Low-frequency Breakdown Research Articles

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.

On the Evaluation of the 4-D Reaction Integral for the Scalar Potential in Galerkin’s Method of Moments

Despite the great progress made in the evaluation of the 4-D reaction integrals arising in the method of moments applied to the surface integral equation formulations, there are still some aspects that deserve to be explored further. The focus in this paper lies in the evaluation of the 4-D reaction integral for the scalar potential. We demonstrate that, unless this integral is computed to machine precision in the whole impedance matrix, it is imperative to maintain the trianglewise balance in the evaluation of the scalar potential part. A small imbalance, even at very small error, results in a loss of degrees of freedom (DOFs) for the charge distribution, violating the null-space property of the divergence operator and yielding potentially spurious solutions. The effect, although present, may go unnoticed at the conventional frequency regimes, where the usual mesh sizes can be applied. However, it becomes more pronounced as we approach the quasi-static limit, eventually rendering spurious solutions at the frequencies higher than the classical low-frequency breakdown would.

Enhancing the convergence of multipole expansions at intermediate frequency

In this work, we examine near-field acoustic wave scattering from media having a spatial variation in compressibility contrast. Typically, the scattered acoustic pressure can be expressed as a convergent Neumann series when the compressibility contrast is relatively small. However as the magnitude of the compressibility contrast increases a resonant scattering condition ensues, yielding a divergent series. It has been shown that Padé Approximants method can be used in these cases, thereby extending the range of utility of the Neumann series solution. The proposed research explores hybrid methods that combine multipole-methods, and Padé Approximant approaches to calculate the near-field scattered pressure. As part of this work, we will examine the low-frequency breakdown in the plane wave, and monopole-monopole expansion approaches as it affects the numerical stability of spatial translation operations.

Accurate and Efficient Low-Frequency Solution of Partial Element Equivalent Circuit Models

This paper presents an accurate and efficient low-frequency solution of 3-D partial element equivalent circuit models (PEEC). The cause of the low-frequency breakdown is identified from a circuit point of view and an effective approach is proposed to eliminate it in a rigorous way. Further, based on the previous step and on a proper choice of the unknowns, an effective preconditioner for PEEC formulation is proposed to analyze problems involving both lossy conductors and ideal dielectrics, which is well suited to be used along with iterative solvers. The results demonstrate the effectiveness of the proposed formulation and preconditioner in solving two pertinent problems.

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.

A Discontinuous Galerkin Augmented Electric Field Integral Equation for Multiscale Electromagnetic Scattering Problems

A discontinuous Galerkin (DG) augmented electric field integral equation method based on the domain decomposition is proposed in this paper for full-wave solution of multiscale targets. The conventional surface integral equation-based DG method allowing both conformal and nonconformal discretizations for multiscale structures suffers from low-frequency breakdown. By augmenting the DG-EFIE with current continuity equation, the proposed scheme can alleviate the low-frequency breakdown. In the augmented system, the electric field integral equation and the current continuity equation are discretized by using hybrid basis functions including Rao–Wilton–Glisson (RWG) and half RWG basis functions. Since the half RWG basis is not divergence conforming, line charge degrees of freedom on the adjoining edges are introduced in this paper. It is observed that the resulting linear system is well conditioned at low frequencies, which leads to a rapid convergence over wide frequency band. Numerical examples demonstrate the accuracy and efficiency of the augmented system.

A Discretization Method With the $H_{\mathrm{ div}}$ Inner Product for Electric Field Integral Equations

A discretisation method with the $H_{\rm div}$ inner product for the electric field integral equation~(EFIE) is proposed. The EFIE with the conventional Galerkin discretisation shows bad accuracy for problems with a small frequency, a problem known as the low-frequency breakdown. The discretisation method proposed in this paper utilises the $H_{\rm div}$ scalar product with a scalar coefficient for the Galerkin discretisation and overcomes the low-frequency problem with an appropriately chosen coefficient. As regards the preconditioning, we find that a naive use of the widely-used Calderon preconditioning is not efficient for reducing the computational time with the new discretisation. We therefore propose a new preconditioning which can accelerate the computation successfully. The efficiency of the proposed discretisation and preconditioning is verified through some numerical examples.

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.

