Mei Song Tong

Nyström Method Solution of Volume Integral Equations for Electromagnetic Scattering by 3D Penetrable Objects

The volume integral equations (VIEs) for electromagnetic (EM) scattering by three-dimensional (3D) penetrable objects are solved by Nystrom method. The VIEs are essential and cannot be replaced by surface integral equations (SIEs) for inhomogeneous problems, but they are usually solved by the method of moments (MoM). The Nystrom method as an alternative for the MoM has shown much promise and has been widely used to solve the SIEs, but it is less frequently applied to the VIEs, especially for 3D EM problems. In this work, we implement the Nystrom method for 3D VIEs by developing an efficient local correction scheme for singular and near singular integrals over tetrahedral elements. The scheme first interpolates the unknown functions within the tetrahedral elements and then derives analytical solutions for the resultant singular or near singular integrals after singularity subtraction. The scheme is simpler and more efficient in implementation compared with those based on the redesign of quadrature rules for the singular or near singular integrands. Numerical examples are presented to demonstrate the effectiveness of the proposed scheme and its convergence feature is also studied.

Multilevel fast multipole algorithm for elastic wave scattering by large three-dimensional objects

Multilevel fast multipole algorithm (MLFMA) is developed for solving elastic wave scattering by large three-dimensional (3D) objects. Since the governing set of boundary integral equations (BIE) for the problem includes both compressional and shear waves with different wave numbers in one medium, the double-tree structure for each medium is used in the MLFMA implementation. When both the object and surrounding media are elastic, four wave numbers in total and thus four FMA trees are involved. We employ Nystrom method to discretize the BIE and generate the corresponding matrix equation. The MLFMA is used to accelerate the solution process by reducing the complexity of matrix-vector product from O(N^2) to O(NlogN) in iterative solvers. The multiple-tree structure differs from the single-tree frame in electromagnetics (EM) and acoustics, and greatly complicates the MLFMA implementation due to the different definitions for well-separated groups in different FMA trees. Our Nystrom method has made use of the cancellation of leading terms in the series expansion of integral kernels to handle hyper singularities in near terms. This feature is kept in the MLFMA by seeking the common near patches in different FMA trees and treating the involved near terms synergistically. Due to the high cost of the multiple-tree structure, our numerical examples show that we can only solve the elastic wave scattering problems with 0.3-0.4 millions of unknowns on our Dell Precision 690 workstation using one core.

Multilevel fast multipole algorithm for acoustic wave scattering by truncated ground with trenches.

The multilevel fast multipole algorithm (MLFMA) is extended to solve for acoustic wave scattering by very large objects with three-dimensional arbitrary shapes. Although the fast multipole method as the prototype of MLFMA was introduced to acoustics early, it has not been used to study acoustic problems with millions of unknowns. In this work, the MLFMA is applied to analyze the acoustic behavior for very large truncated ground with many trenches in order to investigate the approach for mitigating gun blast noise at proving grounds. The implementation of the MLFMA is based on the Nystrom method to create matrix equations for the acoustic boundary integral equation. As the Nystrom method has a simpler mechanism in the generation of far-interaction terms, which MLFMA acts on, the resulting scheme is more efficient than those based on the method of moments and the boundary element method (BEM). For near-interaction terms, the singular or near-singular integrals are evaluated using a robust technique, which differs from that in BEM. Due to the enhanced efficiency, the MLFMA can rapidly solve acoustic wave scattering problems with more than two million unknowns on workstations without involving parallel algorithms. Numerical examples are used to demonstrate the performance of the MLFMA with report of consumed CPU time and memory usage.

Efficient evaluation of Casimir force in arbitrary three-dimensional geometries by integral equation methods

Abstract In this Letter, we generalized the surface integral equation method for the evaluation of Casimir force in arbitrary three-dimensional geometries. Similar to the two-dimensional case, the evaluation of the mean Maxwell stress tensor is cast into solving a series of three-dimensional scattering problems. The formulation and solution of the three-dimensional scattering problems are well-studied in classical computational electromagnetics. This Letter demonstrates that this quantum electrodynamic phenomenon can be studied using the knowledge and techniques of classical electrodynamics.

Multilevel Fast Multipole Acceleration in the Nyström Discretization of Surface Electromagnetic Integral Equations for Composite Objects

The multilevel fast multipole algorithm (MLFMA) based on the Nyström discretization of surface integral equations (SIEs) is developed for solving electromagnetic (EM) scattering by large composite objects. Traditionally, the MLFMA is based on the method of moments (MoM) discretization for the SIEs and it usually works well when the robust Rao-Wilton-Glisson (RWG) basis function is enough to represent unknown currents. However, the RWG basis function may not represent both the electric and magnetic current in solving the electric field integral equation (EFIE) and magnetic field integral equation (MFIE) for penetrable objects, and how one represents another current could be a problem in the MoM. In this work, we use the Nyström method as a tool to discretize the SIEs and incorporate the MLFMA to accelerate the solutions for electrically large problems. The advantages of the Nyström discretization include the simple mechanism of implementation, lower requirements on mesh quality, and no use of basis and testing functions. These benefits are particularly desired in the MLFMA because the solved problems are very large and complex in general. Numerical examples are presented to demonstrate the effectiveness of the proposed scheme.

