Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries

Kapl, Mario; Buchegger, Florian; Bercovier, Michel; Jüttler, Bert

doi:10.1016/j.cma.2016.06.002

Cited by 71 publications

(87 citation statements)

References 21 publications

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“…The class of AS-G 1 multi-patch parameterizations includes the subclass of bilinear multi-patch parameterizations (cf. [3,16,20,24]) but the class of AS-G 1 multi-patch parameterization is wider than this subclass [7,20]. However, already for generic biquadratic multi-patch parameterizations we obtain in general C 1 isogeometric spaces with dramatically reduced approximation properties.…”

Section: Introductionmentioning

confidence: 99%

“…[7,12,16,20]. Such spaces are of great interest and importance since they provide a possibility to solve numerically fourth order partial differential equations such as the biharmonic equation, see e.g.…”

Section: Analysis-suitable G 1 Multi-patch Parameterizationsmentioning

confidence: 99%

“…Such spaces are of great interest and importance since they provide a possibility to solve numerically fourth order partial differential equations such as the biharmonic equation, see e.g. [16,20]. To obtain C 1 isogeometric spaces with optimal approximation properties, particular multi-patch geometries, so-called analysis-suitable G 1 multi-patch geometries, are needed, cf.…”

Section: Analysis-suitable G 1 Multi-patch Parameterizationsmentioning

confidence: 99%

“…[26,27,32]), the second strategy generates C 1 isogeometric spaces over multi-patch parameterizations, which are only C 0 at the patch interfaces (e.g. [3,4,7,16,17,20,24,25]). Similarly, [35] presents a construction of C k isogeometric spaces over patches having a polar layout, where they present details for k ≤ 2.…”

Section: Introductionmentioning

confidence: 99%

“…In [16,20] a method was proposed to construct mapped piecewise bilinear parameterizations, which are instances of AS-G 1 multi-patch parameterizations beyond bilinear multi-patch parameterizations. The two coordinate functions of such non-bilinear AS-G 1 parameterizations are constructed via least-squares fitting by means of C 1 isogeometric spaces over preselected bilinear domains.…”

Section: Introductionmentioning

confidence: 99%

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Construction of analysis-suitableG1planar multi-patch parameterizations

Kapl

Sangalli

Takacs

2018

Computer-Aided Design

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Isogeometric analysis allows to define shape functions of global C 1 continuity (or of higher continuity) over multi-patch geometries. The construction of such C 1 -smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations, so-called analysis-suitable G 1 (in short, AS-G 1 ) parameterizations, to ensure that the resulting C 1 isogeometric spaces possess optimal approximation properties, cf. [7]. In this work, we show through examples that it is possible to construct AS-G 1 multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-G 1 multi-patch parameterization possessing the same boundary, the same vertices and the same first derivatives at the vertices, and which is as close as possible to this initial geometry. Our algorithm is based on a quadratic optimization problem with linear side constraints. Numerical tests also confirm that C 1 isogeometric spaces over AS-G 1 multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced.

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Section: Introductionmentioning

confidence: 99%

Section: Analysis-suitable G 1 Multi-patch Parameterizationsmentioning

confidence: 99%

Section: Analysis-suitable G 1 Multi-patch Parameterizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Construction of analysis-suitableG1planar multi-patch parameterizations

Kapl

Sangalli

Takacs

2018

Computer-Aided Design

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Tuned hybrid nonuniform subdivision surfaces with optimal convergence rates

Wei

Zhang

et al. 2021

Numerical Meth Engineering

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This article presents an enhanced version of our previous work, hybrid nonuniform subdivision (HNUS) surfaces, to achieve optimal convergence rates in isogeometric analysis (IGA). We introduce a parameter λ (14<λ<1) to control the rate of shrinkage of irregular regions, so the method is called tuned hybrid nonuniform subdivision (tHNUS). Thus, HUNS is a special case of tHNUS when λ=12. While introducing λ in hybrid subdivision significantly complicates the theoretical proof of G1 continuity around extraordinary vertices, reducing λ can recover optimal convergence rates when tHNUS functions are used as a basis in IGA. From the geometric point of view, tHNUS retains comparable shape quality as HNUS under nonuniform parameterization. Its basis functions are refinable and the geometric mapping stays invariant during refinement. Moreover, we prove that a tHNUS surface is globally G1‐continuous. From the analysis point of view, tHNUS basis functions form a nonnegative partition of unity, are globally linearly independent, and their spline spaces are nested. In the end, we numerically demonstrate that tHNUS basis functions can achieve optimal convergence rates for the Poisson's problem with nonuniform parameterization around extraordinary vertices.

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