Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes

Bercovier, Michel; Matskewich, Tanya

doi:10.1007/978-3-319-63841-6

Cited by 41 publications

(93 citation statements)

References 38 publications

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“…In [MVV16], a dimension formula and explicit basis constructions are given for polynomial patches of degree 4 over a mesh with triangular or quadrangular cells. 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].…”

Section: Introductionsupporting

confidence: 65%

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

Blidia¹,

Mourrain²,

Villamizar

2017

Computer Aided Geometric Design

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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.

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Section: Introductionsupporting

confidence: 65%

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

Blidia¹,

Mourrain²,

Villamizar

2017

Computer Aided Geometric Design

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“…[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%

“…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%

“…error AS-G 1 parametr. Convergence rates Figure 1: L 2 approximation of a given function (exact solution) over a non-AS-G 1 (initial) parameterization and over an AS-G 1 parameterization using C 1 isogeometric spline spaces of bidegree (3,3). The numerical results indicate in case of the AS-G 1 parameterization optimal rates of convergence, and in case of the non-AS-G 1 parameterization no convergence in the L ∞ -norm and minimal rates of convergence in the L 2 -norm.…”

Section: Introductionmentioning

confidence: 99%

“…The numerical results indicate in case of the AS-G 1 parameterization optimal rates of convergence, and in case of the non-AS-G 1 parameterization no convergence in the L ∞ -norm and minimal rates of convergence in the L 2 -norm. For both cases C 1 isogeometric spline spaces of bidegree (3,3) are used for the approximation. See Section 4 for more detail about this example.…”

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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Approximation Power of C 1-Smooth Isogeometric Splines on Volumetric Two-Patch Domains

Birner

Jüttler

Mantzaflaris

2020

Lecture Notes in Computational Science and Engineering

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Bases and dimensions of trivariate spline functions possessing first order geometric continuity on two-patch domains were studied in [4]. It was shown that the properties of the spline space depend strongly on the type of the gluing data that is used to specify the relation between the partial derivatives along the interface between the patches. Locally supported bases were shown to exist for trilinear geometric gluing data (that corresponds to piecewise trilinear domain parameterizations) and sufficiently high degree. The present paper is devoted to the approximation properties of these spline functions.We recall the construction of the basis functions and show how to compute them efficiently. In contrast to the results in [4], which relied on exact arithmetic operations in the field of rational numbers, we evaluate the coefficients by computations with standard floating point numbers. We then perform numerical experiments with L 2 -projection in order to explore the approximation power of the resulting spline functions. Despite the existence of locally supported bases, we observe a reduction of the approximation order for low degrees, and we provide a theoretical explanation for this locking.

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Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes

Cited by 41 publications

References 38 publications

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

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

Construction of analysis-suitableG1planar multi-patch parameterizations

Approximation Power of C 1-Smooth Isogeometric Splines on Volumetric Two-Patch Domains

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