Low-frequency Breakdown Problem Research Articles

Broadband S-Parameter Simulation Based on the Mixed Finite-Element Method and Symmetric Lanczos Algorithm

In this paper, the symmetric Lanczos algorithm and mixed finite element method are combined to improve the efficiency and accuracy of S-parameter simulation in a broad frequency band. The mixed finite element method based on tree-cotree splitting is adopted to overcome the low-frequency (high-density) breakdown problem, thus ensuring the accuracy of low frequency (high integration density) results. The modified symmetric Lanczos algorithm for second order eigenvalue problem is employed for fast frequency sweep by one matrix inversion, then the field in the entire frequency band is obtained rapidly by solving the reduced order matrix equation. The accuracy of the high frequency results is improved by increasing the number of Lanczos vectors. Numerical simulations of two integrated passive filters (IPD filter) and an interconnect model are shown to verify the hybrid method along with commercial software HFSS.

The Dilemma of Resolving the Low-Frequency Breakdown Problem in Microwave Components via Traditional and Improved Finite-Element Time-Domain Techniques

This paper developed an improved numerical methods for calculating electromagnetic field at low frequencies, and compared the performance with the traditional method. Traditional finite-element time-domain (FETD) is known to suffer from low frequency breakdown (LFB) problem, where system matrix becomes ill-conditioned, and leads to instability of small size electrical structures. In this paper, we proposed a mixed FETD (mFETD) method in resolving the LFB anomaly in the traditional FEM, with a particular reference to transmission line, inductor, and coaxial cable. We considered wave equation of E-field with incorporation of divergence constraint equation (DCE) as a function of Lagrange multiplier using current continuity equation (CCE) and Gauss’s law. For spatial discretization, both nodal basis functions and Curl conforming vector basis functions were selected, and we employed implicit Newmark beta algorithm (NBA) for integration of time. We describe how the components constructed via DCE in system matrix mitigate the singularity impact of the stiffness matrix, which consequently led to significant improvement of the system matrix. Numerical experiment and results demonstrate how the mFETD obtains stable numerical solution and attains faster rate of convergence while using an iterative solver, as such, improves the computational efficiency. Therefore, the mFETD method handles the transient problems in transmission line, which causes LFB anomaly in the traditional FETD, but not efficient in terms of computational time and iteration at solving the coaxial cable problem.

Characteristic Mode-Based Efficient Broadband Electromagnetic Scattering Analysis Method for Array Structures

A fast and accurate numerical method based on the theory of characteristic mode (TCM) is proposed for analyzing electromagnetic scattering from repetitive multiscale array structures. In order to overcome the low frequency breakdown problem when interacting at short distances between observation and source basis functions, the CMs are extracted from the impedance matrix generated by the augmented electric field integral equation (AEFIE), which is discretized by the method of moments (MoM). Then the CMs of a single unit are used as the global basis functions to reduce the unknowns of the array structures. In addition, the broadband multilevel fast multipole algorithm (MLFMA) based on approximate diagonalization of the Green’s function is implemented to accelerate the matrix-vector multiplications between the impedance matrix and CM vectors. Several numerical examples are given to demonstrate the high efficiency and accuracy of the proposed scheme.

Integral equation fast solver with truncated and degenerated kernel for computing flaw signals in eddy current non-destructive testing

In this article, the kernel independent H-matrix method is applied as the mathematical framework. The dense matrix generated by the selected integral equation, without low frequency breakdown problem for solving 3D arbitrary shaped eddy current nondestructive evaluation (NDE) forward problems, is compressed by the truncated and degenerated (TD) kernel function method. It results in a significant reduction in computational burden with nearly linear complexity. The kernel functions (Green's function) are degenerated by Lagrange polynomials which leads to two advantages: firstly, the double integrals are separated in two single integrals, and secondly, they are represented in a factorized form with good accuracy control. Meanwhile, due to the nature of the kernel function with exponential decay in the lossy medium, it is truncated to improve the overall performance. The TD kernel function-based boundary element method (BEM) fits well for solving eddy current NDE problems over the adaptive cross approximation based BEM, especially for detections of flaws or slots in planar objects. Several numerical predictions calculated by the proposed method are compared with those achieved by others such as the experiment, analytical and semi-analytical methods from benchmarks. The robustness and efficiency are demonstrated with excellent agreements and reduced complexity.

