Phased array antenna analysis with the hybrid finite element method

A new analysis technique for infinite phased array antennas was developed and demonstrated. It consists of the finite element method (FEM) in combination with integral equation radiation conditions and a novel periodic boundary condition for 3-D FEM grids. Accurate modeling of rectangular, circular and circular-coaxial feeds is accomplished by enforcing continuity between the FEM solution and several waveguide modes across an aperture in the array's ground plane. The radiation condition above the array is enforced by a periodic integral equation in the form of a Floquet mode summation, thus reducing the solution to that of a single array unit cell. The periodic boundary condition at unit cell side walls is enforced through a matrix transformation. That mathematically “folds” opposing side walls onto each other with a phase shift appropriate to the array lattice and scan angle. The unit cell electric field is expanded in vector finite elements. Galerkin's method is used to cast the problem as a matrix equation, which is solved by the conjugate gradient method. A general-purpose computer code was developed and validated for cases of open-ended waveguides, microstrip patches, clad monopoles and printed flared notches, showing that the analysis method is accurate and versatile     

开放区域上的电磁场问题的非截断算法——无限大元素法及数值结果

为了避免出现截断边界及截断边界条件，提出了开放区域上的有限单元和无限大单元相结合的区域剖分方法，并给出了无限大单元上基函数的构造。数值结果表明，该方法具有很好的计算效率和计算精度。     

Computational aspects of finite element modeling in EEG sourcelocalization

A comparison is made of two different implementations of the finite element method (FEM) for calculating the potential due to dipole sources in electroencephalography (EEG). In one formulation (the direct method) the total potential is the unknown that is solved for and the dipole source is directly incorporated into the model. In the second formulation (the subtraction method) the unknown is the difference between the total potential and the potential due to the same dipole in an infinite region of homogeneous conductivity, corresponding to the region where the dipole is located. Both methods have the same FEM system matrix. However, the subtraction method requires an additional calculation of flux integrations along the edges of the elements in the computation of the right-hand side (RHS) vector. It is shown that the subtraction method is usually more accurate in the forward modeling, provided the flux integrations are computed accurately. Errors in calculating the flux integrations may result in large errors in the forward solution due to the ill-conditioned nature of the FEM system matrix caused by the Neumann boundary condition. To minimize the errors, closed-form expressions for the flux integrations are used for both linear and quadratic triangular elements. It is also found that FEM forward modeling errors may cause false extrema in the least-square objective function obtained from the boundary potential, near boundaries between media of differing conductivity. Multiple initial guesses help eliminate the possibility of the solution getting trapped in these false extrema     

Electromagnetic scattering from an arbitrary, inhomogeneous 2-Dobject-a finite and infinite element solution

The application of finite elements to the open region associated with scattering problems necessarily requires the truncation of the solution region. This work investigates the use of infinite elements to extend the solution region beyond that of the finite elements which can be truncated by a noncircular boundary. A finite/infinite element formulated is developed in two dimensions and tested on dielectric, magnetic, and conducting scatterers with known analytic solutions     

Transient two-dimensional heat conduction analysis of electronic packages by coupled boundary and finite element methods

Electronic packages experience large temperature excursions during their fabrication and under operational conditions. Inherent to electronic packages are the presence of geometric and material discontinuities. The regions where adhesive bond lines intersect with convective heat-loss surfaces are the most critical locations for failure initiation due to heat flux singularities and extreme thermo-mechanical stresses. Thus, accurate calculation of the flux field, as well as the temperature field, is essential in transient thermo-mechanical stress analysis. Although the finite element method (FEM) is highly efficient and commonly used, its application with conventional elements suffers from poor accuracy in the prediction of the flux field in these regions. The accuracy of the results from the boundary element method (BEM) formulation, which requires computationally intensive time-integration schemes, is much higher than that of the FEM. However, in this study, a novel boundary element-finite element coupling algorithm is developed to investigate transient thermal responses of electronic packages consisting of dissimilar materials.     

