Efficiency of boundary integral equation techniques for internal resonance problem

This book provides an extensive introduction to the numerical solution of a large class of integral equations. The initial chapters provide a general framework for the numerical analysis of Fredholm integral equations of the second kind, covering degenerate kernel, projection and Nystrom methods. Additional discussions of multivariable integral equations and iteration methods update the reader on the present state of the art in this area. The final chapters focus on the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation, in both two and three dimensions. Two chapters are devoted to planar BIE problems, which include both existing methods and remaining questions. Practical problems for BIE such as the set up and solution of the discretised BIE are also discussed. Each chapter concludes with a discussion of the literature and a large bibliography serves as an extended resource for students and researchers needing more information on solving particular integral equations.

The Numerical Solution of Integral Equations on the Half-Line

In this paper, we present four different formulations for the analysis of electromagnetic scattering from arbitrarily shaped three-dimensional (3-D) homogeneous dielectric body in the frequency domain. The four integral equations treated here are the electric field integral equation (EFIE), the magnetic field integral equation (MFIE), the combined field integral equation (CFIE), and the PMCHW (Poggio, Miller, Chang, Harrington, and Wu) formulation. For the CFIE case, we propose eight separate formulations with different combinations of expansion and testing functions that result in sixteen different formulations of CFIE. One of the objectives of this paper is to illustrate that not all CFIE are valid methodologies in removing defects, which occur at a frequency corresponding to an internal resonance of the structure. Numerical results involving the equivalent electric and magnetic currents, far scattered fields, and radar cross section (RCS) are presented for three canonical dielectric scatterers, viz. a sphere, a cube, and a finite circular cylinder, to illustrate which formulation works and which does not.

The single-source integral equation (SSIE) technique is presented to analyze electromagnetic scattering from an arbitrarily shaped three-dimensional conducting object coated by a homogeneous dielectric body. The external and interior surface equivalence principles combined with the continuity of the tangential components of the total electric and magnetic fields on the surface of the dielectric and zero tangential component of the total electric field on the surface of the conductor are used to formulate the single-source electric and magnetic field integral equations. With a localization procedure, the Calderon preconditioned single-source electric field integral equation (SSEFIE) is constructed and combined with single-source magnetic field integral equation (SSMFIE) to produce a single-source combined field integral equation (SSCFIE), which is free from spurious resonance and dense-mesh breakdown. Numerical results validate the good performance of the proposed algorithm.

Numerical solutions to the impedance boundary condition (IBC) combined source integral equation (CSIE) for scattering from impedance spheres are presented. The CSIE formulation is a well-posed alternative to the IBC electric and magnetic field integral equations which can be contaminated by spurious resonant modes. Compared with the IBC combined field integral equation (CFIE), CSIE solutions have the same accuracy when the combined source coupling admittance is chosen to be the same value as the combined field coupling admittance. However, the CSIE formulation is better suited than the CFIE for creating a general purpose computer code capable of handling aperture radiation problems and/or a scatterer which has a spatially varying surface impedance.

The adaptive integral method (AIM) is applied to the solution of the combined field integral equation (CFIE) of scattering by a three-dimensional (3-D) perfect electric conductor (PEC). The employment of CFIE eliminates the interior resonance problem suffered by both the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE). Furthermore, it significantly improves the efficiency of AIM by reducing the number of iterations for convergence. It is shown that the memory requirement and computational complexity per iteration of the AIM solution of CFIE are O(N1.5) and O(N1.5log N), respectively, for 3-D PEC surface scattering problems. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 19: 321–328, 1998.

The detection of concealed objects in the baggage by non-invasive methods has become a very important issue in the border control process. This paper is concerned with the multiple inverse scattering of the plane wave on various objects or scatterrers by using µ-diff MATLAB open-source toolbox. The acquired information is corelated with the material nature (i.e. steel, salt, neoprene and PVC) for different measured orientation of the scatterers. The non-invasive detection of concealed objects represents a huge challenge, which has attracted particular attention due to a heightened threat of terrorism. The Fourier series and on four integral equations, i.e. Electric Field Integral Equation (EFIE), Magnetic Field Integral Equation (MFIE), Combined Field Integral Equation (CFIE) and Brakhage-Werner Integral Equation (BWIE) generate the approximation framework able to solve the multiple scattering problems. A multi-frequency scattering data approach is adopted and a low frequency domain is proposed for detecting concealed objects in the luggage.

The motion of shallow water, unlike deep water, is influenced by the bottom topology. In order to obtain the velocity potential at any given point of the surface at any given time, we apply boundary integral techniques and solve a system of two integral equations, one expresses the velocity potential at the free surface and on the bottom, the other expresses the streamfunction at the free surface and on the bottom. Boundary integrals can produce very accurate results for shallow water equations but they are too complex to solve analytically and must be solved numerically. In this paper, we use three different numerical methods to compute the solutions to these integral equations: by Gaussian elimination which has high compuational cost; by iteration to construct the Neumann series which converges; and by a preconditioned version of the iteration which effectively lowers the total computation cost.

A new acceleration method is proposed for efficiently solving the problem of electromagnetic scattering from metal targets in half-space. The analysis of electromagnetic problems in any structure can be settled by the electric field integral equation. But the generated matrix condition number is large and the iterative solution has poor convergence. The number of the matrix condition generated by the magnetic field integral equation is small and iterative convergence is good. But only the closed structure problems can be worked out. The combined field integral equation is adopted because of the universality of the electric field integral equation and the convergence of the magnetic field integral equation. The gradient term of Green's function is involved in the integral equation of the mixed field. In order to further enhance the calculation efficiency, an efficient four-dimensional spatial interpolation method is introduced for half-space Green's function. Tabulation and lagrange interpolations are performed in the Sommerfeld integrals for the half-space Green's function. The improved efficiency can be 7.5 times higher than that of the traditional combined field integral equation(CFIE). Numerical results show that the computational time can be reduced significantly by the proposed method with encouraging accuracy.

An alternative method of moments discretization of the magnetic field integral equation (MFIE) uses testing functions inside the target and in a plane normal to the target surface. This approach is adapted to targets modeled with flat-faceted patches. A comparison with traditional numerical solutions of the MFIE that use testing functions on the target surface shows that the normally-integrated MFIE formulation produce far fields that are more accurate than those obtained from the traditional MFIE. The alternate approach can be made free from internal resonances and that approach is often more accurate than the combined field integral equation.

Conjugate gradient methods for the solution of boundary integral equations on a piecewise smooth boundary

A fast multigrid solution of boundary integral equations

Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson’s equation on irregular exterior regions. The method requires $O(N\log N)$operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

The “conventional” finite element‐boundary integral (FE‐BI) method suffers from two major drawbacks, one is the huge computer resources required by the dense BI submatrix; the other is the slow convergence rate of the resulting linear system of equations. In this article, both the drawbacks are investigated. First, the integral equation fast Fourier transform (IE‐FFT) algorithm with a Gaussian interpolation scheme is employed to deal with the BI part. Second, an effective and simple preconditioner is developed for solving the matrix equation obtained from FE‐BI method. Numerical results show that the proposed preconditioner can dramatically speed up the convergence of the FE‐BI system, whatever the combined field integral equation (CFIE), the electric field integral equation (EFIE), and the magnetic field integral equation (MFIE) are employed on the truncation boundary. Some examples are given to demonstrate the accuracy and efficiency of the method. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2065–2071, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24538