Mixed Spectral Element Method for Electromagnetic Secondary Fields in Stratified Inhomogeneous Anisotropic Media

The spectral element method (SEM) is widely used for analyzing high frequency electromagnetic wave propagation and scattering in complex structures. At low frequencies, however, the primary (direct) fields are usually much larger than the secondary fields caused by scattering, thus making it much more difficult to accurately simulate the secondary (scattered) fields because of the source singularity in the primary fields. To more effectively and accurately simulate the low-frequency secondary fields in subsurface structures, in this work the vector Helmholtz equation for the scattered fields is used to directly obtain the secondary field data in inhomogeneous anisotropic media. The computational method for the secondary electromagnetic fields is based on the mixed spectral element method (mixed SEM) for anisotropic objects in a layered anisotropic background medium. The electromagnetic field is separated into the background fields that are evaluated analytically using the dyadic Green's functions (DGFs) of a layered uniaxially anisotropic medium, and the secondary fields caused by anomalous bodies with arbitrary shapes are numerically computed by the mixed SEM. This avoids the source singularity, and allows the transmitters to be located outside the computational domain. By enforcing Gauss' law as the constraint condition, the mixed SEM efficiently eliminates the low frequency breakdown problem. Numerical results verify the proposed method over the traditional SEM in low-frequency scattering from objects in subsurface layered anisotropic media.

A Combination of Mixed FETD Method and SPICE to Simulate Nonlinear Multiport Circuits

A combination of the mixed finite-element time-domain method (MFETD) and SPICE solver is proposed in this letter to analyze the circuit response and crosstalk effects in complex microwave circuits with linear/nonlinear, passive/active components. The MFETD method in the EM part is adopted as an unconditionally stable time integration scheme to account for the low-frequency breakdown problem. SPICE is introduced to solve the nonlinear circuit equations in the circuit part by taking into account the coupling between multiport circuits. A microwave circuit based on an AD633 chip and an MCP3221 board is simulated to verify that this hybrid MFETD-SPICE method not only is superior in mesh flexibility to the finite-difference time-domain (FDTD)-based method but also exploits the well-developed open-source software SPICE to solve the nonlinear equations.

New Mixed SETD and FETD Methods to Overcome the Low-Frequency Breakdown Problems by Tree-Cotree Splitting

Traditional time-domain finite-element and spectral-element methods are known to suffer from the so-called low-frequency breakdown problem, where the system matrix can become ill-conditioned, and results can be unstable for electrically fine structures. In this article, a new mixed spectral-element time-domain (SETD) and finite-element time-domain (FETD) methods are proposed to overcome this low-frequency breakdown problem by constraint equations and tree-cotree splitting. Two forms of gradient matrices or the null space of the stiffness matrix of high-order basis functions are calculated to construct the constraint equations. Then, the constraint equations are directly applied in the time-stepping matrix equation by tree-cotree splitting to improve the properties of system matrices when the implicit Newmark-beta algorithm is adopted in two different forms. In the calculation of the first form of gradient matrix, the electric field is expanded by high-order edge basis functions; in the second form of gradient matrix, high-order edge basis functions associated with cotree edges and nodal basis functions defined on the free nodes are both employed to expand the electric field. Several numerical examples demonstrate that the new mixed SETD and mixed FETD methods can effectively overcome the low-frequency breakdown with high accuracy and without increasing the number of degrees of freedom over the conventional SETD and FETD methods.

Mixed Finite Element Method for Full-Wave Simulation of Bioelectromagnetism From DC to Microwave Frequencies.

