  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.