Internally Combined Volume-Surface Integral Equation for a 3-D Electromagnetic Scattering Analysis of High-Contrast Media

An internally combined volume surface integral equation (ICVSIE) for analyzing three-dimensional (3-D) electromagnetic scattering from heterogeneous high-contrast (HC) dielectric objects is proposed. Iteratively solved (generalized) volume integral equations [(g)VIEs] often converge slowly when applied to the analysis of electromagnetic interactions involving dielectric objects with permittivities much higher than those of the surrounding medium [HC breakdown] or discretized using mesh elements much smaller than the wavelength [low-frequency (LF) breakdown]. The proposed ICVSIE hybridizes a VIE and a surface integral equation (SIE) in a manner that eliminates both breakdown phenomena. Just as in previous hybrid VIE–SIE approaches, the ICVSIE invokes Huygens surface equivalence principle to transform the original problem into coupled equivalent exterior and interior scenarios. In the exterior scenario, surface currents radiate in the original surrounding medium; in the interior problem, surface and volume polarization currents radiate in a medium with permittivity closer to that of the object. By judiciously combining integral equations pertinent to both scenarios, the ICVSIE system is assembled. Numerical results show that the proposed ICVSIE is both HC and LF stable when applied to the analysis of 3-D scattering problems.

Augmented Electric-Field Integral Equation for Inhomogeneous Media

The augmented electric-field integral equation is extended for the analysis of inhomogeneous media at low frequencies. The internal surface integral equation for inhomogeneous region is derived from the vector potential ( ${{\mathbf A}}$ ) and the scalar potential ( ${{\mathbf \Phi }}$ ) equations, which are free of low-frequency breakdown. In the new equations, Green's functions of ${{\mathbf A}}$ and ${{\mathbf \Phi }}$ for inhomogeneous media are incorporated. Due to the absence of the analytic solutions, Green's functions for ${{\mathbf A}}$ and ${{\mathbf \Phi }}$ are solved numerically with the finite-element method and represented in matrix forms. Numerical examples are presented to demonstrate the validity of the proposed scheme in the study of inhomogeneous objects at low frequencies.

Acceleration of Augmented EFIE Using Multilevel Complex Source Beam Method

The computation of the augmented electric field integral equation (A-EFIE) is accelerated by using the multilevel complex source beam (MLCSB) method. As an effective solution of the low-frequency problem, A-EFIE includes both current and charge as unknowns to avoid the imbalance between the vector potentials and the scalar potentials in the conventional EFIE. However, dense impedance submatrices are involved in the A-EFIE system, and the computational cost becomes extremely high for problems with a large number of unknowns. As an exact solution to Maxwell’s equations, the complex source beam (CSB) method can be well tailored for A-EFIE to accelerate the matrix-vector products in an iterative solver. Different from the commonly used multilevel fast multipole algorithm (MLFMA), the CSB method is free from the problem of low-frequency breakdown. In our implementation, the expansion operators of CSB are first derived for the vector potentials and the scalar potentials. Consequently, the aggregation and disaggregation operators are introduced to form a multilevel algorithm to reduce the computational complexity. The accuracy and efficiency of the proposed method are discussed in detail through a variety of numerical examples. It is observed that the numerical error of the MLCSB-AEFIE keeps constant for a broad frequency range, indicating the good stability and scalability of the proposed method.

An Enhanced Augmented Electric-Field Integral Equation Formulation for Dielectric Objects

A full-wave surface integral equation (SIE) method based on the augmented electric-field integral equation (A-EFIE) for dielectric objects with low-frequency stability is presented in this paper. Motivated by the A-EFIE formulation for perfect electric conductor (PEC), the internal and external problems are both augmented with the current continuity equation and renormalized to eliminate the low-frequency breakdown. Although the magnetic-field integral equation operator $\mathcal{K}$ is free of low-frequency breakdown, its matrix form is ill-conditioned and unsolvable if the traditional Rao–Wilton–Glisson (RWG) basis function is used as the testing and basis functions. As a remedy, the Buffa–Christiansen (BC) basis function is introduced to alleviate this testing issue. After this treatment, the matrix form of operator $\mathcal{K}$ is well conditioned. To solve problems with a large number of unknowns, a preconditioning scheme is introduced to accelerate the convergence and the mixed-form fast multipole algorithm (FMA) is adopted to accelerate the matrix vector product.

Spectral Element Method and Domain Decomposition for Low-Frequency Subsurface EM Simulation

Low-frequency subsurface electromagnetic measurements are important tools for characterizing natural resources and environmental wastes. Rapid simulations of low-frequency subsurface electromagnetic measurements are still a challenge because of the large computational domain and low-frequency breakdown phenomenon. We develop an effective method to simulate these low-frequency subsurface electromagnetic measurements by using the spectral element method together with a domain decomposition method (DDM). A specific mesh has been designed based on the traveling wave nature in the air and the diffusion field nature in the underground space to greatly reduce the number of unknowns. The frequency-domain version of the Riemann solver (upwind flux) is used as an effective transmission condition to simulate the interactions between neighboring subdomains in DDM. Several numerical examples demonstrate the efficiency of the proposed approach in low-frequency subsurface electromagnetics simulations.

