Planar Parametrization in Isogeometric Analysis Jens Gravesen 1 , Anton Evgrafov 1 , Nguyen Dang Manh 2 , and Peter Nørtoft 3⋆ 1 DTU Compute, Technical University of Denmark, Denmark, {jgra,aaev}@dtu.dk. 2 Institute of Applied Geometry, Johannes Kepler University, Austria, Manh.Dang Nguyen@jku.at. 3 Applied Mathematics, SINTEF ICT, Norway, peter@noertoft.net. Abstract. Before isogeometric analysis can be applied to solving a par- tial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The chal- lenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It uti- lizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysis- oriented methods. Their performance is illustrated through a few nu- merical examples. Keywords: Isogeometric analysis, shape optimization, parametrization 1 Introduction Isogeometric analysis is a modern computational method for solving partial dif- ferential equations (PDEs), which is based on a successful symbiosis between the variational techniques utilized in isoparametric finite element analysis with the geometric modelling tools from computer aided design [14, 4]. A key ingredi- ent of isogeometric analysis is the parametrization of the physical domain over which the PDE is posed, in many ways analogous to mesh generation in stan- dard finite element analysis. Just as mesh quality affects the accuracy of a finite element approximation, the quality of the parametrization affects the accuracy of isogeometric analysis, see [21, 2, 34, 35]. ⋆ Presently at DTU Compute, Technical University of Denmark, Denmark.