Bioelectromagnetism focuses on the study of electromagnetic fields in biological tissues from direct current (DC) to optical frequencies. It is challenging to develop an electromagnetics (EM) simulation method to cover this entire frequency band due to the electrically small/large scattering problem at extremely low/high frequencies. This paper focuses on the band from DC to microwave frequencies in bioelectromagnetism. Its main research objective is to develop a method that can overcome the low frequency breakdown problem at low frequencies (practically DC) and still stay stable at microwave frequencies. Based on the scattered field vector Helmholtz equation, the mixed finite element method (mixed FEM) is developed for the broadband electromagnetic field simulation in biological tissues. By imposing Gauss' law as the constraint condition, the mixed FEM overcomes the low frequency breakdown problem without resorting to the quasi-static approximation and remains effective and accurate at high frequencies. Extremely low frequency and high frequency numerical results are demonstrated to verify that the mixed FEM is a stable full-wave electromagnetic field simulation method for the full-bandwidth bioelectromagnetism.

Method for Analytically Finding the Nullspace of Stiffness Matrix for Both Zeroth-Order and Higher Order Curl-Conforming Vector Bases in Unstructured Meshes

The stiffness matrix in a vector finite-element method has a nullspace whose size grows linearly with the matrix size. Finding it numerically can be computationally expensive. However, the nullspace is important in many applications from solving the low-frequency breakdown problem to developing fast solvers. In this article, an analytical method to generate the nullspace of the stiffness matrix is developed, not only for the zeroth-order but also for any higher order vector basis function in arbitrary unstructured meshes. Using the mesh information, we are able to analytically construct the nullspace without any difficulty, thus avoiding solving an eigenvalue problem for finding the nullspace. The proposed analytical method has been applied to a variety of 2-D and 3-D irregular meshes for various orders of vector basis functions. Comparisons with the reference nullspace obtained from a brute-force eigenvalue solution have validated the proposed analytical method. The proposed work has also been applied to efficiently solve the low-frequency breakdown problem in a full-wave solver and perform fast layout parasitics extraction.

Study on Characteristic Mode Analysis of Three-Dimensional Conducting Objects in Lossless Layered Medium

We comprehensively study the characteristic-mode analysis (CMA) of arbitrarily shaped 3-D conducting objects embedded in a lossless planarly layered medium. The electric field integral equation (EFIE) with the layered medium Green’s function is applied as the core equation for numerical analysis. The modal behavior of an object in a layered medium is first comparatively studied with that in free space. The low-frequency breakdown problem of the EFIE-based CMA and its remedy is then elaborated. The asymptotic frequency dependence of the characteristic eigenvalues for the low-order modes is numerically investigated. A theoretical analysis is conducted by constructing a frequency-independent generalized eigenvalue problem and applying Taylor expansions. Finally, the modal expansion for scattering problems and parameter extraction for circuit problems are discussed, both in the context of a layered background. Since the electromagnetic property of an electrically small object is mainly governed by a few of low-order modes, this paper may provide an effective way of model order reduction for complicated structures residing in a layered medium.

Spectral-Element Method With Divergence-Free Constraint for 2.5-D Marine CSEM Hydrocarbon Exploration

Rapid simulations of large-scale low-frequency subsurface electromagnetic measurements are still a challenge because of the low-frequency breakdown phenomenon that makes the system matrix extremely poor-conditioned. Hence, significant attention has been paid to accelerate the numerical algorithms for Maxwell’s equations in both integral and partial differential forms. In this letter, we develop a novel 2.5-D method to overcome the low-frequency breakdown problem by using the mixed spectral element method with the divergence-free constraint and apply it to solve the marine-controlled-source electromagnetic systems. By imposing the divergence-free constraint, the proposed method considers the law of conservation of charges, unlike the conventional governing equation for these problems. Therefore, at low frequencies, the Gauss law guarantees the stability of the solution, and we can obtain a well-conditioned system matrix even as the frequency approaches zero. Several numerical experiments show that the proposed method is well suited for solving low-frequency electromagnetic problems.

