Finite Element-boundary Integral Formulation Research Articles

FE-BI Formulations for Characteristic Modes

Two types of finite-element boundary-integral (FE-BI) formulations are developed to determine the characteristic modes (CMs) of general composite material objects. The electromagnetic (EM) fields inside the general objects with inhomogeneous anisotropic material are flexibly formulated using the finite-element (FE) formulations, while the EM fields outside the general material objects are accurately described using the boundary-integral (BI) equations defined on the general objects’ boundary. The developed FE-BI formulations result in two generalized Hermitian eigenvalue (GHE) equations for the solution of CMs. These two novel FE-BI based formulations provide clear physical explanation for the analysis of the CMs of the general objects. It is found that their corresponding eigenvalues indicate the imaginary part of the complex power of CMs, which are zero at resonance. Simulated results for typical dielectric resonators are presented to clearly demonstrate the accuracy of the proposed FE-BI formulations.

Location and Sizing of Defects in Coated Metallic Pipes Using Limited View Scattered Data in Frequency Domain

The inverse problem of reconstructing the location and size of defects in a coated metallic pipe from single frequency limited view electromagnetic scattered field is considered. Specifically, the paper addresses the problem of assessing shape changes in the shadow region entailed by limited view data in 2D by operating in an intermediate size parameter (ka) range. The inverse scattering problem is formulated as a non-linear optimization problem solved through genetic algorithm that seeks to minimize in the least-squared sense the difference between measured data and simulated data through iterations of the solution to a forward problem. The hybrid finite element boundary integral formulations were used to solve the forward problem of coated metallic pipe with defects. The proposed inversion methodology was applied for shadow region shape change assessment to determine whether a metal pipe is coated or not, and to reconstruct the location and size of corrosion-like defect, over a range of size parameters. The study is carried out using transverse electric and transverse magnetic polarized fields. To understand the effect of coating on backscattered fields, parametric studies are conducted using numerical data. A range for size parameter \(2.0\le ka\le 4\) was found to produce the highest contrast between defect free pipe and coated metallic pipes with defects in the deep shadow region. The experiments are carried out using vector network analyzer in an anechoic chamber. The inversion results obtained using measured data were found be in good agreement with inversion results obtained using synthetic data. Estimated extent of corrosion in deep shadow region of a coated metallic pipe was found to be within 9 % of actual extent of corrosion.

A Finite Element-Boundary Integral Formulation for Numerical Simulation of Scattering by Discrete Body-of-Revolution Geometries

A hybrid technique is presented that combines the finite element and boundary integral methods for simulating electromagnetic scattering from discrete body-of-revolution (DBOR) geometries. The inherent angular periodicity present in a DBOR is utilized to decompose the system matrix into several smaller modal problems, which can be solved efficiently with sparse direct solvers. The proposed method has been validated numerically and provides an efficient tool for the simulation of the scattering by DBOR geometries in electromagnetics.

The Implicit, Element Residual Method for a Posteriori Error Estimation in FE-BI Analysis

Hybrid finite element-boundary integral (FE-BI) formulations are widely used for electromagnetic analysis. In such analysis, estimation of the mesh-wide error distribution and global solution quality may be used to drive adaptive refinements. In this paper, an a posteriori error estimator based on the implicit, element residual method is constructed for the cavity FE-BI formulation, with dominant-mode coaxial port excitation. It is based on solving elemental variational boundary value problems of the error, using approximate, localized boundary conditions. Results show that the scheme can be applied to practical problems, yielding significant reductions in computational cost.

Application of the inner–outer flexible GMRES-FFT method to the analysis of scattering and radiation by cavity-backed patch antennas and arrays

The finite element-boundary integral (FE-BI) method is applied for the analysis of scattering and radiation by cavity-backed patch antenna and arrays. In this investigation, the FE-BI formulations have been implemented using brick element volumes and it allows for a particular use of the efficient FFT-based iterative solver. The inner–outer flexible GMRES algorithm is applied to solve the equation with a higher convergence speed when compared with the standard GMRES algorithm.

