Isogeometric finite element methods for shape optimization

In this thesis we develop a shape optimization framework for isogeometric analysis in the optimize first-discretize then setting. For the discretization we use isogeometric analysis (IGA) to solve the state equation, and search optimal designs in a space of admissible B-spline or NURBS combinations. Thus a quite general class of functions for representing optimal shapes is available. For the gradient-descent method, the shape derivatives indicate both stopping criteria and search directions and are determined isogeometrically. The numerical treatment requires solvers for partial differential equations and optimization methods, which introduces numerical errors. The tight connection between IGA and geometry representation offers new ways of refining the geometry and analysis discretization by the same means. Therefore, our main concern is to develop the optimize first framework for isogeometric shape optimization as ground work for both implementation and an error analysis. Numerical examples show that this ansatz is practical and case studies indicate that it allows local refinement.