Accurate Solution of Electromagnetic Scattering by Super-Thin Conducting Objects Based on Magnetic Field Integral Equation

Electromagnetic scattering by super-thin conducting objects is formulated by integral equation approach. It could be difficult to obtain accurate solutions for such a problem because the current density changes dramatically near the edges of such objects and many low-quality meshes exist on the side faces of objects when discretized. Traditionally, the electric field integral equation is used to describe the problem and the three-dimensional (3-D) objects are approximated as a two-dimensional (2-D) open structure with a summation of the current density at two opposite sides. In this communication, the magnetic field integral equation (MFIE) is employed to govern the problem and the super-thin objects are strictly treated as 3-D objects. The MFIE is a second-kind integral equation resulting in a better conditioning and can also release the low-frequency breakdown problem, but it has not been applied to very thin structures. In the method of moments solution, a robust near-singularity treatment for its kernel is developed based on the Green's lemma. The derived formulations are friendly and very suitable for low-quality triangular meshes. Numerical examples are presented to demonstrate the scheme and good results have been obtained.

Augmented combined field integral equation for low frequency scattering analysis

In this study, a new augmented combined field integral equation formulation has been proposed to address the low-frequency breakdown problem. The proposed formulation is developed by adding the magnetic field integral operator to the conventional augmented electric field integral equation (AEFIE). In the meanwhile, a saddle point constraint preconditioning is also introduced to improve the convergence property of the system matrix. In some cases, the conventional AEFIE method cannot achieve the accurate solutions when the frequency is extremely low and a perturbation method is needed to tackle this accuracy problem. However, the new method is able to get the accurate solutions without any other strategy in the same cases mentioned above. Some numerical examples have been presented to demonstrate the validity of the proposed method.

Mixed Spectral-Element Method for Overcoming the Low-Frequency Breakdown Problem in Subsurface EM Exploration

One fundamental difficulty in low-frequency subsurface electromagnetic exploration is the low-frequency breakdown phenomenon in numerical computation. It makes the discretized linear system very poorly conditioned and thus difficult to solve. This issue is present in both integral equation and partial differential equation solution methods, and thus has attracted many researchers who have proposed various methods to overcome this difficulty. In this paper, we propose a new mixed spectral element method (mixed SEM) to eliminate this low-frequency breakdown problem and apply this method to solve the subsurface electromagnetic exploration problem. Since Gauss’ law is now explicitly enforced in the mixed SEM to make the system matrix well-conditioned even at extremely low frequency, we can solve the linear system from dc to high frequencies. With the proposed method, we study the surface-to-borehole electromagnetic system for hydrocarbon exploration. Numerical examples show that the mixed SEM is accurate and efficient, and has significant advantages over conventional methods.

A Frequency Stable Volume Integral Equation Method for Anisotropic Scatterers

We introduce a volume integral equation method for the numerical solution of the electromagnetic scattering problem from electrically anisotropic inhomogeneous objects. The problem’s unknown, that is the sum of the conduction and polarization current densities, is decomposed in terms of its loop, star and facet components. Suitable strategies have been implemented to enforce the uniqueness of this discrete representation. Furthermore, through a convenient scaling of the unknowns, we also make the present method immune to the low-frequency breakdown problem, showing its stability regardless of the operating frequency. This approach has been extensively validated against the null-field method by analyzing the scattering from a uniaxial dielectric sphere and a uniaxial dielectric slab and comparing the scattered electric field in both the near and far zones. The condition number of the scattering problem consisting of a slab with an assigned conductivity tensor is also monitored as a function of the exciting frequency.

