A Numerical Boundary Integral Equation Method for Transient Motions

Citation

Cole, David Martin (1980) A Numerical Boundary Integral Equation Method for Transient Motions. Dissertation (Ph.D.), California Institute of Technology. https://resolver.caltech.edu/CaltechTHESIS:02142024-183355928

Abstract

This thesis presents the results of a study of a numerical technique for the solution of initial-boundary value problems of linear elastodynamics. The numerical method is based on a boundary integral equation (BIE) formulation of the mechanics of bodies of arbitrary shape. These integral equations are discretized and a time stepping technique is used to so1ve the resulting system of linear algebraic equations.

The theoretical basis of the continuous problem and the general interpolation and discretization scheme are described in Chapter 1. The problem is then specialized to the two-dimensional case of antiplane strain and most subsequent calculations and discussions take place in this context. The performance of the numerical method depends entirely on the interpolation scheme used, and on the manner in which boundary shapes are approximated.

The consequences of particular interpolation schemes for boundary value problems on a half-plane are discussed in Chapter 2. The results of several numerical calculations are compared with exact, or much more accurate solutions. This chapter also presents a compari­son of the performance of the numerical BIE method with the performance of other specialized numerical procedures which have been used previously for problems of this nature. The BIE method yields results which are as accurate, or more accurate than the other methods for given discretization parameters.

The method is applied to basic boundary value problems for curved symmetric and nonsymmetric boundaries in Chapter 3. The solutions obtained there are again compared to more accurate or exact solutions produced by independent methods. The general dependence of errors on discretization parameters is discussed.

Chapter 4 gives the solution of a problem in which a Love wave propagates through a limited region of laterally varying structure. The time stepping nature of the BIE method makes feasible certain rearrangements of the numerical equations which yield a representation of the mechanical system in which the incident, unperturbed Love wave arises as an inhomogeneous term. Solution of this localized numerical equation then yields an intermediate variable, the change in the traction boundary value of the layered space surface, which is used to evaluate the scattered displacement wave.

The performance characteristics and unusual properties of the time stepping BIE method are summarized in the General Summary. The appendices deal with several subjects. Appendix A gives an evaluation of singular integrals arising in the general continuous integral equation formulation. Appendix B gives a body force equivalent of nonequilibrium static initial values. Appendix C discusses the con­ vergence and stability of solutions obtained using a particular inter­-polation scheme. Appendix D contains FORTRAN subroutines used in evaluating discrete kernels for the antiplane strain case. Appendix E gives the solution to a diffraction problem which is used to evaluate errors in a BIE solution of the same problem which is given in Chapter 3.

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