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Equivalent polynomials for quadrature in Heaviside function enriched elements
Author(s) -
Ventura G.,
Benvenuti E.
Publication year - 2014
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.4679
Subject(s) - heaviside step function , partition of unity , mathematics , gaussian quadrature , quadrature (astronomy) , classification of discontinuities , gauss–kronrod quadrature formula , tanh sinh quadrature , clenshaw–curtis quadrature , mathematical analysis , finite element method , boundary value problem , structural engineering , physics , nyström method , engineering , optics
Summary One of the advantages of partition‐of‐unity FEMs, like the extended FEM, is the ability of modeling discontinuities independent of the mesh structure. The enrichment of the element functional space with discontinuous or non‐differentiable functions requires, when the element stiffness is computed, partitioning into subdomains for quadrature. However, the arbitrary intersection between the base mesh and the discontinuity plane generates quadrature subdomains of complex shape. This is particularly true in three‐dimensional problems, where quite sophisticate methodologies have been presented in the literature for the element stiffness evaluation. The present work addresses the problem of Heaviside function enrichments and is based on the replacement of the discontinuous enrichment function with the limit of an equivalent polynomial defined on the entire element domain. This allows for the use of standard Gaussian quadrature in the elements crossed by the discontinuity. The work redefines conceptually the first version of the equivalent polynomial methodology introduced in 2006, allowing a much broader applicability. As a consequence, equivalent polynomials can be computed for all continuum element families in one, two, and three dimensions. Copyright © 2014 John Wiley & Sons, Ltd.