Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method

Boissonnat, Jean‐Daniel; Kachanovich, Siargey; Wintraecken, Mathijs

doi:10.1007/s00454-020-00250-8

Cited by 6 publications

(13 citation statements)

References 49 publications

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“…However, we stress that it does not give lower bounds on the quality of the linear pieces in the PL approximation. This is a clear difference with previous methods [13,14,16,23] whose output is a thick triangulation. Although this is an appealing property, it complicates the analysis further and requires unpractical perturbation schemes.…”

Section: Isomanifolds (Without Boundary)mentioning

confidence: 66%

“…The idea to avoid these low-dimensional simplices originates with Whitney [13], with whom Allgower and George [11,12] were apparently unfamiliar. Very heavy perturbation schemes for the vertices of the ambient triangulation T are needed to ensure that the manifold stays sufficiently far from simplices in the ambient triangulation that have dimension less than the codimension of the manifold [13,14]. Various techniques have been developed to compute such perturbations with guarantees.…”

Section: Guarantees For Isosurfacingmentioning

confidence: 99%

“…The approximation of a manifold that is the zero set of a function is an example of the more general question of how to triangulate a manifold. It is known that C 1 manifolds are triangulable, see, for example, [13], and algorithms have been proposed recently to triangulate smooth manifolds [14,17,22,23]. However, all known methods use intricate perturbation schemes to guarantee the correctness of the triangulation algorithms when the intrinsic dimension of the manifold exceeds 2.…”

Section: Triangulating General Manifolds (Without Boundary)mentioning

confidence: 99%

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The Topological Correctness of PL Approximations of Isomanifolds

Boissonnat

Wintraecken

2021

Found Comput Math

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Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$ f : R d → R d - n . A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ T of the ambient space $${\mathbb {R}}^d$$ R d . In this paper, we give conditions under which the PL approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine and thick triangulation $$\mathcal {T}$$ T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary.

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Section: Isomanifolds (Without Boundary)mentioning

confidence: 66%

Section: Guarantees For Isosurfacingmentioning

confidence: 99%

Section: Triangulating General Manifolds (Without Boundary)mentioning

confidence: 99%

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The Topological Correctness of PL Approximations of Isomanifolds

Boissonnat

Wintraecken

2021

Found Comput Math

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“…An alternative approach to generate high-quality pentatope meshes could be based on Coxeter triangulations, 13 with the currently addressed issue of generating boundary-conforming meshes. 14 In most cases, SST meshes are employed to facilitate adaptive mesh refinement in space and time. Suitable pentatope mesh refinement procedures have been explored by Neumüller and Steinbach 15 and Grande.…”

Section: F I G U R Ementioning

confidence: 99%

“…The above‐mentioned strategies have in common that the four‐dimensional mesh is based on an extruded tetrahedral mesh. An alternative approach to generate high‐quality pentatope meshes could be based on Coxeter triangulations, 13 with the currently addressed issue of generating boundary‐conforming meshes 14 …”

Section: Introduction and Problem Definitionmentioning

confidence: 99%

Four‐dimensional elastically deformed simplex space‐time meshes for domains with time‐variant topology

Danwitz

Antony

Key

et al. 2021

Numerical Methods in Fluids

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Considering the flow through biological or engineered valves as an example, there is a variety of applications in which the topology of a fluid domain changes over time. This topology change is characteristic for the physical behavior, but poses a particular challenge in computer simulations. A way to overcome this challenge is to consider the application‐specific space‐time geometry as a contiguous computational domain. In this work, we obtain a boundary‐conforming discretization of the space‐time domain with four‐dimensional simplex elements (pentatopes). To facilitate the construction of pentatope meshes for complex geometries, the widely used elastic mesh update method is extended to four‐dimensional meshes. In the resulting workflow, the topology change is elegantly included in the pentatope mesh and does not require any additional treatment during the simulation. The potential of simplex space‐time meshes for domains with time‐variant topology is demonstrated in a valve simulation, and a flow simulation inspired by a clamped artery.

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Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

Danwitz

Voulis

Hosters

et al. 2023

Numerical Meth Engineering

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We construct four variants of space‐time finite element discretizations based on linear tensor‐product and simplex‐type finite elements. The resulting discretizations are continuous in space, and continuous or discontinuous in time. In a first test run, all four methods are applied to a linear scalar advection‐diffusion model problem. Then, the convergence properties of the time‐discontinuous space‐time finite element discretizations are studied in numerical experiments. Advection velocity and diffusion coefficient are varied, such that the parabolic case of pure diffusion (heat equation), as well as, the hyperbolic case of pure advection (transport equation) are included in the study. For each model parameter set, the L2$$ {L}_2 $$ error at the final time is computed for spatial and temporal element lengths ranging over several orders of magnitude to allow for an individual evaluation of the methods' spatial, temporal, and space‐time accuracy. In the parabolic case, particular attention is paid to the influence of time‐dependent boundary conditions. Key findings include a spatial accuracy of second order and a temporal accuracy between second and third order. The temporal accuracy tends toward third order depending on how advection‐dominated the test case is, on the choice of the specific discretization method, and on the time‐(in)dependence and treatment of the boundary conditions. Additionally, the potential of time‐continuous simplex space‐time finite elements for heat flux computations is demonstrated with a piston ring pack test case and a subtractive manufacturing test case.

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Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method

Abstract: We quantise Whitney’s construction to prove the existence of a triangulation for any $$C^2$$ C 2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.

Cited by 6 publications

References 49 publications

The Topological Correctness of PL Approximations of Isomanifolds

The Topological Correctness of PL Approximations of Isomanifolds

Four‐dimensional elastically deformed simplex space‐time meshes for domains with time‐variant topology

Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

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