Matrix inversion in spatial frequency analysis of electromagnetic scattering and radiation

Abstract: In this paper, a half space adaptive fast multipole boundary facemethod (FMBFM) is presented for solving the three-dimensional half space ex-terior acoustic problems. In the presented method, the Burton-Miller equationbased on the conventional boundary integral equation (CBIE) and its hyper-singularboundary integral equation (HBIE) is used to deal with the fictitious eigenfrequen-cies problem. The half space Green’s function is employed, thus the tree structurein the fast multipole method can be used only for the real domain. The higher orderelements and an adaptive tree structure are used to improve the efficiency of theFMBFM. This half space adaptive FMBFM for half space acoustic problems is anextension of the adaptive FMBFM for full space acoustic problems developed bythe authors. Numerical examples for half space acoustic problems in this paperdemonstrate the efficiency and validity of this method.Keywords: fast multipole boundary face method, Burton-Miller equation, acous-tic problems, half space, modified hyper-singular boundary integral equation1 IntroductionAcoustic problems in half space are a major subject in engineering. Many researchworks have been done on this aspect (Seybert et al., 1988; Secgin et al., 2010; Caoet al., 2010). The half space Green’s function (Seybert et al., 1988) can be appliedby adding the source solution at the image point to the original full-space Green’sfunction for dealing with half space problems. Sound source localization and iden-tification of a refrigerator were accomplished by using the half space boundaryelement method (BEM) (Secgin et al., 2010). Using the half space Green’s func-tion, the discretization of the rigid infinite plane is removed and only the boundaries

The Method of Moments (MoM) is one of the most popular techniques for solving electromagnetic scattering problems. Its main advantage is that, when used to discretise a Boundary Integral Equation (BIE), only the surface of the objects needs to be discretised. Many other techniques require discretisation of the volume as well, leading to an increase of unknowns and requiring an artificial boundary to emulate infinity. The downside of the MoM is that the resulting linear system is dense, due to the Green's function, and often ill-conditioned. The density of the impedance matrix would lead to a prohibitive O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) complexity for both the CPU time and the memory. A number of techniques have been developed in the past, reducing this complexity. Among the most succesful are the Fast Multipole Methods (FMM), which reduce the complexity for a matrix-vector product to O (N log N). This matrix-vector product is the bottleneck of an iterative solution of the linear system. However, an iterative solution can only be efficient if the problem is sufficiently well-conditioned. The choice of BIE influences the conditioning number of the impedance matrix. The Electric Field Integral Equation (EFIE) leads to a highly accurate solution but is notoriously ill-conditioned. In comparison, the Magnetic Field Integral Equation (MFIE) is less accurate (for the same level of discretisation) but is well-posed. Also, on open surfaces the MFIE is not applicable. In order to alleviate the problems with the EFIE, a powerful preconditioner is required to stabilise the breakdown, which occurs in particular at low frequencies (LF). Recently, an approach has been proposed, allowing multiplication of the EFIE impedance matrix with itself, leading to a well-conditioned problem. In the next section, this approach will be briefly revisited. In order to maintain the O (N log N) complexity for the global solution, we employ FMM to accelerate the Calderon Preconditioner as well. A number of difficulties occur, but these can all be overcome. This combination of FMM and Calderon Preconditioning will be dealt with in the remaining sections. At the time of the conference, examples will be shown to illustrate the novel techniques.

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N) operations to compute the system of equations and another O(N) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or

Electromagnetic fields created in a homogeneous medium by a known distribution of electric current are determined by integral equations involving free space Green's functions. In actual situations, however, current distributions are not known beforehand and must be determined, in such a way that the total fields satisfy the boundary conditions on the conductors. The integral equation technique was extended to treat also heterogeneous structures, in order to analyze and design multilayer printed circuits and antennas, such as the broadband strip-slot-foam-inverted patch printed antenna. Sommerfeld integrals provide Green's functions for stratified media, which are evaluated by means of specifically developed, efficient computer algorithms. Structures including coaxial or line feeds and coupling through slots can now be easily analyzed with commercially available software packages.

In the use of the time-domain integral equation (TDIE) method for the analysis of layered media, it is important to have the time-domain layered medium Green's function computed for many source-to-field distances /spl rho/ and time instants t. In this paper, a numerical method is used that computes the mixed potential Green's functions G/sub v/(/spl rho/,t) and G/sub A/(/spl rho/,t) for a multilayered medium for many /spl rho/'s and t's simultaneously. The method is applicable to multilayered media and for lossless or lossy dispersive media. Salient features of the method are: 1) the use of complex /spl omega/ so that the surface wave poles are lifted off the real k/sub /spl rho// axis such that pole extractions are not required; 2) the use of half-space extraction so that the integrand for the Sommerfeld integral decays exponentially along the k/sub /spl rho// axis to obtain fast convergence of the integral; and 3) the use of the fast Hankel transform so that the Green's function is calculated for many values of /spl rho/ simultaneously. For a four-layer medium, we illustrate the numerical results by a three-dimensional plot of /spl rho/G/sub v/(/spl rho/,t) versus /spl rho/ and t and demonstrate the space-time evolution of these Green's functions. For a maximum frequency range of 8 GHz, the method requires only a few CPU minutes to compute a table of 100 (points in /spl rho/) /spl times/ 168 (points in t) uniformly spaced values of G/sub v/(/spl rho/,t) on an 867-MHz Pentium PC.

