Planar Parametrization in Isogeometric Analysis

Gravesen, Jens; Evgrafov, Anton; Nguyen, Dang-Manh; Nørtoft, Peter

doi:10.1007/978-3-642-54382-1_11

Cited by 39 publications

(68 citation statements)

References 26 publications

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“…It is widely used in numerical grid generation [20], and has also been applied in the context of domain parameterization in IgA based on tensor-product B-splines [13,14,24]. The Winslow functional has particularly nice mathematical properties, which can be more easily understood after changing the integration in (4.4) from the reference domain Ω 0 to the physical domain Ω,…”

Section: The Winslow Functionalmentioning

confidence: 99%

“…In [41,42] a good B-spline parameterization of the physical domain is generated by an optimization procedure, and an easy-to-check algorithm is proposed to ensure that the constructed parameterization has no self-intersections. Several domain parameterization methods based on solving a constrained optimization problem are reviewed in [14]. Geometry-oriented methods and more expensive analysis-oriented methods are considered and their performance is illustrated numerically.…”

Section: Introductionmentioning

confidence: 99%

“…Geometry-oriented methods and more expensive analysis-oriented methods are considered and their performance is illustrated numerically. Among the various geometric functionals, the Winslow functional seems to be a natural choice and provides high-quality B-spline parameterizations [13,14,24].…”

Section: Introductionmentioning

confidence: 99%

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Optimizing domain parameterization in isogeometric analysis based on Powell–Sabin splines

Speleers

Manni

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: H. Speleers, C. Manni, Optimizing domain parameterization in isogeometric analysis based on Powell-Sabin splines, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractWe address the problem of constructing a high-quality parameterization of a given planar physical domain, defined by means of a finite set of boundary curves. We look for a geometry map represented in terms of Powell-Sabin B-splines. Powell-Sabin splines are C 1 quadratic splines defined on a triangulation, and thus the parameter domain can be any polygon.The geometry map is generated by the following three-step procedure. First, the shape of the parameter domain and a corresponding triangulation are determined, in such a way that its number of corners matches the number of corners of the physical domain. Second, the boundary control points related to the Powell-Sabin B-spline representation are chosen so that they parameterize the boundary curve of the physical domain. Third, the remaining inner control points are obtained by solving a nimble optimization problem based on the Winslow functional.The proposed domain parameterization procedure is illustrated numerically in the context of isogeometric Galerkin discretizations based on Powell-Sabin splines. It turns out that the flexibility rising from the generality of the parameter domain has a beneficial effect on the quality of the parameterization and also on the accuracy of the computed approximate solution.

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Section: The Winslow Functionalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimizing domain parameterization in isogeometric analysis based on Powell–Sabin splines

Speleers

Manni

2015

Journal of Computational and Applied Mathematics

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“…This can be circumvented by adding det J > 0 as a constraint in the optimisation. Here it is crucial to express det J in B-spline form, see [7,Theorem 1].…”

Section: Obtaining a Near Isometry To Euclidean Spacementioning

confidence: 99%

The metric of colour space

Gravesen

2015

Graphical Models

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The space of colours is a fascinating space. It is a real vector space, but no matter what inner product you put on the space the resulting Euclidean distance does not corresponds to human perception of difference between colours.In 1942 MacAdam performed the first experiments on colour matching and found the MacAdam ellipses which are often interpreted as defining the metric tensor at their centres. An important question is whether it is possible to define colour coordinates such that the Euclidean distance in these coordinates correspond to human perception.Using cubic splines to represent the colour coordinates and an optimisation approach we find new colour coordinates that make the MacAdam ellipses closer to uniform circles than the existing standards.

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“…Several existing methods allow such a parameterization of single patches, e.g. [9,10,11,13,28]. If only the boundary of a multi-patch domain is given, the recent method [5] is able to generate a suitable multi-patch parameterization.…”

Section: Introductionmentioning

confidence: 99%

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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Planar Parametrization in Isogeometric Analysis

Cited by 39 publications

References 26 publications

Optimizing domain parameterization in isogeometric analysis based on Powell–Sabin splines

Optimizing domain parameterization in isogeometric analysis based on Powell–Sabin splines

The metric of colour space

Construction of analysis-suitableG1planar multi-patch parameterizations

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