Time-harmonic Maxwell Equations Research Articles

Preconditioners With Low Memory Requirements for Higher-Order Finite-Element Method Applied to Solving Maxwell’s Equations on Multicore CPUs and GPUs

This paper discusses two fast implementations of the conjugate gradient iterative method using a hierarchical multilevel preconditioner to solve the complex-valued, sparse systems obtained using the higher order finite-element method applied to the solution of the time-harmonic Maxwell equations. In the first implementation, denoted PCG-V, a classical V-cycle is applied and the system of equations on the lowest level is solved exactly. The second variant involves an approximate solution to the system of equations on the lowest level. To this end, auxiliary space preconditioning (ASP) is used instead of a direct solution. In this approach, denoted PCG-V-ASP, the time needed to solve the sparse system of equations is longer, but the memory requirements are smaller. To accelerate the computations, a graphics processing unit (GPU, Pascal P100) was used for both variants of the multilevel preconditioner. As a result, significant speedups were achieved over the reference parallel implementation using a multicore central processing unit (CPU, Intel Xeon E5-2680 v3, twelve cores). The results indicate that the auxiliary space preconditioning does in fact reduce the memory requirements, as compared with the reference PCG-V method, and at the same time performs each iteration faster. However, if symmetry is taken into account and the memory-efficient supernodal $LDL^{T}$ factorization is employed, the savings are less spectacular than anticipated based on previously published results using LU factorization and the multifrontal technique. PCG-V also requires a fewer iterations, so it’s time to solution is ultimately shorter. The difference is more pronounced if both preconditioners are run on a CPU. The use of a GPU as an accelerator for the computations considerably improves the performance of PCG-V-ASP over that of PCG-V.

L2 superconvergence analysis of nonconforming element approximation for 3D time-harmonic Maxwell's equations

In this paper, the superconvergent property is found for the interpolation error of the nonconforming finite element at element centers. Based upon this property, the superconvergence results in the discrete l2 norm for the solutions E→,H→ and curl→E→ are presented for the 3D time-harmonic Maxwell's equations. In order to get the global superconvergence, a new postprocess operator derived from the rotated Q1 element interpolation is constructed, which is based on the superconvergence points. All theoretical results are justified by the provided smoothing and non-smoothing numerical tests.

Effective Maxwell’s equations in general periodic microstructures

ABSTRACTWe study the time harmonic Maxwell equations in a meta-material consisting of perfect conductors and void space. The meta-material is assumed to be periodic with period ; we study the behaviour of solutions in the limit and derive an effective system. In geometries with a non-trivial topology, the limit system implies that certain components of the effective fields vanish. We identify the corresponding effective system and can predict, from topological properties of the meta-material, whether or not it permits the propagation of waves.

An explicit reconstruction method for magnetic resonance electrical property tomography based on the generalized Cauchy formula

This paper presents an explicit reconstruction formula for magnetic resonance electrical property tomography (MREPT). We derive a Dbar problem from the time-harmonic Maxwell equations under the assumptions that , , and , where the z-axis is parallel to the body axis, is the measured magnetic field, and . Then, by using the generalized Cauchy formula, the electrical conductivity and permittivity are explicitly expressed in terms of their boundary values and the measured magnetic field. We also propose an iterative algorithm based on the explicit reconstruction formula without the assumption that . Numerical simulations show that the proposed methods can reconstruct the electrical properties even with noisy data.

Saddle point least squares iterative solvers for the time harmonic Maxwell equations

We introduce a least squares method for discretizing the time-harmonic Maxwell equations written as a first order system. The method belongs to the recently introduced least squares approach called the Saddle Point Least Squares (SPLS) method (see Bacuta and Qirko 2015, 2017). It is related to two known methods: the Bramble and Pasciak least squares method for solving electromagnetic problems (Bramble and Pasciak, 2004; Bramble et al., 2005) and the minimum residual DPG method of Demkowicz and Gopalakrishnan (2010, 2013). The proposed discretization is based on reduction to a symmetric saddle point formulation on spaces of piecewise polynomial functions. The edge elements are avoided. A discrete test space is chosen first among standard conforming finite element spaces and the discrete trial space is chosen second as the image of the first order Maxwell differential operator on the discrete test space. For the proposed iterative processes, a nodal basis for the trial space is not required and the main inversion at each step is done only on the test space. The method is efficient for both convex and non-convex domains and for a various range of frequencies as shown by the numerical results.

Numerical resolution of an electromagnetic inverse medium problem at fixed frequency

The aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency. The governing equations are the time-harmonic Maxwell equations formulated in electric field in a two-dimensional bounded domain. We express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields. Our numerical reconstruction algorithm combines the BFGS method and an iterative process, called the Adaptive Eigenspace Inversion. The unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator. Both the eigenspace and the mesh are iteratively adapted during the minimization procedure. Numerical experiments illustrate the performance of the reconstruction for various configurations.

Magnetic resonance electrical properties tomography for small anomalies using boundary conditions: A simulation study.

