In this thesis, it is explicitly shown that the exact Green's function and the unknown boundary variables on the boundary, in a given boundary value problem (BVP), satisfy the same boundary integral equation (BIE) but with a different known vector. Indeed, it is made explicit here that in using the BIE method to solve a given boundary value problem, one has in fact constructed the Green's function for the domain. This observation provides a way to construct a library of numerical approximations to exact Green's functions (discretized Green's functions) for problems for which analytical Green's functions are not available. This library thus can be used repeatedly by non-experts;The main ingredient of a discretized Green's function, for a given BVP involving one or two separate surfaces, is identified and implemented for specific applications. Some efficient strategies are proposed;In this thesis, two specific classes of problems are considered as applications of the BEM and the discretized Green's function library. One is the application of the BEM to the analysis of 2D micromechanical behavior of fiber-reinforced composites. A BEM model is developed based on models for both perfectly-bonded and imperfectly-bonded materials in a unit cell. The idea of a library of Green's functions and the entries for the library for fiber-reinforced composites are discussed;The other class of problems considered here involves elastodynamic frequency-dependent wave motion in a halfspace. Radiation from a void inside the halfspace and the scattering from a halfspace surface-breaking crack are considered using a conventional BIE (CBIE) and a hypersingular BIE (HBIE) formulations, respectively. Some new insight into this class of problems was gained during the research. As a result, strategies are suggested to exploit the best features of the fullspace Stokes and halfspace Lamb solutions. A 'parallel' scheme is also designed and implemented when Lamb's solution is used in the BIE formulation. The partitioning method, which is closely related to the process of creating and using a region-dependent Green's function, is also implemented and the efficiency of the Green's function library idea is demonstrated.

The direct multi-region boundary element method (BEM) is developed to simulate the interaction of ultrasonic waves with submerged finite elastic bodies. This technique is further specialized to model fluid-solid half-space problems with fluid-filled indentations and air bubbles on the solid surface. These problems usually occur in ultrasonic immersion testing and are of particular interest to the NDE community. The problems are formulated in the Fourier (frequency) domain and are described by the three dimensional acoustic and elastodynamic boundary integral equations (BIE) with pressure and displacement serving as primary variables. The techniques developed are general and may be used with any kind of incident wave, although plane waves are used in all numerical experiments. The equations governing the acoustic region are first converted mathematically to equations like those of an elastodynamic region and the resulting two elastodynamic regions are coupled and solved for the interface displacements. Tractions, pressures, and pressure gradients are subsequently computed using these displacements and the interface conditions. Numerical results are obtained for some typical problems and justified using reciprocity relations and by comparison with solutions available for the half-space elastodynamic problem.

In sensitivity analysis for problems involving thin domains or domains with cracks, conventional boundary integral equations must be supplemented and/or replaced by hypersingular ones. This is due to the fact that the conventional equations become nearly-degenerate for thin domains and actually degenerate for cracks. Such degenerate character follows from the close proximity to each other, or actual coincidence, of two defining surfaces in each case. Hypersingular boundary integral equations for sensitivity analysis are developed in two forms in this thesis, using a global regularization and a local regularization. The regularizations are facilitated by observing that the singularity order of the sensitivity BIE. formulas is no more than that of the ordinary BIE formulas. One motivation for this work is the computation of stress-intensity-factor sensitivities with respect to crack-growth. Other motivations would include optimization and design applications wherein sensitivities would be needed, but would otherwise be unavailable, for any reason, from conventional integral equations alone. In this thesis, examples of stress-intensity-sensitivities with respect to the size of a crack are given. Specifically, sensitivity values for a circular bar with an embedded penny-shaped crack under tension, bending, and torsion loadings are obtained and shown to be accurate. These examples verify the formulas and the codes developed in this dissertation.