An explicit topology optimization method using moving polygonal morphable voids (MPMVs)

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Conventional topology optimization approaches are implemented in an implicit manner with a very large number of design variables, requiring large storage and computation costs. In this study, an explicit topology optimization approach is proposed by moving polygonal morphable voids whose geometry parameters are considered as design variables.

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Science & Technology Development Journal, 23(2):536-540 Open Access Full Text Article Research Article An explicit topology optimization method using moving polygonal morphable voids (MPMVs) Van-Nam Hoang* ABSTRACT Introduction: Conventional topology optimization approaches are implemented in an implicit manner with a very large number of design variables, requiring large storage and computation Use your smartphone to scan this costs. In this study, an explicit topology optimization approach is proposed by moving polygonal QR code and download this article morphable voids whose geometry parameters are considered as design variables. Methods: Each polygonal void plays as an empty-material zone that can move, change its shapes, and overlap with its neighbors in a design space. The geometry parameters of MPMVs consisting of the coordi- nates of polygonal vertices are utilized to render the structure in the design domain in an element density field. The density function of the elements located inside polygonal voids is described by a smooth exponential function that allows utilizing gradient-based optimization solvers. Re- sults & Conclusion: Compared with conventional topology optimization approaches, the MPMV approach uses fewer design variables, ensure mesh-independence solution without filtering tech- niques or perimeter constraints. Several numerical examples are solved to validate the efficiency of the MPMV approach. Key words: Topology optimization, Moving morphable void, Moving morphable bar INTRODUCTION fewer design variables, using an explicit structural de- Topology optimization is typically described by scription that is convenient for the post-processing searching the distribution of a given amount of ma- stage, and straightforward feature size control 7 . In 7 , terial in a prescribed design domain to maximize we introduced an explicit topology optimization ap- the structural performance, i.e., the structural com- proach using moving morphable bars for the design Mechanical Engineering Institute, pliance, buckling load, displacement. Over the past of structural compliance and compliant mechanism Vietnam Maritime University, Hai three decades, topology optimization has undergone problems. The extension of the moving morphable Phong City, Viet Nam a long period of development, contributed by re- bar approach 7 has been applied in several applica- Correspondence searchers around the world. Topology optimization tions in recent years, i.e., coated designs 8 , embedded Van-Nam Hoang, Mechanical has been integrated into commercial software such as components 9 , and cellular structures 10 . Engineering Institute, Vietnam Maritime Comsol, Altair, and Ansys as a powerful tool for struc- University, Hai Phong City, Viet Nam The aforementioned explicit approaches mostly used tural optimization solutions. Until now, the major- non-flexible components like circles, bars, and el- Email: namhv.vck@imaru.edu.vn ity of existing approaches have been implicit, that is, lipses 4–7 or complex flexible components using B- History the structure is implicitly described by element den- splines/NURBS 11,12 . In this study, we will model • Received: 2020-04-10 sity fields (SIMP 1 , ESO 2 ) or level-set functions 3 . One • Accepted: 2020-06-12 simple flexible components using polygonal voids for • Published: 2020-06-27 of the disadvantages of the implicit approaches is that explicit topology optimization of two-dimensional they have a very large number of design variables, DOI : 10.32508/stdj.v23i2.2067 structures. The structural optimization is performed equal to the number of grid elements (or the number by optimizing the positions of the vertices of the of grid nodes) of the design domain. The optimization polygonal voids. requires large storage capacity as well as demanding calculations. MPMV Copyright To reduce the number of design variables as well as © VNU-HCM Press. This is an open- We consider an MPMV with ns connected segments access article distributed under the minimize computational costs, explicit topology opti- terms of the Creative Commons mization approaches have been proposed recently 4–7 , denoted by Ωk as illustrated in Figure 1. The segment Attribution 4.0 International license. in which the structure is explicitly described by geom- i is determined by the coordinates of two adjacent ver- etry parameters of geometric components. The ben- tices xi and x i+1 . Let T be a vector originating from efits of the explicit approaches can be listed as using x i to x i+1 , defined by T=xi+1 - xi . The unit vector t Cite this article : Hoang V. An explicit topology optimization method using moving polygonal mor- phable voids (MPMVs). Sci. Tech. Dev. J.; 23(2):536-540. 536
