Mesh Truncation Research Articles

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.

Comparative study of the PML and C-COM mesh-truncation techniques

The recently developed concurrent complementary operators method (C-COM) for mesh truncation of finite-difference time-domain (FDTD) problems has been shown to produce a substantial improvement in differential equation-based absorbing boundary conditions (ABCs). In this letter, the C-COM and the perfectly matched layer (PML) approaches are compared using two-dimensional problems. Two representative geometries are considered that address the absorption of evanescent waves and waves traveling at near-grazing incidence. It is found that the C-COM yields significant improvements over the PML solution for these geometries.

The concurrent complementary operators method for FDTD mesh truncation

The complementary operators were developed for the purpose of truncating the computational domain of finite-difference time-domain (FDTD) open-region simulations. In the original implementation of the complementary operators method, two independent simulations were performed. The solutions from these two simulations were then averaged to obtain a solution devoid of first-order reflections. The complementary operators are implemented in a concurrent fashion. This new implementation, referred to as the concurrent complementary operators method (C-COM), requires only one simulation, thus reducing the operation count by approximately one half. The implementation for two-dimensional (2-D) space is outlined first with emphasis on reducing reflections from corner regions; then the extension to three dimensions is developed. Numerical results are provided in addition to an analysis of the source of error that arises from this new implementation.

Dynamic analysis of V transmission lines

In this work, a dynamic analysis of the V line is presented. Previous work analyzed the performance of this structure for low frequency applications using quasistatic approximations. Here, we extend the analysis of the V line into the higher frequency range where dispersion becomes significant and where it cannot be predicted by quasistatic methods. We show that the V line provides features and advantages that are not present in the conventional microstrip structures, most notably the appreciable decrease in coupling between adjacent lines in comparison with the conventional microstrip structure. This feature makes the V line well suited for high packaging density applications. The full-wave analysis is carried out using a Yee-cell based finite-difference time-domain (FDTD) method, while enforcing a highly efficient and stable mesh truncation technique. Results are presented for a single and multiconductor structures.

A new finite-element solution for parameter extraction of multilayer and multiconductor interconnects

In this letter, the geometry-independent measured equation of invariance is adopted with the finite-element method for mesh truncation and is successfully used in parameter extraction of interconnects. The major advantage is overcoming the tedious closed-form Green's function deduction and disagreeable Sommerfeld integral calculation for multilayer and multiconductor structures. Moreover, the high-order finite-element is employed to increase the accuracy and save computer resources. Furthermore, an optimizing numerical scheme, which is found to be very efficient, is developed to solve finite-element equations.

Comparison of fast integral mesh truncation schemesfor hybrid FE-BI systems

By virtue of their low operation count, the application of fast integral methods such as the fast multipole (FMM) and adaptive integral methods (AIM) result in a substantial quickening of the boundary integral portion of the hybrid finite element-boundary integral (FE-BI) method, independent of the shape of the BI contour. Recently, various versions of the FMM have been proposed, each introducing a different approximation to the implementation of the boundary integral. The main goal of this Letter is to provide a comparison of the effect of these fast integral algorithms on the boundary integral when used in conjunction with the traditional FE-BI method.

Investigation of nonplanar perfectly matched absorbers for finite-element mesh truncation

We present a detailed theoretical and numerical investigation of the perfectly matched layer (PML) concept as applied to the problem of mesh truncation in the finite-element method (FEM). We show that it is possible to extend the Cartesian PML concepts involving half-spaces to cylindrical and spherical geometries appropriate for closed boundaries in two and three dimensions by defining lossy anisotropic layers in the relevant coordinate systems. By using the method of separation of variables, it is possible to solve the boundary value problems in these geometries. The analytical solutions demonstrate that under certain conditions, outgoing waves are absorbed with negligible reflection, and the transmitted wave is attenuated within the PML. To reduce the white space in radiation or scattering problems, conformal PMLs are constructed via parametric mappings. It is also verified that the PML concept, which was originally introduced for problems governed by Maxwell's equations, can be extended to cases governed by the scalar Helmholtz equation. Finally, numerical results are presented to demonstrate the use of the PML in FEM mesh truncation.

