behaviour of perfect systems from their corresponding imperfect structures should be investigated more. A better theoretical foundation for this method should be established and the possibilities and limitations should be further investigated. • The method for establishing post-buckling behaviour with linear analysis of imperfect systems should be extended to also include coupling between modes. B IBLIOGRAPHY [1] L. Krog and G. Rollema, Application of topology sizing and shape optimization methods to optimal design of aircraft components, - (2011). [2] B.-C. Bian and Y.-K. Sui. [3] M. Zhou, Topology optimization for shell structures with linear buckling responses, Computational Mechanics (2004). [4] B. Budianski, Theory of buckling and post-buckling bahaviour of elastic structures, Advances in applied mechanics 14 (1974). [5] A. Michell, The limits of economy of material in frame-structures, Philosophical Magazine (1904). [6] M. Bendsoe, Optimization of Structural Topology, Shape and Material (Springer-Verlag, New York, 1995). [7] L. Spunt, Optimum Structural Design (Prentice-Hall, Englewood Cliffs, 1971). [8] J. Thompson and W. Supple, Erosion of optimum design by compound branching phenomena, J. Mech. Phys. Solids. (1973). [9] M. Bendsoe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, Berlin, Germany, 2003). [10] M. Zhou, Difficulties in truss topology optimization with stress and local buckling constraints, JSME journal (1996). [11] H. Neves, M. M. an Rodrigues and J. Guedes, Generalized topology design of structures with a buckling load criterion, Structural optimization (1995). [12] S. Rahmatalla and C. C. Swan, Continuum topology optimization of buckling-sensitive structures, AIAA Journal (2003). [13] E. Lindgaard and J. Dahl, On compliance of buckling objective functions in topology optimization of snapthrough problems, Struct. Multidisc. Optim. (2013). [14] K. Ishii and S. Aomura, Topology optimization for the extruded three dimensional structure with constant cross section, JSME journal (2004). [15] U. Kirsch, On singular topologies in optimum structural design, Structural Optimization (1990). [16] N. Pedersen, Maximization of eigenvalues using topology optimization, Struct. Multidisc. Optim. (2000). [17] M. M. Neves, O. Sigmund, and M. P. Bendsoe, Topology optimization of periodic microstructures with a penalization of highly localized buckling modes, International journal for numerical methods in engineering (2002). [18] Z.-D. Ma, N. Kikuchi, and H.-C. Cheng, Topological design for vibrating structures, Comput. Methods Appl. Mech. Engrg. (1995). [19] K. Svanberg, The method of moving asymptotes - a new method for structural optimisation, International Journal for Numerical Methods in Engineering (1987). 51 A D ETAILED RESULTS Table A.1: Parameters and results of the test case with a distributed load on the top edge Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15 Case 16 PARAMETERS Domain Resolution [m] x [m] # of elements 10 x 10 30 x 30 10 x 10 30 x 30 10 x 10 30 x 30 10 x 10 30 x 30 10 x 10 10 x 10 10 x 10 10 x 10 10 x 10 50 x 50 10 x 10 50 x 50 10 x 10 30 x 30 10 x 10 30 x 30 10 x 10 30 x 30 10 x 10 30 x 30 10 x 20 30 x 60 10 x 20 30 x 60 20 x 10 60 x 30 20 x 10 60 x 30 Filter size [m] 0 0 0 0 0 0 0 0 0.4 0.7 0.9 1.1 0 0 0 0 λ∗cr 0 0.4864 0.6079 0.8106 0 2.7861 0 0.4247 0.6079 0.6079 0.6079 0.6079 0 0.3048 0 0.5499 53 λ increase % 0 20 50 100 0 50 0 50 50 50 50 50 0 50 0 50 R ESULTS First λcr Compliance # of iterations 0.40532 0.4864 0.6079 0.8106 1.8574 2.7861 0.2831 0.4247 0.6077 0.6079 0.6077 0.6077 0.2032 0.3048 0.3666 0.5819 0.23883 0.23814 0.22292 0.22349 0.0293 0.0556 