In shape optimization problems, the goal is to find an optimal shape that minimizes some functional. One of the most studied shape optimization problem is that of drag minimization, that is, what is the shape of an object flowing in a fluid that gives the least drag forces. This has applications in the design of ships, racecars, wind turbines and airplane wings. Unfortunately, classical formulations of shape optimization problems are often ill-posed, i.e., the optimal shape may not exist. In this talk, we present a mathematical formulation of the drag minimization problem in a stationary Navier-Stokes flow. The formulation utilizes a phase field approach, which leads to a well-posed optimization problem where we can prove the existence of a minimizer. Furthermore, we can also derive first order necessary optimality conditions rigorously, and together with a gradient flow approach, we use these conditions to numerically solve for the optimal shapes of drag minimization.
Biography
Kei-Fong Lam received his Ph.D. in Mathematics from University of Warwick in 2014. Since September 2014, he has been Research Assistant at the University of Regensburg, Germany. His research interests are: soluble surfactants in two phase flow, tumour growth with chemotaxis and active transport.