A two-dimensional time-domain finite element formulation for periodic structures

A formulation is presented for a two-dimensional time-domain finite-element method (FEM-TD) that incorporates periodic boundaries. The specifics of the method are shown for scattering problems, but it should be straightforward to extend it to radiation problems. The method solves for a transformed field variable (instead of solving directly for the electric field) in order to easily enable periodic boundary conditions in the time domain. The accuracy and stability of the method is demonstrated by a series of examples where the new formulation is compared with reference solutions. Very accurate results are obtained when the excitation (frequency range) and the geometry are such that no higher order Floquet modes are present. The accuracy is degraded in the presence of higher order modes due to the rather simple absorbing boundary condition that is used with the present formulation. The method is found to be stable even for angles of incidence close to grazing.     

A hybrid finite element method for 3-D scattering using nodal andedge elements

A hybrid finite element method for three-dimensional scattering is presented and numerical examples shown. This approach, which couples finite element discretization with the method of moments, is particularly well suited for monostatic radar cross section calculations. The method is based on a scalar and vector potential formulation of Maxwell's equations, the use of nodal elements, and a highly efficient body of revolution implementation of the method of moments. Combined nodal and edge elements are employed to accurately model fields around corners and edges. A curvature-based sampling criterion is derived and shown to ensure accurate answers for highly curved scatterers. Numerical results and Cray computer timings are compared with published results for an edge element code using radiation boundary conditions     

A new finite element formulation for RF scattering by complexbodies 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     

Vector finite element formulation for scattering fromtwo-dimensional heterogeneous bodies

A formulation is proposed for electromagnetic scattering from two-dimensional heterogeneous structures that illustrates the combination of the curl-curl form of the vector Helmholtz equation with a local radiation boundary condition (RBC). To eliminate spurious nonzero eigenvalues in the spectrum of the matrix operator, vector basis functions incorporating the Nedelec constraints are employed. Basis functions of linear and quadratic order are presented, and approximations made necessary by the use of the local RBC are discussed. Results obtained with linear-tangential/quadratic normal vector basis functions exhibit excellent agreement with exact solutions for layered circular cylinder geometries, and demonstrate that abrupt jump discontinuities in the normal field components at material interfaces can be accurately modeled. The vector 2D formulation illustrates the features necessary for a general three-dimensional implementation     

A 3-D hybrid finite element/boundary element method for the unifiedradiation and scattering analysis of general infinite periodic arrays

We present a rigorous frequency domain variational 3-D electromagnetic formulation for the general nonself-adjoint infinite periodic array problem. The hybrid method described combines the vector finite element and Floquet boundary element techniques. It is general in the sense that it is applicable to infinite periodic arrays of the open or aperture-types. It is thus effective for modeling both the scattering and radiation performance of diverse FSS, absorber, and phased-array structures. The technique accurately handles arbitrarily complicated 3-D geometries, lossy inhomogeneous media and internal as well as external excitations. These analyses can be applied to general skewed grids under arbitrary scan and polarization conditions     

An efficient scalar finite element formulation for nonlinearoptical channel waveguides

A self-consistent numerical approach based on the scalar finite element method is described for the analysis of both TE-like and TM-like nonlinear guided waves in optical channel waveguides. In order to improve the convergence and accuracy of solutions, isoparametric elements and numerical integration formulae derived by Hammer et al. are introduced. Numerical results are presented for nonlinear elliptical core optical fibers, and it is confirmed that in this approach, highly accurate solutions can be obtained with small scale computation. Furthermore, graded-index nonlinear optical channel waveguides are also analyzed, and the influence of refractive-index profiles on propagation characteristics of the nonlinear guided waves is investigated     

3-D finite element analysis of induction logging in a dippingformation mark

Electromagnetic induction (EMI) by a magnetic dipole located above a dipping interface is of relevance to the petroleum well-logging industry. The problem is fully three-dimensional (3-D) when formulated as above, but reduces to an analytically tractable one-dimensional (1-D) problem when cast as a small tilted coil above a horizontal interface. The two problems are related lay a simple coordinate rotation. An examination of the induced eddy currents and the electric charge accumulation at the interface help to explain the inductive and polarization effects commonly observed in induction logs from dipping geological formations. The equivalence between the 1-D and 3-D formulations of the problem enables the validation of a previously published finite element solver for 3-D controlled-source EMI     

