Extended integral equation formulation for scattering problems from a cylindrical scatterer

New integral equations are introduced in which the transverse electromagnetic field components are unknown for the two‐dimensional (2‐D) electromagnetic scattering problems and electromagnetic waveguide problems. A numerical analysis by the boundary element method is proposed. The conventional integral equations in which the longitudinal electromagnetic field components are unknown are difficult to determine at certain angular frequencies and at certain longitudinal propagation constants for the electromagnetic scattering problem. In the waveguide problem, the spurious modes may enter as nonphysical solutions.The integral equations proposed in this paper do not have conventional difficulties. Here, the integral equations using the transverse electromagnetic fields are derived and the uniqueness of the solution is proven. Further, for practical examples, the integral equations are analyzed numerically by the boundary element methods so that the validity and effectiveness are investigated.

In this work, we solve the scattering problem of a plane wave by a dielectric cylinder through the Unimoment Method, where a new approach was taken when solving the interior problem: a meshless method based on local boundary integral equations (LBIE). Traditionally, the interior problem is attacked through the Finite Element Method, because one gets sparse matrices. But as in FEM, a meshless LBIE approach also provides sparse matrices, without the drawback of constructing a grid. We develop the formalism and illustrate the application of the aforementioned method to the scattering by a dielectric circular cylinder, a problem which is known to possess analytical solution.

In this work, the problem of transient scattering by arbitrarily shaped two‐dimensional dielectric cylinders is solved using the marching‐on‐in‐time technique. The dielectric problem is approached via the equivalence principle. Three different formulations, namely, the electric field integral equation formulation, the magnetic field integral equation formulation, and the combined field integral equation formulation are considered. Numerical results are presented for two cross sections, namely, a circle and a square, and compared with inverse discrete Fourier transform (IDFT) techniques. In each case, good agreement is obtained with the IDFT solution.

It is well known that at certain discrete frequencies the conventional boundary integral equation formulation of free surface fluid–structure interaction analyses breaks down. At such ‘irregular’ frequencies the BIE method fails to provide either an acceptable or a unique solution. Having established the existence of irregular frequencies, a review of the different approaches adopted to remedy this problem is presented.A very simple modification of the BIE method is also presented to eliminate the irregular frequency problem. The proposed procedure, designated the combined boundary integral equation method (CBIEM), can be categorized as a modified integral domain method. A description of the CBIEM formulation is presented and its ability to provide a unique solution at all frequencies is demonstrated. Predictions of 3D hydrodynamic reactive coefficients of added mass and fluid damping for a Series 60 hull form and an ellipsoid based on the CBIEM procedure are presented. These predictions are compared with results generated using conventional integral equation methods. The numerical studies demonstrate that the CBIEM is both a practical and effective method of suppressing irregular frequencies. In particular, the procedure is easy to implement in existing BIE computer codes with minimal additional computational effort.

The problem considered is that of determining the induced current on, and field scattered by, a two-dimensional, flat, perfectly electrically conducting strip, which is edge loaded by a dielectric cylinder of circular cross-section. The strip is orientated radially with respect to the centre of the cylinder, and can be located outside, inside, or partially inside and partially outside the cylinder. The structure is excited by a plane-wave polarised transverse magnetic to the cylinder axis. A system of two coupled integral equations is derived for the current density on the strip in the presence of the cylinder, and is solved numerically by the method of moments. A series acceleration technique is devised to improve the convergence of the series incorporated in the kernels of the integral equations. Sample results are presented which illustrate how the current induced on the strip, the far field, and the radar cross-section are affected by the presence of the dielectric cylinder.

