Time-harmonic Regime Research Articles

Analysis of 2D Maxwell's Equations in a Time-Harmonic Regime

The variational formulation is an essential tool to analyze the existence and uniqueness of the solution of certain partial differential equations with boundary conditions. We can further approximate this analytical solution by computing a corresponding numerical solution obtained by the finite element method. In this paper, we studied 2D Maxwell's equations in a time-harmonic regime. We established a corresponding variational formulation and proved its well-posedness in certain conditions. We also constructed a corresponding internal approximation and gave an error estimate within some prior assumptions. This theoretical analysis provides a basis to compute the numerical solution of time-harmonic 2D Maxwell's equations and gives physical significance to the transverse magnetic problem.

A Controllability Method for Maxwell's Equations

We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a nonintrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap postprocessing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence toward the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.

Modal approximation for time-domain elastic scattering from metamaterial quasiparticles

This paper aims at quantitatively understanding the elastic wave scattering due to negative metamaterial structures under wide-band signals in the time domain. Specifically, we establish the modal expansion for the time-dependent field scattered by metamaterial quasiparticles in elastodynamics. By Fourier transform, we first analyze the modal expansion in the time-harmonic regime. With the presence of quasiparticles, we validate such an expansion in the static regime via quantitatively analyzing the spectral properties of the Neumann-Poincaré operator associated with the elastostatic system. We then approximate the incident field with a finite number of modes and apply perturbation theory to obtain such an expansion in the perturbative regime. In addition, we give polariton resonances as simple poles for the elastostatic system. Finally, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the resonant modal expansion with sharp error estimates.

Mathematical analysis of Goldstein’s model for time-harmonic acoustics in flows

Goldstein’s equations have been introduced in 1978 as an alternative model to linearized Euler equations to model acoustic waves in moving fluids. This new model is particularly attractive since it appears as a perturbation of a simple scalar model: the potential model. In this work we propose a mathematical analysis of boundary value problems associated with Goldstein’s equations in the time-harmonic regime.

Cloaking strategy for Love waves

Love waves are antiplane elastic waves which propagate along the surface of a heterogeneous medium. Under time-harmonic regime, they are governed by a scalar equation of the Helmholtz type. We exploit the invariance of this governing equation under an in-plane arbitrary coordinate transformation to design broadband cloaks for surface defects. In particular, we apply transformation elastodynamics to determine the anisotropic, position dependent, mechanical properties of ideal cloaks able to hide triangular and parabolic-shaped defects. Dispersion analysis and time-harmonic numerical simulations are employed to validate the proposed strategy. Next, we utilize layered monoclinic materials, with homogenized properties matching those of the ideal cloaks, to design feasible triangular-shaped cloaks. The performance of the layered cloaks is validated via parametric analysis of the dispersion curves, which converge to those of the ideal cloak when the unit cell-wavelength ratio vanishes. Finally, time-harmonic numerical simulations confirm a significant reduction of the defect-generated scattered fields by the layered cloaks.

Stable Determination of a Rigid Scatterer in Elastodynamics

We deal with an inverse elastic scattering problem for the shape determination of a rigid scatterer in the time-harmonic regime. We prove a local stability estimate of $log\,log$ type for the ident...

Linearized electrodynamics and stabilization of a cold magnetized plasma

We consider a linearized Euler–Maxwell model for the propagation and absorption of electromagnetic waves in a magnetized plasma. We present the derivation of the model, and we show its well-posedeness, its strong and polynomial stability under suitable and fairly general assumptions, its exponential stability in the same conditions as the Maxwell system, and finally its convergence to the time-harmonic regime. No homogeneity assumption is made, and the topological and geometrical assumptions on the domain are minimal. These results appear strongly linked to the spectral properties of various matrices describing the anisotropy and other plasma properties.

Uniqueness for time-dependent inverse problems with single dynamical data

In this work, we investigate the shape identification and coefficient determination associated with two time-dependent partial differential equations in two dimensions. We consider the inverse problems of determining a convex polygonal obstacle and the coefficient appearing in the wave and Schrödinger equations from a single dynamical data. With the near-field data, we first prove that the sound speed of the wave equation together with its contrast support of convex-polygon type can be uniquely determined, then establish a uniqueness result for recovering an electric potential as well as its support appearing in the Schrödinger equation. As a consequence of these results, we demonstrate a uniqueness result for recovering the refractive index of a medium from a single far field pattern at a fixed frequency in the time-harmonic regime.

A Benchmark TEAM Problem for Multi-Objective Pareto Optimization in Magnetics: The Time-Harmonic Regime

The study presented in this article reformulates and generalizes the TEAM benchmark, originally proposed for multi-objective optimization of magnetic devices under dc conditions, now extending it to the ac regime. A solution is furnished which has enabled an extensive search and reliable estimation of the shape of the Pareto front. Field uniformity and losses are considered with reference to a class of power inductors. It is argued that the benchmark provides a challenging target for new algorithms, especially those involving numerical modeling based on finite-element codes, where the number of objective function calls needs to be minimized for practical designs.

