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A semi‐analytical scheme for highly oscillatory integrals over tetrahedra
Author(s) -
HospitalBravo R.,
Sarrate J.,
Díez P.
Publication year - 2017
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.5474
Subject(s) - partition of unity , mathematics , tetrahedron , monomial , exponential function , bottleneck , numerical integration , gauss , complex plane , function (biology) , set (abstract data type) , product (mathematics) , algorithm , mathematical analysis , computer science , finite element method , pure mathematics , geometry , physics , quantum mechanics , evolutionary biology , biology , programming language , thermodynamics , embedded system
Summary This paper details a semi‐analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the partition of unity method enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non‐oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared with standard Gauss–Legendre quadratures. Copyright © 2016 John Wiley & Sons, Ltd.