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This dissertation, "Fast and Well-conditioned Integral Equation Solvers for Low-frequency Electromagnetic Problems" by Qin, Liu, 刘{274b4d}, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Inspired by the important low frequency applications, such as the integrate circuits, the nano electromagnetic compatibility and quantum optics, several aspects of the computational electromagnetic low-frequency problems in surface integral equation (SIE) are carefully investigated in this dissertation. Firstly a capacitive model is studied that the convergence of the matrix system is co-determined by both the condition of the matrix and the righthand-side excitation. In a current solution, the weighted contributions from different singular vectors are not only decided by the corresponding singular values but also the right-hand side. The convergence of the capacitive problems is guaranteed by the fact that the singular vectors corresponding to the small singular values are not excited under the delta-gap source. The dominant charge currents are enough to capture the capacitive physics. Detailed spectral analysis with right-hand side effect validates the proposed theory. Secondly, in order to overcome the low-frequency inaccuracy problem for open capacitive structures in CMP-EFIE, a perturbed CMP-EFIE is proposed to extract accurate high-order current at low frequencies. Further study of the capacitive problems in CMP-EFIE utilizes a simplified two-term system by removing the contribution from the hypersingular preconditioned term, which captures the correct physics without doing the perturbation steps. The afore-built right-hand side analysis theory is applied here to explain the stability and accuracy of the simplified CMP-EFIE system. Thirdly, a point testing system is constructed to eliminate the nontrivial nullspaces of the static MFIE systems by enforcing extra zero magnetic flux conditions at the testing points locations. The projection of the current solution onto the magnetostatic nullspaces is truncated accordingly, thus the system convergence can be much improved without losing any accuracy. Finally, the electromagnetic solution is obtained from a potential-based integral equation solver, capturing electrostatic physics from the scalar potential formulation and magnetostatic physics from the vector potential formulation. The combination of the two formulations reveals the correct solution and physics at low frequencies. And the equations, formulated with the potential quantities, make it possible to couple with quantum effects theories. The resulting system appears to be a symmetric saddle point problem, where the efficiency of the iterative solver can be well-solved by a typical appropriate constraint preconditioner. The stability and capability of the new system in solving different kinds of electromagnetic problems are validated over a wide range of frequency range. The research topics in this dissertation cover different aspects of low frequency integral equation solvers, aiming at fast, stable, wide-band and accurate integral algorithms. Subjects: Integral equations Electromagnetic fields - Mathematical models
Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research. Also, learning the fundamentals of linear elastic wave theory does not require a quantum leap for electromagnetic practitioners. Integral equation methods have been around for several decades, and their introduction to electromagnetics has been due to the seminal works of Richmond and Harrington in the 1960s. There was a surge in the interest in this topic in the 1980s (notably the work of Wilton and his coworkers) due to increased computing power. The interest in this area was on the wane when it was demonstrated that differential equation methods, with their sparse matrices, can solve many problems more efficiently than integral equation methods. Recently, due to the advent of fast algorithms, there has been a revival in integral equation methods in electromagnetics. Much of our work in recent years has been in fast algorithms for integral equations, which prompted our interest in integral equation methods. While previously, only tens of thousands of unknowns could be solved by integral equation methods, now, tens of millions of unknowns can be solved with fast algorithms. This has prompted new enthusiasm in integral equation methods. Table of Contents: Introduction to Computational Electromagnetics / Linear Vector Space, Reciprocity, and Energy Conservation / Introduction to Integral Equations / Integral Equations for Penetrable Objects / Low-Frequency Problems in Integral Equations / Dyadic Green's Function for Layered Media and Integral Equations / Fast Inhomogeneous Plane Wave Algorithm for Layered Media / Electromagnetic Wave versus Elastic Wave / Glossary of Acronyms
