Mixed discretization schemes for electromagnetic surface integral equations

A novel surface integral equation (IE) formulation for magneto-quasi-static analysis of current flow in two-dimensional conductors situated in lossy layered substrates is proposed. Traditional approach is based on the volume IE formulated with respect to the unknown current distribution in the conductors cross-section. In this work, we transform the volume IE to a surface IE via representation of the volumetric current density as a superposition of the waves emanating from the conductors boundaries. The resulting surface IE features a single unknown surface current density unlike the traditional surface IEs which require both electric and magnetic surface current densities. Also, the resultant single-source surface IE formulation involves only the electric field Greens functions instead of both electric and magnetic field Greens functions as required by the traditional surface IE formulations. The latter property makes the proposed IE formulation particularly suitable for analysis of current flow in conductors embedded in lossy layered substrates.

Conforming boundary element methods in acoustics

Reconstruction of unknown objects requires efficient inversion for measured electromagnetic scattering data. In frequency-domain integral equation method for reconstructing dielectric objects, the volume integral equations (VIEs) are involved because the imaging domain with both true unknown objects and part of background is inhomogeneous. When solving the forward scattering integral equation (FSIE), the Nyström method is used since the traditional method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function may not be convenient due to the inhomogeneity of the imaging domain. The benefits of the Nyström method include the simple implementation without using any basis and testing functions and lower requirement on geometrical discretization, so it is very suitable for inhomogeneous problems. When solving the inverse scattering integral equation (ISIE), the Gauss-Newton minimization approach (GNMA) with a multiplicative regularization method (MRM) is employed. Numerical examples for reconstructing three-dimensional dielectric objects are presented to illustrate the inversion approach.

Multiscalets in the multiwavelet family are used as the basis and testing functions in Galerkin's method. Since the multiscalets are orthogonal to their translations under the Sobolev inner product, the resulting Galerkin's method behaves like a collocation method but possesses the ability of derivative tracking for unknown functions in solving integral equations. The former makes the method simple in implementation and the latter allows to use coarse meshes in discretization. These robust features have been demonstrated in solving two‐dimensional (2D) electromagnetic (EM) problems, but have not been exploited in three‐dimensional (3D) scenarios. For 3D problems, the unknown functions in the integral equations are dependent on two coordinate variables. In order to preserve the use of coarse meshes for 3D cases, we realize the omnidirectional derivative tracking by tracking the directional derivatives along two orthogonal directions, or equivalently tracking the gradient. This process yields a nonsquare matrix equation and we use the least‐squares method (LSM) to solve it. Numerical examples show that the multiscalet‐based Galerkin's method is also robust in solving for 3D EM integral equations with a minor cost increase from LSM. Copyright © 2007 John Wiley & Sons, Ltd.

Exponentially accurate spectral and spectral element methods for fractional ODEs

The Characteristic Basis Finite Element Method (CBFEM) [1–3] is a new descendant of the Characteristic Basis Function Method (CBFM), which is a non-iterative domain decomposition approach for solving large-scale electromagnetic problems by using characteristic basis functions (CBFs). The CBFM was originally introduced in the context of Method of Moments [4], but has rapidly evolved, with appropriate modifications, into a general-purpose approach that is also applicable to Finite Element Method (FEM). Some of the features that are common to all of the CBFM-based approaches are: (i) utilization of high-level physics-based basis functions — called CBFs — to represent the fields inside each sub-domain; (ii) reduced-matrix that can be handled by using direct solvers; (iii) parallelization to reduce the overall CPU time and memory. The basic steps of the CBFEM algorithm are as follows: (i) computational domain is divided into a number of sub-domains; (ii) CBFs are generated for each sub-domain; (iii) unknowns are expressed as a weighted sum of CBFs; (iv) original matrix is transformed into a reduced-matrix by using the Galerkin procedure, which uses the CBFs as both basis and testing functions; (v) reduced matrix is solved for the weight coefficients, which are substituted into the series expressions to solve for the unknowns in the entire computational domain.

Numerical experiments involving galerkin and collocation methods for linear integral equations of the first kind

In this paper, it is attempted to approach a fast efficient algorithm for solving the famous Hallen's and Pocklington's integral equations, regarding the current distribution on a finite-length linear thin wire antenna. Here, the conventional moment method in conjunction with wavelet basis functions was used to obtain the current distribution of the antenna. The aim of this work is first to introduce the application of wavelet in electromagnetic scattering, secondly a comparison of the two method of analysis the thin wire antenna. By using the wavelet expansion, wavelets as basis and testing functions, a sparse matrix is generated from the previous moment method dense matrix. A sparsely filled matrix is easier to store and invert. The result extracted from Pocklington's integral equation gives better convergence at the feeding point, though it takes more time to be computed because of the complexity in Pocklington's equation. Results are compared to the previous work done and published, excellent results are obtained.

