. Structural Design Optimization of Vibration Isolating Structures by Max van der Kolk in partial fulﬁllment of the requirements for the degree of Master of Science in Mechanical Engineering at Delft University of Technology, to be defended publicly on Monday, 26 October 2015 at 09:45. Student number: Project duration: Thesis committee: 4079035 1 December 2014 – 26 October 2015 Prof. dr. ir. F. van Keulen, TU Delft Prof. dr. ir. K.M.B. Jansen, TU Delft Dr. ir. J.W. van Wingerden, TU Delft Dr. ir. M. Langelaar, TU Delft Dr. ir. G.J. van der Veen, TU Delft Dr. ir. J. de Vreugd, TNO An electronic version of this thesis is available at http://repository.tudelft.nl/. Abstract The design of high performance instruments often involves the attenuation of poorly damped resonant modes. These resonant modes are a limiting factor to the performance of these instruments. Current design approaches typically start from a baseline design and introduce stiﬀening or damping reinforcements to tune and/or damp these modes. However, the inﬂuence on the structural damping of these reinforcements is diﬃcult to predict and often results in trial and error-based design approaches for the design of damping reinforcements. A common solution is to introduce viscoelastic material in baseline designs to increase structural damping. These materials dissipate energy when subjected to deformation and should therefore be located at positions which undergo large deformations during vibration. Typically, the viscoelastic material is placed in conventional (un)constrained layer damping conﬁgurations. However, to achieve optimized damping characteristics both the location as well as the geometry of viscoelastic material should be optimized. In this thesis, a multi-material topology optimization routine is presented as a systematic methodology to develop structures with optimal damping characteristics. The proposed method applies a multi-material, parametric level set-based approach to simultaneously distribute structural and viscoelastic material within the design domain. The developed optimization routine allows for the design of freeform, viscoelastic dampers without the limitation to conventional (un)constrained layer damping conﬁgurations and is thereby able to achieve improved damping characteristics. The structural loss factor is applied as a performance measure to compare the damping between diﬀerent viscoelastically damped structures and as objective function during the optimization. The viscoelastic material behavior is represented by a complex-valued material modulus, which results in a complex-valued eigenvalue problem. The formulation of the structural loss factor is modiﬁed to account for the complex-valued eigensolutions, resulting in accurate assessment of the structural loss factor for designs containing viscoelastic material with high material loss factors. The optimization routine maximizes the structural loss factor for single or multiple selected eigenmodes. An adjoint sensitivity analysis is performed to provide an exact expression for the structural loss factor sensitivity. Based on this sensitivity information, the optimization routine is able to develop structures with optimized loss factors for the speciﬁed eigenmodes. The method is also able to generate damping solutions for existing designs containing badly damped resonant modes. iii Preface This thesis is the ﬁnal result of my Master’s studies at Delft University of Technology. The project has been a collaboration between the Structural Optimization and Mechanics group within the department of Precision and Microsystems Engineering and the Optomechatronics department at the Netherlands Organization for Applied Scientiﬁc Research (TNO) in Delft. I am glad to have gotten the opportunity to work on my graduation project at TNO after completing my internship here. A lot of freedom was available to formulate this project, which allowed me to combine many diﬀerent aspects from the courses I enjoyed most. The result has been an exciting and equally challenging graduation project. The presented thesis would not be the same without my supervisors. I would like to thank Jan de Vreugd for his supervision during my internship and for providing the opportunity to do my graduation work at the Optomechatronics department. Our weekly meetings have been fun and resulted in many interesting discussions. Also, thanks are to Gijs van der Veen for the discussions we had on the theoretical aspects of the work. Furthermore, thanks are to Matthijs Langelaar for his enthusiasm in supervising this project and his critical questioning during our progress meetings. Finally, I would like to thank all of them for providing the opportunity to write a scientiﬁc publication, the contributions they made, and their eﬀorts in revising the draft manuscripts. These past years in Delft have been challenging and exciting, but most of all, they were a lot of fun. I am grateful for the many great friends I got to meet and the time we spend together. I am sure that these years would not have been the same without you! At last, I would like to thank my family, especially my parents, for their support and kindness throughout these years and before. Max van der Kolk Delft, October 2015 v Contents Abstract iii Preface v List of Figures ix List of Tables xi Nomenclature xiii 1 Introduction 1 2 Topology optimization 2.1 Optimization methods . . . . . . . . . . . . 2.1.1 Density-based optimization . . . . . 2.1.2 Evolutionary structural optimization 2.1.3 Level set-based optimization . . . . . 2.1.4 Black and white design . . . . . . . . 2.1.5 Multi-material topology optimization 2.2 Comparison of optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 6 7 7 8 8 3 Parametric level set method 3.1 Level set method . . . . . . . . . . 3.2 Parametric level set method . . . . 3.2.1 Shape functions . . . . . . . 3.2.2 Multi-material formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 14 15 17 . . . . . . . . . . . . . . . . . 19 20 20 22 22 23 24 25 26 26 27 28 28 28 31 33 34 35 . . . . . . . . . . . . . . . . . . . . 4 Topology optimization of viscoelastically damped structures Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modeling of Viscoelastic Material . . . . . . . . . . . . . . . . . . . . 4.2.1 Complex material modulus. . . . . . . . . . . . . . . . . . . . 4.2.2 Structural loss factor . . . . . . . . . . . . . . . . . . . . . . . 4.3 Topology optimization of viscoelastic and structural material . . . . . 4.3.1 Multi-material boundary representation . . . . . . . . . . . . . 4.3.2 Parametric level set method . . . . . . . . . . . . . . . . . . . 4.3.3 Optimization problem for structural loss factor maximization . 4.3.4 Adjoint sensitivity analysis . . . . . . . . . . . . . . . . . . . . 4.3.5 Numerical implementation . . . . . . . . . . . . . . . . . . . . 4.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Case study I: cantilever beam . . . . . . . . . . . . . . . . . . 4.4.2 Design inﬂuences by complex moduli . . . . . . . . . . . . . . 4.4.3 Case study II: existing structure with limited design domain . 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Base excitation and thermal analysis 37 5.1 Base excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.3 Thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 vii viii Contents 6 Discussion 45 7 Conclusions and recommendations 47 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Bibliography 49 53 A Viscoelastic material model A.1 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 A.2 Energy storage and dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 A.3 Loss factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 B Numerical implementation B.1 Finite element model . . . . . . . . . . . . . . B.1.1 Thermal model . . . . . . . . . . . . . B.1.2 Radial basis function placement . . . . B.2 Heaviside function: discrete and approximate. B.3 From Level set function to topology . . . . . . C Loss factor sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 57 59 60 60 64 67 D MATLAB Code 71 D.1 Main ﬁle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 D.2 Optimization loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 D.3 Example case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 E Commercial FEM implementation 75 List of Figures 1.1 1.2 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 Part of GAIA BAM: a high performance optomechatronic instrument developed at TNO for angle measurement between stars. . . . . . . . . . . . . . . . . . . . . . . . . An illustration of the unconstrained and constrained layer damping methods. . . . . . 1 2 Three most common optimization methods in structural design. The optimization methods from top to bottom: (a) size, (b) shape, c topology optimization. . . . . . . 6 Two examples of designs obtained with the optimization routine developed by Robinson. 9 Examples of designs obtained by a single and multi-material BESO method. . . . . . . 9 Examples of designs obtained by a single and multi-material parametric level set method. 10 An illustration of the level set function φ(X) and the corresponding domain Ω and boundary Γ as described by the level set φ = c. . . . . . . . . . . . . . . . . . . . . . . Illustration of a cross section of the level set function and its parameterization. . . . . (a): Illustration showing the cross section of the radial basis function placement. The RBFs are aligned with the elements in the ﬁnite element discretization. The shape functions ξi and the expansion coeﬃcients αi are aligned with the ith element in the design domain. (b): Illustration of a compactly-supported radial basis function as described by Equation (3.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of a multi-material structure deﬁned by two level set functions. . . . . . . Illustration of the clamped cantilever beam containing a constrained viscoelastic layer. The arrow indicates the location of the excitation force and position measurement during dynamic loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Illustration of a multi-material structure deﬁned by two level set functions. The ﬁrst level set function φ1 determines the placement of material, while the second level set function φ2 distinguishes between structural and viscoelastic material. . . . . . . . . . 4.3 The design domain for case study I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The design progression throughout the optimization routine at iteration 2, 5, 20 and 35 and the corresponding objective values. . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The history of the objective values (a) and corresponding loss factors (b) during the optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The history of the volume (top) and frequency (bottom) constraint values during the optimization. The constraints are normalized with respect to the speciﬁed maximum value. Positive values represent unsatisﬁed constraints . . . . . . . . . . . . . . . . . . 4.7 The deformation shapes and corresponding maximum shear strain εmax for the ﬁrst (a) and second (b) eigenmode of the optimized structure. The uncolored elements represent void elements within the design. . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 The energy dissipated for each element for the ﬁrst eigenmode of the structure. The values are normalized with respect to the maximum energy dissipated in an element. The obtained result is directly related to the shear strain given in Figure 4.7a. The uncolored elements represent void elements within the design. . . . . . . . . . . . . . . 4.9 A comparison between the optimized designs and their performance for the application of multiple complex material models: (a): shear, (d): Young’s and (g): bulk modulus. The diﬀerences in performance are highlighted by the maximum principal and maximum shear strains within the structure for the ﬁrst eigenmode. . . . . . . . . . . . . . . . . 4.10 The design and non-design domains for case study II. The non-design domain is clamped at both sides and cannot be modiﬁed by the optimization routine. . . . . . . . . . . . 4.11 The ﬁnal design of the optimization achieved after 58 iterations. The design achieved an objective value of J = 0.3803. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 14 16 17 4.1 ix 24 26 29 30 30 30 31 31 32 33 34 x List of Figures 4.12 The history of the loss factor (a) and normalized volume and frequency constraints (b) during the optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 The deformation shape for the ﬁrst eigenmode of the optimized structure of case study II. The dissipated energy in each element is plotted to illustrate the energy dissipation for the ﬁrst eigenmode. The uncolored elements correspond to void elements in the design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 Four diﬀerent viscoelastic dampers: (a): single element viscoelastic layer, (b): three elements viscoelastic layer, (c): a viscoelastic layer matching the thickness of the viscoelastic layer in the optimized design, and (d): the design generated by the optimization. The presented loss factors correspond to their ﬁrst resonant mode. . . . . . . . . Frequency response of four structures damped using viscoelastic material. The response of the optimized structure is compared to thee other designs implementing a constrained layer damping conﬁguration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The ﬁgures illustrate the dissipated energy per cycle for each elements in the design. In each illustration the values are normalized with respect to the element with maximum energy dissipation. The distributions are given for the ﬁrst and second resonant mode of the optimized design in ﬁgures (a) and (b), and for a CLD conﬁguration in ﬁgures (c) and (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustrations of the heat transfer problem for a generalized structure and the situation applied to the optimized structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalized temperature distribution as result of the energy dissipation due to the deformation of the viscoelastic material. . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Illustration of the phase diﬀerence between strain ε(t) and stress σ(t) for a harmonically excited viscoelastic material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Illustration of the frequency and temperature dependent behavior of the shear modulus and loss factor of a viscoelastic material. The loss factor shows a maximum for certain operating frequency and temperatures. Image modiﬁed from Beards (1997). . . . . . . A.3 (a): Illustration of a hysteresis loop of a viscoelastic material subjected to harmonic excitation. The loop is found by plotting the stress σ(t) as function of the strain ε(t). (b): The complex material modulus drawn in the complex plane to express the modulus as function of the storage or loss moduli using the phase angle θ. . . . . . . . . . . . . B.1 Illustration of a Q4 element applied for the discretization of the design domain. . . . . B.2 One-dimensional illustration of the radial basis function placements. . . . . . . . . . . B.3 Approximations of the discrete Heaviside function and its derivative. Both approximated formulations are plotted for various values of the parameter ∆. For decreasing values of ∆ the approximations approach the discrete formulations. . . . . . . . . . . . B.4 Illustration of the diﬀerence between discrete and approximate Heaviside functions on the material distribution and element densities within the ﬁnite element discretization. B.5 Visualization of both level set functions as presented by the ﬁnal iteration of the optimization of case study I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6 The material distribution at four iterations during the optimization of case study I. . . B.7 The material distribution at four iterations during the optimization of case study II. . E.1 A schematic diagram of the optimization procedure using a dedicated ﬁnite element environment to perform the FE calculations. The green blocks are performed within the FE environment, while the orange blocks are evaluated in MATLAB or similar programming environments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 34 38 39 40 42 43 53 54 55 58 60 61 63 64 65 65 76 List of Tables 4.1 4.2 Comparison between the structural loss factors and Q-factor of a structure containing a constrained viscoelastic layer (Figure 4.1). The loss factors γ̄ and γ are determined using the undamped and damped eigenmodes. The relative diﬀerence is calculated between the structural loss factors and the inverse of the Q-factor to illustrating the improved prediction of damping behavior when using the complex-valued eigenmodes for the loss factor calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A comparison between the loss factors for the optimized designs, when the performance is evaluated with diﬀerent material models than used during the optimization. . . . . . xi 25 33 Nomenclature Symbols αi Expansion coeﬃcient corresponding to shape function ξi αi,max Upper bound applied to the expansion coeﬃcients αi,min Lower bound applied to the expansion coeﬃcients ∆ Parameter for approximate Heaviside formulation δ Kronecker delta ε Strain ε0 Strain amplitude εmax Maximum in-plane shear strain ε1,2 Maximum in-plane principal strains η Material loss factor Γk The boundary described by the kth level set function γ̄r The structural loss factor based on the rth eigenmode of an undamped system γr The structural loss factor based on the rth eigenmode of a damped system γr⋆ Adjoint formulation of the structural loss factor γr κ Bulk modulus or thermal conductivity in context of thermal analysis λr Eigenfrequency of the rth eigensolution µ1 Adjoint variables µ2 Adjoint variable Ωk Domain enclosed by the kth level set function ω Excitation frequency φ Level set function ϕr Eigenmode r of a damped system ψr Eigenmode r of an undamped system σ Stress σ0 Stress amplitude ξ Shape function a Parameter for approximate Heaviside formulation b Parameter for approximate Heaviside formulation D Available domain for level set functions D Diﬀusive term in Hamilton-Jacobi equation E Young’s modulus xiii xiv Notation E Elasticity tensor f Externally applied forces G Shear modulus h Convective heat transfer coeﬃcient H(φ) i Heaviside function of the argument φ √ The complex number: i = −1 K Global stiﬀness matrix Ke Element stiﬀness matrix KTc Conductivity matrix KTh Convective matrix M Global mass matrix Me Element mass matrix Q Q-factor Qcyc Dissipated power per unit volume ri Radius corresponding to shape function ξi R Inﬂuence radius corresponding to the shape function ξ R Reactive term in Hamilton-Jacobi equation Rh Heat load introduced by convective boundary condition Rq Heat load introduced by dissipation in viscoelastic material t Time or pseudo-time variable T∞ Environment temperature T Nodal temperatures V k k Vmax V e Volume enclosed by the domain of the kth level set function Volume constrained applied to the kth level set function Volume of a single ﬁnite element vn Velocity ﬁeld in level set-based methods Wcyc Dissipated energy per unit volume per cycle X Describes a point within the design domain D x Nodal displacements Notation ′ Denotes the storage (real) component of the material modulus, ′′ Denotes the loss (imaginary) component of the material modulus T Transpose of a vector or matrix (superscript only) H Conjugate transpose of a complex vector or matrix (superscript only) R Real component of the vector or matrix (subscript only) I Imaginary component of the vector or matrix (subscript only) Acronyms xv ℜ() Real component of the argument ℑ() Imaginary component of the argument ∇ Gradient div Divergence Acronyms CFL Courant-Friedrichs-Lewy CLD Constrained Layer Damping CSRBF Compactly-Supported Radial Basis Function ERR Element Removal Ratio FE Finite Element FEM Finite Element Method HJ Hamilton-Jacobi LSF Level Set Function LSM Level Set Method MMA Method of Moving Assymptotes ODE Ordinary Diﬀerential Equation PDE Partial Diﬀerential Equation PLSM Parametric Level Set Method RBF Radial Basis Function TNO Netherlands Organisation for Applied Scientiﬁc Research TO Topology Optimization 1 Introduction The design of high performance instruments often involves the attenuation of badly damped resonant modes. These resonant modes are a limiting factor to the performance of these instruments. The Optomechatronics department at the Netherlands Organization for Applied Scientiﬁc Research (TNO) focusses on the development and design of high performance optomechatronic instruments, such as GAIA BAM, which is shown in Figure 1.1, and MSI VNS1 . GAIA BAM measures the angle between two stars using two telescopes and is accurate to 24 microarcsec. To put this measurement in perspective, it is comparable to measuring the diameter of a human hair at a distance of 1000 km. Next to the design challenges to achieve this performance, the instruments are designed to operate in space: a highly challenging environment. The structures are subjected to strict weight constraints to aid the transportation of the instrument and are subjected to large temperature variations during operation. To achieve the required performance the instruments are mostly constructed from thinwalled, monolithic components to maintain dimensional stability with limited mass. The construction of these structures results in low intrinsic damping levels (de Vreugd et al., 2014). If badly damped resonant modes are excited during launch, transportation or operation they might introduce signiﬁcant – possibly damaging – dynamic stresses in the instrument. Figure 1.1: Part of GAIA BAM: a high performance optomechatronic instrument developed at TNO for angle measurement between stars. To limit the inﬂuence of external vibrations attempts are made to isolate the instruments from these external vibrations and by attenuating badly damped resonant modes of the structure. Current design methods typically start from a baseline design and introduce stiﬀening and/or damping reinforcements 1 The reader is referred to Gielesen et al. (2013) and Tabak et al. (2013) for more information on these instruments. 