Despite vast advancements in computational hardware capabilities, full-wave electromagnetic simulations of many multiscale problems continue to be a daunting task. Multiscale problems are encountered, for example, when modeling interconnects in an integrated circuit or when simulating complex electromagnetic structures. In interconnect problems, the main challenge is to model the multiscale skin effect that develops inside the conductors at high frequency. Similarly, complex electromagnetic structures are multiscale because these surfaces are tens of wavelengths large, while each unit cell often contains subwavelength geometrical features. This thesis presents reduced-order integral equation methods to solve complex multiscale problems. For interconnect problems, it proposes a single-source surface integral equation method to model 2-D and 3-D conductors or dielectrics of arbitrary shape. In this approach, electromagnetic fields inside a conductor or a dielectric object are accurately modeled by a differential surface admittance operator and an equivalent electric current density on the object's surface. Since the proposed method does not use any volumetric unknowns, it is more efficient than volumetric methods encountered in the literature and commercial solvers, which require a fine mesh to model the skin effect. Furthermore, since the proposed approach is single-source, it is more efficient than other surface methods in the literature that require both equivalent electric and magnetic current densities. Numerical results show that the proposed method can be over 100x and 20x faster than commercial FEM solvers for 2-D and 3-D problems, respectively, while consuming significantly lower memory. The proposed surface method for conductors and dielectrics is further generalized to develop the so-called macromodeling technique to simulate complex scatterers. In this technique, a heterogeneous scatterer composed of dielectric and PEC objects is accurately modeled by equivalent electric and magnetic current densities that are introduced on a fictitious surface enclosing the element. The crux of the technique is to solve for unknowns only on the fictitious surface, instead of the scatterers, which results in fewer unknowns. Numerical results show that the proposed macromodeling technique can efficiently simulate electrically large reflectarrays composed of square patches and Jerusalem crosses, that are difficult to simulate even with commercial solvers.
Novel integral-equation methods for efficiently solving electromagnetic problems that involve more than a single length scale of interest in complex backgrounds are presented. Such multi-scale electromagnetic problems arise because of the interplay of two distinct factors: the structure under study and the background medium. Both can contain material properties (wavelengths/skin depths) and geometrical features at different length scales, which gives rise to four types of multi-scale problems: (1) twoscale, (2) multi-scale structure, (3) multi-scale background, and (4) multi-scale-squared problems, where a single-scale structure resides in a different single-scale background, a multi-scale structure resides in a single-scale background, a single-scale structure resides in a multi-scale background, and a multi-scale structure resides in a multi-scale background, respectively. Electromagnetic problems can be further categorized in terms of the relative values of the length scales that characterize the structure and the background medium as (a) high-frequency, (b) low-frequency, and (c) mixed-frequency problems, where the wavelengths/skin depths in the background medium, the structure's geometrical features or internal wavelengths/skin depths, and a combination of these three factors dictate the field variations on/in the structure, respectively. This dissertation presents several problems arising from geophysical exploration and microwave chemistry that demonstrate the different types of multi-scale problems encountered in electromagnetic analysis and the computational challenges they pose. It also presents novel frequency-domain integral-equation methods with proper Green function kernels for solving these multi-scale problems. These methods avoid meshing the background medium and finding fields in an extended computational domain outside the structure, thereby resolving important complications encountered in type 3 and 4 multi-scale problems that limit alternative methods. Nevertheless, they have been of limited practical use because of their high computational costs and because most of the existing 'fast integral-equation algorithms' are not applicable to complex Green function kernels. This dissertation introduces novel FFT, multigrid, and FFT-truncated multigrid algorithms that reduce the computational costs of frequency-domain integral-equation methods for complex backgrounds and enable the solution of unprecedented type 3 and 4 multi-scale problems. The proposed algorithms are formulated in detail, their computational costs are analyzed theoretically, and their features are demonstrated by solving benchmark and challenging multi-scale problems.
Compiled by two editors with substantial experience in developing algorithms to numerically solve integral equations in the case of discretized real-life structures, this book explains the main available approaches for the numerical solution of surface integral equations for analysing real-world multi-scale electromagnetic problems.