Unconditionally energy stable schemes for fluid-based topology optimization

We present first- and second-order unconditionally energy stable schemes for fluid-based topology optimization problems. Our objective functional composes of five terms including mechanical property, Ginzburg–Landau energy, two penalized terms for solid, and the volume constraint. We consider the steady-state Stokes equation in the fluid domain and Darcy flow through porous medium. By coupling a Stokes type equation and the Allen–Cahn equation, we obtain the evolutionary equation for the fluid-based topology optimization. We use the backward Euler method and the Crank–Nicolson method to discretize the coupling system. The first- and second-order accurate schemes are presented correspondingly. We prove that our proposed schemes are unconditionally energy stable. The preconditioned conjugate gradient method is applied to solve the system. Several numerical tests are performed to verify the efficiency and accuracy of our schemes.