On an integration rule for products of barycentric coordinates over simplexes in Rn

Journal article (2018)

Authors

F.J. Vermolen Numerical Analysis - EEMCS
A. Segal Numerical Analysis - EEMCS
Research Group
Numerical Analysis (EEMCS) (TU Delft)
Copyright: © 2018 F.J. Vermolen, A. Segal
DOI: https://doi.org/10.1016/j.cam.2017.09.013
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Published Date

2018

Language

English

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Numerical Analysis

Abstract

In finite-element computations, one often needs to calculate integrals of products of powers of monomials over simplexes. In this manuscript, we prove a generalisation of the exact integration formula that was reported and proved for two-dimensional simplexes by Holand & Bell in 1969. We extend the proof to n-dimensional simplexes and to simplexes on d-dimensional manifolds in n-dimensional space. The results are used to develop finite-element and boundary-element simulation tools. The proofs of the theorems are based on mathematical induction and coordinate mappings.

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