Isoparametric Elements and Gaussian Quadrature
Published:January 01, 2001
The BEM involves a large amount of integration because it divides boundaries into many elements, each of which is integrated over. During the computation of the integrals, it is very troublesome if the variables of integration are denoted by global coordinates (i.e., all the elements are based on coordinates of identical origin). Rather, it benefits the computations to use local coordinates within the elements to denote these variables.
The isoparametric element (Rao, 1982) is a powerful tool that uses local coordinates to perform the integration over the element. The integrals expressed by isoparametric elements usually cannot be calculated through analytical methods because they are rather complicated. Thus, numerical methods must be used. This chapter introduces the isoparametric element, followed by the Gaussian quadrature method of numerical integration (Rao, 1982).
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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.