An Efficient Threshold Dynamics Method for Topology Optimization for Fluids (CSIAM Transactions on Applied Mathematics, 2022)
We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based onminimization of an objective energy function that consists of the dissipation power inthe fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimizationproblem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regionsare then updated according to simple convolutions followed by a thresholding step.We prove mathematically that the iterative algorithm has the total energy decayingproperty. The proposed algorithm is simple and easy to implement. Extensive numerical experiments in both two and three dimensions show that the proposed iterationalgorithm converges in much fewer iterations and is more efficient than many existingmethods. In addition, the numerical results show that the algorithm is very robust andinsensitive to the initial guess and the parameters in the model.