Time-harmonic Electromagnetic Problems Research Articles

Hybrid CPU-GPU solution to regularized divergence-free curl-curl equations for electromagnetic inversion problems

The Curl-Curl equation is the foundation of time-harmonic electromagnetic (EM) problems in geophysics. The efficiency of its solution is key to EM simulations, accounting for over 95% of the computation cost in geophysical inversions for magnetotelluric or controlled-source EM problems. However, most published EM inversion codes are still central processing unit (CPU)-based and cannot utilize recent computational developments on the graphic processing units (GPUs). Based on a previously proposed divergence-free algorithm developed on CPUs, this study demonstrates the current limits of the CPU-based inversion procedure. To exploit the high throughput capability of GPUs, we propose a hybrid CPU-GPU framework to solve forward and adjoint problems required for EM inversions. The large sparse linear systems arising from the staggered-grid finite difference approximation of the Curl-Curl equation are solved with a mixed-precision Krylov subspace solver implemented on a GPU. The algorithm is then tested in EM forward and adjoint calculations, with real-world three-dimensional numerical examples. Test results show promising 30× kernel-level speed-ups over the conventional CPU algorithm. This approach may further take the complex frequency domain EM inversions onto the next, practical stage on small affordable GPU platforms.

Local Multiple Traces Formulation for electromagnetics: Stability and preconditioning for smooth geometries

We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple traces formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes (2012) for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators. In particular, we propose a novel second-order inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies proposed.

Analysis of Variational Formulations and Low-regularity Solutions for Time-harmonic Electromagnetic Problems in Complex Anisotropic Media

We consider the time-harmonic Maxwell's equations with physical parameters, namely the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e. with a boundary condition on the electric field or its curl, respectively) is proven using well-suited functional spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analysed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a H(curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark.

Simulation of induction thermography NDT technique using SIBC

PurposeThis paper aims to present a lightened 3D finite element model (FEM) for coupled electromagnetic thermal simulation of the induction thermography non-destructive testing (NDT) technique to reduce the computation time.Design/methodology/approachThe time harmonic electromagnetic problem is expressed in A – ϕ formulation and lightened by using the surface impedance boundary condition (SIBC) applied to both the massive induction coil surface and the surface of conductor workpiece including open cracks. The external circuit is taken into account by using the impressed voltage or the impressed current formulation. The thermal diffusion in the workpiece is solved by using surface electromagnetic power density as thermal source.FindingsThe accuracy and the usefulness of the method for the design of the induction thermography NDT technique have been shown with acceptable deviation compared with a full FEM model. It is also observed that at high frequency, when the ratio between the local radius of the conductor and the skin depth is high, a very good accuracy can be obtained with the SIBC methods. At lower frequency, the effect of the curvature of the surface becomes significant. In this case, the use of the Mitzner’s impedance can help to correct the error.Originality/valueThe SIBC can be used for both massive coil and workpieces with open cracks to alleviate 3D FEMs of the coupled electrothermal model. The implementation in matrix form of the coupled electrothermal formulation is given in details. The comparisons with reference analytical solution and full 3D FEM show the accuracy and performance of the method. In the test case presented, the computation time is 6.6 times lower than the classical model.

Three-Dimensional Time-Harmonic Electromagnetic Scattering Problems from Bianisotropic Materials and Metamaterials: Reference Solutions Provided by Converging Finite Element Approximations

A recently developed theory is applied to deduce the well posedness and the finite element approximability of time-harmonic electromagnetic scattering problems involving bianisotropic media in free-space or inside waveguides. In particular, three example problems are considered of which one deals with scattering from plasmonic gratings that exhibit bianisotropy while the other two deal with bianisotropic obstacles inside waveguides. The hypotheses that guarantee the reliability of the numerical results are verified, and the ranges of the constitutive parameters of the media involved for which the finite element solutions are guaranteed to be reliable are deduced. It is shown that, within these ranges, there can be significant bianisotropic effects for the practical media considered as examples. The ensured reliability of the obtained results can make them useful as benchmarks for other numerical approaches. To the best of our knowledge, no other tool can guarantee reliable solutions.