1 2 1. Introduction (a) Unconstrained. (b) Constrained. Figure 1.2: An illustration of (a): unconstrained and (b): constrained layer damping methods in which the viscoelastic material (green) is combined with structural, rigid material (gray). During resonance the viscoelastic material is subjected to extensional or shear strains for the unconstrained and constrained conﬁgurations. to tune and damp speciﬁc resonant modes. In most cases the stiﬀening reinforcements are applied to increase the resonance frequency of the badly damped resonant modes outside the frequency spectrum of the expected external vibration. By locally introducing damping reinforcements an attempt is made to signiﬁcantly increase the damping, such that no damaging stresses are caused by the external vibration. However, the inﬂuences of these reinforcements are diﬃcult to predict and optimal damping characteristics are hard to achieve. This results in a trail and error based approach until the desired performance is found. A new design methodology is required to develop a design approach for components with speciﬁed or optimized damping performance. Damping reinforcements are typically realized by adding viscoelastic material to the structure. The viscoelastic material dissipates energy when subjected to deformation. Thus, to suppress the response of a resonant mode the viscoelastic material is placed at locations which undergo large deformation when in resonance. The viscoelastic material is cyclicly deformed and thereby dissipates energy from the system reducing the response to the external vibrations and improving the dynamic performance of the instrument. Multi-material structures combining structural and viscoelastic materials are known to provide high structural damping and are widely applied in many diﬀerent ﬁelds (Johnson, 1995). Applications are for example found in automotive and aerospace industries (Rao, 2003; Rittweger et al., 2002; Wang et al., 2008), and to increase structural damping in civil applications (Samali and Kwok, 1995; Wang et al., 2013). The viscoelastic material is commonly applied in two diﬀerent conﬁgurations: unconstrained and Constrained Layer Damping (CLD) (Grootenhuis, 1970). Both methods are visualized in Figure 1.2 and are deﬁned as: • Unconstrained layer damping: the viscoelastic material is bonded to one or both sides of a structural, load-carrying base member. When in resonance the ﬂexing of the structure introduces extensional strains into the damping layers. • Constrained layer damping: one or multiple viscoelastic layers are sandwiched and bonded between layers of structural, load-carrying base members. During resonance the ﬂexing of the multilayer structure will introduce shear strains into the damping layers. Each component within a high performance instrument should show optimal performance to satisfy the strict requirements and speciﬁcations. Optimal performance might be deﬁned diﬀerently for each component. In the case of damping reinforcements the objective is to achieve maximized damping characteristics. The challenge in designing viscoelastic dampers is to reach optimal damping performance with limited viscoelastic material. Also, the introduced viscoelastic material should not inﬂuence the structural integrity of the design. To approach these optimization problems either heuristic methods or mathematical optimization routines are applied. By applying experience and knowledge from previous design iterations, the heuristic methods are capable to generate designs which perform relatively well. However, the mathematical optimization routines provide a rigorous, systematic approach to ﬁnd optimized designs, especially 3 for designs involving challenging, multi-physical problems. In structural optimization problems three common optimization routines are applied: size, shape and topology optimization. The latter method allows to generate complicated, freeform structures with optimized performance for many diﬀerent design objectives. This method has become an active research topic in recent years, especially with improvements made in ﬁnite element analysis, additive manufacturing and the increased availability of computational power. Even though advanced topology optimization packages are becoming available for commercial use, the technology is far from fully developed and continuous eﬀorts are made to improve the optimization routines. Structural optimization has also been applied for the development of viscoelastic dampers. The objective is to maximize the damping characteristics using a combination of viscoelastic and structural material. Initial attempts applied shape optimization to unconstrained and constrained layer dampers (Plunkett and Lee, 1970). Various parameters of the conventional damping conﬁgurations have been used as design variable in the optimization. Besides optimizing the layout of the structural and viscoelastic material, the optimal placement of the viscoelastic dampers have been evaluated using optimization routines. For example to optimize the locations of viscoelastic patches to dampen the response of a frame subjected to external vibrations (Lunden, 1980). Recently, the routines using size and shape optimization have been extended to topology optimization. These implementations mainly investigated the distribution of viscoelastic materials within (un)constrained layer conﬁgurations. Numerous studies are available which investigate the optimal distribution of a limited volume of viscoelastic material, such that maximized damping characteristics are obtained. Studies have investigated vibrating structures, such as simply supported beams, plates and shells (Zheng et al., 2004; El-Sabbagh and Baz, 2013; Zheng et al., 2013). Even though these studies provide optimized results, the designs are still limited to predeﬁned (un)constrained layer damping conﬁgurations. Before the optimization is initialized the designer has to predeﬁne the dimensions and location of the viscoelastic layer within the design domain, possibly limiting the achievable performance. This work presents a multi-material, parametric topology optimization method for the design of viscoelastic dampers to overcome restrictions of existing optimization methods. The viscoelastic and structural material are distributed simultaneously throughout the design domain to achieve optimized damping characteristics. There is no need to predeﬁne a location of the viscoelastic layer, as both materials are distributed simultaneously. Besides, the optimization routine is able to develop arbitrary, freeform designs, since the viscoelastic material is not limited to a certain region within the design domain. The ability to generate freeform viscoelastic dampers is expected to provide higher levels of structural damping compared to conventional (un)constrained layer conﬁgurations. In the next chapter the principles of topology optimization are introduced. Common methods of topology optimization are evaluated and discussed to motivate the decision to apply a parametric level set method for the optimization of viscoelastic dampers. The mathematical background of level set-based methods and the extension to multi-material, parametric methods are presented in Chapter 3. The developed method is applied for the optimization of viscoelastic structures in Chapter 4. It is to be noted that this chapter has been submitted as article to Journal of Vibration & Control and the author apologizes for possible overlapping material discussed in the preceding chapters. The article discusses the viscoelastic material model, the objective function and sensitivity analysis. Also, two case studies are treated to illustrate the performance of the optimization routine. Moreover, diﬀerent material models are applied for the viscoelastic material to illustrate the importance to select a viscoelastic material model which accurately describes its material behavior before initializing the optimization. The thesis continuous with Chapter 5, which investigates the thermal behavior of the viscoelastic dampers during operation. Chapter 6 discusses the results obtained using the optimization routine. Finally, conclusions and recommendations are presented in Chapter 7. 2 Topology optimization This chapter introduces the general principles of topology optimization and its application in structural design. The most common methods applied in structural topology optimization are introduced and brieﬂy discussed. A comparison between the performance of these methods for multi-material applications is presented. Finally, a method is chosen for the optimization of viscoelastically damped structures. 2.1. Optimization methods In structural mechanics three types of optimization are most common: size, shape and topology optimization (Bendsøe and Sigmund, 2003). The methods are distinguished based on the design variables used during the optimization and the freedom provided to the optimizer. For size optimization the routine is initialized with a predeﬁned topology. Design parameters used to parameterize the topology are provided as design variable to the optimization, e.g. truss length or cross-sectional area. Only these parameters are allowed to be modiﬁed during the optimization routine. The top ﬁgure of Figure 2.1 illustrates the shape optimization for a bridge-like structure. Shape optimization extends the size optimization by introducing the shape of boundaries, holes and cross-sectional areas as design variable. A simpliﬁed topology is presented as initial design for the optimization routine. The shape optimization has a larger design freedom compared to size optimization and is able to generate complicated boundaries within the design to achieve optimized performance, as illustrated by the middle ﬁgure in Figure 2.1. In topology optimization the optimization has even more freedom. The density of each element in the ﬁnite element discretization is presented as design variable. During the optimization the element densities are scaled to represent either solid or void material and thereby generate a new topology to show optimized performance for the objective of interest. This method is illustrated in the bottom image in Figure 2.1. Topology optimization is a powerful technique to generate complicated, freeform designs with optimized performance for a speciﬁc objective. The method does not require any conceptual or base-line design to initialize the optimization routine. Therefore, topology optimization will be applied for the synthesis and optimization of vibration isolating structures. To generate viscoelastic dampers, we require a multi-material topology optimization routine to distribute both materials throughout the design domain. Many researchers have worked on the formulation and implementation of topology optimization routines and to provide a complete literature overview on the available methods is outside the scope of this work. However, the following sections provide context regarding common optimization formulations and motivate the decision to develop a multi-material, parametric level set method for the structural optimization of vibration isolating structures. 2.1.1. Density-based optimization In density-based optimization methods a density value is assigned to each element within the ﬁnite element discretization of the design domain. These density, or pseudo-density, values are supplied as design variables to the optimizer. The density of each element is scaled between solid and void 5 6 2. Topology optimization Figure 2.1: Three most common optimization methods in structural design. The optimization methods from top to bottom: a) size, b) shape, c) topology optimization. The initial structure is given on the left with the optimized result on the right. Image source: Bendsøe and Sigmund (2003). material. By scaling the element densities the optimization routine is able to develop topologies with optimized performance with respect to the speciﬁed design objective. Either a discrete or continuous scaling is applied for the element densities. The discrete method results in a raster of solid and void elements. However, due to diﬃculties regarding discrete optimization, the most common methods scale the elements continuously. The continuous scaling allows to apply gradient-based solution methods to iteratively converge towards an optimized design. The pseudo-densities are scaled linearly between zero and one, representing void and solid material. Similar scaling can be applied to the remaining material properties of interest (Bendsøe, 1989): Eijkl (X) = ρ(X)Ēijkl , 0 ≤ ρ ≤ 1. (2.1) Here E describes the elasticity tensor corresponding to point X in the design domain. The elasticity tensor is a scaled version of the constant tensor of the material Ē. The pseudo-density ρ can be interpreted as the local density of the material for the element located at X. The intermediate material properties might provide optimized performance for certain objective functions. However, from a manufacturing point of view, the occurance of intermediate material in ﬁnal designs is unwanted. A penalization is applied to force the design towards solid and void elements. This is realized by applying an exponential scaling of the pseudo-densities as: Eijkl (X) = ρ(X)p Ēijkl , 0 ≤ ρ ≤ 1, p > 1. (2.2) This approach is referred to as Solid Isotropic Material with Penalization (SIMP) (Bendsøe, 1989; Rozvany et al., 1992). Commonly a penalization parameter of p = 3 is applied to obtain designs with minimal intermediate materials. The density-based methods are often combined with gradient-based optimization routines to ﬁnd the optimal topology for the speciﬁed objective. The sensitivities with respect to the objective function are determined on element level. During each iteration the element densities are modiﬁed using the sensitivity information to achieve an updated topology with improved performance. Depending on the complexity of the objective function and the number of design variables the sensitivities are either calculated analytically, semi-analytically or numerically. 2.1.2. Evolutionary structural optimization In many aspects the Evolutionary Structural Optimization (ESO) method is similar to the discussed density-based approaches. The ESO method ﬁrst emerged in publications by Xie and Steven (1993, 1997) and have since then been applied to various structural optimization problems. In contrast to the density-based approaches the ESO method applies discrete element densities: each element is either solid or void. The discrete densities result in a discrete optimization problem. To overcome 2.1. Optimization methods 7 diﬃculties regarding discrete optimization the ESO method applies a heuristic updating step to iteratively update the topology. The sensitivity number of each element is determined with respect to the optimization objective. A ranking of all elements is created based on the value of their sensitivity numbers. In each iteration a fraction of the elements with the lowest sensitivity are removed from the design domain. The number of elements to remove are based on the Evolutionary Removal Ratio (ERR), which speciﬁes the ratio of elements removed with respect to all elements in the ﬁnite element discretization. The ESO method is continued until a minimum volume fraction is achieved. A downside of the ESO method is the ability to only remove elements from the structure, regardless if the sensitivities would indicate otherwise. Especially during later iterations where the reintroduction of material might increase the performance of the design. The possibility to reintroduce elements has been introduced in the Bi-directional Evolutionary Structural Optimization (BESO) method. The rejection ratio and inclusion ratio are used to determine the ratio of elements which are removed and introduced during each iteration. A similar heuristic ranking is made based on the element sensitivities to determine which elements rejected or included. 2.1.3. Level set-based optimization The Level Set Method (LSM) applies a fundamentally diﬀerent approach to describe and update the topology. The method was originally developed by Osher and Sethian (1988) for the numerical computation of front and boundary propagation in surface motion problems. Since then, the level set method has found various applications in diﬀerent areas of research, such as ﬂuid mechanics (Zhao et al., 1998), image processing and segmentation (Li et al., 2011; Vese and Chan, 2002) and recently in structural mechanics and optimization (Allaire et al., 2002; Wang et al., 2003). Rather than scaling element densities, the level set method describes the boundary of the topology and propagates this boundary until the design converges or a maximum number of iterations has been evaluated. The boundary is extracted from a Level Set Function (LSF), which is deﬁned as a three or four dimensional function. From this LSF a level set (isoline) or level surface (isocurve) is extracted. These functions allow to deﬁne complicated boundaries without providing a parameterization for all curves of the boundary. To update the topology the extracted boundary is propagated through the design domain to ﬁnd optimized performance. The update direction of the boundary is determined from a velocity ﬁeld obtained from a sensitivity analysis. The propagation is captured by a Hamilton-Jacobi convection equation, which can be solved for a small time step to update the topology. This method focusses on the movement of the boundary to update the topology. An alternative approach is to directly apply local changes to the level set function and thereby introduce local changes to the structure’s topology. In these Parametric Level Set Methods (PLSMs) a parameterization is applied to describe the level function using shape functions and expansion coeﬃcients (Wang and Wang, 2006; Luo et al., 2007). These expansion coeﬃcients allow local design changes of the topology and are used as the design variables in gradient-based optimization routines. The update direction of the expansion coeﬃcients is determined from a sensitivity analysis linking a change of expansion coeﬃcient to changes in topology and performance. Both the conventional and the parametric level set methods are discussed in detail in Chapter 3. 2.1.4. Black and white design One of the challenges in topology optimization is to develop black and white designs in which only solid or void elements remain. The occurrence of intermediate materials in optimized designs is often not preferred, as the manufacturability of these intermediate material properties are highly challenging or not possible with current manufacturing methods. Similar problems occur for mixtures of structural and viscoelastic material and to create clearly separated material phases is an important aspect for the development of the optimization routine. Commonly applied methods are brieﬂy discussed. For in depth discussion on these topics the reader is referred to Deaton and Grandhi (2013); Sigmund and Maute (2013) and Bendsøe and Sigmund (2003). 8 2. Topology optimization • Penalization. The goal of penalization is to make intermediate density values unattractive for the optimization routine by introducing a non-linear relation between the pseudo-density value and the material properties of interest. Common methods are SIMP (Bendsøe, 1989; Rozvany et al., 1992) or RAMP (Stolpe and Svanberg, 2001) interpolation. • Checkerboard patterns. These patterns describe areas in the mesh with alternating solid and void elements. These regions provide artiﬁcially high stiﬀness due to bad numerical modeling of these patterns (Diaz and Sigmund, 1995; Jog and Haber, 1996) and therefore are most common in optimization problems in which this additional stiﬀness contributes to the objective function. These patterns might appear when a penalization is applied to force the design to solid and void elements. • Filtering. Numerical ﬁlters are applied for multiple reasons. These ﬁlters smooth either the density or sensitivity values in the mesh. This allows to obtain mesh-independent results and remove possible checkerboard patterns from the design domain. However, the ﬁltering might introduce additional intermediate elements along the boundary of the design, which has to be kept in mind. 2.1.5. Multi-material topology optimization The problems encountered in single material optimization are also applicable for multi-material topology optimization. Besides intermediate density values, the occurrence of material mixtures are now also a possibility, especially when the mixed elements provide ideal material properties regarding the optimization objective. This might result in designs with large areas containing mixed, intermediate elements, which are unfavored from a modeling and manufacturing perspective. To separate material phases the density-based methods apply multi-material formulation of penalization methods, while the ESO/BESO and LSMs introduce discretely separated material phases. Examples of multi-material topology optimization are given for density-based (Sigmund, 2001b; Robinson, 2013), evolutionarybased (Huang and Xie, 2009) and level set-based methods (Allaire et al., 2014; Wang and Wang, 2004). 2.2. Comparison of optimization methods The presented methods are all suited to be applied to multi-material topology optimization problems. However, in this work we limit ourselves to the development of a single implementation using one of the available methods. An numerical implementation has been developed for a multi-material optimization using density-based, evolutionary-based and level set-based approaches. By investigating these methods, their performance, complexity and ease of numerical implementation became clear and provided motivation to choice one of these methods for the optimization of vibration isolating structures. The following paragraphs will discuss the experience with each method. The numerical implementations are modiﬁed from available implementations in literature. For density based approaches the well known 99-line and 88-line codes are used as starting point (Sigmund, 2001a; Andreassen et al., 2010), which has been extended towards multi-material optimization for viscoelastic structures in the master thesis of Robinson (2013). A BESO method is evaluated based on the methods described by (Huang and Xie, 2009). A MATLAB implementation of the LSM is presented by (Otomori et al., 2014). To the knowledge of the author, no available implementation of a parametric level set method has been presented in literature. Therefore, the numerical implementation has been developed based on publications which describe this method, i.e. Wang et al. (2014b); Luo et al. (2008b); Pingen et al. (2009); Wang and Wang (2004). The method proposed by Robinson (2013) has been used to evaluate the density-based approaches for multi-material topology optimization. In the work by Robinson multi-material viscoelastically damped structures were generated. The ﬁnal designs show relatively large regions of intermediate and mixed materials. Two examples of the ﬁnal designs are shown in Figure 2.2. The routine seems unable to generate designs with clearly separated boundaries, especially when the routine is initialized without any predeﬁned geometry. However, it has to be noted that a conventional SIMP penalization method has been applied to interpolate the material properties and that the application of more specialized penalization methods might reduce the occurrence of mixed materials. optimality - structure (c) has worse Q-factor and objective value than structure (a) ditional allowed mass. Both structures had the same initial design, however in (a) raint was initially infeasible due=to0.4the; m lower mmax . This (f) might suggest that the ; mvmax = 0.3 (e) msmax msmax = 0.4 ; m vmax = 0.3 ; vmax = 0.3 ; Cdd = 2.2. ;Comparison of optimization methods one in 1.3E6; local optima mass. = 9.0 to becoming trapped Cdd = Q1 = 13.0when adding additional 1.3E6; Q 1 = 2.5 ; dd mvmax = 0.5 C = 5.7E5; = 9.9 9 ;(a) (h) =m3smax m =to 0.5 ; (c) p(b) (i) 0.5optimization ; 7m;vmax 0.5 ; Cdd to = vmax (b) Example pE1 5 ; ;Cdd = 5.7E5; =smax 3 ; p= Cdd = =subjected 6.0E5; E2==0.5 E1 m E2 1:; poptimization subjected Example 2: = C = 7.6E5; Q = 9.9 7.6E5; Q = 9.9 1 1 Q = 10.0 Q = 10.0 dd 1 1 mass constraints. stiﬀness penalizations. Figure 5.4: Optimal structures with varying separate mass constraints Figure 2.2: Two examples of designs obtained with the optimization routine developed by Robinson. The optimization is subjected to constraints limiting the Q-factor of the designs to reach a certain level of structural damping. The method shows multiple regions in which material mixtures occur. The red shows the results a study on the script top opt multimat separate.m. scriptimages uses are taken andof blue colors correspond respectively to elastic and viscoelastic materials.This The shown from Robinson ate mass constraints applied(2013). to each material. The intention of using separate constraints vent the optimiser from generating large amounts of viscoelastic material by accepting nalty on the objective function, as it was seen that in this circumstance the optimiser Cdd = 6.0E5; (e) pE1 = 5 ; pE2 = 5 ; Cdd = 5.9E5; (f) pE1 = 5 ; pE2 = 7 ; Cdd = 6.4E5; etimes achieveIt aislow Q-factor simply by filling thewhich design volume withconvergence damping material, chosen to evaluate a BESO method, allows faster by reintroducing eleQ Q 10.1 1 = 10.1 1 =for n developing aments high-damping topology such as CLD. The results show the optimiser throughout the iterations (Huang and Xie, 2010a, Chapter 5).that The method presented in Huang and Xie (2009, 2010b) been referenceofimplementation. This method performs similar to of producing feasible designs withhas only a taken smallasamount viscoelastic material. However, density-based optimization routines for multi-material problems, especially when pparent that the objective function is determined almost completely by the use ofmesh-independent sti↵ Cdd ﬁltering techniques are applied. However, the heuristic updating methods are a major disadvantage compared to the gradient-based optimization methods applied in density-based optimization. The heuristic updating method requires large number of iterations and is relatively slow to converge. Two examples of designs generated 59by this method are given in Figure 2.3. A single and multi-material example are given for a typically stiﬀness maximization. The multi-material optimization using the BESO method requires a relatively large number of iterations to converge. The selection of the rejec= 6.3E5; tion and (h) p ; pE2 are = 5diﬃcult ; Cdd =to6.1E5; (i) pcan = 7 signiﬁcant ; pE2 = 7 inﬂuence ; Cdd = on 6.6E5; inclusion determine and the behavior E1 = 7 ratios E1 have Q = 17.5 Q = 11.4 1 1 and performance of the optimization routine. Also, the evolutionary methods can have diﬃculties with strict constraints, such as volume or eigenfrequency constraints. uctures with varying sti↵ness penalisations (rows have constant pE1 , columns constant pE2 ) in Figure 5.2 shows the e↵ect of changes in the two RAMP sti↵ness penalisation it is possible also to penalise the mass, this is not physically realistic, so only nalisations were considered. It can be seen immediately that the penalisation has ↵ect on the optimal structures’ objective and constraint values. Furthermore the gies are very similar, though the shape is somewhat di↵erent. However, observing the ots of the objective and constraints shows that the penalisation has a great e↵ect (a) Single material. (b) Multi-material. ation behaviour. Higher penalisation factors seem to be associated with greater ning a feasible design. Figure 2.3: Examples of designs obtained by a single and multi-material BESO method. The designs ws the e↵ects of filter radius onforce theis applied secondat variable. intention arechanging ﬁxed on the the left side, while a vertical the middle ofThe the right boundary. The designs arefiner optimized to maximize theirform stiﬀness. the multi-material optimization two elastic materials are estigate whether structures would andInimprove the optimal performance, used. The red materials has a higher stiﬀness than the blue material. radius was lower. However, it can be seen that the filter radius in fact has a very the topology and performance. All structures are feasible to within an acceptable A level set implementation is presented by Otomori et al. (2014) for a minimum compliance topology achieve the same objective value to within 5%. Looking at the structures formed, optimization. In this method the boundary is propagated by solving the Hamilton-Jacobi convection he same topology, only ﬁnite di↵erence being the decreased sharpness as the equationwith usingthe a upwind diﬀerencing routine. During the iterations a regularization is applied reases. Structure (a)thedisplays additional di↵erences since filter is inactive to keep level set slight function smooth and well deﬁned. The the implementation has been limited to a material phase, due toofthedamping complexitymaterial of the mathematical derivations of the shape and topologre - thereforesingle single-element layers can be seen on the outer ical derivatives required for the boundary propagation and hole nucleation. The method shows good he sti↵ structural members. performance for relatively coarse meshes. However, when decreasing the element sizes the method 58 10 2. Topology optimization quickly becomes computational expensive, even for a single material phase. Finally, the parametric method has been investigated. The numerical formulation is generated based relevant publications by Wang et al. (2014b); Luo et al. (2008b); Pingen et al. (2009); Wang and Wang (2004). The method can be seen as a mid-way between a density and level set-based formulations (Sigmund and Maute, 2013). The parameterization allows to fully describe the level set function using the corresponding expansion coeﬃcients. These coeﬃcients are then presented to a gradient-based optimization routine, similar to the element densities in a density-based implementation. The boundaries of the material domains are extracted from the level set functions and relatively small regions of intermediate, mixed materials are observed along the boundary between both material phases. The computational eﬀort is similar to a density-based implementation, since the same gradient-based optimization routines are applicable. Two examples of designs generated by the parametric level set method are given in Figure 2.4. The method provides multi-material designs with clearly separated boundaries and little intermediate or mixed material regions. It has to be noted that the choice of basis functions, their inﬂuence radius and the initial material distribution can have an inﬂuence on the performance of the optimization method. However, the values mentioned in literature seem to result in well-performing routines. 