Data Structures for Geometric and Topological Aspects of Finite Element Algorithms

Gross, P. W.; Kotiuga, P. R.

doi:10.2528/pier00080106

Cited by 14 publications

(12 citation statements)

References 18 publications

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“…Their approach differs from the one presented in this work in that it is restricted to simplicial complexes. Furthermore, there is the work of Gross et al [18], Dobkin et al [11], a survey on data structures for level-of-detail models by De Floriani et al [10], and the 3-dimensional triangulations which are part of the CGAL library described by Teillaud in [31] that are also restricted to simplicial complexes.…”

Section: Related Workmentioning

confidence: 99%

OpenVolumeMesh – A Versatile Index-Based Data Structure for 3D Polytopal Complexes

Kremer

Bommes

Kobbelt

2013

Proceedings of the 21st International Meshing Roundtable

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Summary. We present a data structure which is able to represent heterogeneous 3-dimensional polytopal cell complexes and is general enough to also represent nonmanifolds without incurring undue overhead. Extending the idea of half-edge based data structures for two-manifold surface meshes, all faces, i.e. the two-dimensional entities of a mesh, are represented by a pair of oriented half-faces. The concept of using directed half-entities enables inducing an orientation to the meshes in an intuitive and easy to use manner.We pursue the idea of encoding connectivity by storing first-order top-down incidence relations per entity, i.e. for each entity of dimension d, a list of links to the respective incident entities of dimension d−1 is stored. For instance, each half-face as well as its orientation is uniquely determined by a tuple of links to its incident halfedges or each 3D cell by the set of incident half-faces. This representation allows for handling non-manifolds as well as mixed-dimensional mesh configurations. No entity is duplicated according to its valence, instead, it is shared by all incident entities in order to reduce memory consumption. Furthermore, an array-based storage layout is used in combination with direct index-based access. This guarantees constant access time to the entities of a mesh.Although bottom-up incidence relations are implied by the top-down incidences, our data structure provides the option to explicitly generate and cache them in a transparent manner. This allows for accelerated navigation in the local neighborhood of an entity.We provide an open-source and platform-independent implementation of the proposed data structure written in C++ using dynamic typing paradigms. The library is equipped with a set of STL compliant iterators, a generic property system to dynamically attach properties to all entities at run-time, and a serializer/deserializer supporting a simple file format. Due to its similarity to the OpenMesh data structure, it is easy to use, in particular for those familiar with OpenMesh. Since the presented data structure is compact, intuitive, and efficient, it is suitable for a variety of applications, such as meshing, visualization, and numerical analysis. OpenVolumeMesh is open-source software licensed under the terms of the LGPL.

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Section: Related Workmentioning

confidence: 99%

OpenVolumeMesh – A Versatile Index-Based Data Structure for 3D Polytopal Complexes

Kremer

Bommes

Kobbelt

2013

Proceedings of the 21st International Meshing Roundtable

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“…First, they naturally split into a metric-dependent part (3) and a purely topological (i.e., metric-free or manifestly invariant under diffeomorphisms) part (4). This is because d is a metric-free operator [12] and, as such, only (3) is modified under a change on the metric (as encoded by the * operator), whereas (4) remains invariant. The second feature is that (3) and (4) can be exactly transcribed onto a irregular lattice because both d and * operators admit well-defined, direct counterpart representations on such lattices.…”

Section: Fundamentalsmentioning

confidence: 99%

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Teixeira¹

2014

PIER

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Abstract-We discuss the ab initio rendering four-dimensional (4-d) spacetime of Maxwell's equations on random (irregular) lattices. This rendering is based on casting Maxwell's equations in the framework of the exterior calculus of differential forms, and a translation thereof to a simplicial complex whereby fields and causative sources are represented as differential p-forms and paired with the oriented pdimensional geometrical objects that comprise the set of spacetime lattice cells (simplices). We pay particular attention to the case of simplicial spacetime lattices because these can serve as building blocks of lattices made of more generic cells (polygons). The generalized Stokes' theorem is used to construct discrete calculus operations on the lattice based upon combinatorial relations depending solely on the connectivity and relative orientation among simplices. This rendering provides a natural factorization of (lattice) 4-d spacetime Maxwell's equations into a metric-free part and a metric-dependent part. The latter is encoded by discrete Hodge star operators which are built using Whitney forms, i.e., canonical interpolants for discrete differential forms. The derivation of Whitney forms is illustrated here using a geometrical construction based on the concept of barycentric coordinates to represent a point on a simplex, and the generalization thereof to represent higher-dimensional objects (lines, areas, volumes, and hypervolumes) in 4-d. We stress the role of the primal lattice, the barycentric dual lattice, and the barycentric decomposition lattice in achieving a complete description of the lattice theory. Lattice Maxwell's equations based on the exterior calculus of differential forms and on the use of Whitney forms as field interpolants inherits the symplectic structure and discrete analogues of conservation laws present in the continuum theory, such as energy and charge conservation. It also provides precise localization rules for the degrees of freedom associated with the different fields and sources on the lattice, and design principles for constructing consistent numerical solution methods that are free from spurious modes, spectral pollution, and (unconditional) numerical instabilities. We also briefly consider the relationship between lattice 4-d Maxwell's equations and some incarnations of discretization schemes for Maxwell's equations in (3 + 1)-d, such as finite-differences and finite-elements.

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“…Combining (1) and (2), we arrive at (17) where matrices , , , and are all sparse. By inverting and inverse Fourier transforming the above equation, one has (18) A leap-frog time discretization of the above leads to an explicit time-domain update that unfortunately is full because is full.…”

Section: F Application To Time-domain Femmentioning

confidence: 99%

Differential Forms, Galerkin Duality, and Sparse Inverse Approximations in Finite Element Solutions of Maxwell Equations

Teixeira

2007

IEEE Trans. Antennas Propagat.

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Abstract-We identify primal and dual formulations in the finite element method (FEM) solution of the vector wave equation using a geometric discretization based on differential forms. These two formulations entail a mathematical duality denoted as Galerkin duality. Galerkin-dual FEM formulations yield identical nonzero (dynamical) eigenvalues (up to machine precision), but have static (zero eigenvalue) solution spaces of different dimensions. Algebraic relationships among the degrees of freedom of primal and dual formulations are explained using a deep-rooted connection between the Hodge-Helmholtz decomposition of differential forms and Descartes-Euler polyhedral formula, and verified numerically. In order to tackle the fullness of dual formulation, algebraic and topological thresholdings are proposed to approximate inverse mass matrices by sparse matrices.Index Terms-Differential forms, finite element methods (FEMs), Maxwell equations, sparse matrices.

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Data Structures for Geometric and Topological Aspects of Finite Element Algorithms

Cited by 14 publications

References 18 publications

OpenVolumeMesh – A Versatile Index-Based Data Structure for 3D Polytopal Complexes

OpenVolumeMesh – A Versatile Index-Based Data Structure for 3D Polytopal Complexes

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Differential Forms, Galerkin Duality, and Sparse Inverse Approximations in Finite Element Solutions of Maxwell Equations

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