Measured Equation Of Invariance Research Articles

Measured equation of invariance with non-uniformly distributed line sources as metrons

In the paper, δ functions are employed as simple yet effective metrons to generate measured equation of invariance (MEI) coefficients in the MEI method. Compared with the widely used sinusoidal functions, δ functions greatly reduce the time for the computing of measuring functions and alleviate the ill-conditioning phenomenon of the matrix equations for the determination of MEI coefficients, hence enabling one to interrelate more nodes in the MEI. In this way, the accuracy of the solutions for electrically large scatterers can be improved progressively. It is also found that non-uniformly distributed δ metrons produce more accurate and stable results compared to uniformly distributed metrons. Numerical results for two-dimensional scattering problems are presented to show the accuracy and efficiency of the proposed method.

Measured equation of invariance with linear tensor relation for three-dimensional scattering problems

The postulate of a linear tensor relation in the measured equation of invariance (MEI) is proposed for three-dimensional (3D) problems. As a result, all three components of the field vector to be solved are coupled in the MEI. Combined with the finite difference method, the present method is applied to the analysis of scattering by 3D conducting objects and results in a significant improvement in the accuracy of numerical results as compared to those obtained with an uncoupled linear relation.

Applicability of Insensitive Properties of Measured Equation of Invariance Coefficients for Modified Scattering Objects

The applicability of insensitive properties of measured equation of invariance (MEI) coefficients for the computation of scattering from modified structured bodies is presented in this letter. During scattering computation by using MEI technique (in IE-MEI or SIE-MEI method), the MEI coefficients of the whole scatterer is calculated in the conventional method. In contrast, using the insensitive properties of the MEI coefficients, the new method calculates the MEI coefficients only around the modified area if some portion of the scatterer is modified and reuse those for the other portion of the scatterer. Thus CPU time for solving the modified problem can be saved by using the new method. The numerical results verified the validity of the new technique which is compared with the available numerical solutions.

Measured equation of invariance based on wavelet transform technique

AbstractWavelets have been applied in conjunction with the measured equation of invariance (MEI) to solve a two‐dimensional scattering problem. The MEI method with wavelet metrons is referred to as the wavelet MEI method. Haar wavelets are used as metrons. The proposed method leads to a significant savings in CPU time compared to the standard MEI method that does not use wavelets as metrons. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 33: 47–49, 2002; DOI 10.1002/mop.10227

Finite difference analysis of electrically large parabolic reflector antennas

A reflector antenna is analyzed using the finite-difference method (FD). The induced current densities on an axially symmetric parabolic reflector are rigorously calculated. The measured equation of invariance (MEI) is used to terminate the FD mesh very close to the reflector surface. To take advantage of the axial symmetry, the theory of coupled-azimuthal potentials (CAPs) is employed. Illustrative results are obtained for reflector antennas with different aperture dimensions. Results by physical optics (PO) approximation are also included for comparison. The purpose of this paper is not to replace ray optics (RO) and PO in the design of reflector antennas, but to demonstrate the advancement in the FD method, which hitherto was limited to low-frequency and closed-boundary regime. The calculated surface current densities of a reflector antenna do show that the normal component of the current densities at the edges exhibits high standing waves which are missing in PO, and which we know should be there. The standing wave of current densities may not affect the main beam, but certainly will have an effect on side lobes and have a major impact in estimating the loss of the antenna.

Implementation of the matrix sparse decomposition technique to the scattering of two‐dimensional homogeneous dielectric cylinders

In this paper, a novel matrix‐thinning technique, matrix sparse decomposition (MSD) [Liu et al., 1998, 1999], has been implemented to solve the scattering of waves by two‐dimensional (2‐D) homogeneous dielectric cylinders for the first time. The MSD technique is a further development of the integral equation formulation of the measured equation of invariance (MEI) (IE‐MEI) [Rius et al., 1996a; Hirose et al., 1999a]. The MSD describes the local relationship between total currents and scattered fields rather than that between the scattered electric fields and the scattered magnetic fields in the IE‐MEI. The MSD directly thins a dense matrix from singular integral equations, such as method of moments (MOM), into two sparse matrices. The IE‐MEI method has difficulty in solving thin wire or thin plate structure problems. However, the MSD can do it without a hitch. Numerical examples for the scattering of 2‐D homogeneous dielectric circular and rectangular cylinders under both transverse magnetic and transverse electric plane wave incidences show that the MSD is a simple and effective technique to thin the MOM dense matrix.