0.6573 0.5960 0.2798 0.3161 0.3431 0.6079 0.4408 0.4222 0.4627 0.4349 33 30 51 161 22 21 40 86 67 80 74 118 33 37 29 34 54 A. D ETAILED RESULTS Table A.2: Parameters and results of the test case clamped on the upper and lower edge and a downward force inside the domain Case 17 Case 18 Case 19 PARAMETERS Domain Resolution [m] x [m] # of elements 10 x 20 10 x 60 10 x 20 10 x 60 10 x 20 10 x 60 Filter size [m] 0 0 0 λ∗cr 0 0.021 0.21 λ increase % 0 50 1400 R ESULTS First λcr Compliance # of iterations 0.014023 0.023437 0.5796 0.10822 0.12828 0.13822 36 33 41 Table A.3: Parameters and results of the test case clamped at the lower edge with a force to the right in the upper right corner Case 20 Case 21 Case 22 PARAMETERS Domain Resolution [m] x [m] # of elements 10 x 10 30 x 30 10 x 10 30 x 30 10 x 10 30 x 30 Filter size [m] 0 2.4613 24.613 λ∗cr 0 300 3900 λ increase % 50 50 50 R ESULTS First λcr Compliance # of iterations 0.61535 2.4614 3.9361 0.20466 0.20197 0.33377 24 71 300 55 (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.9 1.2 1.1 0.8 1 0.9 Buckling load Compliance 0.7 0.6 0.5 0.8 0.7 0.6 0.5 0.4 0.4 0.3 0.3 0.2 0 5 10 15 20 Iteration number 25 30 0.2 35 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 0.81924 Compliance = 0.23883 (g) Buckling mode 3 Figure A.1: Detailed results of test case nr. 1 0 5 10 15 20 Iteration number 25 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 0.40532 Compliance = 0.23883 (e) Buckling mode 1 Buckling mode4 Buckling load = 1.0617 Compliance = 0.23883 (h) Buckling mode 4 Buckling mode2 Buckling load = 0.68738 Compliance = 0.23883 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.0842 Compliance = 0.23883 (i) Buckling mode 5 30 35 56 A. D ETAILED RESULTS (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.9 1.3 1.2 0.8 1.1 1 Buckling load Compliance 0.7 0.6 0.5 0.9 0.8 0.7 0.6 0.4 0.5 0.3 0.4 0.2 0 5 10 15 Iteration number 20 25 30 0 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 1.009 Compliance = 0.23814 (g) Buckling mode 3 Figure A.2: Detailed results of test case nr. 2 5 10 15 Iteration number 20 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 0.4864 Compliance = 0.23814 (e) Buckling mode 1 Buckling mode4 Buckling load = 1.1575 Compliance = 0.23814 (h) Buckling mode 4 Buckling mode2 Buckling load = 0.78848 Compliance = 0.23814 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.16 Compliance = 0.23814 (i) Buckling mode 5 25 30 57 (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.9 1.3 1.2 0.8 1.1 0.7 Buckling load Compliance 1 0.6 0.5 0.9 0.8 0.7 0.4 0.6 0.3 0.2 0.5 0 10 20 30 Iteration number 40 50 0.4 60 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 0.62135 Compliance = 0.22292 (g) Buckling mode 3 Figure A.3: Detailed results of test case nr. 3 0 10 20 30 Iteration number 40 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 0.6079 Compliance = 0.22292 (e) Buckling mode 1 Buckling mode4 Buckling load = 0.85387 Compliance = 0.22292 (h) Buckling mode 4 Buckling mode2 Buckling load = 0.61404 Compliance = 0.22292 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.89613 Compliance = 0.22292 (i) Buckling mode 5 50 60 58 A. D ETAILED RESULTS (a) Optimised design First five buckling loads during optimization Objective value during optimization 0.65 1.4 0.6 1.3 0.55 1.2 Buckling load Compliance 0.5 0.45 0.4 1.1 1 0.9 0.35 0.8 0.3 0.7 0.25 0.2 0 20 40 60 80 100 120 Iteration number 140 160 180 0 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 0.81654 Compliance = 0.22349 (g) Buckling mode 3 Figure A.4: Detailed results of test case nr. 4 20 40 60 80 100 120 Iteration number 140 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 