Magnetic resonance electrical property tomography (MREPT) is an emerging imaging modality using measured B1 maps from magnetic resonance imaging (MRI) to measure a distribution of electric conductivity and permittivity of the subject at the Larmor frequency. Conventional MREPT approaches at single transmit channel system using the Helmholtz equation rely on an assumption that conductivity and permittivity of the subject are locally homogeneous. For small tissue structures and tissue boundaries, in which the assumption of locally homogeneous conductivity and permittivity does not hold, the reconstructed conductivity values deviated from the actual values, so called "Boundary Artifacts." The aim of this study is to propose new reconstruction processes based on time-harmonic Maxwell's equations to reconstruct conductivity for small tissue structures and tissue boundaries. Instead of removing the electric fields from the equations as done in the Helmholtz equation, three key identities of circularly polarized and longitudinal components of electric fields, circularly polarized component of magnetic fields, and electric properties from time-harmonic Maxwell's equations are derived. Based on the three key identities, the proposed reconstruction methods determine conductivity, permittivity, and circularly polarized component and longitudinal component of electric fields using the measured H1+ . In each iterative step, estimated conductivity, permittivity, electric fields, and artifact-free mask region, Ω, where the contribution of the boundary artifacts is small, were updated. Using the estimated values in the artifact-free mask region as boundary conditions, the estimates beyond the mask region were updated. EM simulations were performed on three types of numerical phantoms with very small regions of homogeneous conductivity and permittivity. The performance of the proposed methods was evaluated using the simulated electric and magnetic fields. For the numerical simulation model, the proposed methods significantly reduced the boundary artifacts compared to conventional methods using Helmholtz equations. In addition, previous methods using the Helmholtz equation could measure conductivity of only large anomalies, but the proposed method can measure the conductivity of the small compartments whose size is 2-3 voxels. The proposed approaches are compatible with spatial filtering which can be used to reduce noise. If a good image segmentation is available as a prior information, better initial boundary conditions can be estimated, and thus the proposed approach can be more accurate for small tissue structures. The proposed reconstruction method not only determines electrical properties, but also circularly polarized component and longitudinal component of electric fields using an iterative process. The proposed method can quantitatively detect the conductivity of the small anomalies better than conventional methods.

Comprehensive thematic T-matrix reference database: A 2015–2017 update

The T-matrix method pioneered by Peter C. Waterman is one of the most versatile and efficient numerically exact computer solvers of the time-harmonic macroscopic Maxwell equations. It is widely used for the computation of electromagnetic scattering by single and composite particles, discrete random media, periodic structures (including metamaterials), and particles in the vicinity of plane or rough interfaces separating media with different refractive indices. This paper is the eighth update to the comprehensive thematic database of peer-reviewed T-matrix publications initiated in 2004 and lists relevant publications that have appeared since 2015. It also references a small number of earlier publications overlooked previously.

Numerical framework for the simulation of dielectric heating using finite and boundary element method

SummaryThis paper presents the latest development towards a software agent–based numerical framework for interdisciplinary coupled problems. The software agent framework is capable to incorporate multiple physical effects and combine multiple numerical methods through the usage of a neutral abstraction layer. The principles of software agent–based programming and the application in a numerical example are discussed. This paper focuses on dielectric heating and different methods for solving the time harmonic Maxwell equation. By reusing state of the art finite element method (FEM) and boundary element method (BEM) software tools and managing the interface between the different tools, this approach solves complicated and realistic three‐dimensional problems. The functionality and efficiency of the framework are demonstrated on the dielectric heating inside a human head due to electromagnetic waves. A proper iterative coupling formulation for FEM and BEM is derived and numerically validated. The FEM‐BEM coupling separates from existing approaches, because it takes advantage of new developments in the FEM, namely, the perfect handling of infinite domains through perfectly matched layers. Moreover, the simplicity and flexibility of the coupling method allow implementation in existing numerical codes.

Partial data inverse problems for Maxwell equations via Carleman estimates

In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim–Uhlmann and Kenig–Sjöstrand–Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.

A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations

In this paper we are concerned with plane wave discretizations of nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. To this end, we design a plane wave method combined with local spectral elements for the discretization of such nonhomogeneous equations. This method contains two steps: we first solve a series of nonhomogeneous local problems on auxiliary smooth subdomains by the spectral element method, and then apply the plane wave method to the discretization of the resulting (locally homogeneous) residue problem on the global solution domain. We derive error estimates of the approximate solutions generated by this method. The numerical results show that the resulting approximate solutions possess high accuracy.

Wavenumber explicit analysis for time-harmonic Maxwell equations in a spherical shell and spectral approximations

Wavenumber explicit analysis for time-harmonic Maxwell equations in a spherical shell and spectral approximations

Analysis of the non-reflecting boundary condition for the time-harmonic electromagnetic wave propagation in waveguides

In this paper, we study the non-reflecting boundary condition for the time-harmonic Maxwell's equations in homogeneous waveguides with an inhomogeneous inclusion. We analyze a series representation of solutions to the Maxwell's equations satisfying the radiating condition at infinity, from which we develop the so-called electric-to-magnetic operator for the non-reflecting boundary condition. Infinite waveguides are truncated to a finite domain with a fictitious boundary on which the non-reflecting boundary condition based on the electric-to-magnetic operator is imposed. As the main goal, the well-posedness of the reduced problem will be proved. This study is important to develop numerical techniques of accurate absorbing boundary conditions for electromagnetic wave propagation in waveguides.

