Surface Integral Equation Approach Research Articles

A Procedure for Modeling Material Junctions in 3-D Surface Integral Equation Approaches

Surface integral equations become cumbersome to solve when arbitrary combinations of materials, and complicated arrangements of junctions between those materials, are considered. This paper describes a straightforward approach for generalizing integral equation techniques to handle any combinations of materials with junctions. Particular attention is focused on a unique, easily implemented junction resolution algorithm that maps each basis function coefficient to one or more unknowns. Three examples are presented for validation purposes.

Computation of Input Impedance of Printed Antennas With Finite Size and Arbitrarily Shaped Dielectric Substrate and Ground Plane

The input impedances of a microstrip antenna with finite and curved substrates are calculated using the hybrid volume surface integral equation approach. The results for flat and large substrate cases are compared with the measurement as well as the method of moment solution based on the spectral domain Green's function for infinitely large ground plane. Numerical results show that the input impedance can be significantly influenced when the substrate of a patch antenna is reduced in size or curved in shape.

Comparison of iteration convergences of SIE and VSIE for solving electromagnetic scattering problems for coated objects

The surface integral equation approach and the hybrid surface‐volume integral equation approach are compared for solving the problem of electromagnetic scattering by conducting objects with dielectric coating. The surface integral equation is formulated by the PMCHW method, and it is efficient for modeling bulk material scattering. The hybrid surface‐volume integral equation approach is better conditioned for the volume scattering problems. For a special class of problems in which the dielectric coating is electrically thin (one to two grid sizes), the numbers of unknowns needed by the two approaches are of the same order, and the surface‐volume integral equation approach converges significantly faster than the pure surface integral equation approach. Several numerical examples are given, and the results showed good agreements with the above statement.

Electromagnetic radiation of antennas in the presence of an arbitrarily shaped dielectric object: Green dyadics and their applications

Although numerical solutions to the electromagnetic scattering by an arbitrarily shaped object have been obtained using Waterman's (1971) T-matrix method (TMM), the general electromagnetic radiation due to an antenna of a three-dimensional (3-D) current distribution in the presence of an arbitrarily shaped object has not been well considered. In this paper, the technique of surface integral equations has been employed; and as a result, a terse and analytical representation of the dyadic Green's functions (DGFs) in the presence of an arbitrarily shaped dielectric object is obtained for the antenna radiation. In a form similar to that associated with the electromagnetic radiation in the presence of a dielectric sphere, the DGFs inside and outside of the object of arbitrary shape are expanded in terms of spherical vector wave functions. However, their coefficients are no longer decoupled due to the arbitrary surface of a 3-D object. The coupled coefficients are then determined using the surface integral equation approach, in a fashion similar to that in the T-matrix method. To confirm the applicability and correctness of the approach in this paper a dielectric sphere, as a special case, is utilized as an illustration. It is found that exactly the same expressions as in the rigorous analysis for the inner and outer spherical regions of the object are obtained using the different approaches. As applications of the approach in this paper, radiation problems of an electric dipole in the presence of superspheroids and rotational parabolic bodies are solved.

Analysis of transient scattering from composite arbitrarily shaped complex structures

A time-domain surface integral equation approach based on the electric field formulation is utilized to calculate the transient scattering from both conducting and dielectric bodies consisting of arbitrarily shaped complex structures. The solution method is based on the method of moments (MoM) and involves the modeling of an arbitrarily shaped structure in conjunction with the triangular patch basis functions. An implicit method is described to solve the coupled integral equations derived utilizing the equivalence principle directly in the time domain. The usual late-time instabilities associated with the time-domain integral equations are avoided by using an implicit scheme. Detailed mathematical steps are included along with representative numerical results.

