Solving Electrically Very Large Transient Electromagnetic Problems Using Plane-Wave Time-Domain Algorithms.

The marching-on-in-time (MOT)-based time domain integral equation solvers provide an appealing avenue for solving transient electromagnetic scattering/radiation problems. These state-of-the-art solvers are high-order accurate, rapidly converging and low-/high-frequency stable. Moreover, their computational efficiencies can be significantly improved by accelerators such as the multilevel plane-wave time-domain (PWTD) algorithm. However, practical transient electromagnetic problems involving millions of spatial unknowns and thousands of time steps were barely solved by PWTD-accelerated MOT solvers. This is due to the lack of (i) an efficient parallelization scheme for PWTD’s heterogeneous structure on modern computing platforms, and (ii) a temporal/angular/spatial adaptive PWTD that further improves the computational efficiency. The contributions of this work are as follows: First, a provably scalable parallelization scheme for the PWTD algorithm is developed. The proposed scheme scales well on thousands of CPU processors upon hierarchically partitioning the workloads in spatial, angular and temporal dimensions. The proposed scheme is adopted to time domain surface/volume integral equations (TD-SIE/TD-VIE) solvers for analyzing transient scattering from large and complex-shaped conducting/dielectric objects involving ten million/tens of millions of spatial unknowns. In addition, we developed a single/multiple graphics processing units (GPU) implementation of the PWTD algorithm that achieves at least one order of magnitude speedups compared to serial CPU implementations. Second, a wavelet compression scheme based on local cosine bases (LCBs) that exploits the sparsity in the temporal dimension is developed. All PWTD operations are performed in the wavelet domain with reduced computational complexity. The resultant wavelet-enhanced TD-SIE solver is capable of analyzing transient scattering from smooth quasi-planar conducting objects spanning well over one hundred wavelengths.PhDElectrical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/113642/1/liuyangz_1.pd

Fast, Parallel Techniques for Time-Domain Boundary Integral Equations

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments