Multi-Wavelength Sized Finite Elements for Three Dimensional Elastic Wave Problems
Lead Research Organisation:
Heriot-Watt University
Department Name: Sch of the Built Environment
Abstract
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.
Planned Impact
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.
Publications
Christodoulou K
(2017)
High-order finite elements for the solution of Helmholtz problems
in Computers & Structures
Diwan G
(2015)
Mixed enrichment for the finite element method in heterogeneous media Mixed enrichment for the finite element method in heterogeneous media
in International Journal for Numerical Methods in Engineering
Drolia M
(2017)
Enriched finite elements for initial-value problem of transverse electromagnetic waves in time domain
in Computers & Structures
Iqbal M
(2016)
An a posteriori error estimate for the generalized finite element method for transient heat diffusion problems
in International Journal for Numerical Methods in Engineering
LAGHROUCHE O
(2012)
EXTENSION OF THE PUFEM TO ELASTIC WAVE PROPAGATION IN LAYERED MEDIA
in Journal of Computational Acoustics
Mahmood M
(2014)
Solution of double nonlinear problems in porous media by a combined finite volume-finite element algorithm
in Applied Numerical Mathematics
Mahmood M
(2017)
Implementation and computational aspects of a 3D elastic wave modelling by PUFEM
in Applied Mathematical Modelling
Mahmood M S
(2012)
The Partition of Unity Method for elastic wave problems in 3D
Mahmood M S
(2013)
Error analysis for numerical solution by PUMFEM of 3D elastic wave problems
Description | Implement and validate a full 3D plane wave basis finite element containing information on both body waves (P- and S-waves). Extend the proposed 3D finite element model to deal with elastic wave problems in layered media with the introduction of Lagrange multipliers to ensure compatibility at the interfaces between layers. Develop fast integration scheme for 3D finite elements with flat surfaces and straight edges (tetrahedral type elements). Develop a parallel version of the code when high order numerical integration schemes are used, in the case of finite elements with curved boundaries. Investigate the conditioning of the proposed model, the possibility of developing an efficient preconditioner and use of iterative solvers. Undertake extensive testing to demonstrate computational efficiency and improved accuracy of the results, compared to results of conventional FEM. |
Exploitation Route | The findings have impact on the academic community working on high-order domain-based methods, in particular finite element methods. |
Sectors | Aerospace, Defence and Marine,Construction,Education,Energy,Environment |
Description | This work allowed for the first time the implementation of the Partition of Unity Finite Element Method for elastic wave problems in three dimensions. It was presented at the Society of Petroleum Engineers Research and Development Competition under the title Multi-Wavelength Sized Finite Elements for Subsurface Imaging. It won, in October 2014, the third prize (10,000 USD) for excellence in addressing one of the grand challenges faced by the industry. Contact: Tom Whipple twhipple@spe.org Tel. +1 972 952 9452. |
First Year Of Impact | 2014 |
Sector | Energy |
Impact Types | Cultural |
Description | Collaboration with Durham University |
Organisation | Durham University |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Extending the use of PUFEM to 3D elastic wave problems Extending the use of enrichment techniques, PUFEM, to heat transfer problems |
Collaborator Contribution | Use of fast integration schemes in the 3D PUFEM for elastci wave problems Solution of heat transfer problems with high temperature gradient by the PUFEM |
Impact | 1) DOI: 10.1016/j.jcp.2013.11.005 2) DOI: 10.1016/j.jcp.2013.05.030 3) DOI: 10.1002/nme.4383 |
Start Year | 2011 |
Description | Collaboration with the University of Edinburgh |
Organisation | University of Edinburgh |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Use of enrichment techniques in solving structural analysis problems under thermo-mechanical loading |
Collaborator Contribution | Structural analysis problems solution under thermo-mechanical loading using enriched solutions. |
Impact | 1) DOI: 10.1016/j.compstruc.2014.05.006 2) doi:10.1088/1742-6596/382/1/012022 3) http://dx.doi.org/10.1016/j.compstruct.2015.05.051 |
Start Year | 2011 |