40 40 35 35 30 30 25 25 20 20 15 15 10 10 5 5 10 20 30 40 50 (a) Single material. 60 70 80 10 20 30 40 50 60 70 80 (b) Multi-material. Figure 2.4: Examples of designs obtained by a single and multi-material parametric level set method. The designs are ﬁxed on the left side, while a vertical force is applied at the middle of the right boundary. The designs are optimized to maximize their stiﬀness. The graphs illustrate the material distribution based on a contour plot of the level set functions. In multi-material optimization two elastic materials are used. The red material has a higher stiﬀness than the blue material. The decision is made to develop a multi-material, parametric level set method for the structural design optimization of vibration isolating structures. The close relation to both density and level set-based optimization routines make the parametric level set method a potential improvement on the formulations presented by Robinson. The method captures advantages of both density and level set-based formulations. The formulation allows to apply gradient-based optimization algorithms to update the topology, while still providing an exact boundary description of the level set function. The mathematical framework and the parameterization of the level set-based approaches are discussed in the following chapter. 3 Parametric level set method This chapter presents the mathematical background of the level set-based methods. A parameterization of the level set function is applied to arrive at the parametric level set method, which is extended to simultaneously describe multiple material domains. 3.1. Level set method In structural mechanics the level set function is deﬁned as a three or four dimensional function, which allows to describe a two or three dimensional topology by extracting a level set or level surface. The reference domain and the level set function are mathematically deﬁned as: Let Ω ⊆ Rd (d = 2 or 3) be an open and bounded set occupied by the region of a material phase. Then, the shape boundary is embedded such that Γ = ∂Ω of the region as the zero level set of the implicit function φ(X) : Rd+1 → R as Γ = {X : φ(X) = 0}. (3.1) In many implementations the zero level set is chosen as reference set to extract the structure’s topology. This results in three distinguishable domains described by the level set function: φ(X) > 0, ∀X ∈ Ω\Γ (material), (3.2) φ(X) = 0, ∀X ∈ Γ (interface), φ(X) < 0, ∀X ∈ D\Ω (void). In this notation D corresponds to the complete design domain in which the material domain Ω is located and X speciﬁes any point within the design domain. An illustration of a level set function and the corresponding boundary is presented in Figures 3.1a and 3.1b. In this example the boundary Γ is extracted from the level set c and encloses the domain Ω. The level set function allows us to deﬁne this boundary without providing a parameterization for any curvatures of the boundary, as this boundary is found directly from the extracted level set function. Therefore, changes applied to the level set function will directly modify the topology. This can be illustrated by applying a vertical translation to the level set function. During the translation a position occurs where the left and right side of the domain are connected by a single point. When continuing to translate the level set function, the domain Ω now describes two separated areas within the design domain, as illustrated by Figures 3.1e and 3.1f. No additional descriptions are required to describe the separation of the boundary into separate domains, which allows us to describe complicated, evolving boundaries eﬀortlessly. 11 12 3. Parametric level set method (a) Level set function φ(X). (b) Domain Ω and boundary Γ obtained from the level set function φ(X). (c) Level set function φ(X). (d) Domain Ω and boundary Γ obtained from the level set function φ(X). (e) Level set function φ(X). (f) Domain Ω and boundary Γ obtained from the level set function φ(X). Figure 3.1: An illustration of the level set function φ(X) and the corresponding domain Ω and boundary Γ as described by the level set φ = c. Figures (a,b) give the level set function and boundary for the initial position. The ﬁgures (c-f) illustrate the changing boundary as a result of a vertical translation of the level set function. The translation results in a separation of the boundary in two separated domains. 3.1. Level set method 13 The level set method is an interesting approach for structural topology optimization since the method is able to describe complicated geometries and the boundaries are directly modiﬁed by a change of the level set function. For structural topology optimization we need to track the changes of the boundary of the level set function to iteratively determine design updates to improve the structure’s performance. Therefore, a pseudo-time variable t is introduced to express the boundary changes through time. The zero level set function φ(X, t) = 0 becomes a function of the spatial coordinate X and the pseudo-time variable t. The evolution of the zero-level set function can be described by a Hamilton-Jacobi (HJ) convection equation and is obtained by diﬀerentiation of the level set function (Osher and Sethian, 1988). This results in the following Partial Diﬀerential Equation (PDE): ∂φ(X, t) dX dφ(X, t) = + ∇φ · = 0, dt ∂t dt (3.3) where dX/ dt is obtained from the shape sensitivity analysis and provides the design boundary velocity ﬁeld. This velocity ﬁeld forces the boundary to move in the direction that improves the structure’s performance. Any inﬁnitesimal displacements in the tangential direction to the boundary will not result in any boundary change. Allowing us to write dX/ dt = vn n with vn a scalar normal velocity ﬁeld and n the outward normal to the boundary (Allaire et al., 2004; Wang et al., 2003; van Dijk, 2012). The outward normal to a level set curve is deﬁned as n = −∇φ/||φ||. Substituting these relations in the Hamilton-Jacobi convection equation (3.3) results in: dφ(X, t) ∂φ(X, t) = − ||∇φ||vn (X) = 0. dt ∂t (3.4) The HJ equation is solved by numerical integration of a given pseudo-time step. The solution of this integral is used to update the level set function φ and thereby update the topology of the structure. Transporting the boundaries allows to describe complicated topologies and is able to merge the domain at the boundary. The design updates originate at the boundary of the level set function and the current formulation is therefore not able to nucleate holes within already deﬁned material domains. Often, the optimization is initialized with many small holes to reduce the probability that the creation of additional holes is required during the optimization. A more mathematical approach is to introduce additional terms to the HJ equation, such that nucleation becomes possible within the solid domains. Applying the formulation as presented by van Dijk (2012) the generalized HJ equation is given as: ∂φ(X, t) − ||∇φ||vn (X) − D(φ) − R(φ) = 0, ∂t (3.5) where D is a diﬀusive term and R a reactive term. Since the diﬀusive term is mainly related to the numerical treatment of the HJ equation it is not treated in detail. The reactive term R provides source and sink terms within the solid domain. These reactive terms allow the optimization to nucleate holes within the solid domains and thereby overcome the nucleation problem of the original level set method. A topological derivative is derived, which provides the sensitivity information for a nucleation of inﬁnitesimal holes in the design domain (Burger et al., 2004; Otomori et al., 2014). The required derivation can become involved for complicated objective functions. Based on the sensitivity value a hole might be nucleated. The nucleation can be performed continuously throughout the iterations or at predeﬁned sets of iterations. The topology is updated by integrating the HJ equation for a certain pseudo-time step. The solution provides an new topology, which is used to update the topology of the current iteration. To solve the HJ equation a forward ﬁnite diﬀerencing method is applied (Osher and Ronald, 2003; Zhao et al., 1998; Otomori et al., 2014). The iterative updates are continued until the boundary converges or a maximum number of iterations are evaluated. The implementation of the numerical integration is non-trivial due to constraints to enforce numerical stability. The stability is enforced by the CourantFriedrichs-Lewy (CFL) condition, which limits the allowable time steps of the numerical integration. 14 3. Parametric level set method Figure 3.2: Illustration of a cross section of the level set function and the applied parameterization by the shape functions ξi and expansion coeﬃcients αi . The level set φ = c is drawn to illustrate a level set deﬁned to extract a boundary from the LSF. Image modiﬁed from van Dijk et al. (2013). The stability constraint is a function of the time step as well as the element size of the ﬁnite element discretization. When reducing the mesh size the CFL condition requires a reduction of the pseudotime steps to ensure numerical stability. Thus, decreasing the mesh size will result in a signiﬁcant increase of computational eﬀort for each iteration. 3.2. Parametric level set method The parametric level set method is a variation on the level set-based approaches and can be seen as a midway between density and level set-based optimization routines. Compared to the level set-based approaches the parameterized method does not require complicated formulations for shape and topological derivatives and overcomes diﬃculties involved in the numerical integration of the HJ convection equations. The parameterization describes the level set function with a combination of shape functions and expansion coeﬃcients. The topology is updated by changing the scaling of the diﬀerent expansion coeﬃcients. The parameterization allows to determine the expansion coeﬃcient updates using gradient-based optimization routines, such as the Methods of Moving Asymptotes (MMA) (Svanberg, 1987). The parametric level set method also allows for the nucleation of holes within the domain by allowing negative values for the expansion coeﬃcients. The remainder of this section discusses the parametric level set method for topology optimization in detail and describes the used shape functions and the modiﬁcations required for a multi-material implementation. The level set function is parameterized by shape functions ξi (X) with corresponding expansion coeﬃcients αi (t). The shape functions are only a function of the spatial coordinates X, while the expansion coeﬃcients are only a function of the pseudo-time variable t. The level set function is parameterized by a summation of scaled shape functions for all N elements in the design domain: φ(X, t) = N ∑ ξi (X) αi (t). (3.6) i=1 Figure 3.2 illustrates a cross section of level set function parameterized by shape functions. The parameterization results in a parametric problem in which we iteratively search for optimized values of expansion coeﬃcients to achieve an optimal topology (Wang and Wang, 2006; Luo et al., 2008b). Two methods are available to update the topology using shape functions. The ﬁrst is to substitute the parameterization in the Hamilton-Jacobi convection equation (3.4) to obtain: N ∑ i=1 ξi (X) N ∑ ( ) dαi (t) − vn || ∇ξi (X) αi (t)|| = 0. dt i=1 (3.7) This allows us to ﬁnd an expression for the velocity ﬁeld for each element in the domain, given by: 3.2. Parametric level set method 15 ∑N i (t) ξi (X) dαdt vn = ∑N i=1( . ) i=1 || ∇ξi (X) αi (t)|| (3.8) By discretizing the design domain and applying the parameterization, the set of PDEs is reduced to a set of Ordinary Diﬀerential Equations (ODEs) Wang and Wang (2006). This set of equations can be solved by a variety of diﬀerent ODE solvers to update the expansion coeﬃcients by integrating along the pseudo-time variable. The optimization routine becomes equivalent to transporting the level set function by updating the values of the expansion coeﬃcients by solving the set of ODEs. The second method applies a gradient-based optimization routine to update the value of the expansion coeﬃcients and thereby update the corresponding topology. The sensitivity of each element to the level set function is determined analytically and presented to the gradient-based routine. During each iteration the optimization routine updates the values of the expansion coeﬃcients to improve the performance of the topology. In this work the second method is applied to generate design updates using a gradient-based routine. The formulation of the objective function is discussed in the next chapter. The adjoint sensitivity analysis is mentioned in Section 4.3.4 with additional details presented in Appendix C. 3.2.1. Shape functions Various shape functions are available for the parameterization of the level set function. The choice of shape function to apply is not straightforward, since the application of diﬀerent shape functions might inﬂuence the performance of the optimization routine. These functions inﬂuence the computational eﬀort involved in the parameterization, the radius of inﬂuence of each design variable and also the sensitivity response at the level set. The general diﬀerences between shape functions are discussed in the next sections. A detailed discussion regarding the choice of shape functions is presented by (van Dijk et al., 2013, section 2). Two discretization are required for the parametric level set method: one for the structural design and the other for the placement of the shape functions. In many implementations the same discretization is used for both applications (Wang et al., 2015; Luo et al., 2008b). This simpliﬁes the numerical implementation, since the centers of the shape functions are aligned with the centers of the elements in the ﬁnite element discretization. Figure 3.3a illustrates the parameterization using a single discretization for the ﬁnite element discretization and the shape function placement in which the expansion coeﬃcients αi are aligned with the center of the ith element. Another, more complicated approach is two apply two diﬀerent discretizations, for example by applying a ﬁner mesh for the structural design (Pingen et al., 2009). The ﬁner mesh allows for more detailed analysis of the structural mechanics. However, some form of interpolation has to be performed to relate the values of the nodes in the level set function to the nodes in ﬁnite element discretization. In the presented implementation the same discretization is applied for the structural design and the level set parameterization, as this simpliﬁes the numerical implementation of the parametric level set method. The inﬂuence (or support) radius of the shape functions is one of the most important parameters in the parametric level set methods. This inﬂuence radius determines the radius within the shape function takes on a non-zero value. Depending on this radius, a single shape function might be limited within a single element in the discretization or extend throughout the complete design domain. Three types of level set functions are distinguishable based on their inﬂuence radius: local, mid-range and global shape functions. Local basis function These shape functions are only non-zero in a small, ﬁnite part of the design domain and show minimal overlap with neighboring elements. The inﬂuence of such a basis function is limited to the support of the corresponding node of the ﬁnite element discretization. A change in expansion coeﬃcient will result in a relatively small, localized change of the level set function. These basis functions generally require more iterations to converge towards an optimized design. 16 3. Parametric level set method (a) Radial basis function placement. (b) Compactly-supported radial basis function. Figure 3.3: (a): Illustration showing the cross section of the radial basis function placement. The RBFs are aligned with the elements in the ﬁnite element discretization. The shape functions ξi and the expansion coeﬃcients αi are aligned with the ith element in the design domain. (b): Illustration of a compactly-supported radial basis function as described by Equation (3.9). Mid-range basis function For mid-range basis functions the inﬂuence radius is slightly larger compared to the local shape functions. These functions show substantial overlap a number of neighboring elements. Changes of a single expansion coeﬃcient will therefore have a larger eﬀect on the shape of the level set function. Commonly radial basis functions (or similar) are applied as mid-range basis function with an inﬂuence radius of two to four times the mesh size. Global basis function The inﬂuence radius of global basis functions span almost the entire design domain. Most of the basis functions will overlap in large parts of the domain. Each expansion coeﬃcient has a slight inﬂuence on almost the complete level set function (de Ruiter and van Keulen, 2004; Wang and Wang, 2006). Compared to the local and mid-range shape functions, the global functions allow to introduce large design changes by small changes of design variables as the inﬂuence of the design variables will propagate throughout most of the design domain. The large inﬂuence radius results in high computational eﬀort for both the parameterization and the sensitivity calculation. Each design update has to be determined carefully, as small changes inﬂuence the complete design domain and various combinations of shape functions will result in the same topology. In the presented work a radial basis function with mid-range inﬂuence radius is applied. The implementation is realized using compactly-supported RBFs. These basis functions provide a midway between local and global basis functions. The mid-range inﬂuence radius provides overlap between neighboring elements and extends the inﬂuence of each expansion coeﬃcient to a slightly larger area, while keeping the computational eﬀort limited. Moreover, CSRBFs are shown to perform well in structural optimization (Luo et al., 2008b; Wang et al., 2014b). The implemented CSRBFs are described by Wendland (1995) and are deﬁned as { ξi (X) = 0, for ri (X) ≥ 1, ( )4 (1 − ri (X) (4ri (X) + 1), for ri (X) < 1. (3.9) The distance ri determines the distance of a point Xi with the current position X and thereby limits the non-zero values of the shape function to its inﬂuence radius R. The radius ri is given as ri (X) = ||X − Xi || . R (3.10) 3.2. Parametric level set method 17 Void Viscoelastic material Void Structural material D Figure 3.4: Illustration of a multi-material structure deﬁned by two level set functions. The ﬁrst level set function φ1 determines the placement of material, while the second level set function φ2 distinguishes between structural and viscoelastic material. For both level set functions the zero-level set is used to extract the boundary. The CSRBFs have shown good performance with an inﬂuence radius between two and four times the mesh size. In the presented work, similar radii have provided good performance for the structural optimization. Applying shape functions which extend along multiple parameters has similar eﬀects as ﬁltering techniques in density-based optimization. Figure 3.3b presents an illustration of this radial basis function. The material discretization and the numerical implementation of the parametric level set method are further discussed in Appendix B. 3.2.2. Multi-material formulation For the structural optimization of viscoelastic dampers we have to simultaneously distribute both viscoelastic and structural material in the design domain. Therefore, the presented formulation has to be extended to introduce a second material phase within the domain. Introducing a second level set function allows to describe the additional material phase. The material properties of a speciﬁc point in the domain are then determined based on the values of both level set functions. In this work a multi-material formulation is applied, which is based on the formulation presented by Wang et al. (2015). The formulation is slightly modiﬁed to allow for a structural and viscoelastic material to be deﬁned by both level set functions. In general, a multi-material formulation using k level set functions will describe the following domains: k k k (material), φ (X) > 0, ∀X ∈ Ω \Γ k k (3.11) φ (X) = 0, ∀X ∈ Γ (interface), φk (X) < 0, ∀X ∈ D\Ωk (void). This introduces k boundaries within the design domain. Only two level set functions are required to deﬁne a structural and viscoelastic material phase. The ﬁrst level set function is applied to distribute any material within the domain. Then, the second level set function distinguishes between viscoelastic and structural material inside the material domain speciﬁed by the ﬁrst level set function. The material boundaries are extracted at the zero-level set, which provides the following material domains: 1 void, φ < 0 1 2 (3.12) φ ≥ 0, φ < 0 structural material, φ1 ≥ 0, φ2 ≥ 0 viscoelastic material. Figure 3.4 illustrates a material distribution constructed from two level set functions and its ability to describe both material regions and void within the design domain. In the next chapter the presented parametric level set method is applied for the optimization of viscoelastically damped structures. 