Hybrid finite-element methodologies for antennas and scattering

This paper is an overview of the finite-element method (FEM) as applied to electromagnetic scattering and radiation problems. A brief review of the methodology is given with particular emphasis on new developments over the past five years relating to feed modeling, parallelization, and mesh truncation. New applications which illustrate the method's capabilities, versatility, and utility for general purpose application are discussed.

Error estimation in finite element transient dynamic analysis using modal superposition

Presents a method of obtaining an optimal mesh in the finite element analysis of two‐dimensional linear elastodynamic problems under transient dynamic loading, which is based on a generalization of the Z‐Z criterion for discretization error estimation for time‐dependent problems. The optimal mesh limits the error due to discretization within a prescribed value, and studies of all other possible errors involved in finite element transient dynamic analysis are carried out systematically. Also proposes methods to study and limit the modal truncation error. Numerous examples show the capabilities of the proposed methods and the importance of the optimal mesh and modal truncation error in finite element transient dynamic analysis.

On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer

Recently, Berenger introduced a perfectly matched layer (PML) technique for absorbing electromagnetic waves. In the present paper, a perfectly matched layer is proposed for absorbing out-going two-dimensional waves in a uniform mean flow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative to the mean flow, and vorticity and entropy waves, which travel with the mean flow. The PML equations to be used at a region adjacent to the artificial boundary for absorbing these linear waves are defined. Plane wave solutions to the PML equations are developed and wave propagation and absorption properties are given. It is shown that the theoretical reflection coefficients at an interface between the Euler and PML domains are zero, independent of the angle of incidence and frequency of the waves. As such, the present study points out a possible alternative approach for absorbing out-going waves of the Euler equations with little or no reflection. In actual computations, nonetheless, numerical reflection will still occur due to discretization and mesh truncation, depending on the thickness of the PML domains and absorption coefficients used. Numerical examples that demonstrate the validity of the proposed PML equations are presented.

Synthesis of a wideband, global boundary condition based on the reciprocity theorem for finite-element analysis

Two broad categories of absorbing boundary conditions (ABCs) are typically utilized in the finite-element method (FEM) for mesh truncation. The first class of hybrid techniques, which are global and rigorous in nature, cause the augmented FEM system matrix to exhibit a loss of sparsity. In the second class of ABCs, which are strictly local but approximate in character, the sparsity of the FEM system matrix is preserved at the expense of numerical accuracy. In this article we synthesize a novel wideband boundary condition based on an application of the reciprocity theorem. Although this boundary condition is essentially global in nature, it tends to maintain the sparsity of the FEM matrix to the same extent that the numerical ABCs (NABCs) do. © 1996 John Wiley & Sons, Inc.

How to Design the Imperfect Berenger PML

ABSTRACT In contrast to the previously popular second order Miir absorbing boundary condition (ABC), the recently introduced Berenger Perfectly Matched Layer (PML) ABC can be designed to lower the numerical reflection coefficient associated with mesh truncation by several orders in magnitude. Nonetheless the PML ABC is not perfect; it does have a numerical reflection coefficient. This reflection coefficient is characterized in the time and frequency domains for narrow and broad bandwidth pulses from several points of view including specifically its behavior in terms of the magnitude of the loss tangent in the PML and the thickness of the PML. It is demonstrated that for broad bandwidth pulses the PML ABC exhibits a low frequency increase in its reflection coefficient that is associated with the actual thickness of the PML layer in terms of the longest wavelengths contained in the signal incident upon it. Moreover, it is shown that the effectiveness of the PML ABC saturates when the loss parameters of the ...

Combining PML and ABC for the finite-element analysis of scattering problems

A perfectly matched layer (PML) is combined with an absorbing boundary condition (ABC) for mesh truncation in the finite-element solution of electromagnetic scattering problems. It is shown by one-, two-, and three-dimensional examples that the combined PML and ABC has less undesired reflection than the PML or ABC alone. ©1996 John Wiley & Sons, Inc.

