Structural Topology Optimization Based on the Smoothed Finite Element Method

Transcription

1 378 Strutural Topology Optimization Based on the Smoothed Finite Element Method Astrat In this paper, the smoothed finite element method, inorporated with the level set method, is employed to arry out the topology optimization of ontinuum strutures. The strutural ompliane is minimized sujet to a onstraint on the weight of material used. The ell-ased smoothed finite element method is employed to improve the auray and staility of the standard finite element method. Several numerial examples are presented to prove the validity and utility of the proposed method. The otained results are ompared with those otained y several standard finite elementased examples in order to aess the appliaility and effetiveness of the proposed method. The ommon numerial instailities of the strutural topology optimization prolems suh as hekeroard pattern and mesh dependeny are studied in the examples. Vahid Shoeiri a a Department of Civil Engineering, Yazd University, Yazd, Iran v.shoeiri@gmail.om Reeived Aepted Availale online Keywords Smoothed Finite Element Method; Level Set Method; Topology Optimization; Continuum Strutures. 1 INTRODUCTION The topology optimization design has eome one of the most important approahes in the field of strutural optimization. The purpose of the topology optimization is to ahieve the est performane for a struture while satisfying various onstraints suh as a onstraint on the weight of material used (Xie and Huang (2010)). For topology optimization of ontinuum strutures, the homogenization method (Hassani and Hinton (1999)), the Solid Isotropi Mirostruture with Penalization (SIMP) (Bendsøe and Sigmund (2003)), the evolutionary strutural optimization (ESO) (Xie and Steven (1993)), the i-diretional evolutionary strutural optimization (BESO) (Huang et al. (2006)), the topologial derivative-ased optimization (Amstutz et al. (2012); Lopes et al. (2015)) and the level set method (Dijk et al. (2013)) are often employed. The level set method has reently developed as an attrative alternative for topology optimization of ontinuum strutures without homogenization. The signifiane of level set method is its simpliity and generality.

2 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method 379 To date, the predominant numerial method used for topology optimization is the finite element method. The finite element method enounters some diffiulties when dealing with prolems suh as large deformation or moving oundary prolems. The standard finite element method often suffers from numerial instailities, and its solving is sensitive to element distortion eause of overestimation of the stiffness matrix. To overome these diffiulties, various numerial methods have een developed and ahieved remarkale progress, suh as meshfree methods (Monaghan (1992); Belytshko et al. (1994); Liu et al. (1995); Atluri and Zhu (1998)) and smoothed finite element method (Liu et al. (2007)). The meshfree methods do not require maintaining the integrity and desired shape of elements due to their meshfree nature. Therefore, large deformation and rak propagation prolems an e effetively modelled with meshfree methods (Shoeiri (2015a, 2015)). Though meshfree methods generally exhiit good numerial staility and auray, the omplex field approximation onsideraly inreases the omputational ost. It is lear that methods whih omine finite element method with meshfree methods an exhiit advantages of omputational effiieny and simpliity. The smoothed finite element method is suh a typial method. This new numerial method is rooted in meshless stailized onforming nodal integration and exhiits a numer of attrative properties suh as good numerial staility and auray, exellent onvergene rate, and insensitivity to volumetri loking and mesh distortion (Liu et al. (2007)). The smoothed finite element method has een suessfully applied to large variety of prolems inluding 2D and 3D linear and nonlinear prolems (Nguyen et al. (2009)), dynami analysis (Luong-Van et al. (2014)), plate and shell strutures (Nguyen-Xuan et al. (2008); Nguyen-Thanh et al. (2008)). In this paper, the smoothed finite element method is proposed to arry out the topology optimization of ontinuum strutures using the level set method. The feasiility and effiieny of the proposed method are illustrated with several 2D examples that are widely used in topology optimization prolems. The optimized topologies are ompared with those otained y the standard finite element -ased method in order to aess the appliaility and effetiveness of the proposed method. The ommon numerial instailities of the strutural topology optimization prolems suh as mesh dependeny and hekeroard patterns are studied in the examples. 2 REVIEW OF CELL-BASED SMOOTHED FINITE ELEMENT METHOD (CS-FEM) In this setion, a rief review of the ell-ased smoothed finite element method (as a ranh of the smoothed finite element method) is presented. Full details an e found in Liu et al. (2007). In the ell-ased smoothed finite element method, the total design domain W is first divided into Ne elements as in the finite element method. Depending on the neessity of staility, eah element is then n () sudivided into a numer of smoothing domains suh that W= i= 1W and i ( j) W W =, i ¹ j. ( i or j) Here, W is the domain of i th or j th smoothing domain and n is the total numer of ells inside the design domain. Fig. 1 shows the smoothing domains relating to various numer of ells in the ell-ased smoothed finite element method. For eah smoothing domain W assoiated with ell, the smoothing strain S an e written as:

