Hierarchical B-spline complexes of discrete differential forms

Evans, John A.; Scott, Michael A.; Shepherd, Kendrick M.; Thomas, Derek C.; Hernández, Rafael Vázquez

doi:10.1093/imanum/dry077

Cited by 14 publications

(10 citation statements)

References 55 publications

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“…To fix notation, we first present univariate B-splines and multivariate B-splines, highlighting some of their properties which come in handy for the present work, before introducing the B-spline complex in both the parametric and physical domain. The notation is based mainly on [24].…”

Section: Parametric Domain Physical Domain and Pullback Operatorsmentioning

confidence: 99%

“…For the definition of the B-spline complex it will be needed to use tensor-products of spaces with mixed degree, combining the standard univariate spaces with the spaces of derivatives, in particular using the Curry-Schoenberg spline basis as done in [44]. We follow the notation in [24] and introduce, for a given multi-index α " pα 1 , α 2 , . .…”

Section: 2mentioning

confidence: 99%

“…Isogeometric discrete differential forms were recently introduced to extend finite element exterior calculus to the emerging framework of Isogeometric Analysis (IGA). A discrete de Rham complex was first constructed and analyzed for tensor-product B-splines [12,11], though generalizations based on analysis-suitable T-splines [13], locally refined B-splines [34] and hierarchical B-splines [24] have also been proposed aiming at local refinement. These spaces have been applied in the Galerkin framework for the discretization of Maxwell's equations [44,19] also in the context of plasma physics [38], and for the development of pointwise divergence free methods for incompressible fluid flow [10,21,22,23,47].…”

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confidence: 99%

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High order geometric methods with splines: an analysis of discrete Hodge--star operators

Kapidani¹,

Vázquez²

2022

Preprint

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A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence tX k h u n k"0 , starting from splines of degree p, and a dual sequence t r X k h u n k"0 , starting from splines of degree p ´1. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge-star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.

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Section: Parametric Domain Physical Domain and Pullback Operatorsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

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High order geometric methods with splines: an analysis of discrete Hodge--star operators

Kapidani¹,

Vázquez²

2022

Preprint

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“…For a comprehensive explanation of these polynomial basis functions, we refer to [6]. In isogeometric analysis, tensor-product B-splines with similar properties have been developed, see, for example [5]. For tetrahedral elements, an analogue development can be found in [15].…”

Section: Primal Polynomial Spacesmentioning

confidence: 99%

Discrete Equivalence of Adjoint Neumann–Dirichlet div-grad and grad-div Equations in Curvilinear 3D Domains

Zhang

Jain

Palha

et al. 2020

Lecture Notes in Computational Science and Engineering

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In this paper, we will show that the equivalence of a div-grad Neumann problem and a grad-div Dirichlet problem can be preserved at the discrete level in 3-dimensional curvilinear domains if algebraic dual polynomial representations are employed. These representations will be introduced. Proof of the equivalence at the discrete level follows from the construction of the algebraic dual representations. A 3-dimensional test problem in curvilinear coordinates will illustrate this approach.

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“…In the case of hierarchical B-splines, a subset of hierarchical B-splines needs to be defined to make sure that a stable divergence-conforming discretization is obtained. In [106], a set of local, easy-to-compute, and sufficient conditions are defined that lead to two-dimensional stable divergenceconforming hierarchical B-spline spaces. As mentioned by the authors in [106], appealing directions of future research are to generalize these local conditions to the three-dimensional setting and construct adaptive h-refinement algorithms which yield hierarchical B-spline spaces satisfying these conditions.…”

Section: Future Workmentioning

confidence: 99%

Non-body-fitted fluid–structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

Casquero

Zhang

Bona-Casas

et al. 2018

Journal of Computational Physics

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Immersed boundary (IB) methods deal with incompressible visco-elastic solids interacting with incompressible viscous fluids. A long-standing issue of IB methods is the challenge of accurately imposing the incompressibility constraint at the discrete level. We present the divergence-conforming immersed boundary (DCIB) method to tackle this issue. The DCIB method leads to completely negligible incompressibility errors at the Eulerian level and various orders of magnitude of increased accuracy at the Lagrangian level compared to other IB methods. Furthermore, second-order convergence of the incompressibility error at the Lagrangian level is obtained as the discretization is refined. In the DCIB method, the Eulerian velocitypressure pair is discretized using divergence-conforming B-splines, leading to inf-sup stable and pointwise divergence-free Eulerian solutions. The Lagrangian displacement is discretized using non-uniform rational B-splines, which enables to robustly handle large mesh distortions. The data transfer needed between the Eulerian and Lagrangian descriptions is performed at the quadrature level using the same spline basis functions that define the computational meshes. This conduces to a fully variational formulation, sharp treatment of the fluid-solid interface, and a 0.5 increase in the convergence rate of the Eulerian velocity and the Lagrangian displacement measured in L 2 norm in comparison with using discretized Dirac delta functions for the data transfer. By combining the generalized-α method and a block-iterative solution strategy, the DCIB method results in a fully-implicit discretization, which enables to take larger time steps. Various two-and three-dimensional problems are solved to show all the aforementioned properties of the DCIB method along with mesh-independence studies, verification of the numerical method by comparison with the literature, and measurement of convergence rates.

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Hierarchical B-spline complexes of discrete differential forms

Cited by 14 publications

References 55 publications

High order geometric methods with splines: an analysis of discrete Hodge--star operators

High order geometric methods with splines: an analysis of discrete Hodge--star operators

Discrete Equivalence of Adjoint Neumann–Dirichlet div-grad and grad-div Equations in Curvilinear 3D Domains

Non-body-fitted fluid–structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

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