Kussmaul's (1969) formulation of acoustic scattering problems involves the superposition of a double-layer distribution and a simple-layer distribution on the boundary.This ensured the existence of a unique solution for all wave numbers of the incoming wave. However,the boundary integral equation involves a highly singular kernel which proves difficult to treat numerically. An alternative approach is to place the double-layer distribution on an interior boundary similar (similarly situated) to the given boundary.This avoids the singularity problems and enables excellent numerical results to be achieved.Some numerical results are presented. INTRODUCTION Problems of hard acoustic scattering may be formulated in terms of boundary integral equations.Direct formulations are based upon the Helmholtz formula;indirect formulations utilise layer potentials.This is a Neumann boundary-value problem which has a unique solution subject to regular behaviour at infinity.However the relevant classical integral equations,whether direct or indirect,fail at wave numbers corresponding with the eigenfrequencies of the interior Dirichlet problem.In this paper we show that a relatively simple, nonsingular,supplement to the classical indirect integral equation ensures a a unique solution at all wave numbers. An immediate consequence is that meaningful numerical solutions can be achieved,for a wide range of wavenumbers,by computationally effective boundary-element methods (BEMs). Some results for scattering by a hard sphere are provided.For completeness we mention some recent papers presenting alternative resolutions of the difficulty. Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 282 Boundary Elements Rego Silva et al [1] Numerical Implementation of a Hypersingular BEM formulation for Acoustic Radiation Problems,following the approach of Burton and Miller [2].Here the hypersingular integral is evaluated numerically by an algorithm of Guiggiani et al [3] building essentially upon Hadamard's finite-part integration. Liu and Rizzo [4] A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,again following the approach of Burton and Miller. However here the hypersingular integral is weakened by a regularisation technique,which enables it to be evaluated numerically by straightforward methods. Wu et al [8] Vectorization and parallelization of the acoustic boundary element code BEMAP on the IBM ES/3090 VF,exploits the supercomputing facilities to improve the CHIEF method originated by Schenck [15]. Kirkup and Henwood [5] Computational Solution of Acoustic Radiation Problems by Kussmaul's Boundary Element Method,adopting Kussmaul's representation [6] of the wave function as the superposition of a simple-layer and a double-layer potential generated from sources on the boundary.This formulation also involves a hypersingular integral,which is evaluated numerically by the method of Terai [7],an early exploitation of Hadamard's finite part integration. Kussmaul's approach may be modified by locating the double-layer sources on an interior boundary,similar and similarly situated to the given boundary.This eliminates the hypersingular integral and at the same time provides a unique solution at all wave numbers. Brief details are given in the following section. KUSSMAUL FORMULATION Any solution, ,of the Helmholtz equation existing on the infinite domain B exterior to a closed (simply connected) surface dB (fig. 1) of order 0(r~') as /•co can generally be represented by a simple-layer potential dq ; q € dB , p € B+dB (1) dB where o(q) is the source density at q and dq is the area element at q ; also Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 283 /-',-'» ; r=|f-g| (2) where k is the wave number associated with , defined by

The E/sub s/ and H/sub s/ variational formulation together with the appropriate surface integral equations was used to solve for scattering from arbitrary anisotropic cylinders. The finite element method (FEM) is used on the scatterer and a small region enclosing it. At the boundary of this finite region the surface integral equation is used to impose boundary conditions on the electromagnetic field components. In the process of generating the FEM mesh a special scheme is used so that when integral equation boundary conditions are imposed the bandwidth of the matrix does not increase at random but leads to a well-structured matrix. This matrix is then solved using matrix partitioning techniques. Numerical results are presented; the E/sub /spl rho// and E/sub /spl phi// components of scattered field for a transverse electric (TE) wave incident at 45/spl deg/ to z-axis are shown. >

In this paper the coupling of natural element method with the boundary element method (Hybrid NEM - BEM method) is proposed to solve the electromagnetic scattering problem. A dielectric cylinder is solved by the proposed approach and the results are compared with the analytical solution showing the usefulness and accuracy of the method.

the In this article, we discuss focusing properties of conducting and dielectric cylinders on the basis of the Fresnel zone principle. The radiation patterns when the cylinders are excited with an isotropic source are evaluated using the boundary element method. Helix antennas combined with a Fresnel zone plate are also reported. Enhanced gain characteristics are experimentally demonstrated.