4 Topology optimization of viscoelastically damped structures In this chapter the parametric, multi-material level set is applied for the structural design optimization of vibration isolating structures. The goal of the optimization is to achieve maximum damping characteristics by simultaneous distribution of structural and viscoelastic material using the methods presented in Chapter 3. This chapter has been submitted to Journal of Vibration & Control and might partially overlap with material discussed in the previous chapters. The article discusses the modeling of the viscoelastic material, the formulation of the optimization problem and presents multiple numerical examples. Addition details regarding the applied viscoelastic material model are presented in Appendix A, the details on the numerical implementation are given in Appendix B and the complete derivation of the sensitivity analysis is presented in Appendix C. 19 20 4. Topology optimization of viscoelastically damped structures Multi-Material Topology Optimization of Viscoelastically Damped Structures using a Parametric Level Set Method M. van der Kolk†,‡ , G.J. van der Veen† , J. de Vreugd‡ and M. Langelaar† † ‡ Delft University of Technology, Delft, The Netherlands Optomechatronics, Netherlands Organisation for Applied Scientiﬁc Research, Delft, The Netherlands Abstract The design of high performance instruments often involves the attenuation of poorly damped resonant modes. Current design practices typically rely on informed trial and error based modiﬁcations to improve dynamic performance. In this article, a multi-material topology optimization approach is presented as a systematic methodology to develop structures with optimal damping characteristics. The proposed method applies a multi-material, parametric, level set-based topology optimization to simultaneously distribute structural and viscoelastic material to optimize damping characteristics. The viscoelastic behavior is represented by a complex-valued material modulus resulting in a complex-valued eigenvalue problem. The structural loss factor is used as objective function during the optimization and is calculated using the complex-valued eigenmodes. An adjoint sensitivity analysis is presented that provides an analytical expression for the corresponding sensitivities. Multiple numerical examples are treated to illustrate the eﬀectiveness of the approach and the inﬂuence of diﬀerent viscoelastic material models on the optimized designs is studied. The optimization routine is able to generate designs for a number of eigenmodes and to attenuate a resonant mode of an existing structure. Keywords: viscoelastic damping, topology optimization, multi-material optimization, level set method, loss factor, modal analysis, constrained layer damping 4.1. Introduction This paper addresses the design of structures composed of both viscoelastic and structural material, to achieve optimal damping characteristics. This is implemented using a multi-material level set-based topology optimization. The design of high-performance instruments often involves the attenuation of poorly damped resonant modes. At the Netherlands Organisation for Applied Scientiﬁc Research (TNO) this is encountered during the design of high performance optomechatronic instruments (de Vreugd et al., 2014), such as GAIA BAM (Gielesen et al., 2013) and MSI VNS (Tabak et al., 2013). Current design approaches typically start from a baseline design and introduce stiﬀening or damping reinforcements to tune and damp these modes. However, the inﬂuence of these reinforcements is diﬃcult to predict, resulting in a trial and error approach. We propose an integrated multi-material topology optimization approach as a systematic methodology for the development of these structures. Designs involving multiple appropriately distributed materials, in speciﬁc the combination with viscoelastic materials, are known to provide high structural damping and have been applied in many ﬁelds (Johnson, 1995), for example in automotive and aviation (Rao, 2003), aerospace (Rittweger et al., 2002; Wang et al., 2008), and civil applications (Samali and Kwok, 1995; Wang et al., 2013). Viscoelastic materials dissipate energy when subjected to deformation. To achieve optimal damping the viscoelastic material has to be placed at locations which undergo large deformation during vibration. Besides, the design of the construction itself should promote deformation of the viscoelastic material. The two most common damping geometries for viscoelastic dampers are unconstrained and constrained layer damping (Grootenhuis, 1970). Here, the viscoelastic material is constrained on one or both sides. During vibration the viscoelastic material is forced to deform and thereby dissipate energy. 4.1. Introduction 21 The development of viscoelastic dampers presents a challenging optimization problem, where the goal is to achieve optimal structural damping. Both the location as well as the geometry of the viscoelastic layer can be determined using an optimization routine. In previous studies both analyses have been performed separately, for example, the placement of viscoelastic patches along a vibrating frame (Lunden, 1980) and the shape optimization of individual, unconstrained and constrained viscoelastic layers (Lumsdaine and Scott, 1998; Plunkett and Lee, 1970). The loss factor (Kerwin, 1959) is determined for each structure to quantitatively compare the amount of structural damping for each design. More recent studies have applied topology optimization to ﬁnd optimized damping characteristics by distribution of viscoelastic material in the design domain. The optimization method searches for the distribution of viscoelastic material to obtain the highest loss factors for single or multiple modeshapes. These optimization routines are implemented for unconstrained layer damping on plates (El-Sabbagh and Baz, 2013) and shell structures under harmonic excitations (Kang et al., 2012). Here, an additional layer of viscoelastic material is optimally distributed on top of the plates to dissipate energy during vibrations. Similar approaches are available for constrained layer treatments: for simply supported beams (Zheng et al., 2004) and vibrating plates (Ling et al., 2011; Zheng et al., 2013). In these designs the viscoelastic layer is sandwiched by two beams or plates and its distribution within this layer is optimized. However, these optimization routines limit the design domain of viscoelastic material to the predeﬁned (un)constrained layer. Before the optimization, the location of the viscoelastic layer is to be provided by the designer. Multiple modeling approaches have been presented to model the viscoelastic behavior. The material can be represented using models based on springs and dashpots, for example, the Maxwell or KevinVoigt models (Bert, 1973) and extensions hereof. Alternatively, the elastic and dissipative behavior can be represented by a complex material modulus (Grootenhuis, 1970). For the modeling of harmonically excited structures, this latter method is widely applied, for example, by (Kerwin, 1959; Johnson, 1995) and the previously mentioned optimization routines. The studies apply diﬀerent formulations of the complex modulus. For example, the authors of (Lunden, 1980; Rao, 1978; El-Sabbagh and Baz, 2013) apply a complex Young’s modulus, while the authors (Liu et al., 2013; Ling et al., 2011) only apply a complex shear modulus. When the complex modulus is limited to the shear modulus, only deformations with shear components dissipate energy. For the analysis of beams or plates using constrained layer conﬁgurations, the diﬀerence between using a complex Young’s or shear modulus might not result in signiﬁcant performance changes: the obtained modeshapes show predominately shear deformation within the viscoelastic layer. Yet, when optimizing arbitrary geometries the choice of complex modulus might steer the optimization towards speciﬁc designs. In this paper we devote speciﬁc attention to this point. The implementation of the complex material modulus provides a complex-valued stiﬀness matrix, resulting in a complex-valued eigenvalue problem to determine the structure’s natural modes. In the mentioned studies, a number of approximations are made regarding the calculation of the loss factor and its sensitivities. In some papers the loss factors are calculated using the eigenmodes obtained from the undamped analysis (Kim, 2011; Liu et al., 2013; Ling et al., 2011; Zheng et al., 2013). However, for low structural damping, we have observed that the eigenmodes of the undamped and damped structure are approximately the same. At higher structural damping it will be required to move towards the complex modeshapes for the calculations. Similarly, it is observed that the sensitivities of the loss factor are determined using approximations. For example, by neglecting the contribution of the modeshape sensitivity to changes in the design (Ling et al., 2011), or by using an approximate formulation (Wang et al., 2014a; Zheng et al., 2013). Several multi-material topology optimization routines are available for various optimization methods, such as, the density methods by (Sigmund, 2001b; Yin and Ananthasuresh, 2001), the evolutionary implementation by (Huang and Xie, 2009) and the level set methods by (Wang and Wang, 2004; Allaire et al., 2014). For multi-material or multi-phase optimization, the level set method has the advantage of distinct separation between the diﬀerent materials. To overcome numerical diﬃculties from tracking the boundaries and solving the Hamilton-Jacobi convection equations, the level set functions have been parameterized (Wang and Wang, 2006; Luo et al., 2007). The parametric level 22 4. Topology optimization of viscoelastically damped structures set methods describe the level set function by a set of shape functions and expansion coeﬃcients, while maintaining distinct separation between material phases. A recent overview of the level set methods and parametrization approaches is given in (van Dijk et al., 2013). The parametric level set methods have recently been extended towards multi-material optimization (Wang et al., 2015). In this article a multi-material topology optimization formulation is developed. Here, the viscoelastic and structural material will be distributed simultaneously to achieve optimal damping characteristics. Within the user-speciﬁed bounds the obtained designs can have arbitrary material distributions and geometries, possibly resulting in higher levels of structural damping. Also, there is no need to predeﬁne a location of the viscoelastic layer by the designer. Furthermore, the calculation of the loss factors is performed using the complex-valued modeshapes and an exact formulation is proposed for the loss factor sensitivities. The article is organized as follows. Section 4.2 describes the modeling of the viscoelastic material and investigates the obtained loss factors deﬁnitions for a constrained layer conﬁguration. Then, Section 4.3 provides the formulation of the parametric, multi-material level set method, the objective and constraints, and the adjoint sensitivity analysis. In Section 4.4 the results of two case studies are presented and discussed. The paper ends with conclusions regarding the proposed optimization routine. 4.2. Modeling of Viscoelastic Material In this paper, the viscoelastic material is described using a complex material modulus. The formulation of the material model is discussed in the following sections. Furthermore, the structural loss factor is introduced as measure to compare the structural damping between diﬀerent structures. Diﬀerent methods to calculate the loss factor are shown and their performance to represent the structural damping for viscoelastically damped structures are compared. 4.2.1. Complex material modulus When a viscoelastic material is deformed dynamically the corresponding stresses are not in phase with the applied strains (Tschoegl, 1989). For a harmonically excited structure the stress will lead the applied strain by an angle θ. The strain ε and stress σ are represented as ε(t) = ε0 sin(ωt) ( ) σ(t) = σ0 (ω) sin ωt + θ(ω) . (4.1) For harmonically excited structures the observed phase angle can be represented by applying a complex material modulus. The complex material modulus is found by dividing the stresses by the strains (Tschoegl, 1989; Grootenhuis, 1970). This results in E(ω) = E(ω)′ + iE(ω)′′ . (4.2) √ The Young’s modulus is given by a complex quantity, with i being the complex number i = −1. The material modulus is given by the storage modulus E ′ and the loss modulus E ′′ . These components are responsible for the elastic and dissipative behavior, respectively. Many diﬀerent models are available to describe the viscoelastic behavior. The complex material model is applied for its relative simplicity and ease of modeling responses of viscoelastic structures to harmonic excitations. For a general viscoelastic material the moduli depend on both temperature and excitation frequency. In the presented work the material moduli are assumed constant to simplify the material description. The storage and loss modulus are related through the material loss factor η and the loss angle θ: E ′′ = tan(θ) = η. (4.3) E′ A similar approach can be applied to deﬁne a complex bulk (κ) or shear (G) modulus. Depending on the applied viscoelastic material any complex modulus can be chosen to represent the dissipative behavior: E = E ′ (1 + iη), κ = κ′ (1 + iη), G = G′ (1 + iη). (4.4) 4.2. Modeling of Viscoelastic Material 23 When a complex bulk or shear modulus is applied, the viscoelastic material only dissipates energy when the deformation contains the corresponding bulk or shear components. However, when a complex Young’s modulus is applied the viscoelastic material will dissipate energy for any deformation. The behavior of the continuum body with a structural and viscoelastic material is described by the following equilibrium condition: { div (Cχ ε(u)) + f − ρχ ü = 0 in D, (4.5) u=0 on ΓkD . The constitutive constant is scaled using a Voigt mixing law and is based on the presence of the material phases k: Cχ = C k χk , where χk is a function of phase Ωk . A similar scaling is applied for the material density ρχ . The applied body forces are given by f , the displacement by u and the strains by ε. To account for the phase lag between stresses and strains the elastic coeﬃcients with Cχ can take on complex values. For the remaining analysis the system is discretized in ﬁnite elements providing the following discretized equations of motion. Due to the application of complex-valued elastic coeﬃcients the stiﬀness matrix is complex-valued: M ẍ + (KR + iKI ) x = f , (4.6) with M the global mass matrix, KR and KI respectively the real and imaginary parts of the global stiﬀness matrix, x the nodal displacements and f the forces applied to the structure. The imaginary components of the stiﬀness matrix model the dissipative behavior. The applied viscoelastic material model also results in a complex eigenvalue problem: ( ) −λ2 M + (KR + iKI ) ϕ = 0. (4.7) The obtained eigenfrequencies λ and eigenmodes ϕ are therefore complex-valued. The harmonically excited, viscoelastic structures show a steady-state oscillatory deformation. Part of the excitation energy is stored elastically, while the remainder is dissipated within the viscoelastic material. In a stress-strain diagram, this dissipative oscillation results in a hysteresis loop. The dissipated energy per unit volume during each cycle is determined from the contour integral along the hysteresis loop. The hysteresis loop is constructed by plotting the stress σ(t) as function of the strain ε(t). The dissipated energy per unit volume for a cycle Wcyc has been derived by Tschoegl (1989). It has to be noted that this derivation can be performed for either choice of complex material modulus. To illustrate the obtained expressions, the derivation has been performed with a complex shear modulus, resulting in the following expressions for the dissipated energy: ∮ Wcyc = σ(t) dε(t) = πε20 G′′ . (4.8) By multiplication with the corresponding excitation frequency the dissipated power per unit volume is found. Since the eigenfrequencies are given in rad/s the result is divided by 2π to obtain power: Qdissip = 1 2 ′′ ε G . 2 0 (4.9) With the complex material modulus, the dissipated energy can also be determined from the imaginary part of the stiﬀness matrix in combination with a modeshape corresponding to the deformation. Then, the dissipated energy per cycle is given as Wcyc = πϕH KI ϕ. (4.10) The superscript ϕH indicates the conjugate transpose. 4.2.2. Structural loss factor To quantitively compare the damping between diﬀerent structures the structural loss factors are determined. A general formulation of the structural loss factor has been proposed by Johnson and 24 4. Topology optimization of viscoelastically damped structures Structural Viscoelastic Figure 4.1: Illustration of the clamped cantilever beam containing a constrained viscoelastic layer. The arrow indicates the location of the excitation force and position measurement during dynamic loading. Kienholz (1982). The structural loss factor describes the ratio between dissipated and stored energy. The structural loss factor for eigenmode r is given by γ̄r = ψrT KI ψr . ψrT KR ψr (4.11) In the proposed formulation (4.11) the modeshapes ψr are based on the undamped system and therefore real-valued. The used modeshapes are mass normalized and are determined by leaving out KI from the eigenvalue problem in equation (4.7): ( ) −λ2 M + KR ψ = 0. (4.12) The symbol γ̄ indicates the loss factors based on the undamped eigenmodes ψ. Applying the undamped eigenmodes introduces an approximation in the calculation of the structural loss factors. The imaginary components introduced by the viscoelastic material are neglected. A revised formulation has been proposed by Xu et al. (2002) to provide a better approximation of the loss factor for high material loss factors. However, the revision is still limited to the undamped eigenmodes. Therefore, we propose to perform the structural loss factor calculation using the complex-valued modeshapes ϕ from the complex-valued eigenvalue problem from equation (4.7). Then, the structural loss factor is given as: ϕH K I ϕr γr = Hr . (4.13) ϕr KR ϕr To compare the performance of both formulations, equations (4.11) and (4.13), the obtained structural loss factors are compared to the Q-factor of the system. The structural loss factor and Q-factor are related via the used equivalence: γr = Q−1 (4.14) r . The Q-factor is determined based on the frequency response of the structure. The frequency response is obtained by performing harmonic response analysis in the analyzed frequency range. The 3dB method is applied to extract the Q-factor from the frequency response by dividing the resonant frequency by the half-power bandwidth, as described by (Bert, 1973). A constrained viscoelastic layer is used as example. Here, aluminum is applied as structural material with Young’s modulus 70 GPa, density 2.7 × 103 kg/m3 and Poisson ratio ν = 0.3 and for the viscoelastic material: Young’s modulus 1 GPa, density 1 × 103 kg/m3 and Poisson ratio ν = 0.3. The analyzed beam has a length of 2.25 m, a height of 0.15 m and unit thickness. Figure 4.1 illustrates the structure. The viscoelastic layer is centered along the beam with a total height of 0.05 m. The structure illustrates a rather extreme scenario in which the viscoelastic material has signiﬁcant contribution to the deformation. For the ﬁrst two eigenmodes both structural loss factors and Q-factor are determined. Furthermore, the comparison is performed for diﬀerent values of the material loss factor η. The results are summarized in Table 4.1. The relative diﬀerence between the Q-factor and the structure’s loss factor is signiﬁcantly reduced by application of the complex-valued modeshapes ϕ. This reduction is mainly applicable for designs in which viscoelastic materials with high material loss factors are used and where large parts of the viscoelastic material are subjected to deformation for the analyzed modeshapes. 4.3. Topology optimization of viscoelastic and structural material The simultaneous distribution of both viscoelastic and structural material is realized using a multimaterial topology optimization routine. The optimization method is mainly based on the study 4.3. Topology optimization of viscoelastic and structural material 25 Table 4.1: Comparison between the structural loss factors and Q-factor of a structure containing a constrained viscoelastic layer (Figure 4.1). The loss factors γ̄ and γ are determined using the undamped and damped eigenmodes. The relative diﬀerence is calculated between the structural loss factors and the inverse of the Q-factor to illustrating the improved prediction of damping behavior when using the complex-valued eigenmodes for the loss factor calculation. η Mode Q−1 γ̄r Relative diﬀerence γr Relative diﬀerence 1.00 1 0.079 0.139 0.774 0.079 0.001 2 0.289 0.416 0.434 0.307 0.059 1 0.072 0.105 0.448 0.073 0.007 2 0.249 0.312 0.249 0.259 0.040 1 0.057 0.069 0.204 0.058 0.007 2 0.187 0.208 0.114 0.191 0.021 1 0.034 0.035 0.025 0.033 0.011 2 0.101 0.104 0.025 0.102 0.003 0.75 0.50 0.25 regarding multi-material, parametric optimization routines by (Wang et al., 2015) and brieﬂy outlined here. We have opted for a level-set based approach, in order to obtain clearly distinct material regions. Trials using density-based multi-material topology optimization often resulted in designs containing mixtures of materials that are diﬃcult to interpret. 4.3.1. Multi-material boundary representation The level set method was originally developed by Osher and Sethian (1988) for the numerical computation of front and boundary propagation and has been applied in the context for shape optimization by Allaire et al. (2004); Wang et al. (2003). In level set-based optimization routines the structural boundary is represented by the zero level set of an auxiliary scalar function, the so called Level Set Function (LSF). Multi-material structures require multiple level set functions to describe the required material boundaries. Each level set deﬁnes the subdomains: k k k (material) φ (X) > 0, ∀X ∈ Ω \Γ k k (4.15) φ (X) = 0, ∀X ∈ Γ (interface) φk (X) < 0, ∀X ∈ D\Ωk (void). Here X is a point within the design domain, the index k indicates the use of k level set functions and D corresponds to the design domain. Two level set functions are required to describe two materials and void. The ﬁrst level set function separates material from void, while the second level set function distinguishes the domains of viscoelastic and structural material. Figure 4.2 illustrates a material distribution using two level set functions. Using the formulation proposed earlier for elastic materials (Wang et al., 2015), we can now deﬁne the following stiﬀness and mass matrices for the considered structural and viscoelastic design problem: {( } ) K(φ) = H(φ1 ) 1 − H(φ2 ) Ke1 + H(φ2 )Ke2 {( } (4.16) ) M (φ) = H(φ1 ) 1 − H(φ2 ) Me1 + H(φ2 )Me2 . Here H(φk ) represents the Heaviside function of the k th level set function, Kek and Mek correspond respectively to the elementary stiﬀness and mass of material k. The applied Heaviside function is 26 4. Topology optimization of viscoelastically damped structures Void Viscoelastic material Void Structural material D Figure 4.2: Illustration of a multi-material structure deﬁned by two level set functions. The ﬁrst level set function φ1 determines the placement of material, while the second level set function φ2 distinguishes between structural and viscoelastic material. both approximated and regularized, as discussed in more detail in Section 4.3.5. 4.3.2. Parametric level set method The topology optimization is implemented by a parametric level set method. In parametric methods a parameterization is applied to uncouple the space and time dependencies within the level set method (Wang and Wang, 2006; Luo et al., 2007). The level set functions are constructed by the summation of basis functions: N ∑ φk (X, t) = ξi (X)αik (t). (4.17) i=1 Here N equals the total number of shape functions included in the design domain. The shape functions are given as ξi (X), which describe the ith shape function. The expansion coeﬃcients αik (t) are applied to scale the individual shape functions. The variable t represents a pseudo-time variable and indicates the change of expansion coeﬃcients throughout the iterations. The decoupling of the space and time dependencies allows us to apply gradient-based optimization routines, rather than solve HamiltonJacobi convection equations to update the structure’s boundary. Besides, the parametric level set method allows us to nucleate holes throughout the design domain by allowing negative values of the expansion coeﬃcients (Luo et al., 2007). Diﬀerent approaches are available to parameterize the level set function and to describe the level set functions (van Dijk et al., 2013). In this study we apply a Compactly-Supported Radial Basis Function (CS-RBF) proposed by Wendland (1995) for the parameterization. These shape functions also have been applied in earlier parametric level set studies (Luo et al., 2008a; Wang et al., 2015). The shape function is given as { 0 )4 ξi (X) = ( (1 − ri (X) (4ri (X) + 1) if ri (X) ≥ 1 if ri (X) < 1. (4.18) The radius ri is given as ri (X) = ||X − Xi || , R (4.19) where R refers to the inﬂuence radius of the basis function. Only the neighboring elements within the inﬂuence radius will contribute to the function value of the level set function. 