A hierarchical construction of LR meshes in 2D

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Highlights

  • The paper discusses bivariate LR splines with the non-nested support (N2S) property.

  • Conditions on LR mesh refinement for preserving this property are presented.

  • A hierarchical construction for LR meshes with the N2S property is proposed and analyzed.

  • We prove the completeness of LR splines on the resulting hierarchical meshes.

Abstract

We describe a construction of LR-spaces whose bases are composed of locally linearly independent B-splines which also form a partition of unity. The construction conforms to given refinement requirements associated to subdomains. In contrast to the original LR-paper (Dokken et al., 2013) and similarly to the hierarchical B-spline framework (Forsey and Bartels, 1988) the construction of the mesh is based on a priori choice of a sequence of nested tensor B-spline spaces.

Introduction

In the last decade the use of spline spaces has spread from the field of applied geometry, in particular Computer Aided Design (CAD), to that of numerical analysis of Partial Differential Equations (PDEs). This is largely due to the influence of the seminal paper by Hughes et al. (2005). The use of B-spline generated spaces in Galerkin methods was attempted before by Höllig (2003), but Hughes et al. (2005) recognized it as a possible way to remove the compatibility layer that is in-between the CAD tools and the Finite Element Method (FEM). The compatibility layer contains the mesh generation process and, in some cases, can be computationally more expensive than the simulation itself (Hughes et al., 2005). The method that reduces the compatibility layer proposed by Hughes et al. (2005) is called IsoGeometric Analysis (IGA) and is based on the isoparametric approach: the solution fields of the PDE are in the same B-spline or NURBS space used for the parametrization of the geometry.

IGA sprouted new research in numerical methods due to the availability of basis functions with higher smoothness and with strong algebraic properties that allow for new numerical schemes like compatible discretizations. It had the same effect in the applied geometry field: the numerical simulation of PDEs requires high quality parametrizations of the domain while in CAD it is common to parametrize only the boundary and to allow both for small gaps and singularities.

Both CAD and IGA applications require the use of function spaces that allow for local changes in spatial resolution. This is necessary to obtain a good approximation with fewer degrees of freedom. The standard tensor-product B-spline spaces do not allow for local changes in spatial resolution and thus different generalizations providing adaptive refinement were proposed in the last 25 years. Forsey and Bartels (1988) introduced the hierarchical splines, later studied by Kraft (1997) and more recently by Giannelli et al. (2012) and Mokriš et al. (2014), Sederberg et al., 2003, Sederberg et al., 2004 introduced T-splines of which an Analysis Suitable subset (AST) was described by Beirão da Veiga et al. (2012), Deng et al. (2008) introduced PHT-splines and Dokken et al. (2013) introduced LR-splines whose local linear independence was studied by Bressan (2013). Each of these approaches has their own strengths and weaknesses determined by the focus with which they were developed. In this article we try to combine the LR-splines framework with the hierarchical approach.

Our aim is to obtain a space that has strong properties such as local linear independence and that can be efficiently implemented. Johannessen et al. (2014) applied LR-spline spaces to IGA and explored different refinement techniques. In contrast to their work, we study refinement strategies that are based on theoretical guarantees. In detail we present a method to construct a box mesh M on a domain Ω whose element size is small in a neighborhood of some given regions and for which the associated LR-spline collection LR(M) is a basis composed of locally linearly independent functions. This implies that the basis is also a partition of unity.

In Section 2 we recall LR-spline definitions and results. Compared to the paper of Dokken et al. (2013) we only target the bi-variate case and we can therefore use a simpler notation. In particular we focus on the equivalence for the LR-spline collection to be a partition of unity, to be a set of locally linearly independent B-splines, and the non-nested support property (N2S for short).

In Section 3 we describe a subset of the domain Ω in which it is possible to add vertical segments while preserving the N2S property. We describe another subset that behaves similarly for the addition of horizontal segments.

In Section 4 we define a hierarchical approach to the construction of box meshes. Then we provide sufficient conditions under which the associated LR-spline space has the N2S property.

In Section 5 we study the completeness of the hierarchically constructed LR-space, that is, whether it equals the piecewise polynomial space that is associated to the mesh.

Section 6 describes our construction of LR meshes that guarantees both the N2S property (and thus local linear independence of the basis functions) and completeness. We comment on the locality of the refinement and show some examples in the case of dyadic refinement.

Section 7 compares the proposed space with the truncated hierarchical B-spline space (THB) on the same Bézier mesh.

Section snippets

Notation and LR-spline properties

We use Pd to denote the space of polynomials of degree less than or equal to d. The space of bivariate polynomials of degree dx in the x variable and degree dy in the y variable is denoted using a vector d=(dx,dy) for the degree:Pd=PdxPdy. For our purpose the degree d=(dx,dy) can be considered fixed at the beginning and it will be omitted in the notation.

A knot vector Θ is a monotone non-decreasing sequence of real numbersθ1θn. The number of repetitions of a knot z in a knot vector Θ is

Addition of segments

In this section we describe the set Rx of horizontally refinable rectangles. Our result is that if M is an N2S mesh and γ is a vertical segment “well contained” in Rx then M+γ is an N2S mesh.2 Similarly we define the set Ry of vertically refinable rectangles. Here “well contained” means that only the vertices of γ can be in Rx or equivalently that γ(Rx). We also provide additional conditions that guarantee that the N2S property is preserved for the

Hierarchical box meshes

In this section we introduce a hierarchical construction of box meshes. By hierarchical we mean that it starts from a sequence of box meshes associated with nested tensor-product B-spline spaces (tensor meshes). After the definition we describe sufficient conditions for the N2S property.

Completeness

We are also interested in the completeness of the provided space, i.e. whether LR(H) equals S(H) or not. Describing which refinements preserve completeness was one of the themes of Dokken et al. (2013) and was pursued using homology based techniques. We restrict our attention to hierarchical LR meshes with the N2S property and we prove that if the Ω are “thick enough” in the direction orthogonal to the refinement then the resulting space is LR(H)=S(H). This is made precise in the following

Construction

In this subsection we present a construction for hierarchical box meshes that guarantees both the N2S property and completeness. We assume that the spaces V0,,Vm are fixed and that a minimum refinement level is specified for some regions.

The input of our construction is a sequence of ω1,,ωm of subsets of Ω=[0,1]2 and the output is a mesh such that all basis functions that are active on a point in ω are refinements of basis functions from B. The ω do not need to be nested and can be empty.

Comparison with THB-splines

It is interesting to compare the described approach with the THB-spline approach developed by Giannelli et al. (2012) and Mokriš et al. (2014). We do this in the simplified setting in which Ω is a union of rectangles of level 1. In this setting we can compare the space LR(H) to the THB-spline space TH(H) having the same Bézier elements and defined from the same levels. This means that LR(H) and TH(H) are defined by the same sequence of tensor-product spaces V and domains Ω.

The hierarchical

Conclusions

We restrict our attention to the subset of bi-variate box meshes that have the N2S-property. We describe two subdomains of Ω where respectively vertical and horizontal refinement preserves the N2S property. Using this knowledge we provide an explicit construction that is based on a hierarchy of tensor spaces and domains. The LR-space associated to the constructed mesh H has the N2S property, i.e. it has a basis of locally linearly independent functions. Moreover LR(H) is the whole space S(H) of

Acknowledgements

The financial support of this research by the Austrian Science Fund (FWF) through the NFN “Geometry + Simulation” (S117) is gratefully acknowledged. Special thanks go to Dr. Michael Pauley for commenting on an earlier version of this paper.

References (16)

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This paper has been recommended for acceptance by Thomas Sederberg.

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