Well Posedness and Finite Element Approximability of Three-Dimensional Time-Harmonic Electromagnetic Problems Involving Rotating Axisymmetric Objects

A set of sufficient conditions for the well posedness and the convergence of the finite element approximation of three-dimensional time-harmonic electromagnetic boundary value problems involving non-conducting rotating objects with stationary boundaries or bianisotropic media is provided for the first time to the best of authors’ knowledge. It is shown that it is not difficult to check the validity of these conditions and that they hold true for broad classes of practically important problems which involve rotating or bianisotropic materials. All details of the applications of the theory are provided for electromagnetic problems involving rotating axisymmetric objects.

On perfectly matched layers for discontinuous Petrov–Galerkin methods

In this article, several discontinuous Petrov–Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov–Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two- and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.

Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems

Abstract Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods for the problem for a single frequency. The arising discretized system takes a two-by-two or four-by-four block matrix form. Since the problems are in general three-dimensional in space and hence of very large scale, one must use an iterative solution method. It is then crucial to construct efficient preconditioners. It is shown that an earlier used preconditioner for optimal control problems is applicable here also and leads to very tight eigenvalue bounds and hence very fast convergence such as for a Krylov subspace iterative solution method. A comparison is done with an earlier used block diagonal preconditioner.

Well posedness and finite element approximability of two-dimensional time-harmonic electromagnetic problems involving non-conducting moving objects with stationary boundaries

A set of sufficient conditions for the well-posedness and the convergence of the finite element approximation of two-dimensional time-harmonic electromagnetic boundary value problems involving non-conducting moving objects with stationary boundaries is provided for the first time to the best of authors’s knowledge. The set splits into two parts. The first of these is made up of traditional conditions, which are not restrictive for practical applications and define the usual requirements for the domain, its boundary, its subdomains and their boundaries, the boundary conditions and the constitutive parameters. The second part consists of conditions which are specific for the problems at hand. In particular, these conditions are expressed in terms of the constitutive parameters of the media involved and of the velocity field. It is shown that it is not difficult to check the validity of these conditions and that they hold true for broad classes of practically important problems which involve axially moving media.

REALISTIC AND CORRECT MODELS OF IMPRESSED SOURCES FOR TIME-HARMONIC ELECTROMAGNETIC BOUNDARY VALUE PROBLEMS INVOLVING METAMATERIALS

The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media. It is shown, in particular, that, even if all interfaces are regular, the class of ill-posed problems can be very large in the presence of general square-integrable impressed sources. However, when a simple and realistic constraint is enforced on these sources, requiring that the support of the sources does not include any interface between a traditional medium and a metamaterial, among the problems here considered just those involving an interface between complementary materials remain ill-posed. These considerations have a very significant impact also on the approximability of the solution of well-posed problems since the numerical noise can introduce small fictitious sources even where the sources to be simulated are not present. These effects on finite element simulators are fully analyzed. Finally, we propose an algorithm that allows to obtain much better approximations of the solutions of the most critical well-posed problems.