Biomechanical 3-D finite element modeling of the human breast using MRI data

Breast tissue deformation modeling has recently gained considerable interest in various medical applications. A biomechanical model of the breast is presented using a finite element (FE) formulation. Emphasis is given to the modeling of breast tissue deformation which takes place in breast imaging procedures. The first step in implementing the FE modeling (FEM) procedure is mesh generation. For objects with irregular and complex geometries such as the breast, this step is one of the most difficult and tedious tasks. For FE mesh generation, two automated methods are presented which process MRI breast images to create a patient-specific mesh. The main components of the breast are adipose, fibroglandular and skin tissues. For modeling the adipose and fibroglandular tissues, we used eight noded hexahedral elements with hyperelastic properties, while for the skin, we chose four noded hyperelastic membrane elements. For model validation, an MR image of an agarose phantom was acquired and corresponding FE meshes were created. Based on assigned elasticity parameters, a numerical experiment was performed using the FE meshes, and good results were obtained. The model was also applied to a breast image registration problem of a volunteer's breast. Although qualitatively reasonable, further work is required to validate the results quantitatively.     

A 3-D hp finite/infinite element method to calculate power deposition in the human head

The electromagnetic power deposition and transfer properties of a G1 continuous head model reconstructed from magnetic resonance imaging (MRI) data are investigated by using the coupled hp finite/infinite element (FE/IE) method. The discretization error is controlled by a self-adaptive process driven by an explicit a posteriori error estimate. Based on the benchmark problem of reproducing the Mie series solution, the scattering of a plane wave on the curvilinear head model is used to evaluate the hp FE/IE approach and calibrate the error bound. The radiation pattern from a short dipole antenna modeling a cell phone, is analyzed in terms of the level and distribution of the specific absorption rates (SAR). The numerical experiments show that the hybrid hp FE/IE implementation is a competitive tool for accurate assessment of human electromagnetic exposure.     

三维时域有限元-边界元方法中奇异积分计算

研究三维时域有限元一边界元混合方法中奇异积分的计算.将其中的一些积分进行等价变形以降低奇异性.并给出了积分等价变形过程中所遇到的数值错误的一种解决方法,这一数值错误与时域有限元一边界元方法中的固有处置有关.讨论了频域中格林函数奇异点提取法推广应用于时域混合方法中奇异积分的计算,指出,频域中奇异点提取法的一些处理不适宜于时域奇异积分.数值实验检验了所给处理方法.     

Finite-element method modeling of superconductors: from 2-D to 3-D

A three-dimensional (3-D) numerical modeling technique for solving problems involving superconducting materials is presented. The model is implemented in finite-element method software and is based on a recently developed 3-D formulation for general electromagnetic problems with solid conductors. It has been adapted for modeling of superconductors with nonlinear resistivity in 3-D, characterized by a power-law E-J relation. It has first been compared with an existing and verified two-dimensional (2-D) model: Compared are the current density distribution inside the conductors and the self-field ac losses for different applied transport currents. Second, the model has been tested for computing the current distribution with typical 3-D geometries, such as corner-shaped and twisted superconductors. Finally, it has been used with two superconducting filaments in the presence of external magnetic field for verifying the existence of coupling currents. This effect deals with the finite length of the conductors and cannot be taken into account by 2-D models.     

2-D validation of a formulation for the analysis of 3-D bodies thatare periodic in one direction and finite sized in the others

A numerical scheme to analyze three-dimensional bodies that are periodic in one direction (z) and finite sized in the other ones (x, y) is presented. The geometry and material composition of the body can be arbitrary. A new formulation using the conjugate gradient-fast Fourier transform method (CG-FFT) has been developed. The formulation is based on the discretization and resolution of the electric field integral equation (EFIE) in both the real and spectral domains and leads to an efficient and accurate numerical procedure. Results are presented for RCS, equivalent currents and fields inside 3-D periodic structures (infinitely long cylinders with arbitrary shape and material composition). These results are compared with analytical solutions and the agreement is found to be good     

A modified finite element method for analysis of finite periodicstructures

A modified finite element method with new solving algorithm is proposed to analyze electromagnetic problems of finite periodic structures. Dielectric-loaded parallel-plate waveguides with rectangular and triangular dielectric gratings are tackled as an example of the present approach. Numerical results are checked by the self-convergence test and by comparing with those obtained by other methods. Finally, the dependence of the scattering parameters on the frequency, the period number, and the grating height is analyzed and compared     

A fast higher-order time-domain finite element-boundary integral method for 3-D electromagnetic scattering analysis

A novel hybrid time-domain finite element-boundary integral method for analyzing three-dimensional (3-D) electromagnetic scattering phenomena is presented. The method couples finite element and boundary integral field representations in a way that results in a sparse system matrix and solutions that are devoid of spurious modes. To accurately represent the unknown fields, the scheme employs higher-order vector basis functions defined on curvilinear tetrahedral elements. To handle problems involving electrically large objects, the multilevel plane-wave time-domain algorithm is used to accelerate the evaluation of the boundary integrals. Numerical results demonstrate the accuracy and versatility of the proposed scheme.     

