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
Article, Bibliographie
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 Richard ; Bothe, Dieter |
| Type of entry: | Bibliographie |
| Title: | Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator |
| Language: | English |
| Date: | 8 May 2018 |
| Journal or Publication Title: | arXiv preprint arXiv:1805.03136 |
| URL / URN: | https://arxiv.org/abs/1805.03136 |
| 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. |
| Uncontrolled Keywords: | volume computation;numerical quadrature;Laplace-Beltrami |
| Divisions: | 04 Department of Mathematics 04 Department of Mathematics > Analysis 04 Department of Mathematics > Analysis > Mathematical Modeling and Analysis 04 Department of Mathematics > Mathematical Modelling and Analysis |
| Date Deposited: | 06 Jun 2018 05:45 |
| Last Modified: | 07 Feb 2024 11:55 |
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