Triangular checkerboard control using a wavelet-based method in topology optimization

Topology optimization has been carried out mainly with structured quadrilateral finite elements. However, triangular elements facilitate mesh generation especially for problems having complex geometries that often appear in practical industrial problems. The use of triangular elements, especially low-order triangular elements, causes a serious numerical trouble that is equivalent to the rectangular checkerboard pattern formation. The objective of this investigation is to develop a triangular checkerboard-freeing method that directly restricts the design space. To this end, we use the multiscale design space that is mapped from the standard single-scale density space. To facilitate the mapping, we employ the triangular mesh subdivision and propose a bi-orthogonal wavelet transform suitable for a triangulated domain. For checkerboard-free designs, a shrinkage method based on the wavelet frame appropriate for triangular mesh is proposed. Typical benchmark problems and a simplified roof-reinforcing problem in an automobile body are considered to check the effectiveness of the proposed method