Three‐dimensional finite element eddy current analysis by using high‐order vector elements

Sets of high‐order basis functions of a tetrahedral element are systematically constructed and applied to finite element analysis of eddy current problems. A polynomial space is divided into a lot of subspaces assigned on the edges, faces, and a volume of the tetrahedral element. Lagrange‐type vector basis functions of the subspaces are presented. The effect of the high‐order vector elements is investigated by a cubic conductor model located in AC steady‐state magnetic fields. In the calculations using the fundamental and second‐order elements, no convergent value of the eddy current power loss can be obtained in spite of fine meshes because the eddy current shifts to the surface of the conductor. The higher‐order vector elements give the convergent solutions in the coarse meshes. © 2004 Wiley Periodicals, Inc. Electr Eng Jpn, 147(4): 60–67, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.10306     

Electromagnetic non‐destructive evaluation by vector finite‐element neural networks

This study proposes neural network‐based iterative inverse solutions for non‐destructive evaluation (NDE) in which vector finite elements (VFEM) represent the forward model that closely models the physical process. The iterative algorithm can eventually estimate the material parameters. Vector finite element method global matrix is stored in a compact form using its sparsity and symmetry. The stored matrix elements are employed as the neurons weights, and preconditioning techniques are used to accelerate convergence of the neural networks (NN) algorithm. Detailed algorithm describing this new method is given to facilitate implementation. Combining vector finite elements and NNs offers several advantages over each technique alone, such as reducing memory storage requirements and the easily computed fixed weights of the NN. Various examples are solved to show the performance and usefulness of the proposed method, including lossy printed circuit board and lossy inhomogeneous cylindrical problems with ferromagnetic materials. These solutions compare very well with other published data where the maximum relative error was 5%. Copyright © 2010 John Wiley & Sons, Ltd.     

A non‐conformal FETI‐like domain decomposition approach of FE‐BI‐MLFMA for 3‐D electromagnetic scattering/radiation problems

A non‐conformal finite element tearing and interconnecting‐like (FETI‐like) domain decomposition approach (DDA) of the hybrid finite element–boundary integral–multilevel fast multipole algorithm (FE‐BI‐MLFMA) is presented by integrating a series efficient techniques for computing electromagnetic scattering/radiation problems. The Robin transmission condition is employed to cement the non‐conformal meshes on the interconnected surfaces between the interior and exterior regions and between sub‐domains in the interior region. The FETI‐like technique is applied to reduce the FE‐BI matrix equation. Furthermore, a preconditioner is constructed to accelerate the convergent speed of this non‐conformal FETI‐like DDA. The numerical performance of the presented non‐conformal FETI‐like DDA‐FE‐BI‐MLFMA is studied for scattering/radiation problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Meshfree‐enriched electromagnetic finite element formulation using nodal integration

This paper presents a meshfree‐enriched finite element formulation using nodal integration for electrostatic analysis. The meshfree‐enriched finite element method, originally proposed to solve the incompressible constraint in mechanical problem, is revisited in this paper and applied to the analysis of electrostatic problems to improve the solution accuracy of conventional finite element method. A novel nodal integration scheme based on the meshfree‐enriched finite element mesh is developed for the integration of discrete equation and is shown to pass the linear exactness in the Galerkin approximation. To demonstrate the accuracy of the proposed formulation, two numerical examples are studied and comparisons are made to several other finite element formulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Parallel FETI‐DP algorithm for efficient simulation of large‐scale EM problems

An efficient parallelization of the dual‐primal finite‐element tearing and interconnecting (FETI‐DP) algorithm is presented for large‐scale electromagnetic simulations. As a nonoverlapping domain decomposition method, the FETI‐DP algorithm formulates a global interface problem, whose iterative solution is accelerated with a solution of a global corner problem. To achieve a good load balance for parallel computation, the original computational domain is decomposed into subdomains with similar sizes and shapes. The subdomains are then distributed to processors based on their close proximity to minimize inter‐processor communication. The parallel generalized minimal residual method, enhanced with the iterative classical Gram‐Schmidt orthogonalization scheme to reduce global communication, is adopted to solve the global interface problem with a fast convergence rate. The global corner‐related coarse problem is solved iteratively with a parallel communication‐avoiding biconjugate gradient stabilized method to minimize global communication, and its convergence is accelerated by a diagonal preconditioner constructed from the coarse system matrix. To alleviate neighboring communication overhead, the non‐blocking communication approach is employed in both generalized minimal residual and communication‐avoiding biconjugate gradient stabilized iterative solutions. Three numerical examples are presented to demonstrate the accuracy, scalability, and capability of the proposed parallel FETI‐DP algorithm for electromagnetic modeling of general objects and antenna arrays. Copyright © 2016 John Wiley & Sons, Ltd.     

