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

## 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 2018 Kromer, Johannes Richard ; Bothe, Dieter Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator English 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. arXiv preprint arXiv:1805.03136 volume computation;numerical quadrature;Laplace-Beltrami 04 Department of Mathematics04 Department of Mathematics > Mathematical Modelling and Analysis 06 Jun 2018 05:45 https://arxiv.org/abs/1805.03136 EndNoteMultiline CSVMODSDublin CoreEP3 XMLAtomIBW_RDAT2T_XMLBibTeXASCII CitationSimple MetadataJSONHTML CitationRDF+XMLReference Manager TUfind oder in Google Send an inquiry

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