Boundary effects in a phase-field approach to topology optimization

A phase-field based topology optimization approach is considered for the maximum stiffness or minimum compliance problem. The objective functional to be minimized consists in addition to the compliance a cost for gray solutions and a cost for interfaces between void and full material. Since the interfaces between void and full material are penalized via a volume integral in the original phase-field formulation there is no penalty associated with interfaces along the external boundaries. In the present contribution, an additional term representing the cost of interfaces at external boundaries is added to the functional subject to minimization. It is shown that the new boundary term enters the optimization as a Robin boundary condition. The method is implemented in a finite element setting and numerical simulations of typical structures are considered. The results indicate that the optimal designs are influenced by the cost of interfaces to a large extent.