Isogeometric analysis with geometrically continuous functions on two-patch geometries

Kapl, Mario; Vitrih, Vito; Jüttler, Bert; Birner, Katharina

doi:10.1016/j.camwa.2015.04.004

Cited by 79 publications

(151 citation statements)

References 18 publications

Supporting

Mentioning

150

Contrasting

Order By: Relevance

“…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%

“…This has been exploited in previous works cf. [7,12,20]. However, two different approaches, based on different types of multi-patch parameterizations, have been adopted.…”

Section: Introductionmentioning

confidence: 99%

“…The high smoothness of the isogeometric spaces allows to solve high order partial differential equations directly via their weak form using a standard Galerkin discretization, see e.g. [1,18,19,34].…”

Section: Introductionmentioning

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%

See 3 more Smart Citations

Construction of analysis-suitableG1planar multi-patch parameterizations

Kapl

Sangalli

Takacs

2018

Computer-Aided Design

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

“…This has been exploited in previous works cf. [7,12,20]. However, two different approaches, based on different types of multi-patch parameterizations, have been adopted.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Construction of analysis-suitableG1planar multi-patch parameterizations

Kapl

Sangalli

Takacs

2018

Computer-Aided Design

Self Cite

View full text Add to dashboard Cite

show abstract

“…In [BM14], a similar result is obtained for the space of G 1 splines of bi-degree (4, 4) for rectangular decompositions of planar domains. The construction of basis functions for spaces of C 1 geometrically continuous functions restricted to two-patch domains, has been considered in [KVJB15]. In [CST16], the approximation properties of the aforementioned spaces are explored, including constructions over multi-patch geometries motivated by applications in isogeometric analysis.…”

Section: Introductionmentioning

confidence: 99%

G 1 -smooth splines on quad meshes with 4-split macro-patch elements

Blidia¹,

Mourrain²,

Villamizar

2017

Computer Aided Geometric Design

View full text Add to dashboard Cite

We analyze the space of differentiable functions on a quad-mesh M, which are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges, that satisfy conditions which ensure that the space of differentiable functions is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of G 1 spline functions of bi-degree (k, k) on each quadrangular face of M. We determine the dimension of this space of G 1 spline functions for k big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. New results on their generators and dimensions are provided. Examples of bases of G 1 splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.

show abstract

Mathematics of Isogeometric Analysis: A Conspectus

Hughes

Sangalli

2017

Encyclopedia of Computational Mechanics Second Edition

View full text Add to dashboard Cite

Isogeometric analysis (IGA) is a successful generalization of finite element analysis. The use of smooth splines simplifies the interaction with geometric design, but also opens new frontiers to the numerical solution of partial differential equations. In this chapter we present an overview of the main mathematical properties and results that explain the advantages of IGA. We describe the construction of isogeometric scalar and vector spaces, and their approximation and spectral properties.

show abstract

Isogeometric analysis with geometrically continuous functions on two-patch geometries

Cited by 79 publications

References 18 publications

Construction of analysis-suitableG1planar multi-patch parameterizations

Construction of analysis-suitableG1planar multi-patch parameterizations

G 1 -smooth splines on quad meshes with 4-split macro-patch elements

Mathematics of Isogeometric Analysis: A Conspectus

Contact Info

Product

Resources

About