Science & Technology Development Journal, 23(2):536-540 this will significantly reduce the mapping time com- pared with the case of geometric mapping compo- nents onto the full grid like most of the current ex- plicit approaches. Figure 2 shows plots of function ∅ = e−β dek with respect to the minimum distance dek for different values of β . The larger β results in a narrower band of nonphysical material around the structural boundaries. β should be selected so that there is a transition zone (low- density element zone) between solid material phase and void phase, to ensure the existence of non-zeros derivation of the element density function for em- ploying gradient-optimization solvers. In this work, the selection β = 2 corresponds to about one low- Figure 1: MPMV: an MPMV consists of ns line seg- density element on the boundaries for unit-length el- ments that connect at their vertices of the poly- ement mesh. gon. along the line segment i is given by: T xi+1 − xi t= = (1) ||T|| ||xi+1 − xi || (i) We name dek as the minimum distance from the cen- (i) ter element xe ∈ Ωk to the segment i of the void k, dek can be expressed by { (i) ||a|| = ||xe − xi || i f at ≤ 0 dek = (2) ||b|| = ||a − (at)t|| i f 0 < at < tT Figure 2: The control parameter β : plots of func- where a = xe − xi is the vector originating from xi to tion ∅ = e−β dek with respect to dek for different xe and b = a − (at)t is a perpendicular vector of the values of β . segment i with its length equal to the minimum dis- tance from the element e to the line through x i and xi+1 (see Figure 1). { TOPOLOGY OPTIMIZATION USING ∪ v 1 ∀xe ̸∈ nk=1 Ωk MPMVS ρe = nv − β ∪nv (3) ∏k=1 e d ek ∀xe ∈ k=1 Ωk The compliance minimal problem is considered in { } this work. The objective is searching for an optimal set (i) dek = min dek , i = 1, 2, ..., ns , (4) of geometry parameters to minimize structural com- where dek is the minimum distance from element e to pliance. The optimization problem is formulated as the boundary ∂ Ωk of the void k; nv is the number of mi n c(x) = ∑ne=1 e χ uTe k0 ue x ∫ voids and β is a positive control number to enforce 1 sub ject to ρe dΩ − f ≤ 0 (5) element density to converge to 0 or 1 (see Figure 2). |Ω0 | Ω0 In equation (3), ρe = 1 if the element does not locate xmin ≤ x ≤ xmax inside the void zones (solid material), ρe = 0 if the where c is the structural compliance; ne is the to- element locates inside the void zones (voids), and 0 < tal number of elements, k0 is the element stiffness ρe < 1 responds to the elements around the structural matrix; ue ⊂ u = K−1 F is the element displace- boundaries. ment vector; K, u and F are the global stiffness ma- It is worth noting that only elements located inside trix, global displacement vector, and global force vec- void zones are considered in the calculation of the tor, respectively; |Ω0 | denotes the design-domain vol- element densities and their sensitivities. Of course, ume; f denotes the allowed material volume ratios; 537
Science & Technology Development Journal, 23(2):536-540 x = {xk } , xk = {xi } , i = 1, 2, ..., ns , k = 1, 2, ..., nv is the variable vector; xmin , xmax are the bounds of x Cantilever beam design with a note that it is not necessary to set strict limits for The cantilever beam problem is considered with de- lower and upper bounds of the design variables. This sign definitions given in Figure 3a. An analytical means that the vertex of polygonal voids can move out mesh of elements is employed. Figure 3b shows of the design domain as can be seen in Figure 4b. The the design by SIMP approach (using 99 lines Matlab material interpolation in SIMP 1 is employed, code 13 ), in which a sensitivity filter with a radius of χ = ρmin + ρeη (1 − ρmin ) (6) 1.5 is employed to avoid the checker-board issue. where ρmin = 10−4 is added to ensure a well-posed finite element analysis. Sensitivity analysis of the objective function is ex- pressed by the following equation, ∂c ne ∂ c ∂ pe ∂ dek =∑ (7) ∂ ξ e=1 ∂ ρe ∂ dek ∂ ξ where ξ ⊂ x is an arbitrary design variable in the vari- able vector x, the derivative of the objective function to element density ∂ c/∂ ρe is derived from Eqs. (5-6), ∂ ρc = ηρeη −1 (1 − ρmin )uTe k0 ue , (8) ∂ ρe Figure 3: Cantilever beam design: (a) prob- the derivative of element density function to the lem definitions, (b) optimizeddesign by SIMP minimum distance ∂ ρe /∂ dek is derived from Equa- (c=175.35). tion (3), { ∪ v ∂ pe 0, ∀xe ̸∈ nk=1 Ωk = ∪ v (9) To optimize the structure using the MPMVs ap- ∂ dek −β ρe , ∀xe ∈ nk=1 Ωk proach, an initial design is predefined with 11 polyg- and to determine ∂ dek /∂ ξ , we suppose that the min- onal voids and 12 vertices for each polygonal void imum distance function in Equation (4) is readily as presented in Figure 4a. The beam is optimized (i) by searching the optimal positions of the polygonal known, that means, dek = dek . Derivative expressions (i) of ∂ dek /∂ ξ is derived from Equation (2) as follows, voids. The optimized layout of MPMVs is plotted in  Figure 4b, and the design is shown in Figure 4c. (i) ∂ dek  −a 1 , i f at ≤ 0 (10) It is worth mentioning that the MPMV approach = { ||a|| } ∂ xi  −b + 1 (ta)b 1 , i f 0 < at < tT significantly reduces the number of design variables ||T|| ||b|| compared with conventional approaches. The current design only uses 264 design variables that are much (i) { ∂ dek 0, i f at ≤ 0 } { (11) less than 7500 design variables by SIMP/ESO ap- = proach or 7701 design variables by level set approach. ∂ xi+1 − ||T|| 1 1 (ta)b ||b|| , i f 0 < at < tT We observed that our overall optimum topology is in agreement with that by SIMP approach. Low-density EXAMPLES elements inside structural boundaries may exist in the A benchmark structural optimization problem, the design by SIMP approach (see the middle part of the cantilever beam optimization is explored in this sec- design in Figure 3b) but not exist in the design by MP- tion. For numerical simulation, we assume that the MVs approach. The proposed approach produces a design material is homogeneous with unit Young’s stiffer structure with 2.04% smaller compliance. modulus and Poisson’s ratio v0 = 0.3. The plane- It is worth remarking that the structural boundaries stress four-node elements are used to discretize the are explicitly described by line segments of polygo- design domain. The design problems are solved with nal voids, hence the proposed method allows the abil- the maximum allowed material volume of 50% design ity to accurately capture structural boundaries to ex- domain volume, f = 0.5. tract final designs. The computer-aided design (CAD) 538
Science & Technology Development Journal, 23(2):536-540 model can be obtained directly by keeping line seg- elements. When a finer mesh is employed, the corre- ments on the structural boundaries while deleting sponding compliance increases, i.e., 0.08% when the unnecessary line segments. Hence, the proposed number of mesh elements is increased from 150x50 method allows capturing accurate structural bound- elements to 300x100 elements and 0.05% when the aries in a cheap way compared with SIMP method, number of mesh 300x100 elements is increased from where the structure is implicitly described by the el- 450x150 elements to elements. Another observation ement density field that needs undergoing many steps is that the checker-board issue does not appear in our of post-process for the final design. design although we do not use any other techniques, i.e., filtering. These mean that optimized designs by the MPMVs-based approach depend on geometry pa- rameters of MPMVs rather than the mesh size. Figure 5: Mesh-independence: (a) result with mesh 300x100 (c=171.91), (b) result with mesh 450x150(c=171.99). Figure 4: Cantilever beam design: (a) initial layout of MPMVs with 11 polygonal voids and 12 vertices for each polygonal void, (b) opti- mized layout of MPMVs, (c) optimized design CONCLUSION (c=171.77) For the first time, an explicit topology optimization approach using MPMVs has been proposed for opti- mum structural designs. The MPMV-based approach allows mapping each polygonal void onto a fit sub- Mesh independency domain instead of a full design domain. The den- The above design problem is resolved with different sity function of the elements located inside polyg- meshes while retaining other design parameters. The onal voids is realized by an exponential function optimum results are presented in Figure 5, in which that allows employing gradient-based optimization Figure 5a plots the design with mesh 300x100 with solvers. The proposed approach works effectively for compliance c = 171.91 and Figure 5b plots the design two-dimensional structural optimization with a sig- with mesh 450x150 with compliance c = 171.99. The nificant reduction of design variables. The filtering convergence of all examples in this paper is obtained techniques or perimeter constraints are not neces- after 100 iterations. sary while still ensuring a mesh-independency solu- Through three numerical examples in Figure 4c, Fig- tion. The extension of the current approach for three- ure 5a, and Figure 5b, it is observed as follows. The dimensional problems can be straightforward by re- first observation is that the general optimum topolo- placing polygonal voids with polyhedral voids. gies are the same for three mesh cases: 150x50 ele- ments, 300x100 elements, and 450x150 elements. The ACKNOWLEDGMENTS second observation is small differences in structural This research is funded by Vietnam National Foun- compliance: c = 171.77 for the case of mesh 150x50 dation for Science and Technology Development elements, c = 171.91 for the case of mesh 300x100 el- (NAFOSTED) under grant number 107.01-2019.317. ements, and c = 171.99 for the case of mesh 450x150 539
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