各向异性材料部分涂覆导体的散射特性研究

利用物理光学法(PO)与有限元法(FEM)混合方法,研究了有各向异性媒质局部涂覆的电大导体目标的电磁散射问题.在有复杂媒质涂覆的电小结构区域采用有限元法,在未涂覆媒质的电大导体区域采用物理光学法结合等效电磁流法.将两部分区域的边界进行消隐处理,考虑耦合效应,并利用共同的等相位面,将不同区域产生的散射场矢量叠加得到总散射场.文中的实例结果与传统的FEM很好的一致,说明了该方法的有效性和精确性.     

Edge-based FEM solution of scattering from inhomogeneous andanisotropic objects

This paper presents an application of the edge-based vector finite element method to scattering problems of anisotropic and inhomogeneous objects. Based on conventional FEM functional, a hybrid finite element-surface integral formulation is established by introducing permittivity and permeability tensors. The space domain is divided into interior and exterior regions by an imaginary surface conformal to the scatterer. Edge vector finite elements are used to model the anisotropic and inhomogeneous interior, and a surface integral equation is used to model the unbounded exterior. Compared to other hybrid techniques, the approach here retains the symmetry and sparsity of the FEM matrix and introduces only one type of unknown equivalent current in the moment matrix equation. To validate the theory, typical 2-D numerical results are first presented, which show excellent agreement with exact eigenmode expansion solutions or accurate MoM data     

On the modeling of a gapped power-bus structure using a hybrid FEM/MoM approach

A hybrid finite-element-method/method-of-moments (FEM/MoM) approach is applied to the analysis of a gapped power-bus structure on a printed circuit board. FEM is used to model the details of the structure. MoM is used to provide a radiation boundary condition to terminate the FEM mesh. Numerical results exhibit significant errors when the FEM/MoM boundary is chosen to coincide with the physical boundary of the board. These errors are due to the inability of hybrid elements on the boundary to enforce the correct boundary condition at a gap edge in a strong sense. A much better alternative is to extend the MoM boundary above the surface of the board.     

介质加载圆形槽波导的分析

采用矢量基函数的有限元方法，在开放边界处采用完全匹配层(PML)截断无限区域。分析了在圆形槽波导不同位置加载平板状介质时的模式特点。分析表明介质加载时圆形槽波导的模式次序要发生变化，为设计圆形槽波导微波化学反应器提供了依据。     

An a posteriori error reduction scheme for the three-dimensionalfinite element solution of Maxwell's equations

The accuracy of the finite element method (FEM) depends on the properties of the mesh which covers the problem geometry. The accuracy can usually be improved by increasing the element density in the mesh or the order of the shape functions in the elements at the expense of a significant increase in computation time. Instead, in this paper an a posteriori error reduction scheme is applied to improve the accuracy in the solution of three-dimensional electromagnetic boundary value problems. In this scheme, first the FEM, solution is generated by the use of lower-order shape functions. Then the numerical error is expressed in terms of higher-order shape functions and calculated on an element-by-element basis from information derived from the FEM solution. Finally, this error is added to the FEM solution to improve its accuracy. The degree of error reduction which is achieved with the application of this scheme is demonstrated by means of several simple electromagnetic boundary value problems     

Radiation characteristics of cavity backed aperture antennas infinite ground plane using the hybrid FEM/MoM technique and geometricaltheory of diffraction

A technique using the hybrid finite element method (FEM)/method of moments (MoM) and geometrical theory of diffraction (GTD) is presented to analyze the radiation characteristics of cavity fed aperture antennas in a finite ground plane. The cavity which excites the aperture is assumed to be fed by a cylindrical transmission line. The electromagnetic (EM) fields inside the cavity are obtained using finite element method (FEM). The EM fields and their normal derivatives required for FEM solution are obtained using: (1) the modal expansion in the feed region and (2) the MoM for the radiating aperture region (assuming an infinite ground plane). The finiteness of the ground plane is taken into account using GTD. The input admittance of open-ended circular, rectangular, and coaxial line radiating into free space through an infinite ground plane are computed and compared with earlier published results. Radiation characteristics of a coaxial cavity-fed circular aperture in a finite rectangular ground plane are verified with experimental results     

用组合法分析各种结构的未屏蔽平板线

本文提出用保角变换边界元组合法计算各种结构的未屏蔽平板线特性阻抗问题。     

Infinite phased-array analysis using FDTD periodic boundaryconditions-pulse scanning in oblique directions

