Lumped Mass Modeling for Local-Mode-Suppressed Element Connectivity Parameterized Topology

6th World Congresses of Structural and Multidisciplinary OptimizationRio de Janeiro, 30 May - 03 June 2005, BrazilLumped Mass Modeling for Local-Mode-Suppressed Element Connectivity Parameterized Topology Optimization of Vibrating StructuresYoung Soo, Joung1, Gil Ho, Yoon2 and Yoon Young, Kim3 1. M.S. Candidate, National Creative Research Initiatives Center for Multi-Scale Design and School of Mechanical and Aerospace Engineering, Seoul National University, Korea. (e-mail: ys188@idealab.snu.ac.kr) 2. Assistant Professor, Department of Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark. (e-mail: ghy@mek.dtu.dk ) 3. Director, National Creative Research Initiatives Center for Multi-Scale Design and School of Mechanical and Aerospace Engineering, Seoul National University, Korea (e-mail: yykim@snu.ac.kr) 1. Abstract For successful topology design optimization of crashworthy “continuum” structures, unstable element-free and local vibration modefree transient nonlinear analyses should be ensured. Among these two issues, element instability was shown to be overcome if a recently-developed formulation, the element connectivity parameterization (ECP) is employed. On the way to the ultimate crashworthy structure optimization, we are now developing a local mode-free topology optimization formulation that can be implemented in the ECP method. In fact, the local mode-freeing strategy developed here can be also used directly for the standard element density method. Local modes are artificial, numerical modes resulting from the intrinsic modeling technique of the topology optimization method. Even with existing local mode controlling techniques, the convergence of the topology optimization of vibrating structures, especially experiencing large structural changes, appears to be still poor. In ECP, the nodes of the domaindiscretizing elements are connected by zero-length one-dimensional elastic links having varying stiffness. For computationally efficiency analysis for dynamic problems, every elastic link is now assumed to have two lumped masses at its ends. Choosing appropriate penalization functions for lumped mass and link stiffness is important for local mode-free results. However, unless the objective and constraint functions are carefully selected, it is difficult to obtain clear black-and-white results. It is shown that the present formulation is also successful in design problems involving self-weight. 2. Keywords: Topology optimization, Vibration problem, Parasite local mode control, Lumped mass modeling, End-lumped link model 3. Introduction A topology optimization formulation for structural problems involving nonlinear transient analysis is a key to successful important ingredient for crashworthy continuum structure design. Since nonlinear problems in topology optimization can be effectively handled by the element connectivity parameterization (ECP) method [1], a recently-developed topology optimization method, this investigation is focused on a local-mode-free formulation that can be incorporated in the ECP method. The local mode problem resulting from the presence low-density elements is well-known and some ideas to overcome it have been suggested [2,3]. However, no unified method with firm theoretical foundation seems to be available. It may be worth noting that the numerical difficulty encountered in self-weight structural design is similar to that from the parasite local modes in the local-mode difficulty in vibration problems [4]. In this work, we investigate an efficient method for local-mode-free optimal design that can be implemented in the ECP formulation. Indeed, the method is also applicable to the standard density method. In order to control artificial local modes, the mass matrix is penalized with a higher penalty exponent value than that for the stiffness matrix. This will push the eigenfrequencies of artificial local modes higher than the eigenfrequencies of interest. However, it is difficult to obtain clear solid and void images alone with the suggested penalization scheme. Thus, an explicit condition to guarantee black and white images, i.e., solid and void results is additionally imposed. For easy ECP implementation for dynamic problems, lumped mass modeling is employed. Several benchmark problems were considered to verify the validity of the proposed method. 