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Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator

Kromer, Johannes and Bothe, Dieter (2018):
Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator.
In: arXiv preprint arXiv:1805.03136, [Online-Edition: https://arxiv.org/abs/1805.03136],
[Article]

Abstract

This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and thrid- to fourth-order convergence in space.

Item Type: Article
Erschienen: 2018
Creators: Kromer, Johannes and Bothe, Dieter
Title: Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator
Language: English
Abstract:

This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and thrid- to fourth-order convergence in space.

Journal or Publication Title: arXiv preprint arXiv:1805.03136
Uncontrolled Keywords: volume computation;numerical quadrature;Laplace-Beltrami
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Mathematical Modelling and Analysis
Date Deposited: 06 Jun 2018 05:45
Official URL: https://arxiv.org/abs/1805.03136
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