On the Near-Interaction Elements in Integral Equation Solvers for Electromagnetic Scattering by Three-Dimensional Thin Objects

The accuracy of near-interaction (NI) evaluation is investigated for hypersingular and strongly singular kernels in solving electromagnetic (EM) integral equations. We first define the NI between a field point and a source triangular patch using a nearness factor (NF) and show the accuracy of numerical integrations for the NI integrals with hypersingular or strongly singular kernels. It is found that there exists a transition area or turning point before and after which the convergence behaviors of those integrations are greatly different. We define the NF value at the 1% relative error as the turning point NF 0 and define NF les NF 0 as the NI range. We find that NF 0=1.4 could be a bound of NI range for the triangles with a quality factor q > 0.35 and analytical formulas should be used in evaluating NI elements within the bound. Numerical examples for EM scattering by three-dimensional (3D) thin objects are used to demonstrate the significance of NI treatments for the accurate solutions.

Nyström method for elastic wave scattering by three-dimensional obstacles

Nystrom method is developed to solve for boundary integral equations (BIEs) for elastic wave scattering by three-dimensional obstacles. To generate the matrix equation from a BIE, Nystrom method applies a quadrature rule to the integrations of smooth integrands over a discretized element directly and chooses the values of the unknown function at quadrature points as the systems unknowns to be solved. This leads to a simple procedure to form the off-diagonal entries of matrix by simply evaluating the integrands without numerical integrations. For the diagonal or near diagonal entries corresponding to the integrals over a singular or near-singular element where the kernels are singular or near singular, we develop a systematic singularity treatment technique, known as the local correction scheme, based on the linear approximation of elements. The scheme differs from the singularity regularization or subtraction technique used in the boundary element method (BEM). It applies the series expansion of scalar Greens function to the kernels and derives analytical solutions for the strongly singular integrals under the Cauchy principal value like (CPV-like) sense. Since the approach avoids the need for reformulating the BIE for singularity removal in BEM and solves for the Somiglianas equation directly, it is easy to implement and efficient in calculation. Numerical examples are used to demonstrate its robustness.

Integral equation solvers for real world applications - some challenge problems

This paper presents some real world applications and problems for integral equation solvers (IES). Integral equation solvers are, in general, more complex to implement compared to differential equation solvers (DESs). This is due to the need for the Greens function method, which generally involves the evaluation of singular integrals. Moreover, due to the dense matrix system, acceleration solution methods have to be invoked before IESs are competitive with differential equation solvers. Also, linearity of the media has to be assumed before frequency-domain and Greens function techniques can be used. In contrast to DESs, the advantage of IESs, lies in the smaller number of unknowns and favorable scaling properties for memory and CPU requirements. DESs are simple to implement, but usually exhibit worse scaling properties when applied to surface scattering problems. The presence of grid-dispersion error worsens their scaling properties for large scale computing. On the other hand, DESs in the time domain can easily account for nonlinear phenomena. Hence, for an area replete with nonlinear physics, such as computational mechanics or computational fluid dynamics, DESs outrank integral equation solvers in popularity. The advantages of IESs in EM make them popular for solving a number of scattering problems. This is especially so when they have been accelerated with fast algorithms

A direct approach for solving 3D EFIE with double gradient of the Green's function

In this paper, a direct approach for solving the 3D EFIE including the double gradient of the Greens function (second form) is developed. The direct solution for the equation is thought to be prohibitive due to the existence of super-hyper singularities in the kernel for self-interaction terms. The closed-form expressions for these super-hyper singular integrations are derived in a limiting sense and the solving of the equation directly is made possible. Moreover, analytical formulas for the weakly singular integrations in the limiting sense (i.e. w0 ne 0) are derived. These formulas can be used to calculate the near-interaction terms exactly no matter how close to the source patch the observation point is. This approach also allows one to represent the current using pulse-like bases with simplicity and flexibility, albeit in a lower-order approximation. A higher-order approximation for the current can be considered in the future

E-Field, H-Field, and Combined-Field Based Nyström Method Analysis for Electromagnetic Scattering by Complex-Material Bodies

The Nyström method (NM) is used to solve for electromagnetic scattering by 3-D composite objects based on surface integral equations (SIEs). These SIEs include both equivalent electric and magnetic currents as unknowns since composite media exist. In the method-of-moments (MoM) solution for these SIEs, one may encounter the problem of how to represent the magnetic current using an appropriate basis function if the electric current is represented by Rao-Wilton-Glisson (RWG) basis function. Some choices like RWG, n̂ × RWG, or dual basis function in representing the magnetic current may have the instability, fictitious charge, or high-cost problems, respectively, and thus, are not ideal. Compared with the MoM, the NM is simpler to implement, and most importantly, it can get rid of these problems. We employ this method to solve the E-field, H-field, and combined-field SIEs with efficient local correction schemes. Numerical examples show that the NM can give stable and efficient solutions for both near and far fields, when away from the resonant frequencies in E-field and H-field formulations, even for relatively complicated structures.