ISSN:
1365-246X
Source:
Blackwell Publishing Journal Backfiles 1879-2005
Topics:
Geosciences
Notes:
The Direct Solution Method (DSM) (Geller et al. 1990; Geller & Ohminato 1994) is a Galerkin weak-form method for solving the elastic equation of motion. In previous applications of the DSM to both laterally homogeneous and laterally heterogeneous media the vertically dependent part of the trial functions has been either linear splines (e.g. Cummins et al. 1994a, b) or the vertically dependent part of modal eigenfunctions (e.g. Hara, Tsuboi & Geller 1993). In this paper we formulate the DSM using analytic trial functions which are solutions of the homogeneous (source-free) equation of motion in locally homogeneous portions of the medium. We present explicit formulations for SH and P-SV wave propagation in an isotropic, laterally homogeneous, plane-layered model. The trial functions so that it is easy to satisfy continuity of displacement at internal interfaces. For the laterally homogeneous SH problem we first find a set of R+ 1 analytic trial functions that satisfy continuity of displacement at the R - 1 internal interfaces between the R layers. Each of the trial functions is non-zero in at most two layers. We then solve the weak form of the equation of motion, which in effect enforces the upper and lower boundary conditions, and continuity of traction at the internal interfaces. The trial functions are chosen so that the equation of motion becomes a tridiagonal (R+ 1) X (R+ 1) system of linear equations. For the P-SV problem we define a set of 2R+ 2 analytic trial functions that satisfy continuity of displacement, but not continuity of traction, at internal interfaces; the trial functions are chosen so that the equation of motion then becomes a (2R+ 2) X (2R+ 2) system with a bandwidth of 7. In contrast, previous global solution methods (e.g. Chin, Hedstrom & Thigpen 1984; Schmidt & Tango 1986), which solve simultaneously for both internal continuity of displacement and traction as well as the external boundary conditions, solve a 2R X 2R system of linear equations for the SH problem or a 4R X 4R system for the P-SV problem, each having approximately twice the bandwidth of our systems of equations. We show that through an appropriate choice of the form of the homogeneous solutions in each layer our approach can also be readily incorporated into strong-form global solution methods, thereby leading to exactly the same system of equations obtained by our weak form derivation. We also present the DSM equation of motion for a plane-layered medium composed of a combination of fluid and solid layers. The dependent variable in the fluid layers is a scalar quantity proportional to the pressure change, while the dependent variable in the solid is the displacement. Continuity of displacement and traction at fluid-solid boundaries is enforced by augmenting the weak-form operator by appropriate surface integrals.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1111/j.1365-246X.1995.tb05918.x
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