Scattering of 2-D conducting concave object by MoM matrix decomposition technique

Recently, fast computational methods based on directly thinning the method-of-moments (MoM) matrix have been introduced. In this paper, we propose a simple matrix decomposition technique, based on the concept of the measured equation of invariance (MEI), to thin the MoM matrix numerically. This technique is referred to as matrix decomposition by MEI (MDMEI). Some effort is required to add MDMEI to any of a variety of MoM programs for electromagnetic problems, such as electrostatic problems, wire antennas, and 2-D conducting object scattering. However, in this paper, we only show the results of a 2-D conducting concave object scattering under a plane-wave incidence. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 25: 149–152, 2000.

Capacitance extraction of integrated-circuit interconnects by matrix decomposition based on MEI concept

Recently, fast computational methods based on directly thinning the matrix of Method of Moment (MoM), have been introduced. In this paper, we propose a simply technique, based on the concept of Measured Equation of Invariance (MEI), to thin the MoM matrix numerically. This technique is referred to as Matrix Decomposition by MEI (MDMEI). A little effort is required to add the MDMEI to any of the variety of MoM programs for electromagnetic problems, such as the electrostatic problems, wire antennas, and two-dimensional (2-D) conducting object scattering. However, in this paper we only demonstrate how MDMEI is used in the capacitance extraction in integrated-circuit interconnects. The approach is verified by 2-D and 3-D examples with computing errors within 2–4%. For the present achievement, the sparsity rates of the resultant matrices in MDMEI depend on the problem to be solved. Further research is required to keep the bandwidths unchanged with object size increasing so that the sparsity rates of the MDMEI matrices are expected to improve.

Decomposing Mom Matrix To Sparse Matrices for Wire Antennas and Wire Scattering By Mei Method

Recently, fast computational methods for directly thinning the matrix of Method of Moment (MoM), such as Impedance Matrix Localization (IML) [1], wavelet expansions [2], and Reduced Expansion and Field Testing (REFT) [3], have been introduced. In this paper, we propose a simple approach to thin the MoM matrix by using the concept of Measured Equation of Invariance (MEI) [4]. This approach is referred to as Matrix Decomposition by MEI (MDMEI). A little effort is required to add the resulting method to any of a variety of MoM programs for solving problems of wire antennas, and wire scattering. When sparsity rate (ratio of non-zero elements to total elements) of the MDMEI matrix reaches to 2.8%-5.6%, the results for a thin wire antenna with 8λ length and a thin wire scattering with 16λ length obtained. by the MDMEI is in excellent agreement to the MoM results.

An iterative algorithm based on the measured equation of invariance for the scattering analysis of arbitrary multicylinders

It is known that the measured equation of invariance (MEI) is generally valid for outgoing waves just as other absorbing boundary conditions (ABCs). However, for the scattering problem of multicylinders, the scattered field from one cylinder is just the in-going incident wave to other cylinders. So the MEI cannot be directly applied to the scattering problem of multicylinders. In this paper, an iterative algorithm based on the MEI is first proposed for the scattering problems of multicylinders with arbitrary geometry and physical parameters. Each cylinder is coated with several layers of meshes and the MEIs are applied to the truncated mesh boundaries. It has been demonstrated that the MEI can truncate the meshes very close to the surfaces of the cylinders and then results in dramatically savings in memory requirements and computational time. The MEI coefficients of each cylinder can be stored and reused to form the sparse matrices during each iteration procedure as they are independent of excitations. So more central processing unit (CPU) time is saved as the MEI coefficients are calculated only once in the algorithm. The method can be applied to problems of various kinds of multiple cylinders with arbitrary configurations and cross sections. Numerical results for the scattered fields are in good agreement with the data available. Finally, examples are given to show the iterative algorithm applicable to electrically large multicylinders coated with lossy media.