A versatile integral equation technique for magnetic modelling

A requirement currently exists in both mineral exploration and environmental or engineering geophysics for a technique to model the magnetic fields caused by bodies with large to extreme susceptibilities in which both induced and remanent magnetizations are significant. It is well known that modelling such magnetic fields is not amenable to any known approximation. It is a significantly difficult task that requires the solution of a magnetostatic boundary value problem. Analytical solutions to the problem are extremely useful for providing insight but generally of limited application in practical interpretation due to the geometrical complexity of real situations. Available numerical solutions include both volume and surface integral equation formulations. However neither of these are particularly efficient for the purpose. An alternative surface integral equation formulation is presented here which represents the required magnetic field in terms of a double layer over the surface of the body. The technique accommodates both remanent and induced magnetization and is generally applicable to any 3D body in a magnetic environment for which the Green's function is available. The present technique has significant advantages over other integral equation solutions in the geophysical literature. It is particularly economic in terms of the density of the surface discretization and consequently the computational effort. Moreover, it is extremely robust. It is found to yield accurate solutions for the type of thin bodies that cause numerical instability with other surface integral equation approaches.

Numerical analysis of an aperture coupled rectangular dielectric resonator antenna using a surface formulation and the method of moments

A rigorous analysis of an aperture coupled rectangular dielectric resonator antenna (DRA) is presented. The method is based on a surface integral equation approach by which the dielectric resonator (DR) is treated via a set of combined field integral equations (CFIE). The CFIE and associated coupling is then formulated with sets of mixed potential integral equations (MPIE). The coupled integral equations are solved by the method of moments (MoM) in the spatial domain using Galerkin's procedure. Complex images are employed in the calculation to enhance numerical efficiency. Numerical tests have shown that only a relatively small impedance matrix is needed to compute the parameters of the antenna with good accuracy. The numerical model is demonstrated to be a very versatile and practical design tool which can analyse the DRA with minimal computation effort.

Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures

Previously, the scattering and absorption of a doubly-periodic array of pyramidal absorbers were analyzed using a surface integral equation (SIE) approach. This method is now extended to multilayered absorbers with arbitrarily shaped surfaces between the layers. The developed method is verified on planar stratified absorbers, on wedge absorbers, and rectangular absorbers showing the capability of the method to handle such structures as well. It is also shown that convergence depends critically on the discretization of the equivalent electric and magnetic current. >

Transient scattering from three-dimensional arbitrarily shaped dielectric bodies

A surface integral equation approach is utilized to calculate the transient scattering from arbitrarily shaped, three-dimensional dielectric bodies illuminated by an incident electromagnetic pulse. The surface of the dielectric body is discretized with triangular patches so that fairly arbitrarily shaped objects may be modeled. The Poggio–Miller–Chang–Harrington–Wu [ R. Mittra , Computer Techniques for Electromagnetics ( Pergamon, Oxford, 1973)] formulation, in conjunction with the marching-on-in-time method, is used to derive explicit expressions for the present-time currents as a function of the incident field and the past history. The late-time instabilities are suppressed by use of a simple averaging scheme. Finally, sample results showing various geometries are presented and are compared with other numerical techniques.

A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures

The scattering and absorption of a doubly periodic array of absorbers, either placed in free space, backed by a perfect conductor or by a half-infinite space with the same material properties as the elements forming the array, is analyzed with a surface integral equation approach (SIE). The use of a suitable periodic Green's function as kernel of the SIE reduces the formulation of the problem to a single absorber. A set of equivalent electric and magnetic currents on the surface of the absorber is discretised using Glisson functions and the SIE is solved with Galerkin's method. The validity and flexibility of the SIE approach is exemplified by comparing numerical results with measurement data for a family of commercially available absorbers. >

Modelling spontaneous mineralization potentials with a new integral equation - a reply

A requirement currently exists in both mineral exploration and environmental or engineering geophysics for a technique to model the magnetic fields caused by bodies with large to extreme susceptibilities in which both induced and remanent magnetizations are significant. It is well known that modelling such magnetic fields is not amenable to any known approximation. It is a significantly difficult task that requires the solution of a magnetostatic boundary value problem. Analytical solutions to the problem are extremely useful for providing insight but generally of limited application in practical interpretation due to the geometrical complexity of real situations. Available numerical solutions include both volume and surface integral equation formulations. However neither of these are particularly efficient for the purpose. An alternative surface integral equation formulation is presented here which represents the required magnetic field in terms of a double layer over the surface of the body. The technique accommodates both remanent and induced magnetization and is generally applicable to any 3D body in a magnetic environment for which the Green's function is available. The present technique has significant advantages over other integral equation solutions in the geophysical literature. It is particularly economic in terms of the density of the surface discretization and consequently the computational effort. Moreover, it is extremely robust. It is found to yield accurate solutions for the type of thin bodies that cause numerical instability with other surface integral equation approaches.

