Mathematical Modeling for Determination the Surface Charge Density and Eddy Current Problem Using the Haar Wavelet

Computational electromagnetics has become a real cornerstone in the design of microwave and millimeter wave components and systems. One of the major computational techniques used in these designs is to solve integral equations with the Method of Moments (MoM). Traditionally, MoM has been used extensively for antennas, transmission lines, filters, couplers, etc., mainly in the MHz and GHz range. However, MoM can also be applied at much higher frequencies. It will be shown that at these frequencies, this computational technique is just emerging. One of the reasons is that there are serious differences with the classical implementations, which prohibits the direct use of the well-known commercially available MoM solvers. For example, one of the differences is the fact that the concept of a surface current, confined to a very thin layer at the surface of a conductor, is not valid any more. In this presentation, the different computational techniques will be reviewed and compared, not only in the traditional microwave and millimeter wave range, but especially at THz, IR, and optical frequencies. In concreto, the MAGMAS MoM solver will be compared to a numerical FDTD solver well-known in the plasmonics research community: Lumerical. It will be shown that also at these very high frequencies, MoM has its proper place in the pallet of computational algorithms.

In the spectral domain method of moments (MoM) solution of printed structures on planar grounded dielectric slabs, the infinite double integrals which appear in the asymptotic parts of the MoM impedance matrix and the MoM excitation vector elements, have been previously transformed to one‐dimensional finite integrals, which have been numerically computed using the highly specialized “International Mathematics and Statistics Library” subroutines. In this paper, these one‐dimensional integrals are evaluated in closed‐form, resulting in an improved efficiency and accuracy for the rigorous investigation of printed antennas and complex millimeter and microwave integrated circuits. Numerical results in the form of mutual impedance between two expansion functions and input impedance of various microstrip antennas are presented to assess the accuracy of these closed‐form expressions. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 882–886, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22274

Finite Element Methods for the Solution of 3D Eddy Current Problems

It is well known that the choice of expansion and testing\nfunctions plays an important role in determining the rate of\nconvergence of the integrals associated with the moment method\nmatrix, and that an improper choice can lead to erroneous results.\nThe main objective of this paper is to critically examine this\nconvergence issue and to provide criteria for the choice of these\nexpansion and testing functions. The question of whether these\nfunctions need to satisfy the Holder condition is also investigated\nand the convergence behavior of the integrals involved in the\nspatial and spectral domain moment method is discussed for some\nrepresentative expansion and testing functions.

In the paper, basing on the electromagnetic field analysis in a system with a thin conducting plate, infinitely extended, a set of equations approximately representing the boundary conditions on the plate is presented as well as the scope of their validity is estimated. Developing the idea announced in the eddy current problem is examined in a system with a thin plate having finite surface dimensions and with an external source of any configuration. The problem has been reduced to the Fredholm's equation of the second kind which, for regular‐shaped plates, can be solved in an explicite form Results of calculations are verified by experiment.

A clever way to simulate Numerically Matched Loads is presented and applied in the mathematical framework of 3-D spectral domain analysis. The numerically matched loads already have been developed in the particular case of the spectral domain approach using a set of subsectional rooftop expansion and testing functions in the moment method. In order to properly model more complicated structures, we propose in this letter an extension of the NML simulation where triangular subdomain functions are used as both expansion and testing functions in the moment method. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 373–377, 1997.

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

An eigensolution expansion of dyadic Green's function for cylindrical sector structures is presented. The cylindrical structure is limited in azimuthal direction by electric or magnetic walls. Due to orthogonality of the eigensolution functions between the walls, the derivation of the dyadic Green's function can be done in spectral domain. The Green's dyad is used to set up an electric field integral equation for the microstrip structure. The solution is carried out with the method of moments. Numerical results for input impedance and radiation pattern are shown.

The application of computational electromagnetics to practical EMI/EMC engineering is an emerging technology. Because of the increased complexity in EMI/EMC issues resulting from advancements in electronics and telecommunications, it is no longer possible to rely exclusively on traditional techniques and tools to solve the growing list of electronic engineering design problems. EMI/EMC Computational Modeling Handbook introduces modeling and simulation of electromagnetics to real-world EMI/EMC engineering. It combines the essentials of electromagnetics, computational techniques, and actual EMI/EMC applications. Included are such popular full-wave computational modeling techniques as the Method of Moments, Finite-Difference Time Domain Technique, Finite Element Method, and several others. The authors have included a myriad of applications for computers, telecommunications, consumer electronics, medical electronics, and military uses. EMI/EMC Computational Modeling Handbook is an invaluable reference work for practicing EMI/EMC engineers, electronic design engineers, and any engineer involved in computational electromagnetics.