Efficient analysis of planar microwave circuits with mixed‐order prism vector finite macroelements

This paper introduces a new concept of a mixed‐order prism macroelement, suitable for an efficient analysis of three‐dimensional planar microwave circuits, using two‐dimensional meshes and preprocessors. The mixed‐order concept used here implies arbitrary orders of variation in different directions and differs essentially from the well‐known mixed‐order approximation that is an integral part of every Whitney element. It is the existence of a related systematic theory of higher‐order vector finite elements, previously documented, that facilitates the introduction of such a concept. The second‐ and third‐order elements, derived by this approach, are successfully applied in the analysis of planar microwave circuits, rendering the application of finite element method in such problems still a favorable option. Copyright © 2008 John Wiley & Sons, Ltd.     

Higher‐order CN‐FETD method for analysis of microwave circuit structures

In this paper, the hierarchical high‐order basis functions on tetrahedrons are introduced to the Crank–Nicolson (CN) finite‐element time‐domain (FETD) with the 3D Maxwell equations for analysis of the microwave circuit structures. Whitney 1‐form high‐order hierarchical basis functions are used to expand the electric field and Whitney 2‐form high‐order hierarchical basis functions for the magnetic field. The CN scheme is employed in the FETD method to lead to an unconditionally stable algorithm. Numerical results were presented to demonstrate the accuracy and efficiency of the proposed high‐order CN‐FETD method. Copyright © 2012 John Wiley & Sons, Ltd.     

A high‐order discontinuous Galerkin method with time‐accurate local time stepping for the Maxwell equations

We present an explicit numerical method to solve the time‐dependent Maxwell equations with arbitrary high order of accuracy in space and time on three‐dimensional unstructured tetrahedral meshes. The method is based on the discontinuous Galerkin finite element approach, which allows for discontinuities at grid cell interfaces. The computation of the flux between the grid cells is based on the solution of generalized Riemann problems, which provides simultaneously a high‐order accurate approximation in space and time. Within our approach, we expand the solution in a Taylor series in time, where subsequently the Cauchy–Kovalevskaya procedure is used to replace the time derivatives in this series by space derivatives. The numerical solution can thus be advanced in time in one single step with high order and does not need any intermediate stages, as needed, e.g. in classical Runge–Kutta‐type schemes. This locality in space and time allows the introduction of time‐accurate local time stepping (LTS) for unsteady wave propagation. Each grid cell is updated with its individual and optimal time step, as given by the local Courant stability criterion. On the basis of a numerical convergence study we show that the proposed LTS scheme provides high order of accuracy in space and time on unstructured tetrahedral meshes. The application to a well‐acknowledged test case and comparisons with analytical reference solutions confirm the performance of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

A conformal mapping‐based approach for fast two‐dimensional FEM electrostatic analysis of MEMS devices

In this paper, a methodology is proposed for expediting the coupled electro‐mechanical two‐dimensional finite element modeling of electrostatically actuated MEMS. The proposed methodology eliminates the need for repeated finite element meshing and subsequent electrostatic modeling of the device during mechanical deformation. We achieve this by mapping the deformed electrostatic domain to the reference undeformed domain ‘conformally’. A ‘conformal’ map preserves the form of the Laplace equation and the boundary conditions; thus the electrostatic problem is solved only once in the undeformed electrostatic domain. The conformal map itself is generated through the solution of the same Laplace equation on the undeformed geometry and with displacement boundary conditions dictated by the movement of the mechanical domain. The proposed methodology is demonstrated through its application to the modeling of three MEMS devices with varying length‐to‐gap ratios, multiple dielectrics and complicated geometries. The accuracy of the proposed methodology is confirmed through comparisons of its results with results obtained using the conventional finite element solution. Copyright © 2010 John Wiley & Sons, Ltd.     