An efficient finite element formulation to analyze waveguides withlossy inhomogeneous bi-anisotropic materials

In this paper a finite element formulation in terms of the magnetic field is presented for the analysis of waveguides with bianisotropic media. Such a formulation can deal with lossy inhomogeneous materials characterized by simultaneous permittivity, permeability, and cross-coupling (as in optical activity) arbitrary full tensors. The analysis takes into account arbitrary cross sections, and results in spurious-mode suppression, complex-mode computation, and the possibility of alternatively specifying the frequency or the complex propagation constant as an input parameter. In this way, many novel classes of waveguides with promising applications, such as chirowaveguides and chiroferrite-waveguides, can be analyzed. The formulation leads to a quadratic sparse eigenvalue problem which is transformed into a sparse generalized eigenvalue problem. This eigensystem is solved by the subspace method, the sparsity of the matrices being fully utilized. The proposed method has been validated by analyzing waveguides with biisotropic and bianisotropic materials. The agreement with previously published data is found to be excellent     

DIRECT SOLUTION BY 2-D AND 3-D FINITE ELEMENT METHOD ON FORWARD PROBLEM OF LOW FREQUENCY CURRENT FIELDS IN INHOMOGENEOUS MEDIA

The paper adopts finite element method to analyze the forward problem of low-frequency current fields in inhomogeneous media. Firstly, the direct solution of 2-D and 3-D scalar potential is given. Secondly, the technique of covering finite elements for problems with movement has been presented; namely, when the place of testing point moved, the meshing data will be produced automatically to avoid re-meshing and distortion of the mesh. Thirdly the free and prescribed potential method is used to make the finite element coefficient matrices. Then this paper provides the result of a validity test obtained by simulating the laterolog-3 logging, compared with the numerical model-matching method. Finally, the MLL response is calculated.     

On the formulation of hybrid finite-element and boundary-integralmethods for 3-D scattering

This paper studies, in detail, a variety of formulations for the hybrid finite-element and boundary-integral (FE-BI) method for three-dimensional (3-D) electromagnetic scattering by inhomogeneous objects. It is shown that the efficiency and accuracy of the FE-BI method depends highly on the formulation and discretization of the boundary-integral equation (BIE) used. A simple analysis of the matrix condition number identifies the efficiency of the different FE-BI formulations and an analysis of weighting functions shows that the traditional FE-BI formulations cannot produce accurate solutions. A new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance. Finally, the multilevel fast multipole algorithm (MLFMA) is employed to significantly reduce the memory requirement and computational complexity of the proposed FE-BI method     

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.     

Finite element formulation for lossy waveguides

An efficient computer-aided solution procedure based on the finite-element method is developed for solving general waveguiding structures composed of lossy materials. In this procedure, a formulation in terms of the transverse magnetic-field components is adopted and the eigenvalue of the final matrix equation corresponds to the propagation constant itself. Thus, it is possible to avoid the unnecessary iteration using complex frequencies. To demonstrate the strength of the presented method, numerical results for a rectangular waveguide filled with lossy dielectric are presented and compared with exact solutions. As more advanced applications of the presented method, a shielded image line composed of a lossy anisotropic material and a lossy dielectric-loaded waveguide with impedance walls are analyzed and evaluated     

A highly effective preconditioner for solving the finite element-boundary integral matrix equation of 3-D scattering

A highly effective preconditioner is presented for solving the system of equations obtained from the application of the hybrid finite element-boundary integral (FE-BI) method to three-dimensional (3-D) electromagnetic scattering problems. Different from widely used algebraic preconditioners, the proposed one is based on a physical approximation and is constructed from the finite element method (FEM) using an absorbing boundary condition (ABC) on the truncation boundary. It is shown that the large eigenvalues of the finite element (FE)-ABC system are similar to those of the FE-BI system. Hence, the preconditioned system has a spectrum distribution clustered around 1 in the complex plane. Consequently, when a Krylov subspace based method is employed to solve the preconditioned system, the convergence can be greatly accelerated. Numerical results show that the proposed preconditioner can improve the convergence of an iterative solution by approximately two orders of magnitude for large problems.