Efficient Quadrature Rules for Computing the Stiffness Matrices of Mass-Lumped Tetrahedral Elements for Linear Wave Problems

Sjoerd Geevers, W.A. Mulder, Jacobus J.W. van der Vegt

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Abstract

We present new and efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for wave propagation modeling. These quadrature rules allow for a more efficient implementation of the mass-lumped finite element method and can handle materials that are heterogeneous within the element without loss of the convergence rate. The quadrature rules are designed for the specific function spaces of recently developed mass-lumped tetrahedra, which consist of standard polynomial function spaces enriched with higher-degree bubble functions. For the degree-2 mass-lumped tetrahedron, the most efficient quadrature rule seems to be an existing 14-point quadrature rule, but for tetrahedra of degrees 3 and 4, we construct new quadrature rules that require fewer integration points than those currently available in the literature. Several numerical examples confirm that this approach is more efficient than computing the stiffness matrix exactly and that an optimal order of convergence is maintained, even when material properties vary within the element.
Original languageEnglish
Pages (from-to)A1041 - A1065
Number of pages25
JournalSIAM journal on scientific computing
Volume41
Issue number2
DOIs
Publication statusPublished - 9 Apr 2019

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Quadrature Rules
Stiffness matrix
Stiffness Matrix
Computing
Triangular pyramid
Function Space
Wave propagation
Materials properties
Polynomials
Bubble Function
Finite element method
Order of Convergence
Polynomial function
Efficient Implementation
Material Properties
Wave Propagation
Convergence Rate
Finite Element Method
Vary
Numerical Examples

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Efficient Quadrature Rules for Computing the Stiffness Matrices of Mass-Lumped Tetrahedral Elements for Linear Wave Problems. / Geevers, Sjoerd ; Mulder, W.A.; van der Vegt, Jacobus J.W.

In: SIAM journal on scientific computing, Vol. 41, No. 2, 09.04.2019, p. A1041 - A1065.

Research output: Contribution to journalArticleAcademicpeer-review

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