Abstract
This paper presents an enriched finite element model for three dimensional elastic wave problems, in the frequency domain, capable of containing many wavelengths per nodal spacing. This is achieved by applying the plane wave basis decomposition to the three-dimensional (3D) elastic wave equation and expressing the displacement field as a sum of both pressure (P) and shear (S) plane waves. The implementation of this model in 3D presents a number of issues in comparison to its 2D counterpart, especially regarding how S-waves are used in the basis at each node and how to choose the balance between P and S-waves in the approximation space. Various proposed techniques that could be used for the selection of wave directions in 3D are also summarised and used. The developed elements allow us to relax the traditional requirement which consists to consider many nodal points per wavelength, used with low order polynomial based finite elements, and therefore solve elastic wave problems without refining the mesh of the computational domain at each frequency. The effectiveness of the proposed technique is determined by comparing solutions for selected problems with available analytical models or to high resolution numerical results using conventional finite elements, by considering the effect of the mesh size and the number of enriching 3D plane waves. Both balanced and unbalanced choices of plane wave directions in space on structured mesh grids are investigated for assessing the accuracy and conditioning of this 3D PUFEM model for elastic waves.
Highlights
Growing research activities have been taking place in various wave numerical modelling fields such as acoustics, surface water waves, radar waves, seismology, geophysics and biomedical ultrasound
Plane wave enriched finite elements are developed within the framework of partition of unity finite element method (PUFEM) for the solution of three dimensional elastic wave problems
These elements are capable of containing many wavelengths per nodal spacing and allow the relaxation of the traditional requirement of several nodal points per wavelength, used in low order polynomial based finite element method (FEM)
Summary
Growing research activities have been taking place in various wave numerical modelling fields such as acoustics, surface water waves, radar waves, seismology, geophysics and biomedical ultrasound. For the case of the Helmholtz equation, various methods have been successful in efficiently solving wave problems with coarse mesh grids in both 2D and 3D with reduced computational effort and better accuracy, in comparison to standard polynomial based FEM Among these methods are the least-squares method [9], the partition of unity finite element method (PUFEM) [10,11,12,13,14,15,16], the ultra weak variational formulation (UWVF) [17,18], Plane wave discontinuous Galerkin (PWDG) [19], the generalized finite element method [20] and the discontinuous enrichment method (DEM) [21].
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