Validity of the measured equation of invariance

The measured equation of invariance (MEI) is derived without any postulates. It is shown that the coefficients of the MEI are invariant to the field of excitation. However, the accuracy of the MEI solution is closely related to the number of nodes in the MEI. Coupling more nodes improves progressively the accuracy of the MEI solution. With increasing nodes, the matrix problem for the determination of the MEI coefficients becomes seriously ill conditioned and generally must be solved using multiple precision arithmetic. The consequences of the ill-conditioning phenomenon are discussed.

Time-domain MEI method for radiation of line source

The measured equation of invariance (MEI) method is used in the time domain to numerically generate the absorbing boundary condition (ABC) for a line source radiating in two-dimensional free space. Compared with the real solution and other ABCs, excellent accurate results are obtained when only two space steps are employed. Further research will extend this method to the scattering problems of arbitrarily shaped conducting objects.

High frequency loss and electromagnetic field distribution for striplines and microstrips [MCM packages

A new three-component measured equation of invariance (MEI) boundary condition is developed and applied to the hybrid edge/nodal vector finite element method. The electric field distribution on the cross section of various lossy transmission lines is calculated. The propagation constant of a lossy transmission line with coated conductor strip is also calculated. The three-component MEI boundary condition simulates the field distribution on the artificial boundary for electromagnetic field excited by the surface charge density and the three vector components of the electric current density. Numerical experiments are performed to test the method by comparing calculated transmission loss with the measured data.

Application of on-surface MEI method on wire antennas

The formulas of the on-surface measured equation of invariance (OSMEI) for wire antennas are derived. The unknowns of each node on the antenna surface are expressed by the vector potential function and surface current density. The unknowns in the vicinity of each node are coupled in a linear equation and the coefficients of the linear equation are determined by the measured equation of invariance (MEI) method. The final impedance matrix obtained by the OSMEI is a highly sparse matrix. It demonstrates that the currents on thin wire antennas may also be solved by a differential equation-based formulation in addition to the conventional integral equations.

SLIM: a slender technique for unbounded-field problems

Electrostatic- and electromagnetic-field problems in unbounded regions are often solved using finite differences (FD's) or finite elements (FE) combined with approximate boundary conditions. Inversion of the sparse FD or FE matrix is then required. First-order or higher order absorbing boundary conditions may be used, or one can use more accurate boundary conditions obtained by the measured equation of invariance (MEI) or by iteration. The more accurate boundary conditions are helpful because they permit reduction of the size of the mesh and, thus, the number of unknowns. In this paper, we show that the process can be carried to a maximally simple limit in which the mesh is reduced to a single layer and the matrix-inversion step disappears entirely. This results in the single-layer iterative method (SLIM), an unusually simple technique for unbounded-field problems. Computational experiments demonstrate the effectiveness of SLIM in electrostatics, and also in electrodynamic examples, such as scattering of TM plane waves from a perfectly conducting cylinder. The technique is most likely to be useful in large or complex problems where simplification is helpful, or in repetitive calculations such as scattering of radiation from many angles of incidence.

Application of discrete periodic wavelets to measured equation of invariance

Recently, the wavelet expansions have been applied in field computations. In the frequency domain, the application is focused on the thinning of matrices arising from the method of moment (MoM). The thinning of matrices can best be done by the measured equation of invariance (MEI), which provides sparsity almost without sacrificing accuracy in that the boundary equation it entails is convertible to that of the MoM. The real power of the wavelet expansions is to give high resolution in convolution integrals. High resolution is also needed in the process of finding the MEI coefficients, which are obtained via an integration process almost identical to that of the MoM. In this paper, it is shown that when the fast discrete periodic wavelets (FDPW) are used as metron currents in the MEI method, the resolutions of the MEI coefficients are improved at high-frequency computations or at geometric extremities. The level of sparsity of the MEI is much more favorable than that achievable by the thinning of MoM matrix using the wavelet expansions. The role of FDPW in the MEI happens to be more fitting than its place in the MoM.

