Checkerboard Problem to Topology Optimization of Continuum Structures

The area of topology optimization of continuum structures of which is allowed to change in order to improve the performance is now dominated by methods that employ the material distribution concept. The typical methods of the topology optimization based on the structural optimization of two phase composites are the so-called variable density ones, like the SIMP (Solid Isotropic Material with Penalization) and the BESO (Bi-directional Evolutional Structure Optimization). The topology optimization problem refers to the saddle-point variation one as well as the so-called Stokes flow problem of the compressive fluid. The checkerboard patterns often appear in the results computed by the SIMP and the BESO in which the Q1-P0 element is used for FEM (Finite Element Method), since these patterns are more favourable than uniform density regions. Computational experiments of SIMP and BESO have shown that filtering of sensitivity information of the optimization problem is a highly efficient way that the checkerboard patterns disappeared and to ensure mesh-independency. SIn this paper, we discuss the theoretical basis for the filtering method of the SIMP and the BESO and as a result, the filtering method can be understood by the theorem of partition of unity and the convolution operator of low-pass filter. Introduction A common structural optimization in mechanical structural design field is three categorized into the size, shape and topology optimization of an elastic structure given certain boundary conditions. The size optimization is to determine only the size of the materials, and the shape optimization refers to the outer shape of materials. The purpose of topology optimization is to find the optimal layout of a structure included the holes within a specified region. The only known quantities in the problem are the applied loads, the possible support conditions, the volume of the structure to be constructed and possibly some additional design restrictions such as the location and size of prescribed holes or solid areas. In this problem the physical size and the shape, and the connectivity of the structure are unknown. The area of the topology optimization of continuum structure is now dominated by methods that employ the material distribution concept, like the SIMP (Bendsøe and Sigmund, 2003) and BESO (Huang and Xie, 2010). The topology optimization problem refers to the saddle-point variation one as well as the so-called Stokes flow one of incompressive fluid (Brezzi and Fortin, 1991). The checkerboard pattern often appear in the results computed by the SIMP and BESO in which the Q1-P0 element is used for FEM (Finite Element Method) as well as the Stokes problem, since these patterns are more favourable than the uniform density regions (Diaz and Sigmund, 1995, Jog and Haber, 1996, Sigmund and Petersson, 1998). Computational experiments of the SIMP and BESO have shown that the filtering of sensitivity information of the optimization problem is a highly efficient way that the checkerboard pattern disappeared. Since the filter is usually used as low-pass filter for noise cleaning in the field of the image processing, this type of filter was introduced to the problems of the topology optimization by Sigmund (1994). The function of the filter is explained as the average of the density by the density of neighbourhood, or the convolution operator of the low-pass filter (Sigmund and Petersson, 1998, Hassani and Hinston, 1999 ) . However, the theoretical basis for the filtering method is not yet understood completely. In this paper, we discuss the theoretical basis for the filtering method of SIMP and BESO and as the result, the filtering method can be explained by the theorem of partition of unity and the convolution. The paper is consisted of theory of FEM for elasticity equation of the isotropic material and of checkerboard problem (Sec.1), the calculated procedure of SIMP and BESO (Sec.2), the basic theory for the filtering (Sec.3), the computed Results of SIMP and BESO (Sec.4), and conclusion remarks.