A novel higher order large-domain hybrid computational electromagnetic technique based on the finite element method (FEM) and method of moments (MoM) is proposed for three-dimensional analysis of antennas and scatterers in the frequency domain. The geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders. The fields and currents on elements are modeled using curl- and divergence-conforming hierarchical polynomial vector basis functions of arbitrary approximation orders, and the Galerkin method is used for testing. The elements can be as large as about two wavelengths in each dimension. As multiple MoM objects are possible in a global exterior region, the MoM part provides much greater modeling versatility and potential for applications, especially in antenna problems, than just as a boundary-integral closure to the FEM part. The examples demonstrate excellent accuracy, convergence, efficiency, and versatility of the new FEM-MoM technique, and very effective large-domain meshes that consist of a very small number of large flat and curved FEM and MoM elements, with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -refined field and current distributions of high approximation orders. The reduction in the number of unknowns is by two orders of magnitude when compared to available data for low-order FEM-MoM modeling.

Gauging of vector potential formulations of 3-D eddy current problems is considered from a new viewpoint. Assuming that the formulation in terms of E and B of the eddy current problem under consideration has a unique solution, gauging can be regarded as setting conditions on potentials, say A, V and psi , that ensure a one-to-one correspondence between pairs (E,B) and triples (A,V, psi ). This approach decouples two different issues which are usually mixed: gauging properly said and well-posedness of the field problem. Once the above one-to-one correspondence is proved, the existence and uniqueness of (A,V, psi ) will then automatically follow from that of (E,B) whenever the field problem is well-posed. Proving that the correspondence between (E,B) and (A,V, psi ) is one-to-one is a definitely easier task than proving the existence and uniqueness of (A,V, psi ) directly. The approach presented applies to both A,V, and T, psi formulations and to problems ranging from magnetostatics to high frequency electromagnetics; no additional complexity comes out when dealing with nonlinear materials. Practical impact of the proposed method is discussed.

The paper compares the features of integral and differential formulations for the solution of of transient eddy current problems in nonlinear media. The integral method used reduces the nonlinear eddy current problem to the analysis of an equivalent network. Two dual edge element differential formulations are also used. Their field estimates automatically verify Faraday's and Ampere's laws and furnish the distribution of the constitutive error in the solution domain, which provides a useful tool for the refinement of the discretization. The methods are applied to analyze a massive magnetic circuit characterized by the presence of soft and hard magnetic materials with narrow gaps.

A general numerical solver for analyzing microstrip geometries of rectangular shape is presented in this paper. The analytical foundation of this solver is based on an integral equation approach which we formulate in the spatial domain. The unknown surface current on the microstrip is solved by the method of moments using 2D rectangular pulses as the expansion functions. Transmission‐line modeling is then used to parameterize a given microstrip junction or discontinuity. Aided by a graphics interface, this solver can analyze complex structures without incurring additional analytical complexity. We illustrate the accuracy and versatility of our solver by applying it to several different microstrip discontinuities ranging from a single‐stub to an interdigitated capacitor.

A simple moment solution is given for low frequency electromagnetic scattering and radiation problems. The problem is reduced to the corresponding electrostatic and magnetostatic problems. Each static problem is solved using the Method of Moments. The surface of the perfectly conducting scatterer is modeled by a set of planar triangular patches. Pulse expansion functions and point matching testing are used to compute the charge density in the electrostatic problem. For the magnetostatic current a set of charge-free vector expansion functions is used. The problem is formulated assuming the scatterer to be in an unbounded homogeneous region. Scatterers of various shapes, such as the circular disc, the sphere, and the cube are studied. Special attention is paid to a conducting box with a narrow slot. The computed results are the scattered fields, the induced charge and current distributions, and the induced electric and magnetic dipole moments. These are in close agreement with whatever published data are available.

In the first part of this paper the moment method is employed to solve the nonlinear Boltzmann equation. An expansion about a local Maxwellian distribution is used with the basis functions introduced by Waldmann [in Handbuch der Physik, edited by S. Fl\"ugge (Springer, Berlin, 1958), Vol. 12, p. 295]. Earlier approaches are extended by the inclusion of more expansion functions (Sonine polynomials) in order to obtain an approximation for the velocity distribution function. General relations for the coupling of the moments are derived and the resulting transport relaxation equations are solved for the special Couette geometry. The influence of the higher moments on the viscosity coefficients is small, but the higher moments are essential for the distribution function itself. The inclusion of the quadratic collision matrix elements leads to minor modifications only. In another part of the paper the velocity distribution function is obtained from nonequilibrium molecular dynamics. Excellent agreement with the predictions of the moment method is found provided that the constant temperature constraint of the simulation is taken into account by incorporating a nonconservative external force term into the Boltzmann equation. The effect of this modification is discussed in detail for the viscosity coefficients. The non-Newtonian flow behavior of gases is studied on the microscopic level of the velocity distribution function. In addition an isotropic distortion of the Maxwellian distribution is observed.