Improvement of PUFEM for the numerical solution of high-frequency elastic wave scattering on unstructured triangular mesh grids

• Published in: International journal for numerical methods in engineering 2010-10, Vol.84 (3), p.330-350
• Main Authors: El Kacimi, A., Laghrouche, O.
• Format: Article
• Language: English
• Subjects: Construction; Elastic waves; exact integration; Exact sciences and technology; Finite element method; Fundamental areas of phenomenology (including applications); Integrals; Mass matrix; Mathematical analysis; Mathematical models; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis. Scientific computation; Physics; plane waves; PUFEM; Sciences and techniques of general use; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; unstructured triangular mesh; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Wavelengths
• ISSN: 0029-5981; 1097-0207; 1097-0207
• DOI: 10.1002/nme.2901

The 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. Copyright © 2010 John Wiley & Sons, Ltd.