Level set-based topological shape optimization of nonlinear heat conduction problems using topological derivatives

A level set-based topological shape-optimization method is developed to relieve the well-known convergence difficulty in nonlinear heat-conduction problems. While minimizing the objective function of instantaneous thermal compliance and satisfying the constraint of allowable volume, the solution of the Hamilton-Jacobi equation leads the initial implicit boundary to an optimal one according to the normal velocity determined from the descent direction of the Lagrangian. Topological derivatives are incorporated into the level set-based framework to improve convergence of the optimization process as well as to avoid the local minimum resulting from the intrinsic nature of the shape-design approach.