4. Artificial Local Vibration Mode-Free Formulation 4.1. Problem definition of topology optimization Before considering the local-mode-free formulation for ECP, the local mode issue in the element density method will be considered first. When eigenfrequencies and compliance under self-weight are simultaneously considered, the following topology optimization problem can be set up:Minimize : φ = w1F T U − w2 λr(1a)Subject to:∑∫e =1NEΩeγ e dΩ − M 0 ≤ 0( NE :Number of elements )(α << 1)(1b)⎛ NE pc ⎜ ∑ γ e − γ lower ⎝ e =1() (γupper⎞ − γ epc ⎟ / NE ≤ α ⎠)q(1c)γ lower ≤ γ e ≤ γ upper = 1(1d)The objective function φ involves two weighting factors w1 and w2 to adjust the relative importance between the rth eigenfrequency and the compliance F T U , where F and U are the force vector by self-weight and the displacement vector of a continuum body, respectively. The design variable assigned to the eth element is denoted by γ e which varies between its lower bound γ lower and upper bound γ upper . Eq. (1c) is an explicit condition introduces to ensure that the optimized result is black and white. The role of Eq. (1c) will be clear when the penalization strategy for the mass and stiffness matrix is explained in the next section. The penalty parameters, pc and q in Eq. (1c) can be adjusted to improve solution convergence. To track the desired eigenfrequency λr , a mode-tracking method [5] using the modal assurance criterion (MAC) is employed. 4.2. Artificial local vibration modes In this section, the issue of the local vibration modes associated with low-density elements in the standard element density method is examined and a method to deal with it will be proposed. Since the troublesome local modes appear in a low-density element region, only the dynamic behavior a low-density element region may be investigated. Figure 1 shows a distribution of element densities in a given design region. The stiffness matrix K and the mass matrix M are assumed to be penalized asstructure K = ∑ γ epK K nominal e =1 NEandstructure M = ∑ γ epM M nominal e =1NE(2)structure structure where K nominal and M nominal denote the nominal element matrices (corresponding to γ e = 1 ) and pM and pK are penalization exponents. To investigate the dynamic behavior of the low-density elements region in Figure 1, the following form of the discretized dynamic equations may be used:⎡ K KK K KD ⎤ ⎡φ K ⎤ ⎢K K DD ⎥ ⎢ ⎥ ⎣ DK 244⎦ ⎣φ D ⎦ 14 4 3 {KM ω (λr = , ω = angular speed) 2πThe eigenvectorM KD ⎤ ⎡φ K ⎤ ⎡M = ω 2 ⎢ KK M DK M DD ⎥ ⎢ φ D ⎥ ⎣ 4 14 244 ⎦ ⎣ ⎦ 3 {(3)The eigenvectorIn Eq. (3), the symbols K and D imply “kept” and “deleted”. To delete unwanted degrees of freedom, the Guyan condensation method [6,7] can be employed. Hence, the nodes inside the low-density region in Figure 1 will be kept.Figure 1. A structure consisting of high- and low-density elements Before carrying out the static condensation, the material properties of the low-density elements are assumed to be controlled by a single design variable γ . Furthermore, it is assumed that the stiffness and mass matrices of high-density elements in Figure 1 as a function of the element density. These assumptions will simply the subsequent analysis considerably. Thus, Eq. (3) may be now written as⎡γ pK K 0 KK ⎢ pK 0 γ K DK ⎣γ p K 0 ⎤ ⎡φ K ⎤ KDKK DDpM M0 2 ⎡γ KK ⎥ ⎢ ⎥ = ω ⎢ pM 0 φD ⎦ γ M DK ⎦⎣ ⎣γ p M 0 ⎤ ⎡φ K ⎤ KDM⎥⎢ ⎥ M DD ⎦ ⎣φ D ⎦(4)where K 0α ) and M 0α ) ( α = KK , KD, or DD ) denote the nominal submatrices corresponding to γ = 1 . ( ( By choosing the nodes inside the low-density region as the kept nodes, one may obtain the following reduced system equation:-2-K KK φ K = ω 2 M KK(5a) (5b) (5c)whereK KK = γ pK K 0 − γ 2 pK ( K 0 K −1 K 0 ) → γ pK K 0 (as γ → 0) KK KD KK DK KK M KK = γ pM M 0 − γ 2 pM M 0 K −1 M 0 − γ pK + pM K 0 K −1 M 0 + γ 2 pK K 0 K −1 M DD K −1 K 0 → γ pM M 0 (as γ → 0) KK KD DD DK KD DD DK KD DD DD DK KKIn obtaining Eq. (5a), the Gayan reduction technique [6,7] was used. Using the limiting results of K KK and M KK as γ → 0 , the following approximate system equation for φ R :(γPKK 0 − ω 2γ PM M 0 ) φ K ≅ 0 KK KK(6)From Eq. (6), it is clear that if pM < pK , the eigenfrequencies of the low density region approach zero as γ approaches zero. In this case, the local mode corresponding to the vanishingly small eigenfrequency of the low-density region may become the fundamental eigenfrequency of the whole structural system. Thus, the problem of parasite low-frequency local mode can occur. If pM > pK , however, the eigenvalues approach infinity as γ → 0 so that the local mode problem can disappear. In this case, the mass evaluation formula of Eq. (1b) replaced by∑∫e =1NeΩeγ ep dΩ − M 0 ≤ 0 , but this replacement is not desirable because it con not be used in staticMproblems. Although the condition that pK < pM resolves the local mode problem, it will make it difficult to obtain black and white images at the convergence. Therefore, an explicit condition to push intermediate density values towards either γ lower or γ upper such as Eq. (1c) is introduced. Instead of using the constraint of Eq. (1c), slightly different form from Eq. (1c) may be incorporated into the objective function as a penalty term. The penalization exponents pc and q in Eq. (1c) may be adjusted to improve solution convergence. In this work, q = 2 and pc = 4 were used for all numerical examples considered. 4.4. The mass matrix modeling for the element connectivity parameterization formulation The element connectivity parameterization (ECP) formulation was developed as an alternative formulation to the conventional element density method using the SIMP (Simple Isotropic Materials with Penalization) approach. In [1], the ECP method was proposed to resolve numerical difficulties resulting from the presence of low-density elements for the topology optimization of geometrical nonlinear structures. Figure 2 compares the modeling techniques of the standard density method and the ECP method. To define a topology in Figure 2(a), for instance, the SIMP method assigns the original material property in the region occupied by a structure and an artificially low material property in void regions (see Figure 2(b)). In ECP, the design domain is discretized by solid elements having the original material and the structural regions are connected by stiff (or virtually rigid) zero-length one-dimensional links and the void regions, by very week links (see Figure 2(c)).(a)(b)(c)Figure 2. The comparison of the modeling techniques. (a) A given structural layout, (b) the model by the standard element density method, (c) the model by the ECP method In [1], only static problem was considered by ECP. Therefore some additional work should be done to external ECP for dynamic problem. Let us first express the stiffness matrix of the zero-length one-dimensional link for two-dimensional plane problems:k link = l0 (γ e ) pK k link e nominalwhere(7)-3-k link nominal⎡ 1 0 −1 0 ⎤ ⎢ 0 1 0 −1⎥ ⎥ =⎢ ⎢ −1 0 1 0 ⎥ ⎢ ⎥ ⎣ 0 −1 0 1 ⎦(8)In Eq. (7), l0 and k link are the upper bound of the link stiffness and the nominal stiffness matrix of the link. The stiffness matrix nominal link connects the degrees of freedom of node m and n in Figure 3. Note that node n and node m are connected to different plane ke elements. In Figure 3, the subscripts x and y denote the Cartesian coordinates and u and F, displacement and force. For dynamic analysis, a mass matrix should be considered. Here, a lumped mass matrix modeling technique is used so that the nodes connected by a link have inertia terms both in the x and y axes: see Figure 3. Conceptually, one can imagine that two lumped masses are attached at the end of an elastic link. In this case, one may define the mass matrix of the link m link as em link = e m0 (γ e ) pM m link , nominal 2m0 = ∑ ρ0Ve / NLe =1 NE(9)mlink nominal⎡1 ⎢0 =⎢ ⎢0 ⎢ ⎣00 1 0 00 0 1 00⎤ 0⎥ ⎥ 0⎥ ⎥ 1⎦(10)where m link is the nominal link mass matrix. The symbol NL denotes the number of links that share the same global node. The nominal role of the penalization exponent pM for ECP is the same as that for the standard density approach. Other than the mass modeling related issue, the same ECP formulation used in [1] is also used for dynamic problems.Figure 3. Link model for the dynamic analysis for the ECP formulation. The link connects nodes m and n having lumped masses. 