Application of the LU Recombination Method to the FETI-DP Method for Solving Low-Frequency Multiscale Electromagnetic Problems

The LU recombination method is applied within a dual-primal finite element tearing and interconnecting (FETI-DP) framework to solve low-frequency multiscale electromagnetic problems. The FETI-DP method, a robust domain decomposition method, is adopted to deal with the large-scale linear system resulting from finite element modeling of multiscale problems. When frequency decreases or a highly nonuniform mesh is used, the FETI-DP method encounters a low-frequency breakdown problem. The LU recombination method is adopted to identify and eliminate dependent degrees of freedom (DOFs), so that nearly singular finite element matrices are regularized. Since it deals with matrices directly, the LU recombination method is easily combined with different types and orders of basis functions. A nested scheme is adopted to apply LU recombination within the FETI-DP framework. The proposed algorithm is applied to large-scale eddy-current, switched reluctance motor, and logging-while-drilling tool problems. With the LU recombination method, the FETI-DP method is extended down to DC, making it possible to deal with a variety of problems of practical interest.

Calderón Multiplicative Preconditioned EFIE With Perturbation Method

In this paper, we address the low-frequency breakdown and inaccuracy problems in the Calderon multiplicative preconditioned electric field integral equation (CMP-EFIE) operator, and propose the perturbation method as a remedy for three-dimensional perfect electric conductor (PEC) scatterers. The electric currents at different frequency orders as a power series can be obtained accurately in a recursive manner by solving the same matrix system with updated right hand side vectors. This method does not either require a search for the loops in the loop-tree/-star based method or include charge as additional unknown in the augmented EFIE method. Numerical examples show the far-field pattern can be accurately computed at extremely low frequencies by the proposed perturbation method.

A NEW EFIE METHOD BASED ON COULOMB GAUGE FOR THE LOW-FREQUENCY ELECTROMAGNETIC ANALYSIS

To solve the low-frequency breakdown inherent from the electric fleld integral equation (EFIE), an alternative new form of the EFIE is proposed by using the Coulomb-gauge Green's function of quasi-static approximation. Difierent from the commonly adopted Lorentz-gauge EFIE, the Coulomb-gauge EFIE separates the solenoidal and irrotational surface currents explicitly, which captures inductive and capacitive responses through electrodynamic and electrostatic Green's functions, respectively. By applying existing techniques such as the loop-tree decomposition, frequency normalization, and basis rearrangement, the Coulomb-gauge EFIE also can remedy the low-frequency breakdown problem. Through comparative studies between the Lorentz-gauge and Coulomb-gauge EFIE approaches from mathematical, physical and numerical aspects, the Coulomb-gauge EFIE approach shows the capability of solving low-frequency problems and achieves almost the same accuracy and computational costs compared to the Lorentz-gauge counterpart.

A Highly Efficient Domain Decomposition Method Applied to 3-D Finite-Element Analysis of Electromechanical and Electric Machine Problems

A highly efficient domain decomposition method, called the dual-primal finite-element tearing and interconnecting (FETI-DP) method, is applied to improve the efficiency of three-dimensional (3-D) finite-element analysis of electromechanical and electric machine problems. The FETI-DP method decomposes an original large-scale 3-D problem into many subdomain problems and then treats the subdomain problems concurrently using parallel computing schemes to reduce the total computation time. Higher order hierarchical basis functions are implemented to improve the accuracy of the simulation and also to facilitate the treatment of the low-frequency breakdown problem, which is encountered by the traditional finite-element method with vector basis functions when solving electromechanical problems. With the separation of higher order basis functions from the lowest order ones offered by hierarchical basis functions, the low-frequency breakdown problem can be handled by applying the tree-cotree splitting method to the lowest order basis functions. The resulting accurate finite-element computation of the magnetic field is then used to calculate force and torque by using the Maxwell stress tensor method.

Alternative AEFIE‐EFIE method for broadband CEM simulations

AbstractIt is well known that electric field integral equation (EFIE) has the low frequency breakdown problem due to the decoupling of electric and magnetic field. Augmented electric field integral equation (AEFIE) method overcomes the low frequency breakdown. However, its performance degrades with the increasing frequency. To obtain a unified method that works for both low and high frequencies, a novel method using the EFIE complement process inside AEFIE is proposed. Numerical results demonstrated that this alternative AEFIE‐EFIE method is able to extend integral equations to wide band applications. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1737–1740, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26919