3 380 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method S = S(x) F(x - x )dw ò W (2.1) Figure 1: Division of quadrilateral element into smoothing domains in the ell-ased smoothed finite element method: (a) nsc = 1; () nsc = 2; () nsc = 3; (d) nsc = 4; (e) nsc = 8; (f) nsc = 16. Figure 2: Design domain of the antilever eam. where F is the smoothing funtion written as: ìï 1 / A x ÎW F(x - x ) = ï í ï 0 x ÏW ïî (2.2) where areas: = ò W is the area of smoothing ell W. The element area is the sum of element ell () A d W A e nsc = å A (2.3)

4 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method 381 where nsc is the numer of ells for eah element. Note that this kind of smoothing is also employed in the smoothed partile hydrodynamis method (Monaghan (1992)). Sustituting F into Eq. (2.1), the smoothing strain an e written as: 1 S = ò n (x)u(x)d G = å B (x )d (2.4) A G u ui I IÎN where G is the oundary of the smoothing ell W, N n is the numer of nodes per element, B ui (x ) is smoothing strain matrix of the domain ( W ) (), and n u is the normal outward vetor on the oundary G. The vetors of B (x ) and are otained as: ui () n u n é nx 0 ù nu = 0 ny êny nx ú ë û (2.5) Figure 3: Optimization results otained from different mesh disretizations with different numer of smoothing ells.

5 382 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method B é ê Nnd G 0 I x é g g N (x ) x (x ) 0 I n ù G n 1 1 g g ui (x ) = 0 Nnd 0 I y G = A ò å NI(x ) ny (x ) l ê A G ú = 1 g g g g êni(x ) ny (x ) NI(x ) nx (x ) ú êë ò ò Nnd G I y I x G G ò Nnd G ù ú úû In the aove equations, n is the total numer of oundary setions of G, x g is the midpoint (Gauss point) of the eah smoothing domain oundary segment G, and l is the length of eah segment of G. It an e pointed out from Eq. (2.6) that unlike the finite element method, there is no derivative of shape funtion in the smoothing strain matrix. By employing the urrent formulation of the smoothed finite element method, the disrete equation for the ell-ased smoothed finite element method is given as: ë û (2.6) K CS-FEM U f (2.7) = where U is the vetor of nodal displaements, f is the vetor of nodal fores, and gloal smoothed stiffness matrix: CS-FEM K is the Figure 4: Evolution histories of the ojetive funtion over iterations, example 1. Figure 5: Evolution histories of the ojetive weight over iterations, example 1. nsc CS-FEM T e = å ( u ) u A K B DB (2.8) where D is the stress-strain relationship matrix. It should e noted that nodal shape funtions onstruted in ell-ased smoothed finite element method have the Delta funtion property. Therefore, essential oundary onditions are imposed as in the finite element method.

6 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method FORMULATION OF TOPOLOGY OPTIMIZATION PROBLEM 3.1 Optimization Prolem The ojetive of topology optimization is the optimal distriution of material in the design domain to minimize the ost funtionals under various onstraints suh as stress, displaement or weight onstraints. In this study, the aim is to find the stiffest struture sujet to a given strutural weight. Therefore, the optimization prolem an e formulated as: T Minimize C( x)= U K U = u K u Sujet to : W( x)= W req N e CS-FEM T CS-FEM xe e e e e= 1 : x = 0 or 1 " e=1,..., N e å e (3.1) where C ( x ) is the strutural ompliane, W ( x) is the weight of the urrent topology, W req is the presried weight and x =( x1,..., x Ne ) is the vetor of element densities. The design variale x e indiates the presene (1) or asene (0) of an element, where e is the element index. u e is the element CS-FEM displaement vetor and K e is the element stiffness matrix for element e. In this study, the disrete level set method is employed to find the solution for the optimization prolem. The disretized level set funtion y an e defined as: ìï ï y( e) < 0 if xe = 1, í ï y( e) > 0 if xe = 0. ïî (3.2) where e is the enter position of element e. Here, is initialized as a signed distane funtion and an upwind finite differene sheme is employed to aurately solve the evolution equation. The disrete level set funtion is updated to find new struture using the following equation: Figure 6: Results otained y the SIMP method from different mesh sizes without and with sensitivity filtering.

