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.