Refinable bi-quartics for design and analysis

Karčiauskas, Kȩstutis; Peters, Jörg

doi:10.1016/j.cad.2018.05.001

Cited by 17 publications

(19 citation statements)

References 12 publications

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“…The paper [29] presents a new surface construction, where bicubic splines are complemented by biquartic splines in the neighborhood of extraordinary vertices. The methodology [31] can be seen as an extension of the above two techniques, and allows the construction of nested C 1 isogeometric spline spaces for a finite number of refinement steps.…”

Section: The Design Of C 1 Isogeometric Spacesmentioning

confidence: 99%

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

Kapl¹,

Sangalli²,

Takacs³

2019

The SMAI Journal of Computational Mathematics

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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 [26]. 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.

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Section: The Design Of C 1 Isogeometric Spacesmentioning

confidence: 99%

“…See Figure 3 for a possible construction. We refer to [29,30,31,41], where such constructions were employed.…”

Section: Beyond Analysis-suitable G 1 Parameterizationsmentioning

confidence: 99%

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

Kapl¹,

Sangalli²,

Takacs³

2019

The SMAI Journal of Computational Mathematics

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“…This includes subdivision surface constructions Catmull-Clark (1978), macro patch constructions in low degree Loop (1994b), Peters (1995), Prautzsch (1997), Reif (1995), Peters (2002), Ying et al (2004), Fan et al (2008), Hahmann et al (2008), Bonneau et al (2014), manifold based constructions Gu et al (2006), He et al (2006), Tosun et al (2011), Wang et al (2016), constructions using transition maps dened from mesh embeddings Beccari et al (2014), or constructions using guided surfaces Kar£iauskas et al (2016), Kar£iauskas et al (2017a), Kar£iauskas et al (2018). Some of these works focus on the construction of G 1 spline surfaces that interpolate a network of curves Sarraga (1987), Sarraga (1989), Peters (1991), Loop (1994a), Tong et al (2009), Cho et al (2006), Bonneau et al (2014), Kar£iauskas et al (2017b), Kar£iauskas et al (2018). To solve this so-called transnite interpolation problem, vertex enclosure constraints have to be satised by the curves at a vertex of even valency.…”

Section: Introductionmentioning

confidence: 99%

Geometrically smooth spline bases for data fitting and simulation

Blidia

Mourrain

2020

Computer Aided Geometric Design

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Given a topological complex M with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth spline functions that satisfy dierentiability properties across shared edges. We present new and ecient constructions of basis functions of the space of G 1 -spline functions on quadrangular meshes, which are tensor product b-spline functions on each quadrangle and with b-spline transition maps across the shared edges. This new strategy for constructing basis functions is based on a local analysis of the edge functions, and does not depend on the global topology of M. We show that the separability of the space of G 1 splines across an edge allows to determine the dimension and a basis of the space of G 1 splines on M. This leads to explicit and eective constructions of basis functions attached to the vertices, edges and faces of M.This basis construction has important applications in geometric modeling and simulation. We illustrate it by the tting of point clouds by G 1 splines on quadrangular meshes of complex topology and in Isogeometric Analysis methods for the solution of diusion equations. The ingredients are detailed and experimentation results showing the behavior of the method are presented.

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“…[43,48,49], or consists of a special construction, see e.g. [33,34,42]. The methods [43,48,49] use a singular parameterization with patches in the vicinity of an extraordinary vertex, which belong to a specific class of degenerate (Bézier) patches introduced in [45], and that allow, despite having singularities, the design of globally C 1 isogeometric spaces.…”

Section: Introductionmentioning

confidence: 99%

“…The methods [43,48,49] use a singular parameterization with patches in the vicinity of an extraordinary vertex, which belong to a specific class of degenerate (Bézier) patches introduced in [45], and that allow, despite having singularities, the design of globally C 1 isogeometric spaces. The techniques [33,34,42] are based on G 1 multi-patch surface constructions, where the obtained surface in the neighborhood of an extraordinary vertex consists of patches of slightly higher degree [33,42] and is generated by means of a particular subdivision scheme [34]. As a special case of the first approach can be seen the constructions in [41,47], that employ a polar framework to generate C 1 spline spaces.…”

Section: Introductionmentioning

confidence: 99%

Isogeometric analysis with C1 hierarchical functions on planar two-patch geometries

Bracco

Giannelli

Kapl

et al. 2020

Computers & Mathematics with Applications

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Adaptive isogeometric methods for the solution of partial differential equations rely on the construction of locally refinable spline spaces. A simple and efficient way to obtain these spaces is to apply the multi-level construction of hierarchical splines, that can be used on single-patch domains or in multi-patch domains with C 0 continuity across the patch interfaces. Due to the benefits of higher continuity in isogeometric methods, recent works investigated the construction of spline spaces with global C 1 continuity on two or more patches. In this paper, we show how these approaches can be combined with the hierarchical construction to obtain global C 1 continuous hierarchical splines on two-patch domains. A selection of numerical examples is presented to highlight the features and effectivity of the construction.

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Refinable bi-quartics for design and analysis

Cited by 17 publications

References 12 publications

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

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

Geometrically smooth spline bases for data fitting and simulation

Isogeometric analysis with C1 hierarchical functions on planar two-patch geometries

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