Mesh truncation by perfectly matched anisotropic absorbers in the finite‐element method

In this article we develop the concept of a conformal, anisotropic material layer for mesh truncation in the finite-element analysis of electromagnetic wave propagation problems. The layer, which can be as close as λ/10 from the object, absorbs electromagnetic waves of arbitrary frequency, irrespective of the incident angle. © 1996 John Wiley & Sons, Inc.

Mesh truncation by perfectly matched anisotropic absorbers in the finite-element method

In this article we develop the concept of a conformal, anisotropic material layer for mesh truncation in the finite-element analysis of electromagnetic wave propagation problems. The layer, which can be as close as λ/10 from the object, absorbs electromagnetic waves of arbitrary frequency, irrespective of the incident angle. © 1996 John Wiley & Sons, Inc.

Application of anisotropic absorbers to the analysis of MMIC devices by the finite element method

Anisotropic material characteristics can be exploited to construct absorbers that provide reflectionless interfaces for waves at arbitrary incident angles. Since their computer implementation requires the specification of appropriate material characteristics only, these perfectly matched layers (PMLs) allow for simple but accurate mesh truncation schemes for the finite element method. The present paper extends the concept of PMLs to waveguides of inhomogeneous cross-sections and layered media. We describe the use of generalized PMLs in a software package for (M)MIC devices and demonstrate the benefits of the suggested approach by two numerical examples.

Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers

Perfectly matched layers (PMLs), which are employed for mesh truncation in the finite-difference time-domain (FDTD) or in finite element methods (FEMs), can be realized by artificial anisotropic materials with properly chosen permittivity and permeability tensors. The tensor constitutive parameters must satisfy the Kramers-Kronig relationships, so that the law of causality holds. These relations are used to relate the real and imaginary parts of the constitutive parameters of the PML media to deduce the asymptotic behaviors of these parameters at low and high frequencies.

Mesh truncation in the finite‐element frequency‐domain method with a perfectly matched layer (PML) applied in conjunction with analytic and numerical absorbing boundary conditions

AbstractThe perfectly matched layer (PML) concept, introduced by Berenger with the aim of synthesizing an absorbing boundary condition (ABC) for the finite‐difference‐time domain (FDTD) method, is modified and extended to finite‐element frequency‐domain (FEFD) applications. The governing equations that characterize this approach do not require the splitting of the field components of interest, nor do they involve negative conductivity parameters. The PML‐type medium is employed together with analytic and numerical ABCs as its termination to obtain an improved boundary condition for the FEFD method at a limited increase in computational cost. © 1995 John Wiley & Sons, Inc.

A numerical absorbing boundary condition for quasi‐TEM analysis of microwave transmission lines using the finite‐element method

AbstractThis article presents a new numerical absorbing boundary condition useful for quasi‐TEM analysis of open‐type microwave transmission lines employing the finite‐element method (FEM). The standard FEM equations are used for the inner nodes of the mesh, and mesh truncation is achieved by imposing linear relationships for the nodes on the outer boundary and their neighbors, which are derived from the appropriate Green's functions. This approach greatly reduces the mesh size, while preserving the sparsity of the FEM equations. Higher‐order approximations of the absorbing boundary condition are derived through the use of higher‐order finite elements. © 1995 John Wiley & Sons, Inc.

A numerical absorbing boundary condition for 3D edge‐based finite‐element analysis of very low‐frequency fields

AbstractThe numerical absorbing boundary condition (NABC) approach in conjunction with the three‐dimensional edge‐based formulation of the finite‐element method (FEM) is applied to the problem of very low frequency (VLF) electromagnetic radiation and scattering. The key step in the NABC approach is to derive linear relationships that link the values of the field along the boundary edges to those at the neighboring ones. Derived via a robust SVD‐based scheme, these linear relationships are satisfied, to within a certain tolerance, by all of the outgoing weve components. They can thus be used in lieu of the FEM equations employing the usual absorbing boundary conditions for the boundary edges. Although the NABC approach provides a simple way to derive an accurate local boundary condition for mesh truncation at an arbitrary frequency, it appears to be the only viable absorbing boundary condition at VLF, because the conventional absorbing boundary conditions are almost as reflecting as perfectly conducting magnetic walls at these frequencies. © 1995 John Wiley & Sons, Inc.