Kromer, Johannes Richard ; Bothe, Dieter (2018)
Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator.
In: arXiv preprint arXiv:1805.03136
Artikel, Bibliographie
Kurzbeschreibung (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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2018 |
| Autor(en): | Kromer, Johannes Richard ; Bothe, Dieter |
| Art des Eintrags: | Bibliographie |
| Titel: | Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator |
| Sprache: | Englisch |
| Publikationsjahr: | 8 Mai 2018 |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | arXiv preprint arXiv:1805.03136 |
| URL / URN: | https://arxiv.org/abs/1805.03136 |
| Kurzbeschreibung (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. |
| Freie Schlagworte: | volume computation;numerical quadrature;Laplace-Beltrami |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Analysis 04 Fachbereich Mathematik > Analysis > Mathematische Modellierung und Analysis 04 Fachbereich Mathematik > Mathematische Modellierung und Analysis (MMA) |
| Hinterlegungsdatum: | 06 Jun 2018 05:45 |
| Letzte Änderung: | 07 Feb 2024 11:55 |
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