Extending the enlarged cell and uniformly stable conformal techniques to modeling curved conductors in two‐dimensional high‐order finite‐difference time‐domain algorithms

The critical tool of modeling irregularly shaped perfect conductors is developed for the extended‐stencil high‐order two‐dimensional M24 variant of the finite‐difference time‐domain (FDTD) method. Two standard FDTD conformal approaches are analyzed and successfully extended to work accurately with M24. They both afford higher order convergence with respect to mesh density than a previously developed technique, which better matches M24's characteristics. Both approaches rely on borrowing weighted electromotive forces from nearby extended‐stencil cells to ensure accuracy and numerical stability while the overall algorithm is efficiently operated at the maximum allowable time steps by FDTD and M24 theories. Validation examples demonstrate that M24's amplitude and phase accuracies using coarse numerical meshes were not compromised. Copyright © 2012 John Wiley & Sons, Ltd.     

Soft‐fault diagnostic method based on locus circle of steady‐state node‐voltage vector

This study describes a relationship between an element parameter and a fault feature steady‐state node‐voltage vector (SSNVV). The locus of the SSNVV endpoints is proposed and proven to be an arc of circle under certain conditions, whereas the element parameter varies from small to large within a certain range. The arc of circle is called locus circle of SSNVV (LCoSSNVV). An LCoSSNVV‐based soft‐fault diagnostic method is also proposed. The method utilizes three discrete SSNVV endpoints to draw an LCoSSNVV for each element and collects all LCoSSNVVs to produce an LCoSSNVV set for fault diagnosis. In the following diagnostic process, the method measures a fault feature SSNVV of actual faulty circuit and judges which element is faulty on the basis of the relationship between the measured fault feature SSNVV and the LCoSSNVVs in an LCoSSNVV set. Subsequently, the fault can be further subdivided into different types of soft faults on the basis of the relative position of the SSNVV on the LCoSSNVV. A range of the faulty element parameter value can be estimated. Several examples illustrate the method. A measurement solution of SSNVV and a solution of a programming strategy are described. The new LCoSSNVV‐based method would effectively improve the soft‐fault diagnosis process of a linear time‐invariant circuit network. Copyright © 2015 John Wiley & Sons, Ltd.     

A uniformly stable conformal FDTD‐method in Cartesian grids

A conformal finite‐difference time‐domain algorithm for the solution of electrodynamic problems in general perfectly conducting 3D geometries is presented. Unlike previous conformal approaches it has the second‐order convergence without the need to reduce the maximal stable time step of conventional staircase approach. A novel proof for the local error rate for general geometries is given, and the method is verified and compared to other approaches by means of several numerical 2D examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Two‐graph analysis of pathological equivalent networks

Recently, abstract current mirror and voltage mirror elements have been proposed for behavioral modeling of active analog blocks. Such artificial elements and the traditionally used nullor element together are called pathological elements in the literature. Pathological elements are very useful in modeling and analysis of active network. Hence, researchers have been motivated to study symbolic analysis methods for networks containing pathological elements. However, so far, only nodal admittance matrix analysis has been formulated. In this work, an alternative two‐graph method is formulated, which has the advantage of providing a compact intermediate form for subsequent symbolic term generation. With a compact two‐graph representation, either a matrix method or a tree enumeration method can be employed. For completeness, the classical two‐graph theory has been extended in this paper to encompass all four types of dependent sources and all pathological elements. Illustrative examples are presented to demonstrate the principle of compact symbolic term generation by the presented two‐graph method. Copyright © 2014 John Wiley & Sons, Ltd.     

A dual‐primal finite‐element tearing and interconnecting method combined with tree‐cotree splitting for modeling electromechanical devices

The dual‐primal finite‐element tearing and interconnecting (FETI‐DP) method is combined with the tree‐cotree splitting (TCS) method to expand the capability and improve the efficiency of the finite‐element analysis of electromechanical devices. With the FETI‐DP method, an original large‐scale problem is decomposed into smaller subdomain problems and parallel computing schemes are then employed to reduce the computation time significantly. The TCS method is adopted to deal with the low‐frequency breakdown problem, which often accompanies the finite‐element analysis of electromechanical problems. On the basis of the computed magnetic field values, the force is computed with the use of the Maxwell stress tensor method. The proposed technique is applied to solve both high‐contrast magnetostatic problems and eddy‐current problems. Results are compared with both measurement data and brute‐force finite‐element calculations without domain decomposition. Comprehensive tests are conducted to investigate the parallel efficiency and numerical scalability. The results show that the proposed method can achieve a good parallel efficiency and an excellent numerical scalability with respect to the number of subdomains and the size of the problem. Copyright © 2012 John Wiley & Sons, Ltd.     

