Premium
Quadrature rules for triangular and tetrahedral elements with generalized functions
Author(s) -
Holdych David J.,
Noble David R.,
Secor Robert B.
Publication year - 2007
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2123
Subject(s) - mathematics , gaussian quadrature , quadrature (astronomy) , numerical integration , heaviside step function , gauss–kronrod quadrature formula , tanh sinh quadrature , mathematical analysis , gauss–jacobi quadrature , polynomial , boundary value problem , nyström method , electrical engineering , engineering
Quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains. These quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries. Specifically, the integrands considered here consist of a quadratic polynomial multiplied by a Heaviside or Dirac delta function operating on a linear polynomial. This form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function. Exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values. Compared with methods involving explicit integration over subdomains,the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical Jacobian information describing the dependence of the integrals on the nodal level set values. The accuracy of the method is demonstrated for a simple conduction problem with the Neumann and Robin‐type boundary conditions. Copyright © 2007 John Wiley & Sons, Ltd.