Vector Absorbing Boundary Conditions Research Articles

First- and Second-Order ABCs as Applied with Constant Tangential/Linear Normal Edge Elements

In this review paper, the solution of lossy inhomogeneous scattering problems is considered. The solution is given in terms of the transverse field components rather than the axial components. In order to terminate the edge element mesh, several second-order and first-order vector absorbing boundary conditions are enforced and compared. Unexpectedly, the second-order vector ABC did not show a much better performance compared with the first-order. The resulting sparse matrices are solved via the biconjugate gradient method enhanced by incomplete Choleski decomposition. Examples are given where the predicted solutions compare very well with exact solutions and other published data.

A domain decomposition method for the vector wave equation

A nonoverlapping domain decomposition method (DDM) is presented for the finite-element (FE) solution of electromagnetic scattering problems by inhomogeneous three-dimensional (3-D) bodies. The computational domain is partitioned into concentric subdomains on the interfaces of which conformal vector transmission conditions are prescribed and that can be implemented in the inhomogeneous part. The DDM is numerically implemented when a conformal vector absorbing boundary condition (ABC) is utilized on the outer boundary terminating the FE mesh, while employing the standard edge-based FE formulation. Then, numerical experiments are performed on a sphere and a cone sphere that emphasize the advantages of this technique in terms of memory storage and computing times, especially when the total number of unknowns is very large. Also, these numerical experiments serve as a severe test for the performances of the ABC.

A multilevel formulation of the finite-element method for electromagnetic scattering

Multigrid techniques for three-dimensional (3-D) electromagnetic scattering problems are presented. The numerical representation of the physical problem is accomplished via a finite-element discretization, with nodal basis functions. A total magnetic field formulation with a vector absorbing boundary condition (ABC) is used. The principal features of the multilevel technique are outlined. The basic multigrid algorithms are described and estimations of their computational requirements are derived. The multilevel code is tested with several scattering problems for which analytical solutions exist. The obtained results clearly indicate the stability, accuracy, and efficiency of the multigrid method.

A vector absorbing boundary condition for vector potential satisfying the Lorentz gauge

A vector absorbing boundary condition is derived for a vector potential which satisfies the Lorentz gauge, based on a form of the Representation theorem for field quantities having non-zero divergence. The potential is assumed to be the linear superposition of those of arbitrarily arranged Hertzian dipoles, which potentials are individually particular solutions of the Helmholtz equation. Compared with ABCs derived for fields a penalty is introduced by the non-zero divergence and for a first-order ABC the error is shown to be 0(r/sup -2/). A second-order condition is derived, and using a Galerkin formulation both conditions can yield symmetric matrices.

Vector absorbing boundary conditions for nodal or mixed finite elements

We present 2D and 3D vector absorbing boundary conditions for unbounded microwave problems. Coupling with vector formulation leads to a non-symmetric system matrix. We show bow it is possible to symmetrise these conditions with the same degree of precision. Edge and corner conditions are also presented for rectangular boundaries.

Modeling unbounded wave propagation problems in terms of transverse fields using 2D mixed finite elements

We present in this paper an approach for the modeling of open boundary microwave problems in the frequency domain by using high order 2D mixed elements conforming in H(curl). The Galerkin formulation for the vector wave equation in two dimensions is used to discretize the problem. The analysis region is truncated using a 2D vector absorbing boundary condition that satisfies the Sommerfeld radiation condition at infinity. This modeling is applied to scattering problems or to open ended waveguides.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Modeling the electromagnetic field in lossy dielectrics using finite elements and vector absorbing boundary conditions

A new functional for the finite element method is described for the distribution of high-frequency electromagnetic fields in arbitrarily shaped, lossy dielectrics in 3D. The method uses a brick-shaped edge element (covariant-projection element) and a second-order, symmetric, vector absorbing boundary condition. Analytic solutions are available for the case of a plane wave incident on a lossy sphere, and these are used to show that the new method is capable of predicting the fields in and around the sphere to an average accuracy of 1-2%, even when the outer, absorbing boundary is no more than a third of a wavelength from the sphere. >

Conformal absorbing boundary conditions for 3-D problems: derivation and applications

Vector absorbing boundary conditions (ABCs) for doubly curved surfaces are presented, and their applicability to finite elements for scattering calculations are discussed. A performance study of these ABCs is carried out in terms of accuracy and computational requirement, and scattering patterns for several targets are included for validation purposes. It is found that accurate far-field results can be obtained by terminating the finite element mesh a fraction of a wavelength from the scattering structure. >

