Isogeometric analysis with $C^1$ functions  on planar, unstructured quadrilateral meshes

Mario Kapl; Giancarlo Sangalli; Thomas Takacs

doi:10.5802/smai-jcm.52

Isogeometric analysis with

C^{1}

functions on planar, unstructured quadrilateral meshes

Mario Kapl ¹; Giancarlo Sangalli ²; Thomas Takacs ³

¹ Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Austria
² Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, and IMATI-CNR, Pavia, Italy
³ Institute of Applied Geometry, Johannes Kepler University Linz, Austria

The SMAI Journal of computational mathematics, Volume S5 (2019), pp. 67-86.

Abstract

In the context of isogeometric analysis, globally $C^{1}$ isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [28]. There, the construction of a specific $C^{1}$ isogeometric spline space for the class of so-called analysis-suitable $G^{1}$ multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of $C^{1}$ spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable $G^{1}$ by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples.

Published online: 2020-01-29

Zbl

DOI: 10.5802/smai-jcm.52

Classification: 65N30
Keywords: Isogeometric Analysis, $C^{1}$ isogeometric functions, geometric continuity, extraordinary vertices, planar multi-patch domain

Author's affiliations:

Mario Kapl ¹; Giancarlo Sangalli ²; Thomas Takacs ³

¹ Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Austria
² Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, and IMATI-CNR, Pavia, Italy
³ Institute of Applied Geometry, Johannes Kepler University Linz, Austria

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     author = {Mario Kapl and Giancarlo Sangalli and Thomas Takacs},
     title = {Isogeometric analysis with $C^1$ functions  on planar, unstructured quadrilateral meshes},
     journal = {The SMAI Journal of computational mathematics},
     pages = {67--86},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {S5},
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Mario Kapl; Giancarlo Sangalli; Thomas Takacs. Isogeometric analysis with $C^1$ functions  on planar, unstructured quadrilateral meshes. The SMAI Journal of computational mathematics, Volume S5 (2019), pp. 67-86. doi : 10.5802/smai-jcm.52. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.52/

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