Unit cell analysis of infinite phased-arrays in the finite difference time domain (FDTD) is performed by implementation of periodic boundary conditions. The technique allows for pulse excitation and oblique scan directions in both the cardinal and intercardinal planes. To our knowledge, this is the first paper presenting FDTD computations for intercardinal pulse scanning in oblique directions. The ordinary Yee lattice is used, which makes the algorithm easy to incorporate in an already existing FDTD code. Nonperiodic boundaries are truncated by Berenger's (see J. Comput. Phys., vol.127, p.363-79, 1996) perfectly matched layer (PML). Active impedance of an infinite dipole array is calculated with the new method and validation is performed via the “element-by-element” approach, i.e., by a conventional FDTD simulation of a corresponding large finite array. Excellent agreement is found and the technique has been numerically stable in all cases analyzed     

Two-dimensional singular vector elements for finite-elementanalysis

The finite-element method (FEM) exhibits a reduced convergence rate when used for the analysis of geometries containing sharp edges where the electromagnetic field is singular. The convergence of the method can be-improved by introducing singular elements that model analytically predicted singular behavior. A number of authors have developed singular elements that are compatible with the scalar FEM. In this paper, we propose a new singular element that is compatible with edge-based vector finite elements and can cope with any order of singularity while preserving the sparsity of the FEM equations. Edge-based singular elements more correctly model singular fields and thus require fewer unknowns, while avoiding the introduction of spurious modes in the numerical solution. Numerical results verify that the convergence of the FEM is significantly improved     

理想导电圆柱电磁场分布二维有限元方法分析

电磁问题中的边界条件对场的求解精度和效率都有影响。应用吸收边界条件的有限元方程矩阵具有稀疏性,如果确定好吸收边界条件的位置可以得到高精度的解。本文采用有限元方法对理想导电圆柱电磁场分布特性进行分析,编程计算了在不同圆柱散射体半径、不同离散大小、不同吸收边界条件位置时的场分布,并与解析精确解对比。结果表明：离散尺寸不大于0.1波长、圆柱散射体外边沿到吸收边界的距离大于0.5波长时与精确解基本吻合。对进一步分析圆柱散射问题具有参考意义。     

Model of a nonlinear electromagnetic crystal

2D electromagnetic crystal with lumped nonlinear elements is considered. An electrodynamic model with a rectangular grid is developed for a crystal that is infinite in one coordinate and finite in the other. In the case when the structure is excited by a plane wave, linear boundary value problems are formulated for electromagnetic fields at multiples of the fundamental frequency. The nonlinear problem is solved by means of the harmonic balance method. A system of nonlinear equations for the amplitudes of voltage harmonics at nonlinear elements is derived. Results of numerical solution of the system are presented for resistive and capacitive nonlinear elements.     

A hybrid finite element-boundary integral method for the analysisof 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     

Application of finite-elements to the analysis of diffraction gratings

A numerical technique is presented for analysing diffraction gratings of arbitrary groove shape. The method is based on the application of finite element technique. It is suggested that this technique, thus far reserved mostly for the problems of bounded extent, may be used to advantage for analysing this problem.

To this end, the infinite space is divided into two regions, one finite and the other infinite in extent. A series solution is employed in the infinite region, and in the finite one a functional is used. The two solutions are then matched on the common boundary of the two regions. The resulting equation for the field is finally solved by application of the finite element technique. The numerical results obtained for echelette gratings are favourably compared with a number of similar results reported in the literature.     

大型波导缝隙阵与天线罩的一体化高效精确分析

针对带罩大型波导缝隙阵的辐射特性分析, 基于并行区域分解合元极算法, 提出一种多区域的精确高效算法.将每根波导缝隙天线以及天线罩实体目标作为一个有限元计算区域, 各区域之间通过基于各区域表面的边界积分方程进行耦合, 并于天线罩内部应用区域分解技术来降低计算资源实现高效计算.与传统单区域合元极的数值结果比较验证了该多区域方法的精确高效性, 还计算分析了带罩大型波导缝隙阵的频域辐射特性.