This paper deals with an alternative approach to density and sensitivity filtering based on the solution of the linear heat conduction equation which is proposed for eliminating checkerboard patterns and mesh dependence in topology optimization problems. In order to guarantee the existence, uniqueness and stability of the solution of PDE, Neumann boundary conditions are introduced. With the help of the existing computational framework of FEM, boundary points have been extended to satisfy Neumann boundary conditions, and together with finite difference method to solve this initial boundary value. In order to guarantee the stability, stability factor is introduced to control the deviation for the solution of the finite difference method. Then the filtering technique is directly applied to the design variables and the design sensitivities, respectively. Especially, different from previous methods based on convolution operation, filtering iteration is employed to ensure the function to eliminate numerical instability. When the value of stability factor is changed at setting range, the number of times of filtering is manually corresponding set. At last, using different test examples in 2D show the advantage and effectiveness of filtering iteration of the new filter method in compared with previous filter method.
The Solution of Heat Conduction Equation with Mixed Boundary Conditions