The paper presents an integral equation (IE) based scheme to analyze scattering from inhomogeneous bodies with anisotropic electromagnetic properties. Newly emerging metamaterials have both anisotropic /spl epsiv/ and /spl mu/ due to the structured orientation of inclusions. We develop the necessary mathematics to enable the modeling of 3D bodies of arbitrary shape using tetrahedral tessellation and apply Galerkin testing to solve the problem numerically. Evaluation of elements in the resulting matrix equation is considerably more involved with the introduction of generalized /spl epsiv/ and /spl mu/ tensors; computing these integrals is elucidated. Also, schemes for accurately compensating the artificial surface charges for tetrahedral basis functions is demonstrated. The proposed scheme can be easily retrofitted with acceleration techniques like the fast multipole method (FMM) to permit large scale computations. It is shown that incorporation of such techniques is a trivial task. Likewise, periodic Green's functions can be used for the analysis of periodic band-gap structures.

The fast steepest descent path algorithm (FASDPA) is developed as a counterpart to the fast multipole method (FMM) for EM scattering problems. The basic idea of FASDPA is to expand the Green's function as an integral summation of plane waves. Then the steepest descent path (SDP) is introduced to evaluate the integral efficiently. By separating the source and observation points into groups as has been done in FMM the acceleration of matrix-vector multiplication is possible. In observance of the integral representation of the Green's function arising in a layered medium study, it is easy to see that the FASDPA can be applied to accelerate the computation.

The fast multipole method (FMM) is useful for solving electromagnetic scattering problems through integral equations. Some asymptotic laws are given for the truncation of the series involved in the FMM. Those laws were rigorously proved in Carayol (2002). For the first time, the accuracy of the widely used empirical formulas was studied mathematically. Our results also give some elements to understand the difference of FMM error due to the configuration of points in FMM boxes (connected to the term u/spl I.cap//spl middot/ v/spl I.cap/). We also derived in Carayol some consequences on the number of quadrature points needed for the numerical integrations in the FMM.

Fast multipole singular boundary method for Stokes flow problems

The fast multipole method (FMM) is very efficient in solving integral equations. This paper applies the method to solve large solid-solid boundary integral equations for elastic waves in two dimensions. The scattering problem is first formulated with the boundary element method. FMM is then introduced to expedite the solution process. By using the FMM technique, the number of floating-point operations of the matrix-vector multiplication in a standard conjugate gradient algorithm is reduced from O(N 2) to O(N 1.5), where N is the number of unknowns. The matrix-filling time and the memory requirement are also of the order N 1.5. The computational complexity of the algorithm is further reduced to O(N 4/3) by using a ray propagation technique. Numerical results are given to show the accuracy and efficiency of FMM compared to the boundary element method with dense matrix.

Though computational molecular dynamics is an effective tool for nano-scale phenomenon analysis, computational costs associated with its computer simulation are extremely high. There are two major computational steps associated with computer simulation of dynamics of molecular structures. They are calculation of interatomic forces and formation and solution of the equations of motion. Currently, these two computational steps are treated separately in most commonly-used methods. For example, Fast Multipole Method (FMM) and Cell Multipole Method (CMM) have been used for calculation of interatomic forces, and Cartesian Coordinate Method (CCM) and Internal Coordinate Molecular Dynamics Method (ICMD) have been created for the formation and solution of equations of motion of an atomistic molecular system. In this paper, a new procedure is presented through a proper integration between multibody molecular algorithms (MMA) and fast multipole methods to improve computational efficiency for computer simulation of the dynamical behaviors of multibody molecular structures in polymers and biopolymers. For the computational costs associated with interatomic forces, a fast multipole method is used to calculate the interatomic forces due to the potentials. For the computational costs associated with formation and solution of equations of motion, a multibody molecular algorithm developed by the author in his previous work will be utilized to integrate with fast multipole methods. The algorithm significantly improves computational efficiency when comparing with its counterpart procedures. The fast multipole method begins by scaling all atoms into a box with coordinate ranges to ensure numerical stability of subsequent operations. The parent box is then divided into half in the direction of each Cartesian axis and each child box is then subdivided to form a computational family tree. The flow of calculations is carried out along the tree structure with five passes. The fast multipole method has been improved and modified to achieve better effectiveness and higher efficiency since it was created. The multibody molecular algorithm starts with numbering subsets, forming bond graph, and developing three computing passes along the tree structure of an atomistic molecular system. Computing data flows in the fast multipole method and the multibody molecular algorithm will properly line up with the parent-child recursive relationship along the configuration of the tree structure due to linear recursive natures of both fast multipole method and multibody molecular algorithm. Then the time spent on the recursive simulation passes in the fast multipole method for computing forces may overlap with the time spent on the three recursive computational passes in the multibody molecular algorithm for forming and solving equations of motion.

