A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifolds

Based on [1] we introduce and study a mathematical framework for analysissuitable unstructured B-spline spaces. In this setting the parameter domain has a manifold structure which allows for the definition of function spaces that have a tensor-product structure locally, but not globally. This includes configurations such as B-splines over multi-patch domains with extraordinary points, analysissuitable unstructured T-splines, or more general constructions. Within this framework, we generalize the concept of dual-compatible B-splines (developed for structured T-splines in [2]). This allows us to prove the key properties that are needed for isogeometric analysis, such as linear independence and optimal approximation properties for h-refined meshes. arXiv:1507.08477v1 [math.NA] 30 Jul 2015 c ≥ det(∇G T (x)∇G(x)) ≥ c > 0for all x ∈ Q and for all Q ∈ T h0 . Note that, in general, we cannot define Sobolev spaces of any order on Σ, due to the lack of smoothness of the manifold itself. However the L 2 space on Σ can be defined as L 2 (Σ) = {f : f • G ∈ L 2 (Ω)} and the corresponding norm is given via f L 2 (Σ) = f • G (det(∇G T ∇G)) 1/4 L 2 (Ω) .

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“…Remark 1. The last condition in Definition 1 is taken from [19]. It guarantees that the proto-manifold actually represents a manifold, i.e.…”

Section: Definition 1 (Proto-manifold) a Proto-manifold Of Dimensionmentioning

confidence: 99%

“…During h-refinement G is kept unchanged. This gives a sequence of nested isogeometric spaces V h via (19).…”

Section: Approximation Properties Of Spline Manifolds On Uniform Meshesmentioning

confidence: 99%

Unstructured spline spaces for isogeometric analysis based on spline manifolds

Sangalli

Takacs

Vázquez

2016

Computer Aided Geometric Design

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“…Para isso, movemos os vértices da malha fina (o que irá gerar um movimento dos vértices na malha grossa) de forma que a parametrização usada, proposta em (Siqueira (2009)), gere uma distorção menor dos vértices da malha grossa em relação aos vértices da superfície.…”

Section: Modelo De Otimização Não-linearunclassified

“…Esta implementação foi baseada na implementação de (Siqueira (2009)). A seguir, descrevemos alguns detalhes desta estrutura de dados.…”

Section: Halfedgeunclassified