| In this thesis, the boundary integral equation method (BIEM) is studied and applied to electromagnetic and elastic wave problems.;First of all, a spectral domain BIEM called the spectral domain approach is employed for full wave analysis of metal strip grating on grounded dielectric slab (MSG-GDS) and microstrips shielded with either perfect electric conductor (PEC) or perfect magnetic conductor (PMC) walls. The modal relations between these structures are revealed by exploring their symmetries. It is derived analytically and validated numerically that all the even and odd modes of the latter two (when they are mirror symmetric) find their correspondence in the modes of metal strip grating on grounded dielectric slab when the phase shift between adjacent two unit cells is 0 or pi. Extension to non-symmetric case is also made. Several factors, including frequency, grating period, slab thickness and strip width, are further investigated for their impacts on the effective permittivity of the dominant mode of PEC/PMC shielded microstrips. It is found that the PMC shielded microstrip generally has a larger wave number than the PEC shielded microstrip.;Secondly, computational aspects of the layered medim doubly periodic Green's function (LMDPGF) in matrix-friendly formulation (MFF) are investigated. The MFF for doubly periodic structures in layered medium is derived, and the singularity of the periodic Green's function when the transverse wave number equals zero in this formulation is analytically extracted. A novel approach is proposed to calculate the LMDPGF, which makes delicate use of several techniques including factorization of the Green's function, generalized pencil of function (GPOF) method and high order Taylor expansion to derive the high order asymptotic expressions, which are then evaluated by newly derived fast convergent series. This approach exhibits robustness, high accuracy and fast and high order convergence; it also allows fast frequency sweep for calculating Brillouin diagram in eigenvalue problem and for normal incidence in scattering problem.;Thirdly, a high order Nystr"{o}m method is developed for elastodynamic scattering that features a simple local correction scheme due to a careful choice of basis functions. A novel simple and efficient singularity subtraction scheme and a new effective near singularity subtraction scheme are proposed for performing singular and nearly singular integrals on curvilinear triangular elements. The robustness, high accuracy and high order convergence of the proposed approached are demonstrated by numerical results.;Finally, the multilevel fast multipole algorithm (MLFMA) is applied to accelerate the proposed Nystr"{o}m method for solving large scale problems. A Formulation that can significantly reduce the memory requirements in MLFMA is come up with. Numerical examples in frequency domain are first given to show the accuracy and efficiency of the algorithm. By solving at multiple frequencies and performing the inverse Fourier transform, time domain results are also presented that are of interest to ultrasonic non-destructive evaluation. |
