Numerical shape optimization with finite elements: a bit of theory and a bit of practice

Shape optimization is about finding domain geometries that minimize a given objective function. Dido’s isoperimetric problem of finding a geometry with maximal area for a given perimeter is a classical example. In most applications, evaluating the objective function requires solving a boundary value problem on the domain to be optimized. For example, to compute the energy dissipated by a fluid flowing in a pipe one must first compute a solution to a fluid model. The presence of such constraints makes shape optimization problems particularly challenging. Even computing approximate solutions with numerical methods is not straightforward because this requires solving a boundary value problem on a computational domain that changes at each iteration of the optimization algorithm.

In this talk, I will describe how the finite element method enables a natural implementation of the moving-mesh shape optimization method that generalizes straightforwardly to higher-order discretizations. I will also explain how finite element software can automated the evaluation of shape derivatives along finite element directions. Finally, I will present how these aspects have been realized in the automated PDE-constrained shape optimization toolbox Fireshape.

The talk is designed to be accessible to a general academic audience interested in applied mathematics. Prior knowledge of the finite element method is not assumed.