Finite element based inversion for time-harmonic electromagnetic problems

SUMMARY In this paper we address the inverse problem and present some recent advances in numerical methods to recover the subsurface electrical conductivity from time-harmonic electromagnetic data. We rigorously formulate and discretize both the forward and the inverse problem in the finite element framework. To solve the forward problem, we derive a finite element discretization of the first-order system of Maxwell’s equations in terms of the electric field and the magnetic induction. We show that our approach is equivalent to the standard discretization of the vector Helmholtz equation in terms of the electric field and that the discretization of magnetic induction of the same approximation order is hidden in the standard discretization. We implement the forward solver on unstructured tetrahedral meshes using edge elements. Unstructured meshes are not only capable of representing complex geometry. They can also reduce the overall problem size and, thus, the size of the system of linear equations arising from the forward problem such that direct methods for its solution using a sparse matrix factorization become feasible. The inverse problem is formulated as a regularized output least squares problem. We considertworegularizationfunctions.First,wederiveasmoothnessregularizerusingaprimal-dual mixedfiniteelementformulationwhichgeneralizesthestandardLaplacianoperatorforapiecewiseconstantconductivitymodelonunstructuredmeshes.Secondly,wederiveatotalvariation regularizer for the same class of models. For the choice of the regularization parameter we revisit the so-called dynamic regularization and compare it to a standard regularization scheme with fixed regularization parameter. The optimization problem is solved by the Gauss–Newton method which can be efficiently implemented using sparse matrix–vector operations and exploiting the sparse matrix factorization of the forward problem system matrix. A synthetic data example from marine controlled source electromagnetics demonstrates the feasibility of our method and shows that both regularization functions and both regularization parameter selection schemes produce a comparable reconstruction of the subsurface conductivity.

A p-hierarchical error estimator for a fe–be coupling formulation applied to electromagnetic scattering problems in ℝ3

We examine the construction of p-hierarchical local a posteriori error estimators for time-harmonic electromagnetic problems using edge-based finite elements and boundary elements for hexahedral and tetrahedral meshes in ℝ3. The error estimators rely on stable subspace decompositions of Nédélec elements in H(curl, Ω) and Raviart–Thomas elements in .

Integral Equation Based Domain Decomposition Method for Solving Electromagnetic Wave Scattering From Non-Penetrable Objects

The integral equation (IE) method is commonly utilized to model time-harmonic electromagnetic (EM) problems. One of the greatest challenges in its applications arises in the solution of the resulting ill-conditioned matrix equation. We introduce a new domain decomposition method (DDM) for the IE solution of EM wave scattering from non-penetrable objects. The proposed method is a non-overlapping/non-conformal DDM and it provides a computationally efficient and effective preconditioner for the IE matrix equations. Moreover, the proposed approach is very suitable for dealing with multi-scale electromagnetic problems since each sub-domain has its own characteristics length and will be meshed independently. Furthermore, for each sub-domain, we are free to choose the most effective IE sub-domain solver based on its local geometrical features and electromagnetic characteristics. Additionally, the multilevel fast multi-pole algorithm (MLFMA) is utilized to accelerate the computations of couplings between sub-domains. Numerical results demonstrate that the proposed method yields rapid convergence in the outer Krylov iterative solution process. Finally, simulations of several large-scale examples testify to the effectiveness and robustness of the proposed IE based DDM.

Relaxed Incomplete Cholesky Factorization Preconditioned Symmetric LBCG Algorithm for Solving Electromagnetic Problems

The symmetric linear biconjugate gradient (S-LBCG) iterative algorithm combined with modified relaxed incomplete Cholesky (RIC) preconditioner is proposed for solving large sparse complex symmetric and highly indefinite systems, which results from the use of edge-based finite-element method (FEM) on the analysis of electromagnetic problems. The S-LBCG iterative algorithm is derived from the linear biconjugate gradient iterative algorithm. It contains only one matrix-vector product per iteration for solving complex sparse symmetric systems. The preconditioner is derived from the relaxed incomplete Cholesky factorization with diagonal compensation. More robust and efficient RIC preconditioner can be achieved by employing preprocessing scheme of scaling and reordering before the incomplete factorization. Numerical tests on analysis of time-harmonic electromagnetic field problems show that this RIC preconditioned S-LBCG algorithm performs excellently both on reducing the memory requirement and on convergence rate.

Trefftz discontinuous Galerkin methods for time-harmonic electromagnetic and ultrasound transmission problems

An efficient strategy is described for the solution of time-harmonic electromagnetic and elastic wave problems. The strategy is based on the framework of the discontinuous Galerkin (DG) method, thereby using approximation bases that are locally conservative. Additionally, in the Trefftz DG method each approximation function satisfies the underlying partial differential equation in each finite element. The Trefftz DG method is similar to the ultra weak variational formulation (UWVF) but the local bases are constructed differently. Electromagnetic wave, eddy current and elastic wave problems are solved in order to analyze the performance of the method. The ultrasonic testing (UT) example is a two-dimensional elastic wave model of the Inconel welding area in reactor vessels.

