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00925njm 22002777a 4500 dc Hospital Bravo, Raúl author Sarrate Ramos, Josep author Díez, Pedro author 2017-08 This is the peer reviewed version of the following article: [Hospital-Bravo, R., Sarrate, J., and Díez, P. (2017) A semi-analytical scheme for highly oscillatory integrals over tetrahedra. Int. J. Numer. Meth. Engng, 111: 703–723. doi: 10.1002/nme.5474], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5474/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. 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 (PUM) 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 to standard Gauss-Legendre quadratures. Peer Reviewed Postprint (author's final draft) Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals Integral equations--Numerical solutions highly oscillatory integral semi-analytical integration rule Lagrange polynomials Helmholtz equation partition of unity method (PUM) plane waves Equacions integrals -- Solucions numèriques A semi-analytical scheme for highly oscillatory integrals over tetrahedra