Hierarchical Bases Preconditioner to Enhance Convergence of CFIE With Multiscale Meshes

A hierarchical quasi-Helmholtz decomposition, originally developed to address the low-frequency and dense-discretization breakdowns for the EFIE, is applied together with an algebraic preconditioner to improve the convergence of the CFIE in multiscale problems. The effectiveness of the proposed method is studied first on some simple examples; next, test on real-life cases up to several hundreds wavelengths show its good performance.

A DC-Stable, Well-Balanced, Calderón Preconditioned Time Domain Electric Field Integral Equation

The marching-on-in-time (MOT) solution of the time domain electric field integral equation (TD-EFIE) has traditionally suffered from a number of problems, including: 1) instability; 2) spurious static contributions plaguing the solution; 3) low-frequency breakdown; and 4) dense discretization breakdown. The first issue can be resolved by employing proper space-time Galerkin discretization schemes and accurate quadrature methods. The second and the third issue have been resolved by the quasi-Helmholtz Projected TD-EFIE (qHP-TDEFIE). This contribution introduces a multiplicative preconditioner which can be applied to the qHP-TDEFIE, without further modifying the original scheme. This preconditioner is based on Calderon techniques and guarantees that the MOT system can be solved efficiently using iterative methods, not only for large time step sizes but also for dense spatial discretizations, and for both simply and multiply connected geometries.

Domain Decomposition Method Using Integral Equations and Adaptive Cross Approximation IE-ACA-DDM for Studying Antenna Radiation and Wave Scattering From Large Metallic Platforms

In this paper, we present a new methodology for a nonoverlapping domain decomposition method using boundary integral equations and adaptive cross approximation in the frequency domain (IE-ACA-DDM). This paper proposes two main novelties. First, the IE-DDM condition number is improved with the development of a formulation based on electric field integral equation (EFIE) implementation over the entire subdomains and on a simplification of the magnetic field integral equation (MFIE) on the interfaces. As a consequence, the number of matrix–vector products is significantly reduced during the iterative resolution. Then, this paper proves that the ACA can be applied and efficiently implemented in IE-DDM methods. Moreover, a hierarchical LU (HLU) decomposition is proposed to accelerate the ACA solver. The IE-ACA-DDM is very suitable for the resolution of multiple excitations and to bypass the low-frequency breakdown. The flexibility, accuracy, and efficiency of IE-ACA-DDM are demonstrated by comparing with reference integral equation solutions and scaled mock-up measurements.

Solution of the Electric Field Integral Equation When It Breaks Down

With a method developed in this work, we find the solution of the original electric field integral equation (EFIE) at an arbitrary frequency where the EFIE breaks down due to low frequencies and/or dense discretizations. This solution is equally rigorous at frequencies where the EFIE does not break down and is independent of the basis functions used. We also demonstrate, both theoretically and numerically, the fact that although the problem is commonly termed low-frequency breakdown, the solution at the EFIE breakdown can be dominated by fullwave effects instead of just static or quasi-static physics. The accuracy and efficiency of the proposed method is demonstrated by numerical experiments involving inductance, capacitance, RCS extraction, and a multiscale example with a seven-orders-of-magnitude ratio in geometrical scales, at all breakdown frequencies of an EFIE. In addition to the EFIE, the proposed method is also applicable to other integral equations and numerical methods for solving Maxwell's equations.

Nonconforming Discretization of the Electric-Field Integral Equation for Closed Perfectly Conducting Objects

Galerkin implementations of the method of moments (MoM) of the electric-field integral equation (EFIE) have been traditionally carried out with divergence-conforming sets. The normal-continuity constraint across edges gives rise to cumbersome implementations around junctions for composite objects and to less accurate implementations of the combined field integral equation (CFIE) for closed sharp-edged conductors. We present a new MoM-discretization of the EFIE for closed conductors based on the nonconforming monopolar-RWG set, with no continuity across edges. This new approach, which we call “even-surface odd-volumetric monopolar-RWG discretization of the EFIE”, makes use of a hierarchical rearrangement of the monopolar-RWG current space in terms of the divergence-conforming RWG set and the new nonconforming “odd monopolar-RWG” set. In the matrix element generation, we carry out a volumetric testing over a set of tetrahedral elements attached to the surface triangulation inside the object in order to make the hyper-singular Kernel contributions numerically manageable. We show for several closed sharp-edged objects that the proposed EFIE-implementation shows improved accuracy with respect to the RWG-discretization and the recently proposed volumetric monopolar-RWG discretization of the EFIE. Also, the new formulation becomes free from the electric-field low-frequency breakdown after rearranging the monopolar-RWG basis functions in terms of the solenoidal, Loop, and the nonsolenoidal, Star and “odd monopolar-RWG”, components.