4.3.3. Optimization problem for structural loss factor maximization In the parametric level set method the shape of the level set function is completely determined by the given expansion coeﬃcients. By changing the values of the expansion coeﬃcients local shape changes 4.3. Topology optimization of viscoelastic and structural material 27 are realized. Therefore, the expansion coeﬃcients are used as design variables during the optimization. The optimization problem is formulated as: ﬁnd αik , i = 1, 2, . . . , N, k = 1, 2 ∑ ϕH KI ϕr r max J = γr = HK ϕ ϕ R r r r r ( ′ ) K + iK ′′ ϕ = λ2 M ϕ s.t. ∑ (4.20) ϕH j M ϕl = δjl λ21 ≥ λ2min k V k ≤ Vmax k k ≤ αik ≤ αi,max . αi,min The optimization routine searches for the combination of expansion coeﬃcients αik to maximize the summation of the ﬁrst r loss factors. In the presented work all loss factors have equal weight, however, it is also possible to deﬁne diﬀerent weight factors for each modeshape. The optimization is subjected to the eigenvalue problem. The δ represents the Kronecker delta, furthermore, a minimum eigenfrequency of the structure is required to prevent development of low-frequency localized modes in viscoelastic material regions. Additionally, constraints can be added for the volume of either structural or viscoelastic material V k . Finally, the design freedom of the optimizer is limited by specifying k k a lower αi,min and upper αi,max bound for the design variables. 4.3.4. Adjoint sensitivity analysis An adjoint sensitivity analysis is performed to derive an exact formulation of the loss factor sensitivities. In this analysis we assume that all modes are of multiplicity one. The adjoint sensitivity analysis for discretized systems is discussed by Adelman and Haftka (1991) and is applied in the context of multi-material level set methods by Allaire et al. (2014). The formulation of the adjoint problem including a complex-valued eigenvalue problem is based on the adjoint sensitivity analysis given in (van der Veen et al., 2014, Appendix B). A similar approach is applied to add the adjoint variables µ1 and µ2 to the loss factor: γr⋆ = ( ( ) ) ( ) ϕH r K I ϕr H 2 + ℜ µ K − λ M ϕ + µ2 ϕH M ϕ − 1 . 1 r H ϕr K R ϕr (4.21) Taking the gradients with respect to the design variables αik results in ) ( ) ( H ) ∂ϕr ∂ϕr H ∂KI H H ∂KR H 2ϕ K + ϕ ϕ − ϕ K ϕ 2ϕ K + ϕ ϕ I ∂αk I r R ∂αk r r r ∂αk r r r r ∂αk i i i i ∂γr⋆ = ( )2 k ∂αi ϕH r KR ϕr ) ( ) ∂ϕ ( 2 ∂λ ∂K ∂M r r 2 − λ2r k − M ϕr + µH + ℜ µH 1 K − λr M 1 ∂αik ∂αi ∂αik ∂αik ( ) ∂ϕr H H ∂M + µ2 2ϕr M k + ϕr ϕr . ∂αi ∂αik ( ϕH r K R ϕr ) ( The adjoint multipliers µ1 and µ2 are chosen such that the modeshape sensitivities ∂λ2r ∂αk i ∂ϕr ∂αk i (4.22) and eigenfre- quency sensitivities are not required to be calculated. The following adjoint problem has to be solved to determine the adjoint variables: ( − ϕH K ϕ (2K ϕ )+ ϕH K ϕ (2K ϕ ) ) ( r R r) I r ( r I r) R r 2 K − λ2r M 2M ϕr µ (ϕHr KR ϕr ) 1 = (4.23) . H 2ϕr M 0 µ2 0 28 4. Topology optimization of viscoelastically damped structures When the adjoint variables are known, the loss factor sensitivities are determined from the remaining terms in Equation (4.22): ( ) ( ) ( H ) ( H ) H ∂KR H ∂KI ϕr KR ϕr ϕr ∂αk ϕr − ϕr KI ϕr ϕr ∂αk ϕr i i ∂γr⋆ = ( )2 k ∂αi ϕH r KR ϕr (4.24) ) ( ∂K ∂M ∂M + ℜ µH − λ2r k ϕr + µ2 ϕH ϕr . 1 r ∂αik ∂αi ∂αik To determine the loss factor sensitivity with respect to all design variables αik requires to solve the adjoint problem (4.23) once per iteration for each considered mode. Then, equation (4.24) is solved for each design variable. The required sensitivities with respect to the stiﬀness and mass matrices can be derived from equations (4.16) and (4.17). 4.3.5. Numerical implementation For the numerical implementation of the presented method, the design domain is discretized in fournode quadrilateral (Q4), square plane stress ﬁnite elements. The ersatz material model is applied to scale the material properties around the level set boundaries. This formulation is applied for its simplicity, but note that mixed materials can still appear in the domain near boundaries between material regions. This local eﬀect can be reduced by mesh reﬁnement by reducing the minimal size of the transition region between material regions. The Heaviside function and its derivative are implemented with the approximations (Wang et al., 2003): φk ≤ −∆ a, ( ) (φk )3 3(1−a) φk k 1+a H(φ ) = + 2 , −∆ < φk < ∆ 4 ∆ − 3∆3 1, φk ≥ ∆ (4.25) ( ) ( k )2 3(1−a) 1 − φ , |φk | ≤ ∆ ∂H 4∆ ∆ = ∂φk b, |φk | > ∆. This implementation of the smooth Heaviside can introduce some mixed regions within the domain. However, it has been observed that the problem has a tendency to result in clearly separated material regions. In the following numerical examples, the parameters are implemented as: a = 0.001, b = k k 0.0005, ∆ = 1, αi,min = −2, αi,max = 2. The parameter b is kept to a non-zero value to allow possible design changes after a point in the design has exceeded ∆. Otherwise, the sensitivity information equals zero and the design cannot be changed once the absolute value of the LSF has exceeded ∆. For each element in the discretization a shape function is introduced. Hence, the centers of the shape functions are aligned with the centers of the elements in the mesh. The inﬂuence radius R is kept between two and four times the mesh size. The optimization problem is solved using the Method of Moving Asymptotes (MMA) as given in (Svanberg, 1987). This method iteratively solves nonlinear programming problems and is suited for general and structural optimization problems. In each step of the iterative process a strictly convex approximating subproblem is generated and solved. The resulting topologies are visualized based on the level set functions. 4.4. Numerical examples This section presents a number of case studies to study the optimal design of damped structures for maximum dissipation, and to illustrate the performance of the proposed optimization routine. First, a cantilever beam is optimized. Secondly, a comparison is made between the application of diﬀerent complex material moduli and their inﬂuence on the ﬁnal designs. Finally, the optimization routine is applied to an existing structure with a limited design space. 4.4.1. Case study I: cantilever beam The design domain of the cantilever beam is given in Figure 4.3. The complete cantilever beam is provided as design domain for the optimization. The cantilever is ﬁxed on the left side, the remaining 4.4. Numerical examples 29 Design domain H L Figure 4.3: The design domain for case study I. boundaries are unconstrained. The domain is discretized with L = 70 and H = 20 elements. The ﬁrst level set function describes the structural material. For the structural material steel is applied with Young’s modulus E1 = 200 GPa, Poisson ratio ν1 = 0.3 and density ρ1 = 7.85 × 103 kg/m3 . Any damping introduced by the structural material is neglected, as this is much smaller than the damping provided by the viscoelastic material. The second level set function models the viscoelastic material and general material properties are applied to model the viscoelastic behavior with Young’s modulus E2 = 1 GPa, Poisson ratio ν2 = 0.3 and density ρ2 = 1.0 × 103 kg/m3 . A complex shear modulus with a material loss factor of ηshear = 1.0 is applied to model the dissipative properties. The value of the shear modulus is derived from the Young’s modulus and Poisson ratio as G= E . 2(1 + ν) (4.26) The optimization is initialized with two constant level set functions. The initial values are chosen slightly above the zero level set, such that the design is initially ﬁlled with material which is a mixture of structural material with a small amount of viscoelastic material available. Since the level set functions are initialized as constant values above the zero level set, the design domain is completely ﬁlled by elements with scald material properties with respect to the values of both initial LSFs. The objective of the optimization is to maximize the average of the structural loss factors corresponding to the ﬁrst two eigenmodes. This represents a simpliﬁed form of optimizing the dissipation within a certain frequency band which contains both eigenfrequencies. The optimization is constrained such that only 40% of the design domain may be occupied by viscoelastic material. Furthermore, a frequency constraint is applied to keep the ﬁrst eigenfrequency within 50% of the eigenfrequency of the initial design. We mainly apply the frequency constraint to prevent the creation of low-frequency internal modes in the viscoelastic material regions with high structural loss factors. Therefore, the chosen percentage can be chosen to match any practical frequency requirements. The optimization is terminated when all constraints are satisﬁed and when the ﬁnal objective value is within 1% of the previous three objective values. Figure 4.4 shows the progression of the design throughout the iterations, with the ﬁnal design given in Figure 4.4d. The general layout of the viscoelastic material resembles a constrained-layer damping conﬁguration, as the viscoelastic material is mainly distributed along the centerline of the cantilever. After 35 iterations the ﬁnal design achieved an objective value of J = 0.5075. The result can be compared with a cantilever with the same aspect ratio with a conventional constrained layer damping conﬁguration, which provides a structural loss factor between roughly 0.26 and 0.39 for a centered, viscoelastic layer between respectively 3 and 7 elements. The optimized design is able to achieve higher structural loss factors compared to the conventional conﬁgurations. The history of the objective values and the corresponding loss factors of the two modeshapes are given in Figure 4.5. Similarly, Figure 4.6 presents the progression of the volume and frequency constraints. The constraint values are normalized with respect to the maximum value of the constraint. For this formulation positive values represent unsatisﬁed constraints, while negative values correspond to satisﬁed constraints. All constraints remain satisﬁed at the end of the optimization. 30 4. Topology optimization of viscoelastically damped structures (a) Iteration 2, J = 0.1651. (b) Iteration 5, J = 0.3761. Structural (c) Iteration 20, J = 0.4553. Viscoelastic (d) Iteration 35, J = 0.5075. Figure 4.4: The design progression throughout the optimization routine at iteration 2, 5, 20 and 35 and the corresponding objective values. 1 0.7 0.9 Mode 1 Mode 2 0.6 0.8 0.5 Loss factor (−) Objective (−) 0.7 0.6 0.5 0.4 0.3 0.4 0.3 0.2 0.2 0.1 0.1 0 0 5 10 15 20 Iteration 25 30 0 0 35 5 10 (a) Objective. 15 20 Iteration 25 30 35 (b) Loss factors. Frequency constraint (−) Volume constraint (−) Figure 4.5: The history of the objective values (a) and corresponding loss factors (b) during the optimization. 1.5 Structural material Viscoelastic material 1 0.5 0 −0.5 0 5 10 15 20 25 30 35 5 10 15 20 Iteration 25 30 35 0 −0.5 −1 −1.5 −2 −2.5 0 Figure 4.6: The history of the volume (top) and frequency (bottom) constraint values during the optimization. The constraints are normalized with respect to the speciﬁed maximum value. Positive values represent unsatisﬁed constraints 4.4. Numerical examples (a) Shear strain εmax mode 1. 31 (b) Shear strain εmax mode 2. Figure 4.7: The deformation shapes and corresponding maximum shear strain εmax for the ﬁrst (a) and second (b) eigenmode of the optimized structure. The uncolored elements represent void elements within the design. The application of the complex shear modulus results only in energy dissipation when the viscoelastic material is subjected to shear deformation. Therefore, the performance of the obtained designs can be visualized by plotting the shear deformation of each element. The maximum in-plane shear strain is determined as: √( )2 εx − εy εmax = + ε2xy . (4.27) 2 To compare the deformation between both modeshapes a base excitation is applied. The eigenmodes of the structure only provide information on the deformation shape and not on the actual amplitude of the deformation. Therefore, the ﬁrst and second eigenmodes are excited with the same harmonic acceleration proﬁle at the clamped side of the cantilever. Figure 4.7 shows the resulting deformation shapes and the maximum shear strain. The shear strain is mostly visible in the viscoelastic material. In both eigenmodes the viscoelastic material is eﬀectively used, as almost all viscoelastic elements are subjected to shear strain. In the second mode, slightly higher strain values are observed, corresponding to the slightly higher structural loss factor of the second modeshape. Figure 4.8 illustrates a contour plot of the energy dissipated per element for the ﬁrst modeshape of the structure. The obtained energy is directly related to the shear deformation observed for this modeshape, as illustrated in Figure 4.7a. The dissipated energy is calculated by Equation (4.10) and is normalized with respect to the element with highest dissipation. The direct relation between shear deformation and dissipated energy is observed when comparing the ﬁgures of shear strain and dissipated energy (Figures 4.7a and 4.8). 4.4.2. Design inﬂuences by complex moduli In the previous case study the viscoelastic material has been described with a complex shear modulus. However, as discussed in Section 4.2 the viscoelastic material can also be represented with either a complex Young’s or complex bulk modulus. To illustrate the inﬂuence of the diﬀerent modeling Figure 4.8: The energy dissipated for each element for the ﬁrst eigenmode of the structure. The values are normalized with respect to the maximum energy dissipated in an element. The obtained result is directly related to the shear strain given in Figure 4.7a. The uncolored elements represent void elements within the design. 32 4. Topology optimization of viscoelastically damped structures (a) Complex shear modulus, JG = 0.6002. (b) Max principal strain. (c) Max shear strain. (d) Complex Young’s modulus, JE = 0.8823. (e) Max principal strain. (f) Max shear strain. (h) Max principal strain. (i) Max shear strain. Structural Viscoelastic (g) Complex bulk modulus, Jκ = 0.6940. Figure 4.9: A comparison between the optimized designs and their performance for the application of multiple complex material models: (a): shear, (d): Young’s and (g): bulk modulus. The diﬀerences in performance are highlighted by the maximum principal and maximum shear strains within the structure for the ﬁrst eigenmode. approaches, the optimization of case I will be evaluated separately for a complex shear, Young’s and bulk modulus. For each complex modulus a corresponding material loss factor of η = 1 is applied. The remaining material parameters and the design space are kept the same. However, in this optimization only the structural loss factor corresponding to the ﬁrst eigenmode is considered as the objective. The optimizations were terminated after the objective value converged and all constraints were satisﬁed. The optimization with a shear, Young’s and bulk modulus provide the objective values: JG = 0.6002, JE = 0.8823, Jκ = 0.6940. The obtained designs are given in Figure 4.9. The material layout of the three designs show signiﬁcant diﬀerences. For the complex shear modulus the material is mainly located in the center of the design, similar to conventional constrained layer conﬁgurations. For the complex bulk modulus, the material is mainly located at the outer edge of the domain, where the normal strains are the highest. In the design using a complex Young’s modulus, the viscoelastic material is mainly located at the right side of the cantilever, where it is subjected to both normal and shear strains. The performance of the designs are illustrated by plotting the in-plane maximum shear strains, equation (4.27), as well as the maximum in-plane principal strains within the structure. ε1,2 εx + εy = ± 2 √( εx − εy 2 )2 + ε2xy . (4.28) The obtained ﬁgure are presented in Figure 4.9. The design based on a complex shear modulus shows almost identical principal strains as maximum shear strains, illustrating that most of the viscoelastic material is subjected to shear deformation. On the other hand, the design using the complex bulk modulus shows higher maximum principal strains, illustrating that the viscoelastic material is mainly subjected to normal strains. The complex Young’s modulus contains a combination of both shear and normal deformations. The higher loss factor achieved by the complex Young’s modulus is understood, since any deformation of the viscoelastic material will result in energy dissipation for the complex Young’s modulus, while in the complex shear and bulk modulus only the shear or normal strain components dissipate energy. 4.4. Numerical examples 33 Table 4.2: A comparison between the loss factors for the optimized designs, when the performance is evaluated with diﬀerent material models than used during the optimization. Modulus for performance check Modulus during optimization G E κ G 0.6002 0.6298 0.0341 E 0.1838 0.8823 0.4834 κ 0.0206 0.7168 0.6940 Performance cross-check To illustrate that the obtained designs only show optimized performance for the material model used during the optimization, an additional veriﬁcation is performed. The structural loss factor for each optimized design is evaluated with the other two complex material models. This is to show the impact of the used material model on the optimization process. This is done for all combinations of material moduli and results are given in Table 4.2. Since the complex Young’s modulus dissipates energy for any deformation, it is observed that all designs evaluated with this modulus provide reasonable or even improved performance. However, a signiﬁcant performance diﬀerence is observed between the complex shear and bulk modulus. 4.4.3. Case study II: existing structure with limited design domain The second case study illustrates the ability to generate a design to damp a particular mode of an existing structure. In this case study a clamped beam is studied as existing structure. The beam is clamped at both sides and cannot be modiﬁed by the optimization. A limited design domain is deﬁned on top of the cantilever. The optimization routine is only allowed to develop designs within this domain. An illustration of the design and non-design domains are given in Figure 4.10. The objective of the optimization is to achieve maximum loss factor for the ﬁrst eigenmode of the structure. The material properties are the same as applied in Section 4.4.1 and a complex shear modulus with a material loss factor of ηshear = 1 is applied for the viscoelastic material. Design domain Non design domain Figure 4.10: The design and non-design domains for case study II. The non-design domain is clamped at both sides and cannot be modiﬁed by the optimization routine. The optimization is subject to two constraints: the ﬁnal design should utilize more than half of the total design domain and the volume fraction of the viscoelastic material is limited to half of the total design domain. The frequency constraint is omitted in this optimization routine, since the existing structure prevents the creation of very low frequency resonant modes. The ﬁnal design achieved after 58 iterations is given in Figure 4.11. The optimization results in J = 0.3803. Figure 4.12a presents the history of the loss factor and Figure 4.12b shows the history of the volume and frequency constraints. All applied constraints remain satisﬁed after 58 iterations. The optimization has developed a design which resulted in a loss factor of 0.3803 for the ﬁrst eigen- 34 4. Topology optimization of viscoelastically damped structures Structural Viscoelastic Figure 4.11: The ﬁnal design of the optimization achieved after 58 iterations. The design achieved an objective value of J = 0.3803. 0.4 0 Structural material Viscoelastic material −0.1 0.35 −0.2 Volume constraint (−) Loss factor (−) 0.3 0.25 0.2 0.15 −0.3 −0.4 −0.5 −0.6 −0.7 0.1 −0.8 0.05 0 0 −0.9 10 20 30 Iteration 40 (a) Loss factor. 50 −1 0 10 20 30 Iteration 40 50 (b) Volume and frequency constraints. Figure 4.12: The history of the loss factor (a) and normalized volume and frequency constraints (b) during the optimization. mode. The performance of the design is illustrated by plotting the dissipated energy per element during oscillation in Figure 4.13. During the ﬁrst eigenmode the structure within the design domain resonates in phase with the beam. The relative displacement of the top part is larger compared to the displacement of the beam, straining the viscoelastic material and dissipating energy during oscillation. The behavior of the generated design is similar to tuned mass damper systems and attenuates the ﬁrst resonant mode of the structure. Figure 4.13: The deformation shape for the ﬁrst eigenmode of the optimized structure of case study II. The dissipated energy in each element is plotted to illustrate the energy dissipation for the ﬁrst eigenmode. The uncolored elements correspond to void elements in the design. 4.5. Discussion A systematic design approach has been presented for the development of damped structures using a combination of viscoelastic and structural materials. The optimization routine has shown to be capable to generate structures with high loss factors and improve the damping of a given structure. While the presented method provides promising results, a number of diﬃculties and assumptions remain. The current method optimizes the structures for a speciﬁed eigenmode. However, by the introduction 4.6. Conclusions 35 of structural and viscoelastic material the possibility exists to introduce new modeshapes at lower frequencies. Also, the change of the design might switch the order of modeshapes, especially for designs where eigenfrequencies are close together. A mode tracking procedure could be added to the proposed method to handle such events. Moreover, when multiplicity of eigenvalues is encountered additional measures are required to determine the eigenmode sensitivities, since the eigenmodes can change discontinuously during a crossing of eigenvalues. Moreover, the applied viscoelastic material model has a signiﬁcant inﬂuence on the performance of the optimization routine. As illustrated in Section 4.4.2, the application of diﬀerent complex moduli results in large diﬀerences between the optimized designs. Therefore, an accurate material model is required to describe the viscoelastic behavior. Of particular importance are its dissipative properties, as the optimization routine will exploit any dissipative behavior included in the material model. Finally, the implemented viscoelastic material model contains two assumptions: both temperature and excitation frequency dependencies are neglected. This considerably simpliﬁes the behavior of the viscoelastic material. However, depending on the applied viscoelastic material, both parameters can have signiﬁcant inﬂuence on the dissipative properties of the material. To extend the model to capture the thermal behavior requires a separate thermal analysis to determine the steady state temperature during oscillation. If temperature peaks occur within the viscoelastic material, its dissipative properties are reduced, resulting in lower structural loss factors than predicted. This results in a fully coupled thermomechanical problem that must be solved iteratively. A workaround is to introduce additional thermal constraints within the optimization routine to keep the temperature of the viscoelastic material between speciﬁed boundaries. The coupling with thermal behavior is left as future work. 4.6. Conclusions A well-performing topology optimization approach has been presented that can generate multi-material designs with optimized damping properties. The employed level-set based formulation yields a clear separation between various material phases. Structural loss factors using complex-valued eigenmodes have been found to provide an accurate assessment of damping characteristics, even for designs with high volume fractions of viscoelastic material combined with high material loss factors. In various examples, we have demonstrated that designs were obtained achieving higher structural loss factors than conventional constrained layer conﬁgurations. Also, the optimization method is successfully applied to attenuate a resonant mode of an existing structure. The generated design shows behavior similar to tuned mass damper systems and adds signiﬁcant damping to the resonant mode of interest. The method has the potential to be applied to complicated structures and can be extended towards three-dimensional designs. The inﬂuence of diﬀerent viscoelastic material models is investigated and signiﬁcantly diﬀerent results are found. To achieve suitable designs, the chosen viscoelastic material has to adequately reﬂect the behavior of the actual material. 