Fundamental aspects of shape optimization in the context of isogeometric analysis

• We provide a framework for shape optimization problems with isogeometric analysis.
• For that, we express shape gradients in isogeometric terms.
• Equivalence of discrete systems from discretize-first and optimize-first is shown.
• We formulate shape optimization with respect to NURBS in the optimize-first ansatz.

We develop a mathematical foundation for shape optimization problems under state equation constraints where both state and control are discretized by B-splines or NURBS. In other words, we use isogeometric analysis (IGA) for solving the partial differential equation and a nodal approach to change domains where control points take the place of nodes and where thus a quite general class of functions for representing optimal shapes and their boundaries becomes available. The minimization problem is solved by a gradient descent method where the shape gradient will be defined in isogeometric terms. This gradient is obtained following two schemes, optimize first–discretize then and, reversely, discretize first–optimize then. We show that for isogeometric analysis, the two schemes yield the same discrete system. Moreover, we also formulate shape optimization with respect to NURBS in the optimize first ansatz which amounts to finding optimal control points and weights simultaneously. Numerical tests illustrate the theory.