7 384 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method y =- u y - wg t (3.3) where t indiates time, u is salar field on the design domain, w is a positive parameter that speifies the influene of g, and g is a foring term that determines the nuleation of new holes within the struture. Note that to satisfy the weight onstraint, two parameters u and g are otained ased on the shape and topologial sensitivities of the Lagrangian framework. 4 NUMERICAL EXAMPLES In this setion, three widely studied examples in the field of topology optimization are presented to show the effetiveness of the proposed method. Poisson s ratio m = 0.3 and Young s modulus of E = 1 are used for all examples. 4.1 Example 1 Fig. 2 shows the design domain of a antilever eam with a length to height ratio of 2:1. The ojetive funtion is to minimize the ompliane and the ojetive weight is 45% of the total weight of the design domain. Figure 7: Design domain of eam with fixed support. Figure 8: Evolution histories of the ojetive funtion over iterations, example 2. Figure 9: Evolution histories of the ojetive weight over iterations, example 2.

8 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method 385 To determine the optimal strutural layout, the design domain is disretized using 36 18, 48 24, and quadrilateral elements. And to study the effets of the numer of the smoothing ells, eah element is sudivided into different numer of smoothing ells using nsc = 2, nsc = 4, nsc = 8 and nsc = 16. The optimization results otained from different mesh sizes with different numer of smoothing ells are shown in Fig. 3, from whih it an e seen that the final solutions otained from nsc = 4, nsc = 8 and nsc = 16 with different mesh sizes are almost idential, and are different from those otained from nsc = 2. The optimization results otained using nsc = 2 show the so-alled mesh dependeny effet, for whih different optimal topologies are generated from different mesh sizes. It an e found that the use of four ( nsc = 4 ) or more than four smoothing domains an e good hoies for topology optimization prolems to overome numerial instailities suh as mesh dependeny phenomenon. Figs. 4 and 5 illustrate the history of ojetive funtion and ojetive weight using nsc = 4 ased on different mesh sizes over iterations, respetively. It an e seen from Figs. 4 and 5 that the numer of iterations for mesh sizes of 36 18, 48 24, and are 65, 57, 53 and 47, respetively and their orresponding omplianes are alulated as 70.71, 71.35, and 68.96, respetively. It an e also seen from these results that for different mesh sizes, their onvergene harateristis are very similar. Figure 10: Results otained y the present method with different mesh disretizations. To verify the present method, the aove prolem using the same mesh sizes is solved y the Solid Isotropi Mirostruture with Penalization (SIMP) method (Sigmund (2001)) (standard finite element-ased method). The optimal strutural layouts without and with sensitivity filtering are shown in Fig. 6, from whih it an e seen that with and without using a filter, the topologies otained from the SIMP method are quite different. Note that the SIMP method using a filter generates similar topologies to the designs otained y the present method using nsc = 4, nsc = 8 and nsc = 16. It an e also oserved that numerial instailities suh as hekeroard patterns and mesh-dependeny exist in the results of the SIMP method if no filtering is employed, while for the present method no suh prolem an e seen.

9 386 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method Figure 11: Evolution of topology at various mesh disretizations: (a) 40 20; () 80 40; () Example 2 Fig. 7 shows the design domain of a eam with fixed supports. The eam length to height size ratio is 2:1. The ojetive funtion is to minimize the ompliane, and the target weight is 30% of the total weight of the design domain. The division of the element into four smoothing domains ( nsc = 4 ) is used as default in this example. Figure 12: Results otained y the SIMP method from different mesh sizes without and with sensitivity filtering. Figure 13: Design domain for Mihelle type strutures.

10 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method 387 In order to show that the optimum strutural layout is mesh independent and hekeroard free, the design domain is disretized using 40 20, and quadrilateral elements. Fig. 8 shows the evolution history of the ojetive funtion over iterations ased on different mesh sizes. It an e oserved from Fig. 8 that the numer of iterations for mesh sizes of 40 20, and are 50, 53 and 46, respetively and their orresponding omplianes are alulated as 13.75, and 14.92, respetively. Fig. 9 shows the urves of onvergene of the ojetive weight ased on different mesh sizes. The almost monotoni and uniform onvergene an e seen from this figure. The optimization results otained y the present method are shown in Fig. 10, the topology optimization history at various iterations is shown in Fig. 11, and the optimization results otained y the Solid Isotropi Mirostruture with Penalization (SIMP) method (Sigmund (2001)) (standard finite element-ased method) without and with sensitivity filtering are shown in Fig. 12. From these results it an e seen that the SIMP method using a filter produes similar topologies to the present method, and the present method an effetively remove numerial instailities suh hekeroard pattern and mesh dependeny phenomena. The numer of iterations of the SIMP method for mesh sizes of 40 20, and are 41, 96 and 123 respetively and their orresponding omplianes are respetively alulated as 15.14, and It is onfirmed that the numer of iterations of the present method is less than the SIMP method, and a smoother optimization result an e otained y the present method. 4.3 Example 3 The design domain of a simply supported Mihelle type struture with a length to height ratio of 6:1 is shown in Fig. 13. The ojetive funtion of this example is to minimize the ompliane and the ojetive weight is 50% of the total weight of the design domain. Figure 14: Evolution histories of the ojetive funtion over iterations, example 3. Figure 15: Evolution histories of the ojetive weight over iterations, example 3.