5 Base excitation and thermal analysis In this chapter the viscoelastic structures are excited by a base excitation to investigate their response for a range of excitation frequencies. The performance of the optimized structure is compared with conventional damping solutions to illustrate the achieved performance improvement. Besides, a steady-state thermal analysis is performed to investigate the thermal response of the viscoelastic damper when subjected to an external vibration. 5.1. Base excitation The designs generated by the optimization have optimized loss factors for speciﬁc eigenmodes. In practice the structures are most likely to be excited by a broad frequency spectrum. These vibrations are introduced by varying external forces or by displacements imposed by their supporting structure. The excitation by support motion is chosen to investigate the structure behavior to external vibration. The displacement of the support motion is imposed on the ﬁxed DOFs of the optimized designs. This represents various scenarios in which the vibration is imposed at a structure’s support, such as vibration carried by spacecraft, vehicles or the response to ambient vibrations. The DOFs x of the discretized system are partitioned in two sets to model the imposed support motion. The external displacements are imposed on the support DOFs x2 . The remaining DOFs x1 are left unconstrained. The equations of motion of the structure M ẍ + Kx = f , with K = KR + iKI , is partitioned accordingly: ẍ M11 M12 K 1 + 11 M21 M22 ẍ2 K21 x 0 1 = . K22 x2 r K12 (5.1) (5.2) The forces r in the right hand side represent the reaction forces between the moving structure and the moving support. The partitioning allows us to ﬁnd an expression for the response of the free degrees of freedom from the ﬁrst equation: M11 ẍ1 + M12 ẍ2 + K11 x1 + K12 x2 = 0. (5.3) A harmonic displacement is imposed at the support and therefore both the displacements x2 as well as the corresponding accelerations ẍ2 are known. In steady-state the structure is assumed to show a harmonic response at the same frequency. The amplitude of the response is found by rearranging Equation (5.3): ( )−1 ( ) x1 = −ω 2 M11 + K11 ω 2 M12 − K12 x2 , (5.4) 37 38 5. Base excitation and thermal analysis (a) Single element γ = 0.2280. (b) Three elements γ = 0.3142. (c) Thick layer γ = 0.4108. (d) Optimized, γ = 0.4858. Figure 5.1: Four diﬀerent viscoelastic dampers: (a): single element viscoelastic layer, (b): three elements viscoelastic layer, (c): a viscoelastic layer matching the thickness of the viscoelastic layer in the optimized design, and (d): the design generated by the optimization. The presented loss factors correspond to their ﬁrst resonant mode. with ω being the frequency of the external vibration. The viscoelastic material results in complexvalued response for the displacements x1 , which are converted towards real-valued displacements by taking in account the phase diﬀerences of each DOF. When the response of all degrees of freedom are known, the reaction forces as result of the imposed motion can be obtained by solving the second equation from Equation (5.2): r = M21 ẍ1 + M22 ẍ2 + K21 x1 + K22 x2 . (5.5) In practice the structures are mostly excited by a wide frequency spectrum. A support excitation is applied to the optimized design to investigate its frequency response around its ﬁrst eigenmode. This also allows us to compare the performance of the optimized design with conventional constrained layer damping conﬁgurations. The optimized design is compared with three other designs. These designs contain a constrained viscoelastic layer of various thicknesses, ranging from a very think, single layer of viscoelastic material to a thick layer, that matches the width observed in the optimized design. Figure 5.1 illustrates the diﬀerent viscoelastic dampers. All structures have the same dimensions and only the internal distribution of viscoelastic material diﬀers. The loss factors of the three designs are: 0.2280, 0.3142 and 0.4108, which are all below the loss factor of the optimized design: 0.4858. The structures are subjected to a harmonic support motion in the vertical direction, which is applied at the left side of the design. The response are measured at the top right corner of the design domain and the resulting frequency response functions are given in Figure 5.2. A unit displacement is applied for illustrative purposes. Therefore, the frequency response function will show unit amplitude at low frequencies, as the tip will follow the applied displacement. All designs show a magniﬁcation of the applied excitation near their ﬁrst resonant frequencies. All three conventional constrained layer damping conﬁgurations show larger amplitudes compared to the optimized design. Even the CLD conﬁguration with the thickets viscoelastic layer, shown in Figure 5.1c, shows larger amplitudes. The frequency response clearly illustrates the performance increase realized by the material distribution generated by the optimization routine. The optimized design shows a lower resonance frequency compared to CLD damping of a single and three layers thick viscoelastic layer. However, the relatively thick layer shows and even lower ﬁrst resonance frequency. The eigenfrequency of the optimized design is roughly 25% lower compared to the design using a single layer of viscoelastic material. When we are interested to achieve the highest damping for a broad frequency range, the deviation of resonance frequency might not be of speciﬁc importance. For a case in which a design should be optimized to damp an excitation with a speciﬁc frequency, the applied frequency constraint should be enforced more strictly. This would make sure 5.2. Energy dissipation 39 that the frequency of the obtained design corresponds with the speciﬁc excitation. However, in most practical cases the external vibration contains a relatively wide frequency band and the resonance frequency of the viscoelastic damper is not of speciﬁc interest. 1 10 Amplitude (−) Single element Three elements Thick layer Optimized 0 10 2 3 10 10 Frequency (Hz) 0 Phase (deg) −50 −100 −150 −200 −250 2 3 10 10 Frequency (Hz) Figure 5.2: Frequency response of four structures damped using viscoelastic material. The corresponding designs are shown in Figure 5.1. The designs are subjected to a support motion with unit amplitude at the left side of the designs and the response of the structure is measured at the top right corner of the cantilever. The response of the optimized structure (dark blue) is compared with three structures containing a conventional constrained layer damping conﬁguration. These designs contain diﬀerent thicknesses of viscoelastic material: a single layer (red), three layers (green) and a layer matching the thickness observed in the optimized design (cyan). The optimized design shows an amplitude reduction in comparison to the designs with conventional damping treatments. 5.2. Energy dissipation The viscoelastic material dissipates energy when subjected to deformation. The imposed support motion will result in deformation of the structure and especially when the structure is excited in its eigenfrequency. The dissipated energy is determined using the deformation shape of the structure when subjected to a steady, harmonic excitation. For this analysis the deformation shape obtained in Equation (5.4) is used. The dissipated energy is given by the following two expressions: Wcyc = πxH ℑ (K) x, ′′ Wcyc = πεT xy G εxy V, (5.6) with V being the volume of viscoelastic material, since the expression given in Equation (4.9) provides the dissipated energy per unit volume. The second expression can be evaluated based on the chosen material modulus to describe the dissipative behavior. Similar expression can be written for the bulk and Young’s modulus. Equation (5.6) provides the total energy dissipated for a single cycle for the investigated deformation mode. It has to be noted that these expressions are based on a single harmonic response with constant amplitude. If resonances with decreasing amplitudes or multiple 40 5. Base excitation and thermal analysis (a) Optimized design excited at its ﬁrst eigenfrequency. (b) Optimized design excited at its second eigenfrequency. (c) CLD conﬁguration excited at its ﬁrst eigenfrequency. (d) CLD conﬁguration excited at its second eigenfrequency. Figure 5.3: The ﬁgures illustrate the dissipated energy per cycle for each elements in the design. In each illustration the values are normalized with respect to the element with maximum energy dissipation. The distributions are given for the ﬁrst and second resonant mode of the optimized design in ﬁgures (a) and (b), and for a CLD conﬁguration in ﬁgures (c) and (d). harmonics are studied, the complete integral along stress and strain needs to be evaluated to determine the dissipated energy. To illustrate the energy dissipation the structures are subjected to a steady, harmonic oscillation. The resulting deformation modes are used to determine the dissipated energy per cycle. The equations are evaluated at element level to investigate the dissipation of each individual element and their relative contributions to the total energy dissipation. The element displacements or strains are determined and multiplied with either the element matrix or material modulus. The material parameters are scaled based on the level set function values at the investigate locations. The expressions to evaluate the dissipation on element level become: {( } ( )) ( ) e H 1 2 1 2 2 Wcyc = πx(X) H(φ (X)) 1 − H φ (X) Ke + H φ (X) Ke x(X), {( e Wcyc 1 = πεxy (X)H(φ (X)) )) ( ) 1 2 1 − H φ (X) G′′e + H φ2 (X) G′′e ( 2 } (5.7) εxy (X). The dissipated energy per element provides the same result for both methods. The element dissipation is illustrated for the optimized structure when a base excitation is applied in its ﬁrst and second eigenfrequency. The structures are subjected to a unit input and the obtained dissipations are normalized with respect to the maximum dissipation for a single element. Figures 5.3a and 5.3b show the obtained results. The highlighted elements are all located within the viscoelastic elements and most of the elements contribute to the total energy dissipation in both the ﬁrst and second resonant modes. The obtained results clearly correspond to the shear deformation as illustrated in Figure 4.7. The diﬀerence in the observed distribution between the shear strain and the element dissipations are explained by the squared relation of the element strains within the expression of the dissipated energy, as given in Equation (5.6). 5.3. Thermal analysis 41 To investigate the improvements the same analysis is performed for a structure with a CLD conﬁguration. The dissipated energy is presented in Figures 5.3c and 5.3d. In the CLD conﬁguration the dissipation is limited to the thickness of the viscoelastic layer. The optimization has improved its damping characteristics by locally extending the viscoelastic layer to locate more viscoelastic material on locations which are subjected to shear deformation. Besides, the optimization routine has slightly modiﬁed the modeshapes of the structure to allow larger shear deformation within the viscoelastic material. The maximum energy dissipation has thereby been moved towards the center of the design, rather than at the tip of the cantilever as observed for the CLD conﬁguration. 5.3. Thermal analysis The obtained distribution of dissipation energy provides a steppingstone towards a thermal analysis of the design. If we make the assumption that all dissipated energy is converted into heat, the energy dissipated by each element now represents a thermal load for each element within the domain. This thermal load is applied to the structure in a thermal analysis to determine the steady-state temperature distribution of the designs. From this analysis the temperature of the viscoelastic material is mainly of interest, since the dissipative properties can deteriorate by local temperature changes. Furthermore, the viscoelastic material is often concentrated within the designs and limits the conduction of heat by its low thermal conductivity. This results in both localized heat loads and localized temperature increases. The remainder of this chapter presents a thermal analysis to investigate the steady-state temperatures of the designs as a result of a harmonic base excitation. The thermal analysis is simpliﬁed by assuming isotropic thermal behavior of both structural and viscoelastic materials and the thermal coeﬃcients are assumed to be constant. The thermal behavior of the viscoelastic material is not yet included within the model, as this allows to perform a single linear analysis to obtain the temperature distributions. However, if the temperature dependent behavior of the viscoelastic material is included, a fully coupled thermomechanical analysis has to be solved iteratively to determine the steady-state temperatures. The thermal behavior of the continuum body with structural and viscoelastic material is described by the following equilibrium and boundary conditions in D, −div(κχ ∇T ) + Q − cχ ρχ Ṫ = 0 (5.8) q = hχ (T − T∞ ) on Γkconv , k T = T∞ on Γﬁx , which are illustrated in Figure 5.4a. The ﬁrst line describes the thermal conduction within the continuum. Note that in the context of this thermal analysis κ refers to the thermal conductivity in stead of the bulk modulus of the material. A similar notation is used as in Equation (4.5) to scale the conductivity κχ , convective heat transfer coeﬃcient hχ , speciﬁc heat cχ and material density ρχ . This scaling is based on the presence of the material phases k: κχ = κk χk , where χk is a function of phase Ωk . The equilibrium is subjected to two boundary conditions: a convective boundary condition on Γkconv and an imposed temperature at Γkﬁx . It is chosen to keep the temperature at the support of the structure equal to the environment temperature T∞ . The problem is discretized using ﬁnite elements to ﬁnd a solution for the temperature distribution. A detailed description of the formulation of the thermal problem is presented in Appendix B, Section B.1.1. The resulting problem to solve becomes: (KTc + KTh ) T = RQ + Rh , (5.9) with KTc the conductivity matrix, KTh the convective matrix, RQ the heat load introduced by viscoelastic material and Rh the convective boundary condition. A similar partitioning is applied as in the base excitation analysis to partition the free and imposed DOFs. In the analysis the heat conduction of the viscoelastic material is signiﬁcantly lower than the structural material: κ1 = 10κ2 . For the convective boundary condition natural convection is assumed with a similar convective boundary coeﬃcient for both material phases. The temperature of the ﬁxed boundary is assumed to remain 42 5. Base excitation and thermal analysis (a) General heat transfer problem. (b) Boundary conditions for optimized geometry. Figure 5.4: Illustrations of the heat transfer problem. Figure (a) provides a sketch of the domain and boundary deﬁnitions. The domain Ωk contains both two material phases Ω1 and Ω2 . Its boundary is separated in two parts: Γkﬁx and Γkconv which respectively impose a ﬁxed temperature and apply a convective boundary condition to the corresponding boundary. In ﬁgure (b) the boundary conditions are applied for the optimized structure. constant and equal to the environment temperature T∞ . All remaining boundaries are subjected to a convective boundary condition with an assumed convective heat transfer coeﬃcient for free convection in air equal to hχ = 5 W/(m2 K). Figure 5.4b provides an illustration of the boundary conditions applied to the optimized structure. The energy dissipation obtained in the previous section is applied as heat load to determine the steady-state temperature distribution corresponding to the applied excitation. The temperature distribution as result of the excitation at the ﬁrst resonance frequency of the structure is given in Figure 5.5a. It is chosen to visualize the temperature distribution using a normalized temperature diﬀerence with respect to the environment temperature T∞ . We are mostly interested to investigate the temperature distribution of the structure, since the actual obtained temperature distribution is highly dependent on the applied excitation and its amplitude. Furthermore, the temperature problems scales linearly with the heat load RQ , which is expected to change most signiﬁcantly in this problem. Even though the convective problem does not scale equally to RQ , its eﬀect is limited for free convection in air, especially for smaller structures. Therefore, the investigation of the normalized temperature distribution initially provides enough information regarding the observed temperature distribution and possible localized temperature increases within the optimized design. The obtained results provide a temperature distribution as expected. The boundary Γkﬁx remains equal to the chosen environment temperature an the most pronounced temperature increases are located within the center of the viscoelastic layer. This is in correspondence with the distribution of dissipated energy in the designs, as shown in Figure 5.3a. The convective boundary condition results in small reduction of the temperature near the top, right and bottom edges of the domain. The low thermal conductivity of the viscoelastic material prevents the generated heat from distributing throughout the design and peak temperature values are observed in relatively conﬁned areas within the viscoelastic material. The second resonant mode provides similar results, as illustrated in Figure 5.5b. The applied distribution of dissipated energy shows relatively more energy dissipation on the right half of the domain compared to the energy distribution in the ﬁrst resonant mode, as illustrated in Figure 5.5b. These diﬀerences are observed in the resulting steady-state temperature distribution in Figure 5.3b. Again, peak temperature values are achieved within the center of the viscoelastic material. The presented thermal analysis provides brief insights into the expected steady-state temperature distributions of the optimized design for a harmonic excitation in its ﬁrst and second resonant mode. The analysis conﬁrmed that the temperature increases are mostly to be expected within the viscoelastic 5.3. Thermal analysis 43 material and that temperature peaks occur at relatively small areas of the viscoelastic material. Depending on the frequency and magnitude of the excitation, the type of viscoelastic material, and the boundary conditions of the structure, these temperature peaks could deteriorate the damping properties of the viscoelastic material. As this is very case dependent, a careful analysis has to performed to identify the temperature distribution within the viscoelastic material to determine if any possible deterioration of its damping characteristics might occur. (a) Mode 1. (b) Mode 2. Figure 5.5: Normalized temperature distribution as result of the energy dissipation due to deformation of the viscoelastic material. The temperature distribution is normalized with respect to the highest temperature diﬀerence in the structure. The structure is subjected to an imposed temperature at the left edge, while the remaining edges are subjected to a convective boundary condition. The results are shown for the ﬁrst (a) and second (b) resonant mode of the optimized structure. 6 Discussion A systematic design approach has been presented for the development of viscoelastically damped structures. The optimization routine is able to develop structures with high structural loss factors and increases the structural damping for a given structure. Wile the proposed optimization routine provides promising results, a number of diﬃculties and assumptions remain. The chosen material model to represent the viscoelastic material behavior has signiﬁcant inﬂuence on the performance of the optimization routine. As illustrated in Section 4.4.2, the application of diﬀerent complex material moduli results in large diﬀerences between the optimized designs. An accurate description of the viscoelastic behavior is therefore required to achieve realistic, well-performing designs. Of particular importance are the dissipative properties included in the material model, since the optimization routine will exploit any dissipative behavior to achieve higher structural loss factors. Moreover, the current implementation has neglected the temperature and frequency dependencies of the viscoelastic material. These assumptions simplify the behavior of the viscoelastic material. To gain accurate descriptions of the structural loss factor it is important to include both temperature and frequency dependencies within the material model, since both parameters can deteriorate the dissipative properties of the viscoelastic material. The thermal analysis mainly illustrates localized temperature increases within the viscoelastic material, which might locally deteriorate the dissipative capabilities. By including the temperature and frequency dependencies a fully coupled thermomechanical problem is obtained that has to be solved iteratively to determine the steady-state temperatures and the corresponding structural loss factor. In addition, the material description by the level set functions neglects any interaction forces between both materials. It assumes that the displacements of the viscoelastic material are imposed by the structural material and that both displacements are therefore identical. However, in practice both materials are connected by an adhesive interface, which possibly reduces the obtainable structural loss factor of the design. More advanced ﬁnite element modeling allows to study these interface problems in more detail. Methods such as (interface enriched) generalized ﬁnite element methods provide means to accurately study interface eﬀects, while maintaining a relatively simple, structured mesh (Soghrati et al., 2010). 45 7 Conclusions and recommendations 7.1. Conclusions The goal of this research has been to develop a topology optimization routine for the structural design of vibration isolating structures. In this routine the optimization is able to distribute both structural and viscoelastic material simultaneously throughout the design domain to achieve optimized damping characteristics. Previous research by TNO and TUDelft presented a density-based topology optimization routine that is able design viscoelastic dampers using both materials. Even though promising results were obtained, large regions of the design were occupied by material mixtures. Therefore, the research has been continued to investigate alternative methods for the optimization of vibration isolating structures using viscoelastic materials. The performance of three widely applied topology optimization methods have been compared for multi-material optimization problems. The material separation, complexity and ease of numerical implementation were evaluated. The choice was made to adopt a variation on the level set-based approaches and apply a parametric level set method for this multi-material optimization problem. The parametric method performs as a midway between density-based and level set-based formulations and shows advantages of both methods. Gradient-based optimization is possible, which reduces the involvement required for shape and topological derivatives in classical level set methods. Besides, the parametric method still provides an exact description of the boundary between both material domains. The parametric level set method is applied for the multi-material optimization of viscoelastic dampers. This allows to simultaneously distribute structural and viscoelastic material without any requirements of the designer to specify the location and dimensions of a viscoelastic layer. The implementation shows a well-performing topology optimization routine that generates multi-material designs with optimized damping characteristics. The optimization routine maximizes the structural loss factor of the design for resonant modes of interest. In this formulation a modiﬁed structural loss factor is implemented by including the imaginary components of the stiﬀness matrix in the eigenvalue analysis. The full complex modeshapes are obtained and used for the calculation of the structural loss factor. It has been shown that this method agrees much better to the damping characteristics compared to previously proposed measures. The optimization routine is able to generate designs with optimized damping characteristics for a speciﬁc number of eigenmodes. The optimization of a cantilever beam clearly shows improved performance compared to conventional constrained layer conﬁgurations. The freeform distribution of the viscoelastic material allows to achieve higher structural loss factors compared to conventional design approaches. Moreover, the optimization is able to improve the damping characteristics of existing designs. The design of a tuned-mass-damper like system is presented, in which the optimization develops structures to improve the damping of a speciﬁc resonant mode of a given system. 