Electrostatic solution for three-dimensional arbitrarily shaped conducting bodies using finite element and measured equation of invariance

Differential equation techniques such as the finite element (FE) and finite difference (FD) have the advantage of sparse system matrices that have relatively small memory requirements for storage and relatively short central processing unit (CPU) time requirements for solving electrostatic problems. However, these techniques do not lend themselves as readily for use in open-region problems as the method of moments (MoM) because they require the discretization of the space surrounding the object where the MoM only requires discretization of the surface of the object. A relatively new mesh truncation method known as the measured equation of invariance (MEI) is investigated augmenting the FE method for the solution of electrostatic problems involving three-dimensional (3-D) arbitrarily shaped conducting objects. This technique allows truncation of the mesh as close as two node layers from the object. The MEI views sparse-matrix numerical techniques as methods of determining the weighting coefficients between neighboring nodes and finds those weights for nodes on the boundary of the mesh by assuming viable charge distributions on the surface of the object and using Green's function to measure the potentials at the nodes. Problems in the implementation of the FE/MEI are discussed and the method is compared against the MoM for a cube and a sphere.

An iterative measured equation technique for electromagnetic problems

An iterative measured equation technique (IMET) is presented for a numerical solution of electromagnetic problems. This technique is an extension and improvement of the method of measured equation of invariance (MEI). In this technique, an iterative scheme is designed in such away that a new set of metrons used to generate the measured equations is formed in each iteration based on the solution of the previous iteration. The new metrons are more meaningful in that they converge to the physical quantity of interest such as the surface current density for electrodynamic problems and the surface charge density for electrostatic problems. The IMET offers several advantages over the MEI method because it requires only two mesh layers, resulting in a significant reduction in the memory requirement and computing time. More importantly, it provides a means for a systematic improvement of the accuracy of solution. The IMET is applied successfully to two-dimensional (2-D) electrodynamic and three-dimensional (3-D) electrostatic problems. Numerical results show that the technique is highly accurate and the iterative process converges very quickly, usually within two iterations.

Sidewall inclined slot in a rectangular waveguide: Theory and experiment

A novel analysis to compute the admittance characteristics of the slots cut in the narrow wall of a rectangular waveguide, which includes the corner diffraction effects and the finite waveguide wall thickness, is presented. A coupled magnetic field integral equation is formulated at the slot aperture which is solved by the Galerkin approach of the method of moments using entire domain sinusoidal basis functions. The externally scattered fields are computed using the finite difference method (FDM) coupled with the measured equation of invariance (MEI). The guide wall thickness forms a closed cavity and the fields inside it are evaluated using the standard FDM. The fields scattered inside the waveguide are formulated in the spectral domain for faster convergence compared to the traditional spatial domain expansions. The computed results have been compared with the experimental results and also with the measured data published in previous literature. Good agreement between the theoretical and experimental results is obtained to demonstrate the validity of the present analysis.

A novel exact two-point field equation (2PFE) for solving electromagnetic scattering problems

The finite-difference (FD) method is a basic technique for solving differential equations. The disadvantage of it for electromagnetic (EM) problems of an open region is that the mesh needs to be terminated with the application of a proper boundary condition. In this paper, a novel exact two-point field equation (2PFE) is derived from rigorous analysis of the radiation field and it is proposed to be used as the termination boundary condition (BC) for solving EM scattering problems in the open region by the iterative FD method. This 2PFE-BC approaches its exact solution through the iteration process and, at the same time, the scattered field and the induced current density approach their exact solutions. The novel 2PFE is simple in concept and easy to apply. The validity of the 2PFE and the iterative FD method has been tested. Several two-dimensional (2-D) scattering problems have been successfully solved. The results agree very well with those obtained by method of moments (MoM) or measured equation of invariance (MEI).