Mathematical studies of a phase field approach to shape optimization

Project: Research project

Project Details

Description

In shape optimization problems, the aim is to find a class of shapes that optimizes some prescribed goals under a partial differential equation (PDE) constraint. One example is drag minimization, seeking the shape of an impermeable object exhibiting the least drag while flowing through a fluid. The classical formulation for such problems involves representing the unknown shape as a mathematical surface, and seeks to derive optimality conditions (equations/inequalities involving nonlinear PDEs). The optimal shapes can then be realized numerically by solving the optimality conditions. However, shape optimization problems are often severely ill-posed (as the existence of optimal shapes is not guaranteed in general) and can be computationally intensive (due to the requirement of a new approximation of the computational domain at every numerical iteration).

We propose to use a phase field approach to overcome the above problems, and for drag minimization, it has several important mathematical and numerical advantages over the classical approach; namely the phase field method can be justified (in the sense that existence of optimal shapes can be proved rigorously) and in addition both the PDE constraint and the optimality conditions are solved on the same fixed domain throughout the optimization procedure. Furthermore, by sending a small parameter to zero, the optimality conditions from the classical formulation are recovered. Hence, one may view the phase field approach as a consistent approximation for shape optimization problems.

The goal of this proposal is to develop phase field approximations to study nonlinear inverse problems that can be casted as shape optimization problems. In light of the mathematical advantages outlined above, we believe that the proposed phase field approach may offer novel and alternative resolution strategies for such problems. Firstly, we consider an abstract shape optimization problem with the aim of providing analytical results for the PDE constraint and establish the existence of optimal shapes, the derivation of optimality conditions, as well as the connection with the classical formulation. Then, we apply this theory to two important examples of inverse problems: inverse acoustic scattering and electrical impedance tomography. Numerical simulations will be carried out to compare the proposed phase field approach with current state-of- the-art methods, and successful investigations will serve to enhance the applicability of the phase field approach.
StatusFinished
Effective start/end date1/01/19 β†’ 30/06/22

Fingerprint

Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.