A level set method for topological shape optimization of 3D structures with extrusion constraints

This paper proposes a new level set method for topological shape optimization of 3D structures considering manufacturing constraints. First, the boundary of structure is implicitly represented as the zero level set of a higher-dimensional level set function, and the implicit surface is parameterized through the interpolation of a given set of compactly supported radial basis functions. In this way, the original Hamilton-Jacobi partial differential equation is transformed into a system of algebraic equations. Correspondingly, the topological shape optimization is changed to the easiest size optimization in structural optimization. Many more efficient gradient-based optimization algorithms can be directly applied to the size optimization. Second, to save the expensive computational cost in the 3D large-scale optimization problems, the discrete wavelet transform is introduced into the level set method to compress the size of the coefficient matrix in compactly supported radial basis function interpolant. The discrete wavelet transform converts the original matrix into a set of wavelet basis and therefore eliminates the “noise” elements in the new matrix, so that the linear system can be replaced by a sparser one. Finally, a cross section projection strategy is utilized to ensure the satisfaction of the extrusion constraint and reduce the number of the design variables, simultaneously. Several numerical examples in 3D structures are employed to demonstrate the effectiveness of the proposed method.