A second-order unconditionally energy stable scheme for phase-field based multimaterial topology optimization

A multimaterial topology optimization problem is solved by introducing a numerical scheme with fast convergence, second-order accuracy and unconditional energy stability. The modeling is based on the energy of multi-phase-field elasticity system including the classical Ginzburg–Landau, the elastic potential and some constraints. The material layout is updated by using the volume constrained gradient flow of the system. In this work, we transform the traditional objective functional of multimaterial topology optimization into the energy functional of multi-phase-field elasticity system, and transform the optimal material layout into the solutions of volume constrained Allen–Cahn type equations. For these Allen–Cahn type equations, we propose the second-order unconditionally energy stable numerical scheme which combines linearly stabilized splitting method and Crank–Nicolson scheme. For the proposed second-order scheme, we give a theoretical proof of unconditional energy stability. Numerical results show that the scheme converges fast compared to traditional Cahn–Hilliard type equations. Some classical benchmarks are performed to verify the feasibility and efficiency of our method.