Time domain boundary integral equations analysis

The present research study mainly involves a survey of diverse time-domain boundary element methods that can be used to numerically solve the retarded potential integral equations. The aim is to address the late-time stability, accuracy, and computational complexity concerns in time-domain surface integral equation approaches. The study generally targets the transient electromagnetic scattering of three-dimensional perfect electrically conducting bodies. Efficient algorithms are developed to numerically solve the time-domain electric, derivative electric, magnetic, and combined field integral equation for the unknown induced surface current. The algorithms are mainly categorized into three major discretization schemes, namely the marching-on-in-time, the marching-on-in-degree, and the convolution quadrature methods or finite difference delay modeling. Possible choices of space-time integration are examined and the results are successfully compared with the high-resolution finite integration technique's solution to perform the converge study for practical applications where exact solutions are not available. First consistent temporal interpolations with common time integrators are sought based on stability analysis of the delay differential equations. Besides, the higher orders of Lagrange and B-Spline time basis functions are employed to handle the time derivatives analytically. The orthogonal entire-domain but causal weighted Laguerre or Hermite polynomials are then employed to provide unconditionally stable marching-on-in-degree schemes. Moreover, the convolution quadrature methods which use a mapping from the Laplace domain to the z-domain based on the first or second finite difference approximation are investigated. In the convolution quadrature methods, the discretization is accomplished in the bilinear transform domain and the result is inverse transformed to create a time domain method in a marching style. The outcome of this research study is applied to the non-dispersive modeling of the propagation of electromagnetic fields in particle accelerator structures, namely calculating the generated fields when the travelling bunches of charged particles passes through the beam line elements. The application of flexible and widely used Rao-Wilton-Glisson vector basis functions on flat triangular patches, particularly on the cylindrical beam pipes, causes in turn artificial fields in the commonly used barycentric approximation for the testing integrals due to misalignment of the surface normal vectors. To avoid such deficiency, first the cylindrical parts of the scatterers are supplanted by the rectangular ones whose unit normal vectors coincide with the real radial direction of the underlying cylindrical coordinate system. The linearly-varying divergence-conforming spatial basis functions on triangular and quadrilateral meshes are then combined. Additionally, in order to render symmetric interaction matrices complying the reciprocity theorem in the Galerkin's testing method while controlling the precision of numerical quadratures on the refining source and observation subdomains, the adaptive concurrent partitioning of planar patches is exploited. Furthermore, the eigenvalue spectrum of the system iteration matrix reveals that many stabilization techniques pull energy out of the system, and thus, symplectic space-time integration methods that fully conserve the energy are invoked. A one-dimensional discrete fast Fourier transform-based algorithm is proposed to expedite the spatial convolution products of the Toeplitz-block-Toeplitz retarded interaction matrices. Additional saving owing to the system periodicity is linked with the Toeplitz properties due to the uniform space discretization in multi-level sense. In addition to the space-Fourier transformation algorithms, the time-Fourier transform routines are augmented to perform the recursive temporal convolution products for the Toeplitz block aggregates of the retarded interaction matrices in the outermost possible nested Toeplitz levels by array multiplications in spectral domain. Thus, the total computational cost and storage requirements scale down significantly in all the marching-on-in-time and marching-on-in-degree schemes or convolution quadrature methods. The temporal translation invariance properties of the time-tested Green's function are grouped in hybrid fixed and varying-size blocks to boost the efficiency of aggregate matrix-vector products in the diverse time-domain integral solver. Adaptive projection of triangular source elements on an auxiliary uniform grid is implemented for generalization of the algorithm to non-uniformly meshed scatterers. Novel summation reduction techniques are proposed to eliminate the most inner time-order loop in the marching-on-in-degree methods. Closed-form expression are presented for the discretized kernels when the convolution quadrature methods are applied for the time integration. Comparison of the exact near-field evaluation by the analytical integration on time-varying source subdomains with that of the polar integration is investigated as well. Cancelation of 1/R^2 integrals in the magnetic field integral equations are explained to halve the computational cost of the marching-on-in-time schemes. It is shown that the solution procedure for several ten thousands spatial degrees of freedom and hundreds of time steps takes couple of days on a single quad-core machine.