A surface integral equation method for the finite element solution of waveguide discontinuity problems

The surface integral equation method, which is typically used in the finite-element solution of open-region scattering problems, is applied to the solution of waveguide discontinuity problems. The major advantage offered by the surface integral equation approach over other methods is that it allows the mesh-truncating boundaries to be brought as close to the discontinuity as possible, thus helping to reduce the size of the system matrix. Unlike the mode-matching technique, the surface integral equation formulation does not require the solution of any auxiliary matrix system. Numerical results are presented to illustrate the validity of the formulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A hybrid method for the calculation of the resistance and inductance of transmission lines with arbitrary cross sections

The frequency-dependent resistance and inductance of uniform transmission lines are calculated with a hybrid technique that combines a cross-section coupled circuit method with a surface integral equation approach. The coupled circuit approach is most applicable for low-frequency calculations, while the integral equation approach is best for high frequencies. The low-frequency method consists in subdividing the cross section of each conductor into triangular filaments, each with an assumed uniform current distribution. The high-frequency method expresses the resistance and inductance of each conductor in terms of the current normal to the surface. An interpolation between the results of these two methods gives very good results over the entire frequency range, even when few basis functions are used. Results for a variety of configurations are shown and are compared with experimental data and other numerical techniques. >

Computation of cutoff wavenumbers of TE and TM modes in waveguides of arbitrary cross sections using a surface integral formulation

A procedure is described for obtaining the cutoff wave numbers of transverse electric (TE) and transverse magnetic (TM) modes in waveguides of arbitrary cross section. A surface integral equation approach is used in which the E-field equation has been transformed into a matrix equation using the method of moments. An iterative technique is used to pick the eigenvalues of the solution matrix which corresponds to the waveguide cutoff wave numbers. The salient features of this technique are its speed, its simplicity, and the absence of any spurious modes when waveguides of arbitrary cross section are treated. The first four modes are tabulated for various waveguides, and the results are in very good agreement with published data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Magnetic field of a direct current in a cylindrical-shaped conductor imbedded in a resistive half-space beneath a conductive surface layer

New applications of Fourier series and integral equation methods have yielded results of significance for electrical prospecting. This paper presents the first model calculation in three dimensions of the magnetic field due to a direct current flow in a conductor imbedded in a half-space beneath a conductive surface layer. The conductor is in the shape of a cylinder of semi-infinite length, and the current flow is from a point source in the surface layer. The vertical component of the magnetic field is given here by a Fourier series integral over a function which satisfies a Cauchy singular integral equation. For the two special cases, namely, an outcropping cylinder with no surface layer and a nearly invisible (in the sense of negligible conductivity contrast with its surrounding) cylinder, this integral equation can be solved exactly to yield closed-form expressions for the vertical component of the magnetic field. The exact solution in the first special case is based on a new generalization of the Hankel integral transform (published separately), and it extends the class of exactly solvable singular integral equations. This exact solution has also yielded as useful byproducts two new standard integrals involving Bessel functions. The integral equation approach used here, in contrast to the well-established surface integral equation approach, is suitable for numerical calculations, and typical magnetic field profiles are presented. The magnetic field results given here would be valuable for interpretation in electrical prospecting (based on the measurement of magnetic fields of direct currents), and the mathematical techniques used here would be useful for future calculations in electromagnetism.

Finite element-boundary integral formulation for electromagnetic scattering

A new formulation is described which combines the most robust attributes of the volume finite element and surface integral equation approaches to electromagnetic boundary value solutions. The result is a numerical technique which may be applied to scattering problems involving configurations having metallic surfaces and inhomogeneous penetrable material situated in open spatial regions. This is accomplished by way of coupling internal region finite element modal field solutions to equivalent currents on the surrounding boundary surface through an appropriate surface integral equation. The method is demonstrated for the special case of scattering by axisymmetric inhomogeneous penetrable objects. Example numerical calculations are presented for validation of the procedure and potential problem areas are discussed.