Herbert B. Keller

Bio: Herbert B. Keller is an academic researcher. The author has contributed to research in topics: Biharmonic equation & Classification of discontinuities. The author has an hindex of 2, co-authored 2 publications receiving 33 citations.

The Linear Theory of Elasticity

Abstract: Linear elasticity is one of the more successful theories of mathematical physics Its pragmatic success in describing the small deformations of many materials is uncontested The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness

Computational methods of linear algebra

Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Solving two-point seismic-ray tracing problems in a heterogeneous medium. Part 1. A general adaptive finite difference method

Abstract: A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

Fast Seismic Ray Tracing

Abstract: New methods for the fast, accurate and efficient calculation of large classes of seismic rays joining two points x+s and x_R in very general two-dimensional configurations are presented. The medium is piecewise homogeneous with arbitrary interfaces separating regions of different elastic properties (i.e., differing wave speeds c_P and c_S). In general there are 2^(N+1) rays joining x_S to x_R while making contact with N interfaces. Our methods find essentially all such rays for a given N by using continuation or homotopy methods on the wave speeds to solve the ray equations determined by Snell’s law. In addition travel times, ray amplitudes and caustic locations are obtained. When several receiver positions x^(j)_(R) are to be included, as in a gather, our techniques easily yield all the rays for the entire gather by employing continuation in the receiver location. The applications, mainly to geophysical inverse problems, are reported elsewhere.

Interfaces in micromorphic materials: Wave transmission and reflection with numerical simulations:

Abstract: Reflection and transmission of elastic waves at the interface between two distinct micromorphic media are considered in the one-dimensional setting. A dual internal variable approach is used for th...