0.8106 Compliance = 0.22349 (e) Buckling mode 1 Buckling mode4 Buckling load = 0.88456 Compliance = 0.22349 (h) Buckling mode 4 Buckling mode2 Buckling load = 0.81076 Compliance = 0.22349 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.92202 Compliance = 0.22349 (i) Buckling mode 5 160 180 59 (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.08 5.5 5 0.07 4.5 4 Buckling load Compliance 0.06 0.05 0.04 3.5 3 2.5 2 0.03 1.5 0.02 0 5 10 15 Iteration number 20 1 25 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 2.7739 Compliance = 0.029306 (g) Buckling mode 3 Figure A.5: Detailed results of test case nr. 5 0 5 10 15 Iteration number 20 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 1.8574 Compliance = 0.029306 (e) Buckling mode 1 Buckling mode4 Buckling load = 4.5685 Compliance = 0.029306 (h) Buckling mode 4 Buckling mode2 Buckling load = 2.2847 Compliance = 0.029306 (f) Buckling mode 2 Buckling mode5 Buckling load = 5.018 Compliance = 0.029306 (i) Buckling mode 5 25 60 A. D ETAILED RESULTS (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.075 5 4.5 4 Buckling load Compliance 0.07 0.065 3.5 3 2.5 0.06 2 0.055 0 5 10 15 Iteration number 20 1.5 25 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 3.1367 Compliance = 0.055574 (g) Buckling mode 3 Figure A.6: Detailed results of test case nr. 6 0 5 10 15 Iteration number 20 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 2.7782 Compliance = 0.055574 (e) Buckling mode 1 Buckling mode4 Buckling load = 3.7094 Compliance = 0.055574 (h) Buckling mode 4 Buckling mode2 Buckling load = 2.789 Compliance = 0.055574 (f) Buckling mode 2 Buckling mode5 Buckling load = 3.7648 Compliance = 0.055574 (i) Buckling mode 5 25 61 (a) Optimised design First five buckling loads during optimization Objective value during optimization 2 1 1.8 0.9 0.8 Buckling load Compliance 1.6 1.4 1.2 1 0.7 0.6 0.5 0.4 0.8 0.3 0 5 10 15 20 25 Iteration number 30 35 0.2 40 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 0.48044 Compliance = 0.65728 (g) Buckling mode 3 Figure A.7: Detailed results of test case nr. 7 0 5 10 15 20 25 Iteration number 30 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 0.28312 Compliance = 0.65728 (e) Buckling mode 1 Buckling mode4 Buckling load = 0.88188 Compliance = 0.65728 (h) Buckling mode 4 Buckling mode2 Buckling load = 0.34685 Compliance = 0.65728 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.92715 Compliance = 0.65728 (i) Buckling mode 5 35 40 62 A. D ETAILED RESULTS (a) Optimised design First five buckling loads during optimization Objective value during optimization 2 0.9 0.8 0.7 Compliance Buckling load 1.5 0.6 0.5 1 0.4 0.3 0.5 0 10 20 30 40 50 Iteration number 60 70 80 0.2 90 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 0.43566 Compliance = 0.59601 (g) Buckling mode 3 Figure A.8: Detailed results of test case nr. 8 0 10 20 30 40 50 Iteration number 60 70 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 0.42471 Compliance = 0.59601 (e) Buckling mode 1 Buckling mode4 Buckling load = 0.46031 Compliance = 0.59601 (h) Buckling mode 4 Buckling mode2 Buckling load = 0.42534 Compliance = 0.59601 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.46663 Compliance = 0.59601 (i) Buckling mode 5 80 90 63 (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.75 1.4 0.7 1.3 0.65 1.2 1.1 Buckling load Compliance 0.6 0.55 0.5 0.45 0.9 0.8 0.4 0.7 0.35 0.6 0.3 0.25 1 0 10 20 30 40 Iteration number 50 60 0.5 70 (b) Objective during optimisation Deformation (d) Linear deformation Buckling mode3 Buckling load = 0.75501 Compliance = 0.27976 (g) Buckling mode 3 Figure A.9: Detailed