Practical isogeometric shape optimization: Parametrization by means of regularization

Abstract

Shape optimization based on isogeometric analysis (IGA) has gained popularity in recent years. Performing shape optimization directly over parameters defining the computer-aided design (CAD) geometry, such as the control points of a spline parametrization, opens up the prospect of seamless integration of a shape optimization step into the CAD workflow. One of the challenges when using IGA for shape optimization is that of maintaining a valid geometry parametrization of the interior of the domain during an optimization process, as the shape of the boundary is altered by an optimization algorithm. Existing methods impose constraints on the Jacobian of the parametrization, to guarantee that the parametrization remains valid. The number of such validity constraints quickly becomes intractably large, especially when 3D shape optimization problems are considered. An alternative, and arguably simpler, approach is to formulate the isogeometric shape optimization problem in terms of both the boundary and the interior control points. To ensure a geometric parametrization of sufficient quality, a regularization term, such as the Winslow functional, is added to the objective function of the shape optimization problem. We illustrate the performance of these methods on the optimal design problem of electromagnetic reflectors and compare their performance. Both methods are implemented for multipatch geometries, using the IGA library G+Smo and the optimization library Ipopt. We find that the second approach performs comparably to a state-of-the-art method with respect to both the quality of the found solutions and computational time, while its performance in our experience is more robust for coarse discretizations.