Solution to the topology optimization problem using a time-evolution equation

This article describes a numerical solution to the topology optimization problem using a time-evolution equation. The design variables of the topology optimization problem are defined as a mathematical scalar function in a given design domain. The scalar function is projected to the normalized density function. The adjoint variable method is used to determine the gradient defined as the ratio of the variation of the objective function or constraint function to the variation of the design variable. The variation of design variables is obtained using the solution of the time-evolution equation in which the source term and Neumann boundary condition are given as a negative gradient. The distribution of design variables yielding an optimal solution is obtained by time integration of the solution of the time-evolution equation. By solving the topology optimization problem using the proposed method, it is shown that the objective function decreases when the constraints are satisfied. Furthermore, we apply the proposed method to the thermal resistance minimization problem under the total volume constraint and the mean compliance minimization problem under the total volume constraint.