Boundary/Finite Element Modeling of Three-Dimensional Electromagnetic Heating During Microwave Food Processing

A three-dimensional 3D finite element-boundary integral formulation is presented for the analysis of the electric and magnetic field distribution, power absorption, and temperature distribution in electrically conductive and dielectric materials. The hybrid finite/boundary method represents an optimal approach for modeling of large-scale electromagnetic-thermal materials processing systems in which the volume ratio of the sample over the entire computational domain is small. To further improve the efficiency, the present formulation also incorporates various efficient solvers designed specifically for the solution of large sparse systems of linear algebraic equations. The resulting algorithm with a compressed storage scheme is considered effective and efficient to meet the demand of 3D large scale electromagnetic/thermal simulations required for processing industries. Examples of 3D electromagnetic and thermal analysis are presented for induction and microwave heating systems. Numerical performance of the computer code is assessed for these systems. Computed results are presented for the electric field distribution, power absorption, and temperature distribution in a food load thermally treated in an industrial pilot scale microwave oven designed for food sterilization. Computed temperature distribution in a food package compares well with experimental measurements taken using an infrared image camera.

Adaptive finite element-boundary integral analysis for electromagnetic fields in 3-D

This paper presents a complete adaptive finite element-boundary integral (FE-BI) analysis scheme for the time-harmonic, electromagnetic analysis of three-dimensional inhomogeneous scatterers/radiators in free-space. The adaptive scheme is based on an FE-BI formulation which yields electric and magnetic field solutions simultaneously. It employs a posteriori error estimates which exploit the availability of both field solutions and estimates error distributions and global solution quality for the electric and magnetic fields separately. It automatically determines which elements should be refined in order to equi-distribute the estimated error, based on the type of refinement requested (h,p or hp). This automatic determination is based on extrapolating the elemental error estimates. The algorithm terminates when specified tolerance levels are reached by the electric and/or magnetic field global solution quality estimates. The only required user specifications within the algorithm are the termination tolerances and the types of refinements to effect. Results are presented which show that within the scope of the presented error measures significant reductions in computational cost may be achieved. The proposed scheme could be used with other types of error estimates and it could be adapted to other FE or FE-BI formulations.

On the Variational Formulation of Hybrid Finite Element-Boundary Integral Techniques for Electromagnetic Analysis

This paper addresses the time-harmonic, electromagnetic analysis of a three-dimensional inhomogeneous radiator/scatterer in free-space. Such analysis can be carried out by combining the finite element method (FEM) with the method of moments (MoM), which yields finite element-boundary integral (FE-BI) formulations. A general framework is presented, within which stationary FE-BI formulations can be established (variational boundary-value problems), which relate to equivalent, underlying variational principles (stationary functionals). The formulations are shown to be accurate, robust and computationally efficient. They avoid the problem of interior resonances without resorting to the combined field integral equation and they result in symmetric system matrices, which preserve reciprocity explicitly. Thus, the stationary FE-BI framework combines the FEM and MoM on the continuous level, as opposed to the usual approach of hybridization after discretization, which generally leads to asymmetric matrices. The stationary FE-BI framework allows one to solve either for the electric and magnetic fields on the volume of the problem domain, or for one volume and one exterior surface field quantity, with only marginal differences in computational cost. The volume-surface formulations have the same storage requirements as previous FE-BI formulations and can be more efficiently solved. The volume-volume formulations provide simultaneous solutions of the electric and magnetic fields, which could for instance be used to construct error estimators directly based on Maxwell's equations.

