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An extrapolative approach to integration over hypersurfaces in the level set framework
Univ Dayton, Dept Math, 300 Coll Pk, Dayton, OH 45469 USA..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Univ Texas Austin, Dept Math, 2515 Speedway, Austin, TX 78712 USA.;Univ Texas Austin, Inst Computat Engn & Sci, 2515 Speedway, Austin, TX 78712 USA..ORCID iD: 0000-0001-8441-3678
2018 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 313, p. 2365-2392Article in journal (Refereed) Published
Abstract [en]

We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g., curves in two dimensions that contain a finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.

Place, publisher, year, edition, pages
AMER MATHEMATICAL SOC , 2018. Vol. 87, no 313, p. 2365-2392
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-230388DOI: 10.1090/mcom/3282ISI: 000432383400013Scopus ID: 2-s2.0-85041675917OAI: oai:DiVA.org:kth-230388DiVA, id: diva2:1221629
Note

QC 20180620

Available from: 2018-06-20 Created: 2018-06-20 Last updated: 2018-06-20Bibliographically approved

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Tsai, Yen-Hsi Richard

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