5. Numerical examples 5.1. Maximization of the first eigenfrequency The maximization of the first eigenfrequency shown Figure 4(a) is a well-known test problem. The target mode shape of the nominal structure (having uniform density distribution) is shown in Figure 4(b).(a)(b)Figure 4. Topology optimization for the first eigenfrequency maximization. (a) Problem description ( Cv = 0.35 implies a 35% mass constraint ratio), (b) the target mode shape (the fundamental mode shape of the nominal structure in Figure 4(a)) The optimization histories by the standard density method and the ECP method with pK = 3 and pM = 1 (without the explicit image-4-constraint, Eq. (1c)) are shown in Figure 5(a) and Figure 5(b), respectively. The history by ECP with pK = 3 and pM = 4 is shown in Figure 5(c). When pK > pM , local modes appear in the region of low-density or weakly-connected regions. However, the penalization with pK = 3 and pM = 4 yielded local-mode free results not only at the final stage but also during optimization iterations. The successful result in Figure 5(c) also confirms the effectiveness using lumped masses.(a)(b)(c)Figure 5. The comparison of the optimization history for the maximization of the first eigenfrequency. (a) By the standard density method with pK = 3 and pM = 1 , (b) by the ECP method with pK = 3 and pM = 1 , (c) by the ECP method with pK = 3 andpM = 45.2. Combined optimization of compliance minimization under self-weight and the first eigenfrequency maximization Not presented in this paper, one can see that the compliance minimization problem under self-weight causes numerical difficulties similar to the artificial local mode in dynamic problems. In this example, the weighting factors ( w1 , w2 ) in Eq. (1a) are varied among ( w1 = 1, w2 = 0 ), ( w1 = 1, w2 = 1 ) and ( w1 = 0, w2 = 1 ) where the meaning of the weighting factors w1 , w2 may be clearly understood from Eq. (1a). The design problem considered is shown in Figure 6.Figure 6. Combined optimization of compliance minimization under self-weight and the first eigenfrequency maximization Figure 7 shows the optimized results for the problems defined in Figure 6. During the optimization iterations to obtain the results in Figure 7, no numerical instability was observed. For the results, the ECP method with pK = 3 and pM = 4 was used. If pK = 3 and pM = 1 were used, numerical instability would occur.-5-(a)(b)(c)Figure 7. Optimized results by the ECP method with pK = 3 and pM = 4 for the problem defined in Figure 6. (a) For compliance minimization under self-weight ( w1 = 1 , w2 = 0 ), (b) for the first eigenfrequency maximization ( w1 = 0 , w2 = 1 ), (c) for the combined optimization ( w1 = 1 , w2 = 1 )6. ConclusionsIn this work, the mass and stiffness penalization scheme that can be implemented for the element connectivity parameterization formulation was examined for artificial local-mode free topology optimization. For artificial local-mode free results, the mass matrix was penalized with a higher penalty exponent than the stiffness matrix. To obtain clear black and white images, an explicit constraint to suppress intermediate images was also introduced. For the dynamic implementation of the ECP method, a lumped mass modeling technique was developed where the element connecting zero-length links were assumed to have lumped end masses. Numerical examples verified the effectiveness of the developed method in suppressing artificial local-modes associated with void regions. Furthermore, the developed method was also effective for compliance minimization problems under self-weight.7. References1. Yoon G. H. and Kim Y. Y. Element connectivity parameterization for topology optimization of geometrically nonlinear structures. International Journal of Solid and Structures, 2005, 42(7): 1983-2009 2. Tcherniak D. Topology Optimization of Resonating Structures Using SIMP Method. International Journal for Numerical Methods in Engineering, 2002, 54: 1605-1622 3. Pedersen N. L. Maximization of Eigenvalues Using Topology Optimization. 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