A PRACTICAL APPROACH TO MODELING DOUBLY CURVED CONFORMAL MICROSTRIP ANTENNAS - ABSTRACT

Designers are increasingly integrating conformal microstrip antennas into the curved structures of either air or land vehicles. Quite often, these structures are doubly curved (e.g. curved along two orthogonal surface directions). This practice necessitates the development of accurate codes versatile enough to model conformal antennas with arbitrarily shaped apertures radiating from doubly curved surfaces. Traditional planar-structure-based design techniques are not well suited for this application. A hybrid finite element-boundary integral formulation appropriate for the high-frequency analysis and design of doubly curved conformal antennas is introduced in this paper. The novelty of this approach lies in its use of an asymptotic prolate spheroidal dyadic Green's function to model the physics of curved surface diffraction. To demonstrate the utility of this approach, the effects of curvature on the resonant frequency and input impedance of both a doubly curved conformal square and circular patch antenna are investigated. Different feed positions are also considered. Due to a paucity of published experimental data, the numerical results are benchmarked by comparison with the results for planar square and circular patch antennas. The planar results are obtained by using an experimentally validated planar finite element-boundary integral code.

A Practical Approach to Modeling Doubly Curved Conformal Microstrip Antennas

Designers are increasingly integrating conformal microstrip antennas into the curved structures of either air or land vehicles. Quite often, these structures are doubly curved (e.g. curved along two orthogonal surface directions). This practice necessitates the development of accurate codes versatile enough to model conformal antennas with arbitrarily shaped apertures radiating from doubly curved surfaces. Traditional planar-structure-based design techniques are not well suited for this application. A hybrid finite element-boundary integral formulation appropriate for the high-frequency analysis and design of doubly curved conformal antennas is introduced in this paper. The novelty of this approach lies in its use of an asymptotic prolate spheroidal dyadic Green’s function to model the physics of curved surface diffraction. To demonstrate the utility of this approach, the effects of curvature on the resonant frequency and input impedance of both a doubly curved conformal square and circular patch antenna are investigated. Different feed positions are also considered. Due to a paucity of published experimental data, the numerical results are benchmarked by comparison with the results for planar square and circular patch antennas. The planar results are obtained by using an experimentally validated planar finite element-boundary integral code. 296 Macon, Trott, and Kempel

Incorporate Approximate Boundary Conditions Into the Fast and Accurate Finite-Element Boundary-Integral Formulation for Scattering By Complex Targets

In this paper, two useful approximate boundary conditions, impedance boundary condition and resistive boundary condition, are incorporated into the newly-developed fast and accurate finite-element boundary-integral formulation for computing scattering by complex targets. Several representative numerical results are given, demonstrating the validity of this incorporated fast and accurate FE-BI formulation.

Scattering and radiation from planar structures containing small features using a finite element-fast integral method

Fast integral equation algorithms such as the adaptive integral method (AIM) have been demonstrated to reduce memory and execution time associated with moment method solutions for computing electromagnetic scattering and radiation from arbitrarily shaped three-dimensional geometries. The authors examine the efficiency of AIM in modelling planar structures that contain small and intricate details as is the case with spirals and slot antennas. Such geometries require high tessellation due to the inclusion of very small features resulting in a large number of unknowns. The AIM, with its ability to translate the original grid into an equivalent sparser uniform grid, is uniquely suited to the analysis of such geometries. The application of the AIM in connection with finite element-boundary integral formulation for cavity-backed antennas is also presented.

A coupled FE and boundary integral equation method based on exterior domain decomposition for fluid-structure interface problems

A coupled finite element and exterior domain decomposition-based boundary integral formulations for the solutions of two- or three-dimensional time-harmonic fluid-structure interaction problems is described in this paper. It is known that the memory limitation of computers has been one of the major obstacles for solving large scale high frequency fluid-structure interface systems using various existing nonlocal finite element and boundary integral equation coupling techniques due to the fully populated resultant matrix generated from the boundary integral equation representation. The essence of this study is to decompose, through domain decomposition of the exterior region, the original exterior problem into arbitrary subproblems with data sharing only at the interfaces. By decomposing the exterior infinite domain into an appropriate number of infinite subdomains, this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized, blockwise-banded system of discretized equations. The size of an individual submatrix (i.e. a block) that is associated with an exterior subdomain may be decided by the user, and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated. In addition, the method is suited for parallel processing since the data associated with each subdomain (impedance matrices, load vectors, etc.) may be generated in parallel, and the communication needed will be only for the interface values. Most significantly, unlike the existing coupled finite element and boundary integral equation techniques that are valid for all frequencies, our method avoids the use of both the hypersingular operator and the double integrals, therefore reducing the computational complexity. Numerical experiments have been performed for elastic cylindrical shells subjected to a plane incident wave. The results have demonstrated the accuracy of the method for wavenumbers ranging from 0 to 30, both directly on the shell and in the far field, and have confirmed that the procedure is valid for all frequencies.

