XFEM schemes for level set based structural optimization

doi:10.1007/s11465-012-0351-2

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 LI^1,2, 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 Author(s): 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:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-012-0351-2
https://academic.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

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

Tab.2 Time cost of XFEM schemes with and without quadrature sub-cells

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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