results of test case nr. 9 0 10 20 30 40 Iteration number 50 (c) Buckling loads during optimisation Buckling mode1 Buckling load = 0.60774 Compliance = 0.27976 (e) Buckling mode 1 Buckling mode4 Buckling load = 0.94283 Compliance = 0.27976 (h) Buckling mode 4 Buckling mode2 Buckling load = 0.67496 Compliance = 0.27976 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.95561 Compliance = 0.27976 (i) Buckling mode 5 60 70 64 A. D ETAILED RESULTS (a) Optimised design First five buckling loads during optimization 1.4 0.7 1.3 0.65 1.2 0.6 1.1 Buckling load Compliance Objective value during optimization 0.75 0.55 0.5 1 0.9 0.45 0.8 0.4 0.7 0.35 0.6 0 10 20 30 40 50 Iteration number 60 70 0.5 80 (b) Objective during optimisation Buckling mode1 Buckling load = 0.60788 Compliance = 0.3161 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.73239 Compliance = 0.3161 (g) Buckling mode 3 10 20 30 40 50 Iteration number 60 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 0.9982 Compliance = 0.3161 (h) Buckling mode 4 Figure A.10: Detailed results of test case nr. 10 Buckling mode2 Buckling load = 0.66099 Compliance = 0.3161 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.0131 Compliance = 0.3161 (i) Buckling mode 5 70 80 65 (a) Optimised design First five buckling loads during optimization 1.4 0.7 1.3 0.65 1.2 0.6 1.1 Buckling load Compliance Objective value during optimization 0.75 0.55 0.5 1 0.9 0.45 0.8 0.4 0.7 0.35 0.6 0 10 20 30 40 50 Iteration number 60 70 0.5 80 (b) Objective during optimisation Buckling mode1 Buckling load = 0.60773 Compliance = 0.34314 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.86585 Compliance = 0.34314 (g) Buckling mode 3 10 20 30 40 50 Iteration number 60 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 1.2534 Compliance = 0.34314 (h) Buckling mode 4 Figure A.11: Detailed results of test case nr. 11 Buckling mode2 Buckling load = 0.72481 Compliance = 0.34314 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.3075 Compliance = 0.34314 (i) Buckling mode 5 70 80 66 A. D ETAILED RESULTS (a) Optimised design First five buckling loads during optimization Objective value during optimization 1.4 0.75 1.3 0.7 1.2 0.65 0.6 Compliance Buckling load 1.1 1 0.9 0.55 0.5 0.8 0.45 0.7 0.4 0.6 0.5 0 20 40 60 Iteration number 80 100 0.35 120 (b) Objective during optimisation Buckling mode1 Buckling load = 0.60767 Compliance = 0.35921 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.88464 Compliance = 0.35921 (g) Buckling mode 3 20 40 60 Iteration number 80 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 1.295 Compliance = 0.35921 (h) Buckling mode 4 Figure A.12: Detailed results of test case nr. 12 Buckling mode2 Buckling load = 0.74504 Compliance = 0.35921 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.3596 Compliance = 0.35921 (i) Buckling mode 5 100 120 67 (a) Optimised design Objective value during optimization First five buckling loads during optimization 1.8 1.1 1 1.6 0.9 0.8 Buckling load Compliance 1.4 1.2 1 0.7 0.6 0.5 0.4 0.8 0.3 0.6 0.2 0.4 0 5 10 15 20 Iteration number 25 30 0.1 35 (b) Objective during optimisation Buckling mode1 Buckling load = 0.20318 Compliance = 0.44075 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.49264 Compliance = 0.44075 (g) Buckling mode 3 5 10 15 20 Iteration number 25 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 0.52807 Compliance = 0.44075 (h) Buckling mode 4 Figure A.13: Detailed results of test case nr. 13 Buckling mode2 Buckling load = 0.32814 Compliance = 0.44075 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.53688 Compliance = 0.44075 (i) Buckling mode 5 30 35 68 A. D ETAILED RESULTS (a) Optimised design Objective value during optimization First five buckling loads during optimization 1.8 1.2 1.1 1.6 1 0.9 Buckling load Compliance 1.4 1.2 1 0.8 0.7 0.6 0.5 0.8 0.4 0.6 0.3 0.4 0 5 10 15 20 25 Iteration number 30 35 0.2 40 (b) Objective during optimisation Buckling mode1 Buckling load = 0.3048 Compliance = 0.42222 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.30515 Compliance = 0.42222 (g) Buckling mode 3 5 10 15 20 25 Iteration number 30 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 0.52866 Compliance = 0.42222 (h) Buckling mode 4 Figure A.14: Detailed results of test case nr. 14 Buckling mode2 Buckling load = 0.30503 Compliance = 0.42222 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.55755 Compliance = 0.42222 (i) Buckling mode 5 35 40 69 (a) Optimised design Objective value during optimization First five buckling loads during optimization 1.4 1.3 1.3 1.2 1.2 1.1 1 Buckling load Compliance 1.1 1 0.9 0.8 0.9 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0 5 10 15 Iteration number 20 25 30 0 (b) Objective during optimisation Buckling mode1 Buckling load = 0.36662 Compliance = 0.46272 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.56209 Compliance = 0.46272 (g) Buckling mode 3 10 15 Iteration number 20 (c) Buckling loads during optimisation Deformation (d) Linear deformation 5 Buckling mode4 Buckling load = 0.75132 Compliance = 0.46272 (h) Buckling mode 4 Figure A.15: Detailed results of test case nr. 15 Buckling mode2 Buckling load = 0.39389 Compliance = 0.46272 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.84799 Compliance = 0.46272 (i) Buckling mode 5 25 30 70 A. D ETAILED RESULTS (a) Optimised design Objective value during optimization First five buckling loads during optimization 1.4 1.3 1.3 1.2 1.2 1.1 1 Buckling load Compliance 1.1 1 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.9 0 5 10 15 20 Iteration number 25 30 0.4 35 (b) Objective during optimisation Buckling mode1 Buckling load = 0.58193 Compliance = 0.43495 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.67809 Compliance = 0.43495 (g) Buckling mode 3 5 10 15 20 Iteration number 25 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 1.1172 Compliance = 0.43495 (h) Buckling mode 4 Figure A.16: Detailed results of test case nr. 16 Buckling mode2 Buckling load = 0.67784 Compliance = 0.43495 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.1397 Compliance = 0.43495 (i) Buckling mode 5 30 35 71 (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.9 1.4 0.8 1.2 0.7 1 Buckling load Compliance 0.6 0.5 0.4 0.6 0.4 0.3 0.2 0.2 0.1 0.8 0 5 10 15 20 25 Iteration number 30 35 0 40 (b) Objective during optimisation Buckling mode1 Buckling load = 0.014023 Compliance = 0.10822 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.17864 Compliance = 0.10822 (g) Buckling mode 3 5 10 15 20 25 Iteration number 30 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 0.34684 Compliance = 0.10822 (h) Buckling mode 4 Figure A.17: Detailed results of test case nr. 17 Buckling mode2 Buckling load = 0.082124 Compliance = 0.10822 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.50813 Compliance = 0.10822 (i) Buckling mode 5 35 40 72 A. D ETAILED RESULTS (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.9 1 0.9 0.8 0.8 0.7 0.7 Buckling load Compliance 0.6 0.5 0.4 0.6 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 5 10 15 20 Iteration number 25 30 0 35 (b) Objective during optimisation Buckling mode1 Buckling load = 0.023437 Compliance = 0.12828 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.088333 Compliance = 0.12828 (g) Buckling mode 3 5 10 15 20 Iteration