On Shape Optimization in the Context of Isogeometric Analysis
In shape optimization, the communication between the geometric description and the analysis suitable model of the domain plays an important role as updating the geometric design in one optimization step results in a new domain for the analysis, and vice versa. By means of Isogeometric Analysis (IGA) the same model can be used for design and analysis, in contrast to classical Finite Element Methods. So IGA skips conversion between meshes, which may lead to significant benefits. Since the advent of Isogeometric Analysis it has been shown that IGA is suitable for shape optimization in structural mechanics and electromagnetism. In this presentation, we discuss some fundamental issues related to shape optimization based on IGA and specifically the representation of shape gradients. We use Isogeometric Analysis to solve the state equation, and gradient-based methods for the optimization. This involves shape sensitivities that are defined in terms of the abstract framework of shape calculus and that are computed by means of the same basis functions as for the analysis, B-splines or NURBS. In this way, a quite general class of functions for representing optimal shapes and their boundaries becomes available. Moreover, it is possible to re-use the data from the analysis for the gradient computation, which leads to an efficient implementation.