Electromagnetic (EM) problems with conducting and penetrable media are formulated by volume-surface integral equations (VSIEs) which allow inhomogeneous and anisotropic materials. Traditionally, the VSIEs are solved by the method of moments (MoM) with Rao-Wilton-Glisson (RWG) and Schaubert-Wilton-Glisson (SWG) basis functions. We propose a point-matching scheme to discretize the VSIEs. The point-matching scheme does not rely on any basis and testing functions and allows the use of nonconforming meshes. Also, the integral kernels are free of material parameters by selecting current densities as unknowns. To accelerate the solution for electrically large problems, we incorporate the scheme with the multilevel fast multipole algorithm (MLFMA) and a typical numerical example is presented to validate its effectiveness.

SummaryGenetic algorithm (GA) is one of the most popular algorithms of evolutionary computation. To evaluate the performance of GAs, test functions with different levels of epistasis have been included in most of the commonly used benchmark platforms. Such test functions have been produced in general by assuming an underlying linear model for the fitness of a string, in which variable interaction should be known beforehand. This paper proposes to compose epistasis‐tunable test functions via linear combinations of simple basis functions so as to avoid explicit dependence on the knowledge of variable interaction. It is remarked that, for a GA with a binary encoding, a linearly separable fitness function, ie, a zero‐epistasis fitness function, can be decomposed into a superposition of periodical basis functions whose frequencies are exponential to 2. A kind of epistasis‐tunable test functions is accordingly produced by summing up sinusoidal basis functions with such frequencies and proper phases. The merits of thus constructed test functions are: Both the locations and values of the global maxima are trivially known; their degrees of epistasis are smoothly tunable simply by changing the locations of the global maxima. Finally, illustration examples are studied, and the results confirm the claims about the merits.

A hybrid mode-matching/finite-element/method-of-moment (MM/FE/MoM) technique is presented for the rigorous analysis of longitudinally corrugated horns. The horn taper is described by the mode-matching method where the cross-section eigenvalue problem is solved by a two-dimensional (2D) finite-element technique. For the exterior horn surface including the radiating aperture of arbitrary shape, the application of the Kirchhoff-Huygens principle yields an expression for the modal admittance matrix based on the electric field integral equation (EFIE). The equations are solved numerically by the method of moments. The eigenvectors of the last waveguide taper section and Rao-Wilton-Glisson (RWG) functions for triangular patches are utilized as basis-functions for the magnetic or electric surface current densities, respectively. The reported hybrid procedure combines the efficiency of the MM method with the flexibility of the FE and MoM techniques. The accuracy and efficiency of the technique are verified by reference measurements; its flexibility is demonstrated with a longitudinally corrugated horn example.

A novel implementation of the characteristic basis function method (CBFM) is given, in which the high-level basis functions, called characteristic basis functions (CBFs), are represented in terms of curved rooftops generated from nonuniform rational B-splines (NURBS) in the parametric (u,v) domain. The associated macro-testing functions are defined by using curved razor-blade functions corresponding to each rooftop. The underlying objective of the CBFM is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments (MoM). The result is, therefore, an approach which can handle many complex cases via direct solvers, without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill-conditioned matrices. As a result of the combination of the CBFM with the special class of low-level basis and testing functions directly located over NURBS surfaces, complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the CPU time as well as required memory resources.

The electric current volume integral equation formulation is discretized with piecewise linear basis and testing functions. The resulting weak formulation is more complicated, and contains additional terms compared with the previous discretization utilizing piecewise constant approximations. However, the use of fully linear functions provides superior solution accuracy while keeping good convergence properties of the original integral equation.

Absolute charging analysis of space-craft bodies using MoM has been presented. Integral equations are formed by relating the unknown charge density on the metallic conductor. Pulse functions are used as basis functions and the delta functions are used as testing functions. The surfaces of the conducting structure are meshed using non-uniform triangular sub-areas. The matrix equations are solved by using iterative algorithm. The numerical data on the capacitance matrix and charge distribution of these structures have been presented. The temporal behavior of body potential of the satellite is computed.

The nonsingular, nonlinear Volterra integral equation of the second kind for the surface temperature of a cylinder with a continuous heat sink is solved numerically by a simple, but robust, procedure based on a composite formula. The present scheme is capable of handling stiff and nonstiff kernels expressed in terms of infinite series that contain standard functions. The numerical method is tested by imposing linear and nonlinear boundary conditions at the cylinder surface. The boundary conditions and source terms are treated explicitly without resorting to a linearization technique. To show the versatility of the numerical scheme, two classes of nonlinear boundary conditions are considered, the Stefan Boltzmann boundary condition and the time-dependent Newton's cooling law. The accuracy and robustness of the present method for solving Volterra equations of the second kind are verified considering two different forms of the source term. Two additional methods of solution are used in this investigation as benchmarks. One of them is a pseudospectral approach and the other is a standard finite difference calculation. The integral equation scheme is found to work well over the entire nondimensional time domain. Numerical results are presented for a wide range of Stark numbers and steady-state Biot numbers. The numerical scheme is computationally efficient and yields stable solutions. Changes in the geometry alter the form of the kernel in the integral equation, but the general principles used should be applicable to solve nonlinear integral equations.