The importance of the surface divergence term in the finite element-vector absorbing boundary condition method

The vector absorbing boundary condition (ABC) is an effective way of truncating the infinite domain of a 3-D scattering problem, and thereby permitting its solution with a finite element method. One of the terms of the ABC is a surface divergence term. It is shown that due to its presence, the normal continuity of the field must be enforced on the surface where the ABC is applied. Numerical analysis of scattering by a conducting sphere demonstrates that if normal continuity is not enforced, the maximum error in the near held may more than double. A similar error occurs if the surface divergence term is omitted from the formulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Vector one-way wave absorbing boundary conditions for FEM applications

In the paper a derivation is presented which leads to a new and general class of vector absorbing boundary conditions (ABCs) for use with the finite element method (FEM). The derivation is based on a vector one-way wave equation and a polynomial approximation of the vector radical. It is shown that wide-angle absorbing boundary conditions, as proposed in Halpern and Trefethen (1988) for optimal absorption of out-going waves, can be obtained in vector form. Vector plane waves are used to evaluate the accuracy and the reflection performance of these boundary conditions in a wide range of incidence angles. The implementation of the vector ABCs in a FEM formulation is also provided to show how up to the fifth-order absorbing accuracy can be achieved with derivatives only up to the second-order. A possible formulation is described which not only yields a third-order accuracy with first-order derivatives, but also retains the symmetry of the FEM matrix. >

Radar cross section computation of inhomogeneous scatterers using edge‐based finite element methods in frequency and time domains

This paper presents a flexible, body‐conforming finite element modeling technique, using Whitney's edge and face basis functions, for efficient computation of radar cross section (RCS) of inhomogeneous scatterers, both in the frequency and time domains. The vector absorbing boundary conditions, originally developed for a spherical absorbing boundary, are generalized for planar and cylindrical boundaries. The computational resources required for the proposed frequency and time domain techniques for RCS computation are compared with those needed in the finite difference time domain (FDTD) and the nonorthogonal FDTD methods. Illustrative numerical results, that demonstrate the accuracy of the technique, are presented.

A new finite element formulation for RF scattering by complex bodies of revolution

A formulation for and results of solving electromagnetic scattering from complex inhomogeneous axisymmetric bodies are presented. The approach presented uses the finite element method in the frequency domain. A node-based approach is used to solve for the three components of the electric or magnetic field. The finite element mesh is truncated using a three-dimensional vector absorbing boundary condition based on the Wilcox expansion theorem. A harmonic expansion of the near-field solution obtained from the finite element solution is used to compute the far fields and radar cross section. >

3D finite element analysis of a metallic sphere scatterer comparison of first and second order vector absorbing boundary conditions

A 3D vector analysis of plane wave scattering by a metallic sphere using finite elements and Absorbing Boundary Conditions (ABCs) is presented. The ABCs are applied on the outer surface that truncates the infinitely extending domain. Mixed order curvilinear covariant-projection elements are used to avoid spurious corruptions. The second order ABC is superior to the first at no extra computational cost. The errors due to incomplete absorption decrease as the outer surface is moved further away from the scatterer. An error of about 1% in near-field values was obtained with the second order ABC, when the outer surface was less than half a wavelength from the scatterer

Edge-based finite elements and vector ABCs applied to 3-D scattering

An edge-based finite element formulation with vector absorbing boundary conditions is presented for scattering by composite structures having boundaries satisfying impedance and/or transition conditions. Remarkably accurate results are obtained by placing the mesh a small fraction of a wavelength away from the scatterer.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust finite element solution for three-dimensional scattering

An edge-based finite element formulation with vector absorbing boundary conditions is presented for scattering by composite structures having boundaries satisfying impedance and/or transition conditions. Remarkably accurate results are obtained by placing the mesh a small fraction of a wavelength away from the scatterer.

A numerical study of vector absorbing boundary conditions for the finite-element solution of Maxwell's equations

It is reported that in 3-D vector solutions to Maxwell's equations, boundary conditions of the Bayliss-Turkel kind can be used to absorb outgoing radiation. The boundary conditions were used with curvilinear brick finite elements to analyze spherical test problems for which the exact fields are known. Errors due to incomplete absorption decrease as the outer boundary is moved farther away. The second-order boundary condition is appreciably more accurate than the first-order, at the same cost. >