A hybrid finite element-boundary integral method for the analysis of cavity-backed antennas of arbitrary shape

An edge-based hybrid finite element-boundary integral (FE-BI) formulation using tetrahedral elements is described for scattering and radiation analysis of arbitrarily shaped cavity-backed patch antennas. By virtue of the finite element method (FEM), the cavity irregularities, the dielectric super/substrate inhomogeneities, and the diverse excitation schemes inside the cavity may be readily modeled when tetrahedral elements are used to discretize the cavity. On the aperture, the volume mesh reduces to a triangular grid allowing the modeling of nonrectangular patches. Without special handling of the boundary integral system, this formulation is typically applicable to cavity-backed antenna systems with moderate aperture size. To retain an O(N) memory requirement, storage of the full matrix due to the boundary integral equation is avoided by resorting to a structured triangular aperture grid and taking advantage of the integral's convolutional property. If necessary, this is achieved by overlaying a structured triangular grid on the unstructured triangular grid and relating the edge field coefficients between the two grids via two narrow banded transformation matrices. The combined linear system of equations is solved via the biconjugate gradient (BICG) method, and the FFT algorithm is incorporated to compute the matrix-vector product efficiently, with minimal storage requirements. >

A combined finite element-boundary integral formulation for solution of two-dimensional scattering problems via CGFFT

A novel technique is presented for computing the scattering by two-dimensional structures of arbitrary inhomogeneity. The proposed approach combines the usual finite-element (FE) method with the boundary-integral equation to formulate a discrete system. This is subsequently solved via the conjugate gradient (CG) algorithm. A particular characteristic of the method is the use of rectangular boundaries to enclose the scatterer. Several of the resulting boundary integrals are then convolutions and can be evaluated via the fast Fourier transform (FFT) in the implementation of the CG algorithm. The solution approach presented here offers the principal advantage of having O(N) memory demand and employs a one-dimensional FFT, as against the two-dimensional FFT required in a traditional implementation of the proposed CG-FFT algorithm. The speed of the proposed solution method is compared with that of the traditional CG-FFT algorithm. Results are presented for several rectangular composite cylinders and one perfectly conducting cylinder. These are shown to be in excellent agreement with the moment method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Force calculation by Maxwell stress tensor in 3D hybrid finite element-boundary integral formulation

The calculation of magnetic force by the Maxwell stress tensor in 3-D hybrid finite element-boundary integral formulation is presented. The Maxwell stress tensor is calculated on the boundary of the domain considered. One field component (tangential or normal depending on the use of the magnetic or the electric formulation) is calculated from the finite element interpolation function, while the other is derived from the boundary integral method by means of a matrix product. The discretization of a hybrid magnetic formulation and a hybrid electric formulation is described. The symmetry between the different boundary integral methods is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A hybrid finite element-boundary integral formulation of the eddy current problem

A method is presented for the solution of the two dimensional eddy current problem in terms of the magnetic vector potential. In the formulation presented, any region in the problem may be represented by either finite elements or a boundary integral. This information is specified through use of an interactive graphics pre-processor which automatically generates the boundary elements and the finite elements. Examples are given.

A hybrid finite element-boundary integral formulation of Poisson's equation

A method is presented for the solution of two-dimensional static fields in which any region in the plane can be represented by finite elements or by a boundary integral. The form of the set of simultaneous equations obtained is studied and several illustrative examples of the technique are given.