Smooth multi-patch discretizations in Isogeometric Analysis

Thomas J.R. Hughes, Giancarlo Sangalli, Thomas Takacs, Deepesh Toshniwal

Research output: Chapter in Book/Conference proceedings/Edited volumeChapterScientificpeer-review

Abstract

With the aim of a seamless integration with Computer-Aided Design, Isogeometric Analysis has been proposed by Hughes et al. (2005) as a numerical technique for the solution of partial differential equations. Indeed, isogeometric analysis is based on splines, the same functions that are adopted for geometry parametrizations in CAD. Smooth splines yield two important benefits when compared to C0 piecewise polynomial approximations: superior accuracy and stability, and the possibility to directly discretize high-order differential equations, such as the ones arising in thin-shell theory, in fracture models, in phase-field based multiphase flows, and in geometric flows on surfaces. In this chapter we review three different methods to construct C1 isogeometric spaces on multi-patch domains or unstructured quadrilateral meshes. The first, proposed in Collin et al. (2016), is based on the concept of geometric continuity, well-known in geometric design. The second, from Toshniwal et al. (2017c), uses a specific singular construction (the D-patch construction) at extraordinary points. The third is a polar construction from Toshniwal et al. (2017a). Such constructions possess properties that make them suitable for performing both computational analysis and geometric modeling.

Original languageEnglish
Title of host publicationHandbook of Numerical Analysis
Chapter8
Pages467-543
Number of pages77
Volume22
DOIs
Publication statusPublished - 2021

Publication series

NameHandbook of Numerical Analysis
ISSN (Print)1570-8659

Keywords

  • C approximation
  • Computer-Aided Design
  • extraordinary points
  • geometric continuity
  • Isogeometric Analysis
  • multi-patch
  • NURBS
  • polar parametrization
  • singular parametrization
  • splines

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