In Floater, M., Lyche, T., Mazure, M., Mørken, K., & Schumaker, L., editors, *Mathematical Methods for Curves and Surfaces*, volume 8177, of *Lecture Notes in Computer Science*, pages 189–212. Springer Berlin Heidelberg, 2014.

doi abstract bibtex

doi abstract bibtex

Before isogeometric analysis can be applied to solving a partial 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 challenge 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 utilizes 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 numerical examples.

@InCollection{ Gravesen_2014aa, abstract = {Before isogeometric analysis can be applied to solving a partial 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 challenge 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 utilizes 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 numerical examples.}, author = {Gravesen, Jens and Evgrafov, Anton and Nguyen, Dang-Manh and Nørtoft, Peter}, booktitle = {Mathematical Methods for Curves and Surfaces}, doi = {10.1007/978-3-642-54382-1_11}, editor = {Floater, Michael and Lyche, Tom and Mazure, Marie-Laurence and Mørken, Knut and Schumaker, LarryL.}, file = {Gravesen_2014aa.pdf}, isbn = {978-3642543814}, keywords = {isogeometric,optimization,parametrization}, langid = {english}, pages = {189--212}, publisher = {Springer Berlin Heidelberg}, series = {Lecture Notes in Computer Science}, title = {Planar Parametrization in Isogeometric Analysis}, volume = {8177}, year = {2014} }

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