Multi-Wavelength Sized Finite Elements for Three Dimensional Elastic Wave Problems

Elastic wave propagation modelling arises in many engineering applications, including traffic vibrations from roads and railways, seismic induced vibrations and foundation construction, etc. The numerical modelling of these problems, in frequency domain by the conventional Finite Element Method (FEM), requires finite element grids sufficiently fine in comparison with the wavelengths, to get accurate results. When typically, the piecewise linear finite element is implemented, around ten nodal points per lower wavelength are needed, to ensure adequate resolution of the wave pattern. However, in the case of high frequency (small wavelength) and/or large domain of interest, the finite element mesh requires a large number of elements, and consequently the procedure becomes computationally expensive and impractical. The aim of the proposed work is to accurately model three-dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The resulting improvement in computational efficiency will enable problems of practical interest to be simulated using computing facilities available in most engineering design offices.

Many existing finite element codes offer the possibility to model elastic wave problems through polynomial elements which require the use of many nodal points per wavelength and therefore lead to huge computing time and memory requirements. This renders such problems computationally expensive, and sometimes unfeasibly large, for solution using today's computer technology. Let us present a simple example; in geophysical exploration (e.g. for hydrocarbon sites) it is common to consider waves of wavelength 20m propagating through a 3D space of physical dimension 20 x 10 x 7 km^3. If a traditional finite element approach were used it would require over 10^11 unknowns. The proposed research promises to reduce the required problem size for any given short wave problem. When this research field reaches maturity, it will benefit the oil and civil engineering industries through practical computer analysis to support the design in cases like the above example. Applications include not only geophysical prospecting and location of hydrocarbon reserves, but also problems involving vibrations caused by roads and railways, elastic waves caused by piling of foundations, earthquake wave propagation and aseismic design. The proposed research will allow considering large domains of interest covering many wavelengths while using a lot fewer finite elements than current available models. Companies such as Jacobs and Halcrow will be potential beneficiaries and we have received letters of support from both. Also the PI has links to companies such as Network Rail, Balfour Beatty, Rail Research UK, Holequest, AECOM and WTB Geotechnics, through representation on a current advisory board. There are important medical applications for the elastic wave simulations when coupled with fluid domains, including ultrasound imaging and HIFU (high intensity focused ultrasound) therapy. An example would consist of a model dealing with the propagation of short waves through the human skull, as part of a therapeutic system. In the long term, the resulting improvement in computational efficiency will enable a number of problems of practical interest to be simulated using computing facilities available in most engineering design offices. It should be noted that the implementation of the proposed plane wave basis finite element model for industrial scale elastic wave problems is not proposed at this stage. But it is anticipated that this research project could be followed by another of a more commercial nature oriented towards practical implementation on a large scale for industrially relevant problems.