Robust GMRES recursive method for fast finite element analysis of 3D electromagnetic problems

AbstractA robust generalized minimal residual recursive (GMRESR) iterative method is proposed to solve a large system of linear equations resulting from the use of an un‐gauged vector‐potential formulation of finite element method (FEM). This method involves an outer generalized conjugate residual (GCR) method and an inner generalized minimal residual (GMRES) method, where the inner GMRES acts as a variable preconditioning for the outer GCR. The efficient implementation of symmetric successive overrelaxation (SSOR) preconditioned GMRESR (SSOR‐GMRESR) algorithm is described in details for complex coefficient matrix equation. On several three‐dimensional electromagnetic problems, the resulting SSOR‐GMRESR approach converges in CPU time, which is 14.2–71.3 times shorter with respect to conventional conjugate gradient (CG) approach. By comparison with other popularly preconditioned CG methods, the results demonstrate that SSOR‐GMRESR is especially effective and robust when the A‐V formulation of FEM is applied to solve large‐scale time harmonic electromagnetic field problems. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1010–1015, 2007; Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.22333

Geophysical modelling of 3D electromagnetic diffusion with multigrid

The performance of a multigrid solver for time-harmonic electromagnetic problems in geophysical settings was investigated. With the low frequencies used in geophysical surveys for deeper targets, the light-speed waves in the earth can be neglected. Diffusion of induced currents is the dominant physical effect. The governing equations were discretised by the Finite-Integration Technique. The resulting set of discrete equation was solved by a multigrid method. The multigrid method provided excellent convergence with constant grid spacings, but not on stretched grids. The slower convergence rate of the multigrid method could be compensated by using *bicgstab2*, in which case multigrid acted as a preconditioner. Still, the overall performance was less than satisfactory with substantial grid stretching.

Application of Diagonally Perturbed Incomplete Factorization Preconditioned Conjugate Gradient Algorithms for Edge Finite-Element Analysis of Helmholtz Equations

The diagonally perturbed incomplete factorization preconditioning scheme is applied to the conjugate gradient (CG) method for solving a large system of linear equations resulting from the use of edge-based finite-element method (FEM). This scheme contains more global information about the coefficient matrix when compared with banded-matrix schemes. The efficient implementation of this preconditioned CG (PCG) algorithm is described in detail for complex coefficient matrix equation. On several electromagnetic problems the PCG approach converges in CPU time, which is 8.6-19.5 times shorter with respect to the CG approach. By comparison with other preconditioned techniques, the results demonstrate that incomplete factorization preconditioning strategy is especially effective for CG iterative method when edge-FEM is applied to solve large-scale time-harmonic electromagnetic field problems

A multigrid solver for time harmonic three-dimensional electromagnetic wave problems

A geometric multigrid algorithm for time harmonic electromagnetic wave problems including lossy material is presented. A finite element method with edge elements and nodal elements is used to describe the problems. For the multigrid smoother, complex conjugate gradient method with Gauss-Seidel preconditioning is used

Generalised method of lines for Helmholz equations by using discretisation technique of pseudospectral method

A novel semi-analytical higher-order numerical approach to solve electromagnetic field boundary-value problems is described. This pseudospectral-based method of lines is developed by combining a pseudospectral technique with the method of lines so that its solution is not only analytical along the line direction but also maintains high accuracy in the discrete direction. The pseudospectral-based discretisation strategy, distribution of collocation nodes, the global Fourier series interpolation of differential quadrature, a second-order difference matrix for various homogeneous boundary conditions, and the decoupling of coupled ordinary differential equations are all discussed in detail. Numerical results demonstrate that this pseudospectral method of lines can efficiently solve the boundary-value problems with higher accuracy when compared with the conventional method of lines. This method should be a competitive technique for modelling time-harmonic complex electromagnetic problems.