47 48 7. Conclusions and recommendations The improvements of the design are visualized by a frequency response analysis of the structure. This analysis conﬁrmed the improved damping characteristics as predicted by the increased structural loss factors. The optimized design shows a reduced amplitude compared to the conventional constrained layer damping conﬁgurations for the same excitation. The frequency response analysis also allowed to determine the local energy dissipation by the viscoelastic material and thereby determine the corresponding steady-state temperature distribution within the designs. The thermal analysis indicated localized temperature increases, which potentially result in local deterioration of the viscoelastic material and thereby reduce its damping characteristics. 7.2. Recommendations The proposed optimization routine has provided promising results for the optimization of viscoelastic dampers. The method inspires further research regarding the development of the parametric level set approach in topology optimization, the extension towards advanced ﬁnite element modeling and the veriﬁcation and validation of the presented designs. The following list proposes some topics for future research and developments. • The veriﬁcation and validation of the proposed optimization method and the applied material models to represent the viscoelastic behavior. Experimental work could verify the predicted behavior of the optimization routine and validate its eﬀectiveness for the design of viscoelastic dampers. Besides, the improvements by the modiﬁed structural loss factor can be veriﬁed by experimental frequency response analysis. • The current implementation only speciﬁes a minimum frequency constraint and limits the available material volumes for both materials. In the presented investigation these constraints have presented the design of structures completely formed by viscoelastic material, by requiring a certain stiﬀness enforced by the frequency constraint. However, for industrial applications it might be beneﬁcial to include additional strength, stiﬀness or stress constraints within the optimization routine. • Besides, the eﬀectiveness of the optimization routine can be illustrated by applying this method for a high performance, mechatronic system to show the applicability of the optimization routine in industry. This would require extension of the current optimization towards three-dimensional structures to fully cope with industrial design challenges. The three-dimensional optimization could be realized by applying commercial ﬁnite element packages (e.g. ANSYS or Comsol) to perform the ﬁnite element analysis, while the optimization routine is performed within Matlab or similar environments Some thoughts on this approach are given in Appendix E. • Furthermore, the current implementation should be extended to handle temperature and frequency dependencies of the viscoelastic material. The challenge of this research is to include a fully coupled thermomechanical analysis within the optimization routine. Iterative solution methods are required to ﬁnd the steady-state temperature and structural loss factor and the temperature and frequency dependencies have to be included within the sensitivity analysis regarding the structural loss factor and possible thermal constraints to limit the temperature within the viscoelastic material. • Finally, the optimization routine itself might be extended towards advanced ﬁnite element modeling. 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A Viscoelastic material model This appendix provides a detailed discussion on the material model, which represents the elastic and dissipative properties of the viscoelastic material. Besides, additional material and derivations are presented to supplement the implemented theory in Section 4.2. A.1. Material model If a viscoelastic material is subjected to deformation the corresponding stresses are not in phase with the applied strains (Tschoegl, 1989). This is in contrast to purely elastic or viscous materials, in which there is either no phase diﬀerence between the stresses and strains or a phase diﬀerence of exactly 90 degrees. In the case of a harmonically excited structure the stress will lead the applied strain by a phase angle θ. The strain ε and stress σ are represented as ε(t) = ε0 sin(ωt), σ(t) = σ0 (t) sin(ωt + θ(ω, T )). (A.1) An illustration of this relation is show in Figure A.1. The behavior of the harmonically excited structure and the observed phase diﬀerence can be represented by introducing a complex material modulus. This modulus is found by dividing the stresses by the strains (Tschoegl, 1989; Grootenhuis, 1970), resulting in: E(ω, T ) = E(ω, T )′ + iE(ω, T )′′ . (A.2) The storage E ′ and loss modulus E ′′ are related to the material loss factor η and the loss angle θ: E(ω, T )′′ = tan(θ(ω, T )) = η(ω, T ). E(ω, T )′ (A.3) 1 Strain Stress 0.5 0 −0.5 −1 0 0.1 0.2 0.3 0.4 0.5 time (s) 0.6 0.7 0.8 0.9 1 Figure A.1: Illustration of the phase diﬀerence between strain ε(t) and stress σ(t) for a harmonically excited viscoelastic material. 53 54 A. Viscoelastic material model Figure A.2: Illustration of the frequency and temperature dependent behavior of the shear modulus and loss factor of a viscoelastic material. The loss factor shows a maximum for certain operating frequency and temperatures. Image modiﬁed from Beards (1997). Figure A.2 illustrates the loss factor as function of both temperature and frequency. The loss factor signiﬁcantly decreases near the extremities of both frequency and temperature. This behavior has to be kept in mind when selecting viscoelastic materials which show maximal damping characteristics for the required operating frequency and temperature. The proposed method introduced in Section 4.2 simpliﬁes the material model by neglecting the temperature and frequency dependencies of the viscoelastic material and assuming the viscoelastic material remains close to its intended operating frequency and temperature. Single DOF model To illustrate the implementation of the hysteretic model a single degree of freedom, mass spring damper system is compared for viscous and viscoelastic damping. Each structure is subjected to a harmonic excitation, which results in the following equations of motion: mẍ + cẋ + kx = F0 eiωt , mẍ + k(1 + iη)x = F0 eiωt . (A.4) The complex stiﬀness introduces a damping term to the equation of motion, similar to the viscous damping coeﬃcient. It becomes even more clearly visible when comparing both transfer functions: Hc = Hv = 1 (k − ω 2 m) + iωc , 1 . (k − ω 2 m) + ikη (A.5) The viscoelastic damping is not scaled by the excitation frequency and shows a phase diﬀerence, which is not necessarily aligned with the velocity, as in the viscous model. The viscoelastic model will also provide damping for very low frequencies, which is not an accurate representation of the material model. However, when the viscoelastic material is subjected to an excitation, the complex material model provides a relatively straightforward method to model both elastic and dissipative behavior. A.2. Energy storage and dissipation Viscoelastic material that is subjected to deformation will dissipate energy. For a harmonically excited structure, the dissipated energy is determined from a hysteresis loop. Figure A.3a illustrates the elliptical hysteresis loop when plotting the stresses σ(t) as function of the strains ε(t). The area A.2. Energy storage and dissipation 55 (a) Hysteresis loop. (b) Material modulus in complex plane. Figure A.3: (a): Illustration of a hysteresis loop of a viscoelastic material subjected to harmonic excitation. The loop is found by plotting the stress σ(t) as function of the strain ε(t). (b): The complex material modulus drawn in the complex plane to express the modulus as function of the storage or loss moduli using the phase angle θ. captured by the hysteresis loop describes the dissipated energy in the viscoelastic material during each cycle. The dissipated energy is found by solving the following contour integral: ∮ Wcyc = σ(t) dε(t). (A.6) The strain and stress deﬁnitions from Equation A.1 are applied and the integral is solved for one cycle: ∮ ( ) Wcyc (ω) = σ0 (ω) sin ωt + θ(ω) ε0 ω cos(ωt) dt (A.7) ( ) = πε0 σ0 (ω) sin θ(ω) . The relation between the stress and strain is applied to simplify the obtained result: σ0 = E . ε0 (A.8) Also, the material modulus E is expressed in its corresponding storage or loss modulus by the relations obtained from the complex plane, as illustrated in Figure A.3b. The equation is rewritten to obtain: Wcyc (ω) = πε20 E ′′ (ω). (A.9) The power dissipated by the viscoelastic material is captured by multiplying the dissipated energy per cycle with the corresponding excitation frequency. Note that the excitation frequency is converted to Hertz by dividing by 2π: 1 Qdissip (ω) = ε20 E ′′ (ω)λ2 . (A.10) 2 The dissipated power corresponds to a constant value for a single oscillation. This allows to perform steady-state calculations on the achieved temperatures within the viscoelastic material. However, if a thermal analysis performed on time scales smaller than a single period additional calculations are required to determine the dissipations as function of time during a single period. The viscoelastic material also stores energy elastically during each cycle. Applying similar calculations the elastically stored energy per cycle is found. The stored energy is a function of the strain amplitude and the storage modulus. 1 (A.11) Wstored (ω) = ε20 E ′ (ω). 2 56 A. Viscoelastic material model The expressions for the dissipated and stored energy are expressed as function of the Young’s modulus. However, it should be noted that these expressions can also be given as function of the bulk or shear modulus of the material. These formulations can be applied when its assumed the viscoelastic material only shows energy dissipation during speciﬁc types of deformation, e.g. shear or elongation. A.3. Loss factor The material loss factor, Equation (A.3) is commonly applied to describe the dissipative behavior of the viscoelastic material. To describe a structure containing both structural (elastic) and viscoelastic material a formulation is required to quantify a structure’s loss factor. The structural loss factor has been introduced by Johnson and Kienholz (1982) and is expressed as a fraction of dissipated energy by stored energy during each cycle Carfagni et al. (1998), which is similar to the inverse deﬁnition of the Q-factor: γ= Energy dissipated per cycle Stored energy per cycle (A.12) Since the loss factor is mostly determined at a certain resonance frequency of the structure it is often expressed as a function of the corresponding modeshape and the global stiﬀness matrix. These expressions are similar to Equations (A.7) and (A.11) derived in the previous section. This provides the following expression for the structural loss factor: γr = ϕH r K I ϕr . H ϕr KR ϕr (A.13) Notice that the scaling of the modeshapes can be chosen arbitrarily, since the modeshape only provide information on the shape of the deformation and no information regarding the actual displacements. In the formulation of the loss factor the modeshapes appear in both numerator and denominator, which makes the expression of the loss factor independent of the chosen scaling of the modeshapes. B Numerical implementation This appendix provides information on the numerical implementation of the presented multi-material, parametric level set method. The development of the ﬁnite element model is discussed and viscoelastic material behavior is introduced in the element stiﬀness matrices. The placement of the shape functions within the ﬁnite element discretization is illustrated. Moreover, the consequences of applying an approximated Heaviside function are discussed to illustrate the possible introduction of intermediate materials. B.1. Finite element model The design domain is discretized using two-dimensional, plane-stress, four-node quadrilateral ﬁnite elements (Q4). Each element contains four nodes with each two degrees of freedom. The nodes are located at the corners of the element. A single element is illustrated in Figure B.1. The numerical implementation only applies square elements with a = b. Linear shape functions are applied: 1 (a − x)(b − y), 4ab 1 N2 = (a + x)(b − y), 4ab 1 N3 = (a + x)(b + y), 4ab 1 N4 = (a − x)(b + y). 4ab The deformation matrix B contains the derivatives of the shape functions: N1,x 0 N2,x 0 N3,x 0 N4,x 0 B= 0 N1,y 0 N2,y 0 N3,y 0 N4,y , N1,y N1,x N2,y N2,x N3,y N3,x N4,y N4,x N1 = (B.1) (B.2) where the subscript ,x or ,y illustrates the derivative with respect to x or y. For the structural, isotropic material the plane-stress constitutive matrix D is used: 1 ν 0 E D= (B.3) ν 1 0 . 2 1−ν 0 0 1−ν 2 For the viscoelastic material the constitutive matrix is given as function of the bulk κ and shear G modulus. The Young’s modulus is related to the bulk and shear modulus (Tschoegl, 1989, Chapter 1): E = 3κ(1 − 2ν), E = 2G(1 + ν). (B.4) 57 58 B. Numerical implementation Figure B.1: Illustration of a Q4 element applied for the ﬁnite element discretization of the design domain. The method only applies square elements with a = b. This allows to deﬁne diﬀerent material responses as result of bulk or shear deformations by specifying a separate material loss factor for the bulk and shear modulus. A complex Young’s modulus is realized by specifying the same material loss factor for both moduli. The relations are substituted in the constitutive matrix D: 3κ(1 − 2ν)ν 3κ(1 − 2ν) 0 (1 − ν 2 ) (1 − ν 2 ) 3κ(1 − 2ν)ν 3κ(1 − 2ν) D= (B.5) . 0 (1 − ν 2 ) 2 (1 − ν ) 0 0 G The element stiﬀness matrix is found by solving the integral: ∫ Ke = B T DB dVe , (B.6) Ve with Ve the volume of the element. In general these integrals are solved numerically. However, this work applies the same elements throughout the complete design domain. Therefore, the integration is only required to be performed once and has been evaluated numerically. The material properties of the elements are scaled according to the element densities obtained from the optimization routine. Similarly, the element mass matrix is found by evaluating: ∫ Me = ρN T N dVe , (B.7) Ve with N being a vector containing the shape functions Ni and ρ being the density of the material. In the multi-material implementation an element stiﬀness and mass matrix are calculated for each material phase. The kth element matrix is illustrated by the notation Kek , Mek . An assembly routine is applied to position and sum the components of the element matrices within the global matrices. The elements are placed such that connections between neighboring elements are realized. The material properties of each element in the design domain are based on the value of the LSF. A Heaviside function is evaluated to determine which material phase is represented by the element and performs the required scaling of the material properties. The global matrices are given as function the element stiﬀness matrices and both level set functions: {( } ) k 1 2 1 2 2 K(φ ) = H(φ ) 1 − H(φ ) Ke + H(φ )Ke , {( } (B.8) ) k 1 2 1 2 2 M (φ ) = H(φ ) 1 − H(φ ) Ke + H(φ )Ke . B.1. Finite element model 59 The volume of each material phase is determined using the Heaviside functions and by integrating over the design domain: ∫ V1 = H(φ1 ) dΩ, ∫D (B.9) 2 V = H(φ1 )H(φ2 ) dΩ. D To investigate the damping characteristics of the structure it is possible to investigate the strains or stresses available within the viscoelastic material. The optimized structures are expected to show most strain within the viscoelastic material, as this will directly dissipate energy. The strain components are obtained from the nodal displacements u and the deformation B as: [ ε = εx ]T εy = Bu, (B.10) = Dε = DBu. (B.11) εxy with the stress distribution given as: [ σ = σx ]T σy σxy A strain transformation is applied to arrive at coordinate independent measures for the maximum in-plane principal strains ε1,2 and the maximum in-plane shear strain εmax : ε1,2 εmax √( )2 εx + εy εx − εy = ± + ε2xy , 2 2 √( )2 εx − εy = + ε2xy . 2 (B.12) B.1.1. Thermal model The strains calculated during a steady harmonic oscillation allow to calculate the energy dissipated during a singel cycle of the viscoelastic damper. The strains within each ﬁnite element describe the local energy dissipated by each element. To analyse the resulting steady-state temperature, the described ﬁnite element is extended to include element and nodal temperatures. Each node in Figure B.1 is given a additional DOF to describe its temperature. The temperature at the center of the element is then given by the average of the four surrounding nodal temperatures. The discretized equation to describe the thermal problem is given by: CTT Ṫ + KTT T = R. (B.13) Since we are only interested in the steady-state temperature the time derivatives of the temperatures Ṫ become zero and the thermal capacity matrix C drop out of the equation. To ﬁnd the steady-state temperature distribution the following problem is solved: KTT T = R. (B.14) Here KTT being the thermal conduction matrix (also referred to as the thermal stiﬀness matrix) and R the heat load imposed on the structure. In this analysis we limit ourself to convective boundary conditions and imposed boundary temperatures. To include the convective boundary conditions it is required to modify the thermal stiﬀness matrix and the boundary loads accordingly. When including the convective boundary conditions the steady-state problem is given by (KTc + KTh ) T = RQ + Rh . (B.15) The thermal conduction matrix is here given by KTc , the convective matrix KTh , the internal heat sources (or sinks) RQ and the convective boundary conditions Rh . The global matrices are assembled from the corresponding element matrices, which are found by integration of the shape functions for a single element. The matrices are found by evaluating the following integrals. Note that the convective 60 B. Numerical implementation boundary conditions only inﬂuence the nodes located on the boundary. The shape functions regarding the convective boundary conditions have been modiﬁed accordingly. ∫ ∫ KTc = B T κB dV, RQ = N T Q dV, V V ∫ ∫ (B.16) T T KTh = N N h dS, Rh = N hT∞ dS. S S B.1.2. Radial basis function placement Various methods are available to place the RBFs within the discretized design domain. In this implementation it is chosen to align the centers (knots) of the RBFs with the nodes of the ﬁnite element discretization. This means that for each node in the ﬁnite element model a RBF is placed and scaled with the corresponding expansion coeﬃcient αi . To align the RBF knots with the nodes greatly simpliﬁes the numerical implementation of the parametric level set method. The alignment of the knots with the nodes allow to directly extract the function value after the level set function has been constructed by summation. The values of the LSF are known directly above the nodes in the discretization. For implementations with diﬀerent placement of the RBF knots, an interpolation has to be performed to extract the level set function value in between the location of the expansion coeﬃcients. Figure B.2: One-dimensional illustration of the radial basis function placement. The RBFs are aligned with the centers of the elements in the ﬁnite element discretization. The shape functions ξi and the expansion coeﬃcients αi are aligned with the centers of the ith element in the design domain. The placement of the RBFs are illustrated for a one-dimensional situation in Figure B.2. The expansion coeﬃcients α are directly above the center of the ith element in the discretization. The level set function is constructed by summing the contributions of all scaled shape functions. Since this only results in a vertical shift of the LSF, the values at the center of the elements are directly extracted. The material properties of each element are related to the level set function values at the center of each element. The simplicity of using a single discretization for both the level set function and the structural design domain results in possible interplay between the level set function and the structural model (van Dijk et al., 2013). However, these eﬀects have not been observed in the case studies analyzed within this thesis. B.2. Heaviside function: discrete and approximate A Heaviside function is applied to the LSFs to determine the material properties of the elements in the material discretization as stated in Equation (B.8). In general the Heaviside function represents a discontinuous function whose values equal zero for negative arguments and equal one for positive arguments. The Heaviside function, or unit step function, is represented by the discrete form as: { 0, φk < 0, k H[φ ] = (B.17) 1, φk > 0. B.2. Heaviside function: discrete and approximate 61 3 1 Derivative of Heaviside function 0.9 Heaviside function 0.8 0.7 0.6 0.5 0.4 ∆ =1.25 ∆ =1 ∆ =0.75 ∆ =0.5 ∆ =0.25 0.3 0.2 0.1 0 −1.5 −1 −0.5 0 0.5 Level set function value 1 (a) Heaviside function H(φ). 1.5 2.5 2 1.5 1 0.5 0 −1.5 −1 −0.5 0 0.5 Level set function value 1 (b) Derivative of Heaviside function 1.5 dH(φ) . dφ Figure B.3: Approximations of the discrete Heaviside function and its derivative. Both approximated formulations are plotted for various values of the parameter ∆. For decreasing values of ∆ the approximations approach the discrete formulations. The implementation of a discrete Heaviside function results in a discrete optimization problem, where the design variables are either void or solid material. To overcome diﬃculties related to discrete optimization problems an approximated Heaviside function is implemented to continuously scale the material density between void and solid. The material description of a parametric level set method using an approximated Heaviside function becomes similar to density-based optimization methods with continuous pseudo-density descriptions (Sigmund and Maute, 2013). The Heaviside function is described using the analytical approximation as (Wang et al., 2003): φk ≤ −∆, a, ( k ) k 3 (φ ) 3(1−a) φ k H(φk ) = + 1+a (B.18) 4 ∆ − 3∆3 2 , −∆ < φ < ∆, 1, k φ ≥ ∆. The parameter a is chosen as a small number (≤ 1 × 10−3 ) to represent void material. The parameter ∆ determines the range (width) and slope of the approximated Heaviside function. Decreasing the value of ∆ makes the approximation approach the discrete Heaviside function. Figure B.3a illustrates the Heaviside function for diﬀerent values of ∆. The derivative of the Heaviside function is used during the sensitivity analysis for the optimization. In the direct formulation the derivative corresponds to the Dirac delta function and the approximate formulation is found by taking the derivative of the approximate Heaviside function: ( ( k )2 ) 3(1−a) 1 − φ∆ , |φk | ≤ ∆, dH 4∆ = (B.19) dφk b, |φk | > ∆. The parameter b is again chosen as small number to represent the zero derivative below and above the continuous region of the Heaviside function. In Figure B.3b the derivative of the Heaviside is plotted for various values of ∆. For decreasing values the derivative will approach the Dirac delta function. The approximation of the Heaviside function has consequences for the material distribution in the ﬁnite element discretization. If a discrete Heaviside function is applied the material corresponds to either solid or void elements, since the discrete function allows no intermediate density values. However, the introduction of the approximated Heaviside function does allow intermediate density values to appear near the boundary between material phases. The ersatz material model is applied to scale the material properties of elements with intermediate pseudo-density values with respect to the function value of the approximated Heaviside function. Intermediate densities might provide favorable material properties for the objective of the optimization, resulting in large regions of intermediate material in 62 B. Numerical implementation the ﬁnal design. In multi-material implementations, these intermediate material regions might overlap and thereby introduce elements with mixed material properties. Intermediate and mixed elements are unacceptable for the ﬁnal, optimized designs. These mixtures often provide unrealistic material behavior and should be avoided in the optimized topology. If these mixed elements have a strong presence in the ﬁnal designs additional penalization methods can be implemented. For the current implementation the ﬁnal design have shown little mixed elements and provide crisp material boundaries, as highlighted in Section 4.4 and B.3. Therefore no additional penalization has been applied. Figure B.4 illustrates the diﬀerences between the ﬁnite element material discretization for a discrete and approximated Heaviside function. The ﬁgure shows the diﬀerence for domains containing single and multiple materials. A discrete Heaviside function provides a reasonable approximation of the curves in the boundary, as illustrated by Figure B.4a. The representation of the curved boundaries could be improved with diﬀerent approaches, for example by applying smaller element sizes, by application of ﬁnite elements with diﬀerent shapes or by applying improved meshing algorithms to generate a conforming mesh. The topology is shown for an approximated Heaviside function in Figure B.4b. The crisp boundary between solid and void elements has disappeared and gray, intermediate elements are introduced near material boundary. Similar behavior is observed for multi-material structures. A discrete Heaviside function provides distinct separation between all material phases, as given in Figure B.4c. The approximated Heaviside again loses the crisp boundaries, as illustrated in Figure B.4d. Elements with mixed material properties are obtained due to the overlap of both intermediate materials. The mixed domains are mostly obtained near the boundary between the structural and viscoelastic phase and are illustrated by the purple hue surrounding the boundary in Figure B.4d. B.2. Heaviside function: discrete and approximate (a) Single material, discrete Heaviside function. Black and white elements correspond to solid and void elements. (c) Multi-material, discrete Heaviside function. Blue and red elements represent two diﬀerent material domains. 