Multimaterial Topology Optimization of Contact Problems Using Allen-Cahn Approach

The paper deals with the numerical solution of the multimaterial topology optimization problems for bodies in contact. The unilateral contact phenomenon between the elastic body and the rigid foundation with Tresca friction is governed by the elliptic boundary value problem with inequality boundary conditions. The materials distribution function is chosen as the design variable. The structural optimization problem consists in finding such topology of the domain occupied by the body in terms of the design variable that the normal contact stress along the boundary of the body is minimized. The original cost functional is regularized using the multiphase volume constrained Ginzburg-Landau energy functional rather than the perimeter functional. The first order necessary optimality condition is provided. The derivative of the cost functional is used to formulate the generalized gradient flow equation of Allen-Cahn type. The optimal topology is obtained as the steady state of the phase transition governed by this equation. The optimization problem is solved numerically using the operator splitting approach combined with the gradient projection method. Numerical examples are provided and discussed.