Phase field regularized level set approach in topology optimization of variational inequalities

The combined level set and phase field rather than classical level set approach is used in the paper to analyze and solve numerically the topology optimization problem for system governed by the variational inequalities. Classical level set method allows to evolve a given sharp interface but is not capable to generate holes unless the topological derivative is used. The phase field method indicates the position of the interface in a blurry way but is flexible in the hole generation. In the paper two-phase topology optimization problem is formulated in terms of the modified level set function regularized using Cahn-Hilliard based inter-facial energy term rather than the standard perimeter term. The derivative of the cost functional with respect to the level set function is calculated. Modified reaction-diffusion equation updating the level set function is derived. The necessary optimality condition for this optimization problem is formulated. The finite element and finite difference methods are used to solve the state and adjoint systems numerically. Numerical examples are provided and discussed.