63 (b) Single material, approximate Heaviside function. Gray elements illustrate elements of intermediate density. (d) Multi-material, approximate Heaviside function. Lightly colored elements represent elements of intermediate density, while mixed colors (purple) illustrate a mixture of material properties from each material domain. Figure B.4: Illustration of the diﬀerence between discrete and approximate Heaviside functions on the material distribution and element densities within the ﬁnite element discretization. The cyan lines illustrate the boundaries obtained from the LSF. The ersatz material model is applied to scale the material properties of each element with the corresponding LSF value. For a single material (a,b) the approximation introduces intermediate material in the neighborhood of the boundary. Moreover, mixed material properties are observed near the boundary between two materials for the multi-material implementation (c,d). 64 B. Numerical implementation (a) LSF φ1 . (b) LSF φ2 . Figure B.5: Visualization of both level set functions as presented by the ﬁnal iteration of the optimization of case study I. The ﬁrst LSF φ1 represents the placement of material. The second LSF φ2 distinguishes the viscoelastic material within the material domain as speciﬁed by the ﬁrst LSF. The black contour line highlights the zero-level set. B.3. From Level set function to topology The conversion from level set function towards topology optimization is illustrated based on the results generated in case study, as described in Section 4.4.1. The multi-material topology optimization is extracted from both level set functions. The shape of both level set functions at the ﬁnal iteration are presented in Figure B.5. The steep transition between maximum and minimum values result in crisp transitions between the available material phases. The steep boundaries of the level set function are clearly visible in the second level set function in Figure B.5b. Any mixed material regions are more pronounced during the initial iterations of the optimization. For roughly the ﬁrst 10 iterations mixed elements are observed near the zero-level set. Another region of mixed materials are barely visible on the right side of the structure where some material phases overlap during these iterations. The material distribution at four iterations is visualized in Figure B.6. The mixed materials along the boundary are visible in the initial iterations, while these have disappeared almost completely in the ﬁnal design. Similarly, the mixed material properties on the right side of the structure are not observed in the ﬁnal design. The plots also illustrate that the optimization hardly encounters mixed material phases for the used formulation and description of the objective function. Therefore, no additional penalization method is required to force the optimization towards clearly separated material domains. Figure B.7 illustrates a set of similar ﬁgures created for case study II, as treated in Section 4.4.3. These results again show mixed material domains during the initial iterations and hardly any mixed elements in the ﬁnal design. B.3. From Level set function to topology 65 (a) Iteration 2. (b) Iteration 4. (c) Iteration 6. (d) Iteration 35. Figure B.6: The material distribution at four iterations during the optimization of case study I. Red and blue elements correspond respectively to structural and viscoelastic material. The cyan line highlights the zero-level set extracted from the level set functions. The mixture of these colors illustrates elements with mixed material. The mixed materials are mostly encountered along the zero-level set, as well as on the right side of the domain. (a) Iteration 2. (b) Iteration 3. (c) Iteration 30. (d) Iteration 58. Figure B.7: The material distribution at four iterations during the optimization of case study II. Red and blue elements correspond respectively to structural and viscoelastic material. The cyan line highlights the zero-level set extracted from the level set functions. The mixture of these colors illustrates elements with mixed material. The mixed materials are mostly encountered along the zero-level set, as well as on the right side of the domain. C Loss factor sensitivity analysis This appendix discusses the adjoint sensitivity analysis used in the optimization. The structural loss factor is used as objective function during the optimization. The loss factor is given as function of the complex-valued modeshapes ϕ as obtained from the eigenvalue analysis. The structural loss factor for mode r is given as ϕH K I ϕr , (C.1) γr = Hr ϕr K R ϕr with KI = ℑ (K) and KR = ℜ (K). The optimization applies the expansion coeﬃcients αik used in the parameterization of the level set function as design variables. Therefore, we are interested in the sensitivity of the structural loss factor with respect to inﬁnitesimal change of the expansion coeﬃcients. The presented sensitivity analysis assumes a multiplicity of 1 for all modeshapes in the structure. If higher multiplicity is encountered diﬀerent measures are required to determine the sensitivity of the structure. In this implementation no diﬃculties have been encountered during the optimization as result of eigenmodes with a multiplicity larger then one. The sensitivity is given by the derivative of the γr with respect to the expansion coeﬃcients αik : ( ) ( ) ( H ) ( H ) ∂ϕr ∂ϕr H ∂KI H H ∂KR ϕr KR ϕ 2ϕH K + ϕ ϕ − ϕ K ϕ 2ϕ K + ϕ ϕ I ∂αk I r R ∂αk r r r ∂αk r r r r ∂αk i i i i ∂γr = . (C.2) ( ) 2 ∂αik ϕH r KR ϕr The loss factor sensitivity contains various terms that have to be evaluated: the imaginary and R I real components of the stiﬀness matrix sensitivities: ∂K and ∂K and the mode sensitivity of the ∂αk ∂αk i i r eigenmode of interest: ∂ϕ . The eigenmode sensitivity requires to solve many equations to obtain the ∂αk i change of DOF within the eigenmode by changing a single design variable. This provides matrix of RN ×n with N number of DOFs in the ﬁnite element discretization and n number of design variables. We would prefer a method to ﬁnd the sensitivities of the structural loss factor without evaluation all eigenmode sensitivities. Therefore, an adjoint sensitivity analysis is applied, which provides an eﬃcient method to calculate the sensitivities of low number of response functions to a large amount of design variables. The method of adjoint sensitivities are discussed by Adelman and Haftka (1991). In the adjoint method additional terms are introduced to the objective function, which are multiplied by the adjoint variables. The expressions we add to the objective value are the constraints applied to the structural loss factor: the eigenvalue problem and the mass normalization of the modeshapes γr = ϕH r KI ϕr , ϕH r KR ϕr s.t. (K − λr M ) ϕr = 0, ϕH M ϕ = 1. (C.3) These terms are added to the structural loss factor using the adjoint variables µ1 and µ2 . Notice that µ1 is a vector containing equal number of DOFs as the eigenmodes ϕ. The adjoint formulation then becomes: ( ( ) ) ( ) ϕH r KI ϕr ⋆ H 2 γr = H + ℜ µ1 K − λr M ϕr + µ2 ϕH (C.4) r M ϕr − 1 ϕr KR ϕr 67 68 C. Loss factor sensitivity analysis We only require the real part of the second term to equal zero, which has been illustrated by (van der Veen et al., 2014, Appendix B). By taking the derivative of the adjoint formulation with respect to the design variables results gives: ( ) ( ) ( H ) ( H ) ∂ϕr ∂ϕr H H ∂KI H H ∂KR 2ϕ K ϕ K ϕ + ϕ ϕ − ϕ K ϕ 2ϕ K + ϕ ϕ R r I r I r R r k k k k r r r ∂α r r r ∂α ∂αi ∂αi i i ∂γr⋆ = ( )2 k H ∂αi ϕr KR ϕr ( ) ( ) ∂ϕ 2 ∂K ∂M ∂λ (C.5) r r 2 + ℜ µH − λ2r k − M ϕr + µH 1 1 K − λr M ∂αik ∂αi ∂αik ∂αik ( ) ∂ϕr H H ∂M + µ2 2ϕr M k + ϕr ϕr . ∂αi ∂αik ∂λ2 r In these sensitivities both the modeshape sensitivities ∂ϕ and the eigenfrequency sensitivities ∂αrk ∂αk i i appear. Both of these sensitivities are multiplied by the adjoint variables. Therefore, by correctly choosing the values for µ1 and µ2 , we are able to obtain an expression for the structural loss factor in which the calculation of the modeshape and eigenfrequency sensitivities are not required. To ﬁnd the values for the adjoint variables the following statements must hold: ( ) )( H )( H ) ( H ( ( )) ( ) ∂ϕ K K ϕ 2ϕ K − ϕ K ϕ 2ϕ ϕH R I r I r r r r 2 r R r + ℜ µH + µ2 2ϕH = 0, ( )2 1 K −λ M rM k ∂α ϕH K ϕ i R r r ∂λ2r M ϕr = 0. ∂αik (C.6) A trivial solution for this equation is to have the modeshape and eigenfrequency sensitivities equal zero. However, this is not the solution we are looking for, since we attempt to ﬁnd a value for the adjoint variables which satisﬁes these equations. Therefore, we have to ﬁnd a solution for the following relations: µH 1 ( ϕH r K R ϕr )( ) )( H ) ( H ( ) ( )) ( 2ϕr KR 2ϕH r KI − ϕr KI ϕr H 2 + ℜ µ1 K − λ M + µ2 2ϕH ( )2 r M = 0, ϕH r KR ϕr (C.7) and µ1 ∂λ2r M ϕr = 0. ∂αik (C.8) The adjoint variables are found by solving these two statements simultaneously. After rearrangeing, the adjoint problem becomes: − ϕH K ϕ (2K ϕ )+ ϕH K ϕ (2K ϕ ) ( ) ( r R r) I r ( r I r) R r 2 2 µ K − λr M 2M ϕr (ϕHr KR ϕr ) 1 = . µ 2ϕH M 0 2 0 r (C.9) Notice that in the second equation the eigenfrequency sensitivity is left out. Since this is only a scalar multiplication its value will make no diﬀerence when the equation has to equal zero. Besides, the additional multiplication by 2 is introduced to obtain a symmetric matrix on the left hand side. The symmetry of this matrix allows to apply eﬃcient solvers to numerically solve the system. By applying the values found by this adjoint problem, the statements (C.7) and (C.8) equal zero and will drop from the sensitivity expression (C.5). Only the remaining terms from this expression need to evaluated to determine the loss factor sensitivities: 69 ( ( ) ) ( H ) H ∂KI H ∂KR ϕ K ϕ ϕ − ϕ ϕ ϕ I r r r ∂αk r r r ∂αk i i ∂γr⋆ = )2 ( k ∂αi ϕH r KR ϕr ( ) ∂K ∂M ∂M + ℜ µH − λ2r k ϕr + µ2 ϕH ϕr . 1 r k ∂αi ∂αi ∂αik ( ϕH r K R ϕr ) (C.10) To solve this expression we need to ﬁnd the values of the adjoint variables and the sensitivity of the mass and stiﬀness matrices. These can be obtained by diﬀerentiating Equation (B.8) with respect to the design variables αik . We only need to solve the adjoint variable problem once for each investigate modeshape during each iteration. Using these values of the adjoint variables allows us to eﬃciently compute the structural loss factor sensitivities with respect to all design variables αik . If we would not have applied the adjoint sensitivity analysis, the calculation of the structural loss factor sensitivities would require to solve both the eigenfrequency and eigenmode sensitivities for the sensitivity with respect to all design variables. For this sensitivity analysis the adjoint method provides an exact and eﬃcient method to evaluate the structural loss factor sensitivities. D MATLAB Code This appendix describes the developed MATLAB implementation for the presented topology optimization routine. In this appendix the working principles of the code are explained. The most important components are highlighted here and the implementation is related to formulations and equations as discussed earlier in the thesis. Moreover, a minimal working example is provided for the reader to evaluate the code and experiment with simple case studies. The m-code contains two major parts: the main ﬁle init_VEM.m and the optimization loop opt_VEM_1.m. Important parameters and variables are organized within structures to conveniently pass information between the diﬀerent functions and store the obtained results in an organized manner. The following structures are used throughout the implementation: • O.: mesh information, such as element dimensions, numbering and global coordinates. • fix.: information in boundary conditions, such as ﬁxed DOFs and imposed temperatures. • prop_k.: material properties for the kth material. • rbf.: stores information on the radial basis functions and its dimensions. • opt.: parameters, variable and results involved or obtained from the optimizer. • MMA.: all information required for the MMA optimizer, such as objective values, sensitivities and results from previous iterations. The following sections describes the function of the MATLAB code and discussed the various lines of the code. D.1. Main ﬁle The main ﬁle init_VEM.m applies the settings for the optimization routine and initializes the iterations. After the iterations are ﬁnished or the solution is converged, the code is resumed to perform the postprocessing. Lines 27-52 In these lines the number of elements and dimensions are deﬁned. These elements construct the ﬁnite element discretization using during the structural analysis and for the placement of the RBFs. The element numbering and the corresponding nodal coordinates are generated by elem_def.m. The square elements allow for a simple mesh generation by iteration through all x and y elements. Finally, the required boundary conditions are speciﬁed and the subjected nodes are extracted from the mesh. 71 72 D. MATLAB Code Lines 53-75 The material properties for both material are speciﬁed. The bulk and shear modulus of each material are derived from the speciﬁed Young’s modulus. The amount of viscoelastic damping is introduced by specifying the material loss factor for the bulk or shear modulus using prop_k.eta_bulk and/or prop_k.eta_shear. A complex Young’s modulus is realized by specifying the same loss factor for the bulk and shear modulus. Lines 76-102 These lines deﬁne the RBFs used for the parameterization of the level set functions. In the implementation the RBF knots are aligned with the nodes of the ﬁnite element analysis. For simplicity the knots are aligned with FE nodes. The function csrbf.m determines the value of a shape function given the current location and its distance from the knot of the shape function. A simple for-loop is evaluated to build a single matrix that contains the values of the shape functions within a matrix surrounding the knot of the basis function. The obtained matrix rbf.shape will be used to construct the complete LSF by summation of all these matrices multiplied by their corresponding expansion coeﬃcients αi . It has to be noted that small changes to the initial values of αi might result in diﬀerent performance during the optimization. Lines 103-140 These lines deﬁne optimization parameters, such as the investigated number of eigenmodes, the applied volume constraints and maximum number of iterations to consider. Furthermore, the set of variables required for the MMA optimizer are initialized. Line 136 calls the optimization routine opt_VEM_1.m. Lines > 140 All lines after 140 are evaluated once the optimization is converged or the maximum number of iterations have been evaluated. In these lines some post-processing is performed and can be extended as necessary. D.2. Optimization loop The optimization loop is initialized by calling opt.VEM_1.m. This function provides all the analysis required to build the level set functions, to evaluate the current objective and constraint function, to calculate the sensitivities and to evaluate the MMA optimizer. The iterations are continued until the convergence criteria are achieved or until the maximum number of iterations are evaluated. The individual steps are brieﬂy presented in the following paragraphs. Lines 3-25 These lines preallocate some variables to allow for easy storage during the optimization routine. Besides, the element stiﬀness and mass matrices are requested from elem_matrices.m. This function can also be used to determine the element constitutive matrices as well as the element thermal matrices for conduction and convective heat problems. The iterations are started after these lines are evaluated. Lines 27 - 55 Information of the previous iteration is stored, which is provided to the optimizer for the current iteration. This only takes place after iteration 2. Then, the level set function is constructed by summation of the shape functions, which are scaled with the corresponding value of α. Notice that these shape functions are stored in a slightly larger domain. This allows to easily coupe with the overlap of the shape functions by only extracting the domain of interest. Lines 59 - 62 These lines convert both level set functions to the corresponding Heaviside function and the derivative of the Heaviside function by evaluating Equations (B.18) and (B.19) for each RBF knot. D.3. Example case 73 Lines 65 - 82 The ﬁrst lines assemble the global stiﬀness and mass matrices using the assemble.m function. This function loops through all elements of the design domain, determines their degrees of freedom and places the scaled element stiﬀness or mass matrices at the corresponding locations within the global matrices. Faster assembly methods are possibly by evading these for-loops and using smart functions to construct the sparse matrices. The material properties are scaled using Equation (B.8). These global matrices do not include any boundary conditions yet. Therefore, lines 68 and 69 remove any rows and columns from the global matrices that correspond to ﬁxed elements. Using these matrices the eigenvalue problem is solved. The eigensolutions are sorted based on smallest magnitude to obtain the ﬁrst n complex-valued eigensolutions. The obtained eigenmodes are also stored in vector which correspond to the length of the original global matrices in lines 76 to 82. This is to allow for easier calculations during the calculation of the loss factor sensitivity and also for animation purposes. Lines 85 - 94 This loop determines the loss factor, the adjoint variable problem and the loss factor sensitivities for the eigenmodes of interest. The function LossFactor.m evaluates Equation (4.13) to determine the loss factor corresponding to this modeshape. Then, the functionAdjointProblem.m solves the adjoint variable problem as given in Equation (4.23). Finally, LossFactor_Sensitivity.m determines the loss factor and volume constraint sensitivities using the previously calculated values of the adjoint variables. Lines 96 - 167 If multiple eigenmodes are considered for the optimization, their loss factors are combined to determine the objective value of the current design. Lines 187 - 199 The current objective, constraint and sensitivity values are reshaped to satisfy the requirements of the MMA optimizer. When all values are stored, the MMA optimization is called and the design variables α are updated with their new values. Lines > 200 The convergence statements are evaluated, some results are outputted towards the command window and result_plot is called to visualize the changes made to the level set functions and the material distribution. D.3. Example case The provided code performs an example optimization of a cantilever beam. The optimization aims to maximize the structural loss factor of the ﬁrst two resonant modes of the structure. The reader is encouraged to modify various parameters and investigate the many diﬀerent designs presented by the optimization routine. E Commercial FEM implementation This appendix presents some brief thoughts on how the presented parametric level set method can be implemented with commercial ﬁnite element software. In this example MATLAB and ANSYS illustrate the respective programming and ﬁnite element environments. However, the discussed method can be applied with any programming environment and commercial ﬁnite element software as long as the possibility exist to communicate the required numerical information between both environments. The implementation of commercial software will slightly simplify the steps towards a three dimensional analysis by providing assembly, mesh generation and robust solvers from their toolbox. The diagram in Figure E.1 shows a schematic diagram of the optimization procedure in which two diﬀerent computational environments are used. A dedicated ﬁnite element solver is applied to generate and update the mesh of the domain. A separate programming environment is responsible for the sensitivity calculation and optimization procedure. This part can be written in MATLAB, python, C or similar languages, as long as the language allows eﬃcient use of sparse matrix, vector calculations and is able to interface with the native code of the ﬁnite element environment. Some additional thoughts on this method are given in the following list: • The parametric level set method allows for the implementation of two discretization one for the structural analysis and the other for the construction of the level set functions. Combining a dedicated FE routine with a separate Matlab evaluation for the optimization would be an interesting scenario to apply these two diﬀerent discretizations. The three-dimensional mesh can be created as ﬁne as required, or even have non-uniform element sizes throughout the design. From the Matlab point of view a very careful analysis is required for the construction of both level set functions and the distribution of shape functions. Applying the knots of each RBF to match the FE discretization might not be required and a separate, more coarse discretization of the RBFs could be applied. • Care has to be taken during the sensitivity analysis of the system. In the case multiple discretizations are applied, the FE nodes do not necessarily overlap with the knots in the LSFs. Therefore, an interpolation step is required to determine the LSF value to match the location of the FE node. Similar, the sensitivity analysis might become complicated when the nodes are not aligned with the RBF knots. Therefore, all the nodes captures by a single RBF wil have a certain contribution to the sensitivity of changes in the height of that basis function. • Attention is required for the communication and interfacing between both methods. This is especially of importance to develop a robust method, capable of handling diﬀerent types of ﬁnite elements. Also, the interface between both tools requires clear communication on the construction of the elements and the number of the degrees of freedom. Without information on the construction of the elements and the relations between the degrees of freedom, the elementwise calculation of the sensitivities becomes quite diﬃcult. 75 76 E. Commercial FEM implementation Start Initialize shape functions and optimizer settings Finite element environment Contruct, import design domain MATLAB environment Apply boundary conditions Generate mesh Perform analysis Construct LSF Extract global matrices, eigensolutions, element numbering, list of DOF Calculate: objective, constraint functions and sensitivities Call optimizer to find design update Update mesh Update LSFs with new design variables Modify structure to match updated LSF No Design converged? Yes End Figure E.1: A schematic diagram of the optimization procedure using a dedicated ﬁnite element environment to perform the FE calculations. The green blocks are performed within the FE environment, while the orange blocks are evaluated in MATLAB or similar programming environments.

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