Efficient 3D Shape Optimization with Isogeometric Analysis

The subject of this thesis is the development of methods to efficiently solve 3D shape optimization problems with isogeometric analysis. Shape optimization is the art of finding the best possible shape based on a desired property. Such problems often arise in engineering, where the criterion depends on the solution to a partial differential equation, that models the underlying physical process.

Isogeometric analysis is a numerical method for solving partial differential equations posed on a complicated domain. With this method the shape can be represented exactly, by using splines, and the optimization can be performed directly on parameters controlling the shape of the domain. When using isogeometric analysis for shape optimization one needs to maintain a valid parametrization of the interior of the computational domain during the optimization process.

This is not a trivial task, and often methods rely on non-linear constraints on the validity of the parametrization. When considering 3D shape optimization problems the number of constraints can be huge. In this work we will investigate a regularization based approach to maintaining parametrizations, that avoids the expensive constraints.

As the first contribution we demonstrate that this approach performs at par with a constraint based method, when considering a 2D model problem of designing electromagnetic reflectors. As the second contribution we use this regularization based shape optimization approach to find parametrizations. We demonstrate that the method is able to find parametrizations of complicated shapes both in 2D and 3D, and the result serves as a benchmark that illustrates the capability of the method.

Finally as the third contribution we consider a 3D shape optimization problem of designing reflectors for free surface waves. With the proposed approach we are able to obtain a shape that performs much better than the initial guess, while maintaining a valid parametrization. However to obtain a valid parametrization the optimization had to be terminated prematurely.