Adaptive gridding methods are of fundamental importance both for industry and academia. As one of the computing methods, the Boundary Element Method (BEM) is used to simulate problems whose fundamental solutions are available. The method is usually characterised as constant elements BEM or linear elements BEM depending on the type of interpolation used at the elements. Its popularity is steadily growing because of its advantages over other numerical methods most important of which being that it only involves obtaining data at the boundary and computation of the solution in the domain is merely a case of post processing through the use of an identity. This results in the reduction of the problem dimension by unity. Although there is a reduction in dimension when we use BEM, the method usually results in full matrices which can be expensive to solve. This makes the method costly for problems that require very fine grids like those with hot spots as in the example of impressed current cathodic protection systems (ICCP). The BEM is a global method in nature in that the solution in one node depends on the solutions in all the other nodes of the grid. Hence an error in one node can pollute the solution in all the other nodes. In this thesis, we first focus our attention on defining and studying both the local and global errors for the BEM. This is not a completely new study as the literature suggests, however, our approach is different. We use the basic foundations of the the method to define the errors. Since the method is a global method, first we use the interpolation error on each element to define what we have called a sublocal error. Then using the sublocal error we have defined the local error. Understanding the local errors enabled us study the global error. Theoretical and numerical results show that these errors are second order in grid size for both the constant and linear element cases. Then, having explored errors, we study a method for adaptive grid refinement for the BEM. Rather than using a truly nonuniform grid, we present a method called local defect correction (LDC) that is based on local uniform grid refinement. This method is already developed and documented for other numerical methods such as finite difference and finite volume methods but not for BEM. In the LDC method, the discretization on a composite grid is based on a combination of standard discretizations on several uniform grids with different grid sizes that cover different parts of the domain. At least one grid, the uniform global coarse grid, should cover the entire boundary. The size of the global coarse grid is chosen in agreement with the relatively smooth behaviour of the solution outside the hot spots. Then several uniform local fine grids each of which covers only a (small) part of the boundary are used in the hot spots. The grid sizes of the local grids are chosen in agreement with the behaviour of the continuous solution in that part of the boundary. The LDC method is an iterative process whereby a basic global discretisation is improved by local discretisations defined in subdomains. The update of the coarse grid solution is achieved by adding a defect correction term to the right hand side of the coarse grid problem. At each iteration step, the process yields a discrete approximation of the continuous solution on the corresponding composite grid. We have shown how this discretisation can be achieved for the BEM. We apply the discretisation to an academic example to demonstrate its implementation and later show how to use it for an application as the ICCP system. The results show that it is a cheaper method than solving on a truly composite grid and converges in a single step.

A recent analytical formulation by G.A. Kriegsmann et al. (see ibid., vol.AP-35, p.153-61, Feb. 1987) of electromagnetic wave scattering by perfectly conducting two-dimensional objects using the on-surface radiation boundary condition approach is extended to the case of two-dimensional homogeneous convex dielectric objects. It is shown that a substantial simplification in the analysis can be obtained by applying the outgoing radiation boundary condition on the surface of the object. The analysis procedure decouples the fields in the two regions to yield explicitly a differential equation relationship between the external incident field excitation and the corresponding field distribution in the interior of the dielectric object. The interior fields can be obtained by solving the differential equation using either an analytical approach or a suitable numerical method. Two-dimensional scattering examples along with validations are reported, showing the near-surface field distributions for a homogeneous circular dielectric cylinder and an elliptic dielectric cylinder, with with transverse magnetic plane-wave excitation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Numerical Solution of Integral Equations on the Half-Line

The problem about the distribution of the surface current density in a narrow circular strip antenna as an infinitely thin perfectly conducting ribbon folded in a circle and positioned on the surface of a dielectric cylinder is reduced to a one-dimensional integral equation (IE). A method for solving the obtained IE is proposed. Complex distributions of the azimuthal component of the surface current density over the circular conductor are presented for different values of the dielectric permittivity of the cylinder.

We consider boundary integral equations (BIEs) met in the scattering by dielectric cylinders and compare numerical algorithms based on elementary and Muller's BIEs near the "numerical resonances". A procedure of partial removal of well-known defect of elementary BIEs related to the loss of their unique solvability is discussed.

A face based basis function solutions to thermal conduction problems for 3-D irregular shaped bodies using BEM