Manifold-based surfaces with boundaries

Tosun, Elif Karakurt; Zorin, Denis

doi:10.1016/j.cagd.2010.07.005

Cited by 16 publications

(17 citation statements)

References 23 publications

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“…For manifold construction, around each vertex patches consisting of either one or two-rings of elements are considered. As blending functions we use either Surfaces with boundaries require modified charts for manifold constructions, such as those introduced in [36]. The specialised treatment of elements close to the boundary can be avoided by introducing ghost elements just outside the domain.…”

Section: Examplesmentioning

confidence: 99%

Isogeometric analysis using manifold-based smooth basis functions

Majeed

Cirak

2017

Computer Methods in Applied Mechanics and Engineering

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We present an isogeometric analysis technique that builds on manifold-based smooth basis functions for geometric modelling and analysis. Manifold-based surface construction techniques are well known in geometric modelling and a number of variants exist. Common to all is the concept of constructing a smooth surface by blending together overlapping patches (or, charts), as in differential geometry description of manifolds. Each patch on the surface has a corresponding planar patch with a smooth one-to-one mapping onto the surface. In our implementation, manifold techniques are combined with conformal parameterisations and the partition-of-unity method for deriving smooth basis functions on unstructured quadrilateral meshes. Each vertex and its adjacent elements on the surface control mesh have a corresponding planar patch of elements. The star-shaped planar patch with congruent wedge-shaped elements is smoothly parameterised with copies of a conformally mapped unit square. The conformal maps can be easily inverted in order to compute the transition functions between the different planar patches that have an overlap on the surface. On the collection of star-shaped planar patches the partition of unity method is used for approximation.The smooth partition of unity, or blending functions, are assembled from tensor-product b-spline segments defined on a unit square. On each patch a polynomial with a prescribed degree is used as a local approximant. In order to obtain a mesh-based approximation scheme the coefficients of the local approximants are expressed in dependence of vertex coefficients. This yields a basis function for each vertex of the mesh which is smooth and non-zero over a vertex and its adjacent elements. Our numerical simulations indicate the optimal convergence of the resulting approximation scheme for Poisson problems and near optimal convergence for thin-plate and thin-shell problems discretised with structured and unstructured quadrilateral meshes.

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Section: Examplesmentioning

confidence: 99%

Isogeometric analysis using manifold-based smooth basis functions

Majeed

Cirak

2017

Computer Methods in Applied Mechanics and Engineering

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“…Proof. Notice that H(γ) is isomorphic to the projection of the solution set of system (26), (27) The solutions in q = (q 1 , . .…”

Section: G 1 Splines Around a Vertexmentioning

confidence: 99%

“…Since these initial developments, several works focused on the construction of such G 1 surfaces [18], [16], [24], [23], [5], [32], [11], [13], [10], [27], [26], [3], . .…”

Section: Introductionmentioning

confidence: 99%

Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology

Mourrain

Vidūnas

Villamizar

2016

Computer Aided Geometric Design

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We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework does not require manifold constructions and is general enough to allow non-orientable surfaces. We describe compatibility conditions on the transition maps so that the space of differentiable functions is ample and show that these conditions are necessary and sufficient to construct ample spline spaces. We determine the dimension of the space of G 1 spline functions which are of degree k on triangular pieces and of bi-degree (k, k) on rectangular pieces, for k big enough. A separability property on the edges is involved to obtain the dimension formula. An explicit construction of basis functions attached respectively to vertices, edges and faces is proposed; examples of bases of G 1 splines of small degree for topological surfaces with boundary and without boundary are detailed.

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“…Many works over the last decades have been investigating the problem of constructing G 1 surfaces from (quad) meshes. 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).…”

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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Manifold-based surfaces with boundaries

Cited by 16 publications

References 23 publications

Isogeometric analysis using manifold-based smooth basis functions

Isogeometric analysis using manifold-based smooth basis functions

Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology

Geometrically smooth spline bases for data fitting and simulation

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