Dipole excitation of surface plasmon on a conducting sheet: Finite element approximation and validation

We formulate and validate a finite element approach to the propagation of a slowly decaying electromagnetic wave, called surface plasmon–polariton, excited along a conducting sheet, e.g., a single-layer graphene sheet, by an electric Hertzian dipole. By using a suitably rescaled form of time-harmonic Maxwell's equations, we derive a variational formulation that enables a direct numerical treatment of the associated class of boundary value problems by appropriate curl-conforming finite elements. The conducting sheet is modeled as an idealized hypersurface with an effective electric conductivity. The requisite weak discontinuity for the tangential magnetic field across the hypersurface can be incorporated naturally into the variational formulation. We carry out numerical simulations for an infinite sheet with constant isotropic conductivity embedded in two spatial dimensions; and validate our numerics against the closed-form exact solution obtained by the Fourier transform in the tangential coordinate. Numerical aspects of our treatment such as an absorbing perfectly matched layer, as well as local refinement and a posteriori error control are discussed.

Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

We find solutions E:Ω→R3 of the problem{∇×(μ(x)−1∇×E)−ω2ε(x)E=∂EF(x,E)in Ων×E=0on ∂Ω on a bounded Lipschitz domain Ω⊂R3 with exterior normal ν:∂Ω→R3. Here ∇× denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field ℜ{E(x)eiωt} in an anisotropic material with a magnetic permeability tensor μ(x)∈R3×3 and a permittivity tensor ε(x)∈R3×3. The boundary conditions are those for Ω surrounded by a perfect conductor. It is required that μ(x) and ε(x) are symmetric and positive definite uniformly for x∈Ω, and that μ,ε∈L∞(Ω,R3×3). The nonlinearity F:Ω×R3→R is superquadratic and subcritical in E, the model nonlinearity being of Kerr-type: F(x,E)=|Γ(x)E|p for some 2<p<6 with Γ(x)∈GL(3) invertible for every x∈Ω and Γ,Γ−1∈L∞(Ω,R3×3). We prove the existence of a ground state solution and of bound states if F is even in E. Moreover if the material is uniaxial we find two types of solutions with cylindrical symmetries.

Nonlinear time-harmonic Maxwell equations in domains

The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\begin{aligned} \nabla \times \left( \mu (x)^{-1}\nabla \times u\right) - \omega ^2\varepsilon (x)u = f(x,u) \end{aligned}$$ for the field $$u:\Omega \rightarrow \mathbb {R}^3$$ in a domain $$\Omega \subset \mathbb {R}^3$$ . Here, $$\varepsilon (x) \in \mathbb {R}^{3\times 3}$$ is the (linear) permittivity tensor of the material, and $$\mu (x) \in \mathbb {R}^{3\times 3}$$ denotes the magnetic permeability tensor. The nonlinearity $$f:\Omega \times \mathbb {R}^3\rightarrow \mathbb {R}^3$$ comes from the nonlinear polarization. If $$f=\nabla _uF$$ is a gradient, then this equation has a variational structure. The goal of this paper is to give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions. It also contains refinements of known results and some new results.

A Symmetric Low-Frequency Stable Broadband Maxwell Formulation for Industrial Applications

A new monolithic, symmetric, low-frequency stable, broadband formulation of the time-harmonic Maxwell's equations is presented. The approach is based on a variational formulation of Maxwell's equations which accounts for the frequency dependencies of the underlying physical phenomena and does not require auxiliary variables. The formulation allows for stable evaluation at low frequencies, even if the frequency equals zero. It has a broadband validity and correctly comprises both high-frequency effects and (quasi-)static behaviors. This is demonstrated by using different test examples from scientific to large-scale real-world applications.

Homogenization of time harmonic Maxwell equations: the effect of interfacial currents

We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time‐harmonic current source. We perform the two‐scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Modeling electromagnetism in and near composite material using two-scale behavior of the time-harmonic Maxwell equations

The main purpose of this article is to study the two-scale behavior of the electromagnetic field in 3D in and near composite material. For this, time-harmonic Maxwell equations, for a conducting two-phase composite and the air above, are considered. Technique of two-scale convergence is used to obtain the homogenized problem.

Modified Coulomb and Lorenz gauges in the modeling of low- frequency electromagnetic processes

The boundary value problem for the quasistationary magnetic approximation of the time-harmonic Maxwell equations in inhomogeneous media is studied. The considered problem is reduced to the variational problem of determining vector magnetic and scalar electric potentials. The special gauges are discussed, that generalize the Coulomb and Lorenz gauges and allow to formulate the problems of the independent definitions of the vector magnetic potential. The correctness of the problems are established under general conditions on the coefficients. The relation between solutions of the problems with different gauges is studied. The equivalence of the problems for potentials to the original boundary value problem is proved.