Topology optimization (TO) is a promising numerical technique for designing optimal engineering designs in many industrial applications. It is expected that it might become an unavoidable engineering tool for many new rising technologies such as the additive manufacturing or metal 3D printing. We propose an efficient and robust iterative thresholding method for topology optimization with applications to fluids and heat transfer system. For the Stokes fluid flow, the proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and an incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes problem is approximated by a Brinkman equation and solved with the mixed finite-element method. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. Extensive numerical experiments in both two and three dimensions show that the proposed iteration scheme is robust, efficient and insensitive to the initial guess and the parameters in the model. Generalization to heat transfer system will also be discussed.

Prof. Wang received his BSc Degree in mathematics from Peking University in 1984 and his PhD Degree in mathematics from Courant Institute (NYU) in 1990. He was a postdoctoral at MSRI in Berkeley, University of Colorado in Boulder before he joined HKUST.  He is currently the Head and Chair Professor of Mathematics at HKUST.  Prof. Wang’s research interests are: Modeling and Simulations of Interface Problem and Multi-phase Flow; Image Processing; Numerical Methods for Micromagnetics Simulations.  He received Feng Kang Prize of Scientific Computing in 2007 and was a plenary speaker at the International Congress on Industrial and Applied Mathematics (2019).