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Stability properties of the direct boundary element method applied to the elastodynamic equations Pinder, Shelly Deleen

Abstract

The elastodynamic equations are frequently encountered in the geosciences for assessing the stability of mining excavations or performing seismological analyses of rock bursts and earthquakes. However, the geometric complexity of the problems encountered usually prevents exact solutions from being determined. Thus it is necessary to resort to numerical approximations in order to obtain solutions to elastodynamic problems with general geometries. In previous work, using boundary element methods for approximating solutions of the elastodynamic equations has been limited due to numerical instabilities that appear sporadically when applying this technique. Until recently, finite difference and finite element methods have been used almost exclusively for these approximations. Besides being computationally more expensive since the entire domain needs to be discretized, finite difference and finite element methods also have problems with numerical dispersion. Examining boundary element methods could lead to the development of techniques that are less intense computationally and avoid any numerical dispersion problems. For boundary element methods to become a standard tool for approximating these solutions, there must be some type of criterion established for choosing meshing parameters to ensure stability. A complete analysis of a one-dimensional model problem is performed via the z - transform. For this model problem, the validity of the stability analysis is confirmed from a comparison of the analytic results with numerical experiments. This allows some guidelines to be made to ensure that a particular numerical approximation is stable when applied to the model problem.

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