number 25 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 0.1397 Compliance = 0.12828 (h) Buckling mode 4 Figure A.18: Detailed results of test case nr. 18 Buckling mode2 Buckling load = 0.076157 Compliance = 0.12828 (f) Buckling mode 2 Buckling mode5 Buckling load = 0.14876 Compliance = 0.12828 (i) Buckling mode 5 30 35 73 (a) Optimised design Objective value during optimization First five buckling loads during optimization 1.1 1.4 1 1.2 0.9 1 Buckling load Compliance 0.8 0.7 0.6 0.5 0.4 0.8 0.6 0.4 0.3 0.2 0.2 0.1 0 5 10 15 20 25 Iteration number 30 35 40 0 45 (b) Objective during optimisation Buckling mode1 Buckling load = 0.5796 Compliance = 0.13822 (e) Buckling mode 1 Buckling mode3 Buckling load = 0.92826 Compliance = 0.13822 (g) Buckling mode 3 5 10 15 20 25 Iteration number 30 35 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 0.97088 Compliance = 0.13822 (h) Buckling mode 4 Figure A.19: Detailed results of test case nr. 19 Buckling mode2 Buckling load = 0.75519 Compliance = 0.13822 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.0879 Compliance = 0.13822 (i) Buckling mode 5 40 45 74 A. D ETAILED RESULTS (a) Optimised design Objective value during optimization First five buckling loads during optimization 2 0.7 0.65 1.8 0.6 1.6 Buckling load Compliance 0.55 0.5 0.45 0.4 0.35 1.4 1.2 1 0.3 0.8 0.25 0.2 0 5 10 15 Iteration number 20 25 0 (b) Objective during optimisation Buckling mode1 Buckling load = 0.61535 Compliance = 0.20466 (e) Buckling mode 1 Buckling mode3 Buckling load = 1.3701 Compliance = 0.20466 (g) Buckling mode 3 10 15 Iteration number 20 (c) Buckling loads during optimisation Deformation (d) Linear deformation 5 Buckling mode4 Buckling load = 1.7659 Compliance = 0.20466 (h) Buckling mode 4 Figure A.20: Detailed results of test case nr. 20 Buckling mode2 Buckling load = 1.0804 Compliance = 0.20466 (f) Buckling mode 2 Buckling mode5 Buckling load = 1.8131 Compliance = 0.20466 (i) Buckling mode 5 25 75 (a) Optimised design Objective value during optimization First five buckling loads during optimization 0.7 3.5 0.65 3 0.6 2.5 Buckling load Compliance 0.55 0.5 0.45 0.4 2 1.5 0.35 0.3 1 0.25 0.2 0 10 20 30 40 50 Iteration number 60 70 0.5 80 (b) Objective during optimisation Buckling mode1 Buckling load = 2.4614 Compliance = 0.20197 (e) Buckling mode 1 Buckling mode3 Buckling load = 2.5568 Compliance = 0.20197 (g) Buckling mode 3 10 20 30 40 50 Iteration number 60 (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 2.7752 Compliance = 0.20197 (h) Buckling mode 4 Figure A.21: Detailed results of test case nr. 21 Buckling mode2 Buckling load = 2.5384 Compliance = 0.20197 (f) Buckling mode 2 Buckling mode5 Buckling load = 2.8598 Compliance = 0.20197 (i) Buckling mode 5 70 80 76 A. D ETAILED RESULTS (a) Optimised design Objective value during optimization First five buckling loads during optimization 1 4.5 4 0.9 3.5 0.8 3 Buckling load Compliance 0.7 0.6 0.5 2.5 2 1.5 0.4 1 0.3 0.2 0.5 0 50 100 150 200 Iteration number 250 0 300 (b) Objective during optimisation Buckling mode1 Buckling load = 3.9861 Compliance = 0.33377 (e) Buckling mode 1 Buckling mode3 Buckling load = 4.0629 Compliance = 0.33377 (g) Buckling mode 3 50 100 150 200 Iteration number (c) Buckling loads during optimisation Deformation (d) Linear deformation 0 Buckling mode4 Buckling load = 4.1056 Compliance = 0.33377 (h) Buckling mode 4 Figure A.22: Detailed results of test case nr. 22 Buckling mode2 Buckling load = 3.9923 Compliance = 0.33377 (f) Buckling mode 2 Buckling mode5 Buckling load = 4.1228 Compliance = 0.33377 (i) Buckling mode 5 250 300

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