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SUMMARYIn this paper, we investigate a numerical approach based on the partition of unity finite element method, for the time-harmonic elastic wave equations. The aim of the proposed work is to accurately model two-dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The approximation of the displacement field is performed via the standard finite element shape functions, enriched by superimposing pressure and shear plane wave basis, which incorporate knowledge of the wave propagation. A variational framework able to handle mixed boundary conditions is described. Numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented. The results show the performance of the proposed method in both accuracy and efficiency.

SUMMARYThe purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77:1646-1669, is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two-dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed-form solution for two-dimensional wave-based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries.The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill-in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed.

Citation for published item:il uimiD eF nd vghrouheD yF nd wohmedD wFF nd revelynD tF @PHIWA 9fernstein E f¡ ezier sed (nite elements for e0ient solution of short wve prolemsF9D gomputer methods in pplied mehnis nd engineeringFD QRQ F ppF ITTEIVSF Further information on publisher's website: httpsXGGdoiForgGIHFIHITGjFmFPHIVFHUFHRH Publisher's copyright statement: c 2018 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractIn this work, the Bernstein-Bézier Finite Element Method (BBFEM) is implemented to solve short wave problems governed by the Helmholtz equation on unstructured triangular mesh grids. As for the hierarchical Finite Element (FE) approach, this high order FE method benefits from the use of static condensation which is an efficient tool for reducing the total number of degrees of freedom and bandwidth of high order FE global matrices. The performance of BBFEM with static condensation (BBFEMs) is assessed via three benchmark problems and compared to that of the Partition of Unity Finite Element Method (PUFEM) in terms of accuracy, conditioning and memory requirement. Numerical results dealing with problems of Hankel sources interference and wave scattering by a rigid cylinder on quasi-uniform mesh grids indicate that BBFEMs is able to achieve a better accuracy but PUFEM is slightly better conditioned when the wave is not well resolved. However, with a sufficient wave resolution, BBFEMs is better conditioned than PUFEM. Results from L-shaped domain problem, with non quasi-uniform mesh grids, show that the conditioning of BBFEMs remains reasonable while PUFEM with large numbers of enriching plane waves on mesh grids locally well resolved leads to ill-conditioning.bold Keywords Wave problems; finite elements; Bernstein-Bézier; plane waves; Helmholtz equation; high frequency 1 frequency limitation and high resolution requirement of low order methods. These include polynomial based methods relying on the use of high order shape functions such as Lobatto, Bernstein and Hermite polynomials or spectral elements. Assessment of various polynomial shape functions, including Lagrange Gauss-Lobatto, integrated Legendre and Bernstein polynomials, for interior acoustic problems in [6] have shown the advantage of high order polynomials in reducing the pollution error and the good performance of Bernstein polynomials when combined with Krylov subspace solvers. It was concluded, h...

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