Modeling Elastodynamic and Scalar Wave Equation Problems
The application of the BEM to elastodynamics started in the late 1960s. Cruse and Rizzo (1968) transformed elastodynamic problems into stationary problems using the Laplace transform. Niwa et al. (1971) and Manolis and Beskos (1981) obtained more accurate results with the Fourier transform. The BEM applied directly in the time domain started in 1978 with Cole et al. (1978) and has been developed quickly and adopted widely because of its directness (Antes, 1985; Banerjee et al., 1986; Ahmad and Banerjee, 1988). Nardini and Brebbia (1983a, b) put forward another boundary element technique, which directly applied the fundamental solution of statics to elastodynamics and selected interpolation functions to obtain the invariable system matrices that correspond to the stiffness and mass matrices that arise when the FEM is applied Wong and Jennings (1975) applied the BEM to numerically calculate the effects of canyon topography on seismic SH-waves.
As a subset of elastodynamic problems, the scalar seismic wave equation is widely applied in geophysics. This chapter introduces the two primary methods of the BEM for solving elastodynamic and scalar wave equation problems in both the time and frequency domains. Some practical examples are also given.
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The Boundary Element Method in Geophysics
The boundary element method (BEM) divides only the boundaries of the region under investigation into elements, so it diminishes the dimensionality of the problem, e.g., the 3D problem becomes a 2D problem, and the 2D problem becomes a 1D problem. This simplifies inputting the model into a computer and greatly reduces the number of algebraic equations. The advantage of this is even more evident for some 3D and infinite regional problems that often are encountered in geophysics. Originally published in China, this well-organized book is likely the most comprehensive work on the subject of solving applied geophysical problems. Basic mathematical principles are introduced in Chapter 1, followed by a general yet thorough discussion of the BEM in Chapter 2. Chapters 3 through 7 introduce the applications of BEM to solve problems of potential-field continuation and transformation, gravity and magnetic anomalies modeling, electric resistivity and induced polarization field modeling, magnetotelluric modeling, and various seismic modeling problems. Finally, in Chapter 8, a brief discussion is provided on how to incorporate the BEM and the finite-element method (FEM) together. In each chapter, detailed practical examples are given, and comparisons to both analytic and other numerical solutions are presented. This is an excellent book for numerically oriented geophysicists and for use as a textbook in numerical-analysis classes.