The 2‐D boundary element techniques for capacitance extraction of nanometer VLSI interconnects

This paper presents several techniques to accelerate the two‐dimensional (2‐D) direct boundary element method (BEM) for the capacitance extraction of nanometer very large‐scale integrated interconnects. Among these techniques, the nonuniform discretization technique minimizes the number of unknowns needed for accurate computation. The technique of adding virtual dielectric interface increases the sparsity of the coefficient matrix. With the technique of blocked Gaussian elimination, the memory usage and CPU time for solving the linear equation system are largely reduced. The analytical primitive functions for the 2‐D boundary integrals are also presented. Numerical results show that the presented techniques largely accelerate the 2‐D boundary element method. And finally, our BEM‐based capacitance solver demonstrates five times speedup over an advanced capacitance solver based on finite difference method. Copyright © 2013 John Wiley & Sons, Ltd.     

Efficient computation of Maxwell eigenmodes in axisymmetric cavities using hierarchical vector finite elements

Computation of Maxwell eigenmodes in an axisymmetric cavity using hierarchical vector finite elements is presented. The use of curl conforming vector basis functions, which span the null space of the curl operator, leads to the appearance of spurious modes with zero eigenvalues. Such spurious modes lead to electric flux solution with non‐zero divergence. Constraining the solution space in the variational statement for the eigenvalue problem by weakly enforcing the flux to be divergence‐free leads to the elimination of such modes. Discrete equivalent of such a constraint equation is developed for axisymmetric problems solved using hierarchical vector and scalar basis functions of orders complete to p=2. The discrete constraint equation, developed individually for Fourier modes m=0 and m≥1, is efficiently integrated with a subspace iteration‐based eigenvalue solution technique such as the Lanczos/Arnoldi method. The resulting solution technique is free of spurious modes added with an advantage of seeking a solution of a positive definite matrix during each iteration of the eigenvalue solver. Convergence in solution is demonstrated for orders up to p=2, while the proposed technique can be extended to basis functions of arbitrary order. Copyright © 2009 John Wiley & Sons, Ltd.     

Transient analysis of coupled non‐uniform transmission line using finite element method

This paper presents a finite element time domain model for a numerical solution of a coupled non‐uniform transmission line problem. On the basis of the finite element method, a novel numerical procedure for the solution of a system of the non‐uniform multi‐conductor transmission line equations in the time domain is presented. The results obtained by the proposed method have been compared with the solution obtained using the finite difference time domain method, and an excellent correlation has been demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.     

Uniform approximation of time‐invariant non‐linear systems

An Erratum has been published for this article in International Journal of Circuit Theory and Applications 2004; 32(6):633. It is shown that the elements of a large class of time‐invariant non‐linear input–output maps can be uniformly approximated arbitrarily well, over infinite time intervals, using a certain structure that can be implemented in many ways using, for example, radial basis functions, polynomial functions, piecewise linear functions, sigmoids, or combinations of these functions. For the special case in which these functions are taken to be certain polynomial functions, the input–output map of our structure is a generalized finite Volterra series. Results are given for the case in which inputs and outputs are defined on ?. The case in which inputs and outputs are defined on the half‐line ?₊ is also addressed, and in both cases inputs need not be functions that are continuous. Copyright © 2004 John Wiley & Sons, Ltd.     

Modeling of doubly lossy and dispersive media with the time‐domain finite‐element dual‐field domain‐decomposition algorithm

A numerical scheme is presented for the time‐domain finite‐element modeling of an electrically and magnetically lossy and dispersive medium in the dual‐field domain‐decomposition method. Existing approaches for modeling doubly lossy and dispersive media are extended to the dual‐field case, yielding a general dual‐field domain‐decomposition scheme for modeling large‐scale electromagnetic problems involving such media. A quantitative analysis is performed to estimate the error induced by the modeling of medium dispersion. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient hybrid scheme of finite element method of lines for modelling computational electromagnetic problems

A hybrid scheme called finite element method of lines is proposed and described for modelling and analysis of generalized computational electromagnetic problems with emphasis on a number of irregular waveguide examples. This new technique is developed by combining a finite element method with a method of lines so that it can handle not only irregular composite geometry but also maintain high accuracy enjoyed by semi‐analytical procedures. Analytical and numerical algorithmic building blocks of this new scheme are discussed in detail such as geometry discretization, element mapping, element trial functions, reformulation and computational issues of non‐linear ordinary differential equations. Our results show that this new technique is able to efficiently solve complex problems as compared with the conventional method of lines. Copyright © 2004 John Wiley & Sons, Ltd.