A new 2D numerical model to predict the underwater acoustic propagation is obtained by exploring the potential of the Partition of Unity Method (PUM) enriched with plane waves. The aim of the work is to obtain sound pressure level distributions when multiple operational noise sources are present, in order to assess the acoustic impact over the marine fauna. The model takes advantage of the suitability of the PUM for solving the Helmholtz equation, especially for the practical case of large domains and medium frequencies. The seawater acoustic absorption and the acoustic reflectance of the sea surface and sea bottom are explicitly considered, and Perfectly Matched Layers (PML) are placed at the lateral artificial boundaries to avoid spurious reflexions. The model includes semi-analytical integration rules which are adapted to highly oscillatory integrands with the aim of reducing the computational cost of the integration step. In addition, we develop a novel strategy to mitigate the ill-conditioning of the elemental and global system matrices. Specifically, we compute a low-rank approximation of the local space of solutions, which in turn reduces the number of degrees of freedom, the CPU time and the memory footprint. Numerical examples are presented to illustrate the capabilities of the model and to assess its accuracy.

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 ReviewedPostprint (author's final draft