11 388 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method Figure 16: Results otained y the present method with different mesh disretizations. In order to show that the solution is mesh independent and hekeroard free, the design domain is disretized using 84 14, and quadrilateral elements. Eah element is sudivided into four smoothing domains ( nsc = 4 ). Fig. 14 shows the evolution history of the ojetive funtion over iterations, and Fig. 15 gives the urves of onvergene of the ojetive weight ased on different mesh sizes. The numer of iterations for mesh sizes of 84 14, and are 65, 63 and 55, respetively and their orresponding omplianes are alulated as 96.15, and 95.82, respetively. It should e noted that the oasional jumps in Fig. 14 may e attriuted to a remarkale alteration of topology due to the elimination of one or more ars in a single iteration. The optimum strutural layouts and the topology optimization history at various mesh sizes are shown in Figs. 16 and 17, respetively. The optimization results otained y the Solid Isotropi Mirostruture with Penalization (SIMP) method (Sigmund (2001)) (standard finite element-ased method) are given in Fig. 18. A omparison etween the final solutions shown in Figs. 16 and 18 shows that the two different optimization methods generate similar topologies and the present method an avoid numerial instailities suh as hekeroard pattern and mesh dependeny phenomena. The omplianes of the solutions of the SIMP method for mesh sizes of 84 14, and are respetively alulated as , and whih are higher than those of the present method. These differenes may e due to the over-estimated strain energy of elements in the solutions of the SIMP method. Figure 17: Evolution of topology at various mesh disretizations: (a) 84 14; () ; ()

12 V. Shoeiri / Strutural Topology Optimization Based on the Smoothed Finite Element Method 389 Figure 18: Results otained y the SIMP method from different mesh sizes without and with sensitivity filtering. 4 CONCLUSIONS In this paper, the smoothed finite element method is omined with the level set method to develop an effiient approah for topology optimization of ontinuum strutures. The ell-ased smoothed finite element method is employed to improve the auray and staility of the standard finite element method. Several numerial examples were presented to show the validity and feasiility of the proposed method. The examples have shown the effetiveness of the proposed method to overome numerial instailities suh as hekeroard patterns and mesh dependeny phenomena. As a future researh work, the proposed approah an e effiiently used to solve strutural topology optimization prolems with other ojetive funtionals and onstraints suh as stress or displaement onstraints. Referenes Amstutz, S., Novotny, A.A., de Souza Neto, E.A., (2012). Topologial derivative-ased topology optimization of strutures sujet to Druker Prager stress onstraints. Computer Methods in Applied Mehanis and Engineering : Atluri, S.N., Zhu, T., (1998). A new meshless loal Petrov-Galerkin (MLPG) approah in omputational mehanis. Computational Mehanis 22: Belytshko, T., Lu, Y.Y., Gu, L., (1994). Element-free Galerkin method. International Journal for Numerial Methods in Engineering 37: Bendsøe, M.P., Sigmund, O., (2003). Topology Optimization: Theory, Methods, and Appliations. Springer, New York, USA. Burger, M., Hakl, B., Ring, W., (2004). Inorporating topologial derivatives into level set methods. Computational Physis 194: Dijk, N.P., Maute, K., Langelaar, M., Keulen, F.V., (2013). Level-set methods for strutural topology optimization: a review. Strutural and Multidisiplinary Optimization 48(3):1-36. Hassani, B., Hinton, E., (1999). Homogenization and Strutural Topology Optimization: Theory, Pratie and Software. Springer, New York, USA. Huang, X., Xie, Y.M., Burry, M.C., (2006). A new algorithm for i-diretional evolutionary strutural optimization. JSME International Journal 49(4):

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