Linear FREDHOLM integral equations are derived for the STOKES vector of the radiation emerging from a scattering plane parallel medium of finite optical thickness. The integral equations are obtained by means of imbedding the slab in an infinite medium. They are formulated in terms of GREEN's function matrices and renormalized for the asymptotic eigenmode. Explicitly, linear integral equations are given for the reflection and transmission matrices. The reciprocity principle is employed to obtain integral equations also for the mean intensity of the inner radiation field in the case of the slab albedo problem.

A numerical solution of the boundary integral equation for the exterior Helmholtz problem in three dimensions, leads to the solution of a dense linear system. In order to have a well conditioned system, we consider the Despres integral equations (see J. Electromag. Waves and Appl., vol.13, p.1553-68, 1999). An iterative method is given by Despres to solve this system. If Niter is the number of the iterations, the complexity of this resolution is of order Niter k/sup 4/, where k is the wave number. In order to speed up the iterative solution of the system, we have considered the coupling of two methods, the microlocal discretization method and the fast multipole method (FMM). The microlocal discretization method of T. Abboud, J.-C. Nedelec and B. Zhou (1995), enables one to consider a new system whose size is of order k/sup 2/3//spl times/k/sup 2/3/ instead of k/sup 2//spl times/k/sup 2/ for convex geometries. However, due to the geometrical approximation of the surface, the fine mesh of the standard case is still considered. Another method, the fast multipole method, is one of the most efficient and robust methods used to speed up the calculation of matrix-vector products, with a cost of order k/sup 3/ instead of k/sup 4/ for the one level FMM. In this paper, a coupling of these two methods is presented. It enables one to reduce the CPU time very efficiently for large wave number, with a complexity of order k/sup 3/+Niter k/sup 4/3/.

An accelerated integral equations (IE) field solver for determining scattered fields from electrically large electromagnetic metasurfaces using fast multipole method (FMM) is proposed and demonstrated in 2-D. In the proposed method, practical general metasurfaces are expressed using an equivalent zero thickness sheet model described using surface susceptibilities, and where the total fields around it satisfy the generalized sheet transition conditions (GSTCs). While the standard IE-GSTC offers fast field computation compared with other numerical methods, it is still computationally demanding when solving electrically large problems, with a large number of unknowns. Here, we accelerate the IE-GSTC method using the FMM technique by dividing the current elements on the metasurface into near- and far-groups, where either the rigorous or approximated Green’s function is used, respectively, to reduce the computation time without losing solution accuracy. Using numerical examples, the speed improvement of the FMM IE-GSTC method { <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{3/2})$ </tex-math></inline-formula> } over the standard IE-GSTC { <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{3})$ </tex-math></inline-formula> } method is confirmed. Finally, the usefulness of FMM IE-GSTC is demonstrated by applying it to solve electromagnetic propagation inside an electrically large radio environment with strategically placed metasurfaces to improve signal coverage in blind areas, where a standard IE-GSTC solver would require prohibitively large computational resources and long simulation times.

An efficient algorithm for computing 4M x24 point DFT, called radix-4-24p efficient FFT algorithm (EFFT), is developed. To convert the computation of DFT of length N=4M x 24 in M steps to the computation of 4 DFTs of length 24 by radix-4 decimation in time algorithm and from Mth to first , each step converts four DFTs of length L x 24 into one DFT of length (4 x L)x24 are the basic mentality, and an efficient algorithm for computing the 24 -point DFT, which requires only 24 real multiplications, is the core module of radix-4-24p EFFT algorithm. The total number of computational requirements for implementing N=4M x 24- point DFT in the algorithm is (1+3 M)N-16(4M -1) real multiplications, 4M x 20 -4 real right- shiftings 1 bit and 4M x 376+3 MN/2-124 real additions. The equation and block diagram of performing the radix-4-24p EFFT algorithm and a highly effective algorithm for computing directly 24- point DFT are represented in the text. The computing efficiency and computation amount of the radix-4-24p EFFT algorithm are improved than those of radix-4 FFT algorithm as a result of using effectivly cos(pi/3)=sin(pi/6)=1/2 andcos(pi/4)=sin(pi/4)= 0.707.