The coefficients in a mathematical model of a physical problem typically correspond to the material parameters of the problem. In heterogeneous media the material parameters may vary with position, dependent variable value and/or time. The governing equation of a physical problem in heterogeneous media is therefore likely to involve variable coefficients. For this reason the solution of variable coefficient partial differential equations (PDEs) is an important engineering problem. In this thesis ways of solving linear variable coefficient PDEs using the boundary element method have been investigated. The application of the boundary element method to these equations is hampered by the difficulty of finding a fundamental solution. In the literature several methods have been proposed to overcome this problem. A survey of these methods has been undertaken in this study from which it is concluded that the most promising approach is the dual reciprocity boundary element method (DR-BEM). The DR-BEM is tested in this thesis for a range of elliptic variable coefficient PDEs. The results of these test problems indicate that the DR-BEM is a promising method for solving elliptic variable coefficient PDEs. However, in some cases, such as problems in highly heterogeneous media, it is found that a large number of internal solution nodes are necessary to ensure accurate results. This can make the DR-BEM computationally expensive. Some new approaches for improving the efficiency of the DR-BEM are proposed. For problems in highly heterogeneous media a subregion approach is recommended. The use of the DR-BEM for linear parabolic variable coefficient PDEs is also investigated. It is found that by combining the DR-BEM with the coupled finite difference – boundary element method a wide range of parabolic problems can be solved without requiring domain integration. This time-stepping approach can become expensive for variable coefficient PDEs (particularly for large-time solutions) as it requires the solution of a large number of associated elliptic problems with large numbers of internal nodes. Also, it is found that for some problems in highly heterogeneous media the error at each time-step can accumulate leading to poor large-time solutions. To avoid these limitations semi-analytic approaches for solving parabolic equations are investigated. A new semi-analytic method - the separation of variables dual reciprocity method (SOV-DRM) - is proposed which constructs the solution as an eigenfunction expansion. The eigenvalues and eigenvectors are determined using the DR-BEM. This method allows parabolic problems to be solved without requiring time-stepping or domain integration. This method is found to produce accurate results for a range of problems including some problems involving heterogeneous media. Two other semi-analytic methods are also investigated. These methods are implemented and compared with the SOV-DRM. It is concluded that each method has specific strengths and weaknesses and that the choice of method is largely problem dependent.
| Identifer | oai:union.ndltd.org:ADTP/275359 |
| Date | January 1996 |
| Creators | Brunton, Ivan |
| Publisher | ResearchSpace@Auckland |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated., http://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm, Copyright: The author |
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