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Frontiers of Mechanical Engineering

Front Mech Eng    2012, Vol. 7 Issue (4) : 335-356     https://doi.org/10.1007/s11465-012-0351-2
RESEARCH ARTICLE |
XFEM schemes for level set based structural optimization
Li LI1,2, Michael Yu WANG1(), Peng WEI3
1. Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China; 2. School of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China; 3. State Key Laboratory of Subtropical Building Science, School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, China
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Abstract

In this paper, some elegant extended finite element method (XFEM) schemes for level set method structural optimization are proposed. Firstly, two- dimension (2D) and three-dimension (3D) XFEM schemes with partition integral method are developed and numerical examples are employed to evaluate their accuracy, which indicate that an accurate analysis result can be obtained on the structural boundary. Furthermore, the methods for improving the computational accuracy and efficiency of XFEM are studied, which include the XFEM integral scheme without quadrature sub-cells and higher order element XFEM scheme. Numerical examples show that the XFEM scheme without quadrature sub-cells can yield similar accuracy of structural analysis while prominently reducing the time cost and that higher order XFEM elements can improve the computational accuracy of structural analysis in the boundary elements, but the time cost is increasing. Therefore, the balance of time cost between FE system scale and the order of element needs to be discussed. Finally, the reliability and advantages of the proposed XFEM schemes are illustrated with several 2D and 3D mean compliance minimization examples that are widely used in the recent literature of structural topology optimization. All numerical results demonstrate that the proposed XFEM is a promising structural analysis approach for structural optimization with the level set method.

Keywords structural optimization      level set method      extended finite element method (XFEM)      computational accuracy and efficiency     
Corresponding Authors: WANG Michael Yu,Email:yuwang@mae.cuhk.edu.hk   
Issue Date: 05 December 2012
 Cite this article:   
Li LI,Michael Yu WANG,Peng WEI. XFEM schemes for level set based structural optimization[J]. Front Mech Eng, 2012, 7(4): 335-356.
 URL:  
http://journal.hep.com.cn/fme/EN/10.1007/s11465-012-0351-2
http://journal.hep.com.cn/fme/EN/Y2012/V7/I4/335
Fig.1  Level set representation of a 2D design structure. (a) The level set model; (b) design domain
Fig.2  The basic idea of the 2D XFEM scheme
Fig.3  Partition cases of the solid parts in a four-node quadrilateral element
Fig.4  The basic idea of the 3D XFEM scheme
Fig.5  Decomposition methods of a hexahedral element into multiple tetrahedra. (a) Decomposition into 6 tetrahedra; (b) decomposition into 5 tetrahedra
Fig.6  Partition cases of the solid parts in a tetrahedral element
Fig.7  Different decomposition methods for a hexahedral element
Fig.8  Model of circular hole in a plate loaded in tension
Fig.9  Mesh models of different methods for one quadrant of the plate with a hole. (a) XFEM mesh; (b) ANSYS mesh
Fig.10  Maximum stress of the four methods with different mesh scale
Fig.11  Mesh models of different methods for a 3D cantilever beam. (a) XFEM mesh; (b) ANSYS mesh
Fig.12  The comparison of the four methods with different mesh scale. (a) Maximum displacement; (b) maximum stress
Fig.13  Integral cases of a four-node quadrilateral element without quadrature sub-cells
Case 1Case 2Case 3Case 4Case 5
Error5.0299e-170.06062.1539e-161.6008e-169.0132e-17
Δ10%14%13%14%11%
Tab.1  Comparison of five integral cases of the solid parts in a quadrilateral element crossed by structural boundary with and without quadrature sub-cells
Fig.14  The third order Gauss integration of XFEM without quadrature sub-cells
Fig.15  Model of a 2D filet in tension. (a) The whole 2D filet model; (b) a quarter part of the 2D filet model
Fig.16  Mesh models of XFEM scheme and ANSYS for evaluating the higher order elements. (a) XFEM mesh and the local zoom figure; (b) ANSYS mesh and the local zoom figure
Fig.17  The strain energy density distribution along the boundary of different methods. (a) ANSYS; (b) XFEM ; (c) XFEM ; (d) XFEM
Fig.18  Comparison of strain energy density along the boundary of different methods. (a) Strain energy density; (b) local zoom figure
Fig.19  The performance of XFEM using different element types with different mesh scale. (a) Relative error of strain energy density; (b) time cost
Fig.20  Model of a 2D short cantilever beam
Fig.21  The optimization results at different stagesof the 2D cantilever beam with XFEM. (a) Initial design; (b) step 5; (c) step 10; (d) step 20; (e) step 30; (f) step 100
Fig.22  The iteration history of strain energy and volume ratio of the 2D cantilever beam
Fig.23  The final results with the cut element mesh of XFEM integral schemes. (a) With quadrature sub-cells and the local zoom figure; (b) without quadrature sub-cells and the local zoom figure
Analysis modelJ(Ω)(objective)Untilstep 25Untilstep 50Untilstep 75Untilstep 100
XFEM withquadrature sub-cells39.962785 s171 s258 s346 s
XFEM withoutquadrature sub-cells39.970174 s149 s225 s302 s
Tab.2  Time cost of XFEM schemes with and without quadrature sub-cells
Analysis modelJ(Ω)(objective)Untilstep 25Untilstep 50Untilstep 75Untilstep 100
XFEM Q439.962785 s171 s258 s346 s
XFEM T639.960164 s131 s198 s264 s
Tab.3  Time cost until different iteration steps of XFEM with and type of element
Fig.24  Model of a 3D short cantilever beam
Fig.25  The optimization results at different stages of the 3D cantilever beam with XFEM. (a) Initial design; (b) step 50; (c) step 100; (d) step 200; (e) step 300; (f) step 500
Fig.26  The iteration history of strain energy and volume ratio of the 3D cantilever beam
Fig.27  Model of a 3D Michell-type structure
Fig.28  The optimization results at different stages of the 3D Michell-type structure with XFEM. (a) Initial design; (b) step 70; (c) step 100; (d) step 150; (e) step 200; (f) step 400
Fig.29  The iteration history of strain energy and volume ratio of the 3D Michell-type structure
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