Natural Domain Decomposition Algorithms for the Solution of Time-Harmonic Elastic Waves

We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time-harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove that the classical Schwarz method is not convergent when applied to the Navier equations and can thus not be used as an iterative solver, only as a preconditioner for a Krylov method. We then introduce more natural transmission conditions between the subdomains and show that if the overlap is not too small, this new Schwarz method is convergent. We illustrate our results with numerical experiments, both for situations covered by our technical two subdomain analysis and situations that go far beyond, including many subdomains, cross points, heterogeneous materials in a transmission problem, and Krylov acceleration. Our numerical results show that the Schwarz method with adapted transmission conditions leads systematically to a better solver for the Navier equations than the classical Schwarz method.

On Well-Posedness of Scattering Problems in a Kirchhoff--Love Infinite Plate

We address scattering problems for impenetrable obstacles in an infinite elastic Kirchhoff-Love two-dimensional plate. The analysis is restricted to the purely bending case and the time-harmonic regime. Considering four types of boundary conditions on the obstacle, well-posedness for those problems is proved with the help of a variational approach: (i) for any wave number k when the plate is clamped, simply supported or roller supported; (ii) for any k except a discrete set when the plate is free (this set is finite for convex obstacles).

Numerical Modelling of Boundary Layers and Far Field Acoustic Propagation in Thermoviscous Fluid

To model linear acoustics in a thermoviscous fluid in open domain and time-harmonic regime, a Finite Element formulation in a bounded meshed domain is combined with the integral representation of the field for the propagative solution. The integrals are non-singular and involve the only Finite Element node values for temperature variation and particle velocity variables. To overcome the non-uniqueness of solutions at fictitious resonant frequencies, a Burton-Miller combination of integral representation is used. This formulation is suitable to compute acoustic radiation, scattering and diffraction by objects or mutual interaction between transducers. Two-dimensional computational experiments are presented in an infinite, open domain (exterior), showing that the model can be achieved in meshing only a thin domain surrounding the physical boundaries of a device.

Approximate cloaking for electromagnetic waves via transformation optics: Cloaking versus infinite energy

We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device only consists of a layer constructed by the mapping technique. Due to the fact that no-lossy layer is required, resonance might appear and the analysis is delicate. We analyze both non-resonant and resonant cases. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is achieved in both cases. Moreover, the degree of visibility depends on the compatibility of the source inside the cloaked region and the system. These facts are new and distinct from known mathematical results in the literature.

Equivalent boundary conditions for acoustic media with exponential densities. Application to the atmosphere in helioseismology

We present equivalent boundary conditions and asymptotic models for the solution of a transmission problem set in a domain which represents the sun and its atmosphere. This problem models the propagation of an acoustic wave in time-harmonic regime. The specific non-standard feature of this problem lies in the presence of a small parameter δ which represents the inverse rate of the exponential decay of the density in the atmosphere. This problem is well suited for the notion of equivalent conditions and the effect of the atmosphere on the sun is as a first approximation local. This approach leads to solve only equations set in the sun. We derive rigorously equivalent conditions up to the fourth order of approximation with respect to δ for the exact solution u. The construction of equivalent conditions is based on a multiscale expansion in power series of δ for u. Numerical simulations illustrate the theoretical results. Finally we measure the boundary layer phenomenon by introducing a characteristic length that turns out to depend on the mean curvature of the interface between the subdomains.

Microwave tomographic imaging of cerebrovascular accidents by using high-performance computing

The motivation of this work is the detection of cerebrovascular accidents by microwave tomographic imaging. This requires the solution of an inverse problem relying on a minimization algorithm (for example, gradient-based), where successive iterations consist in repeated solutions of a direct problem. The reconstruction algorithm is extremely computationally intensive and makes use of efficient parallel algorithms and high-performance computing. The feasibility of this type of imaging is conditioned on one hand by an accurate reconstruction of the material properties of the propagation medium and on the other hand by a considerable reduction in simulation time. Fulfilling these two requirements will enable a very rapid and accurate diagnosis. From the mathematical and numerical point of view, this means solving Maxwell’s equations in time-harmonic regime by appropriate domain decomposition methods, which are naturally adapted to parallel architectures.

On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model

We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle.

Space–time Trefftz-DG approximation for elasto-acoustics

ABSTRACTDiscontinuous finite element methods have proven their numerical accuracy and flexibility, but they are criticized for requiring high number of degrees of freedom for computation. There have been some advances with Hybridizable Discontinuous Galerkin approximations, but essentially in the time-harmonic regime, because the time integration requires implicit schemes. Another possibility to explore is Trefftz method, which distinguishes itself by the choice of basis functions: they are local solutions of initial equation. Thus, in case of time-dependent problems, space-time meshes are required. Classical Trefftz-type approximation uses the exact solutions of acoustic and elastodynamic systems taken with different frequencies, in order to obtain better numerical errors. We have computed a polynomial basis using the Taylor expansions of generating exponential functions that are the exact solutions of the initial acoustic and elastodynamic systems. This basis, compared to the classical one based on trigonometric functions, requires less degrees of freedom for the same level of accuracy. In the present work, we develop the theory for coupled elasto-acoustic system. We confirm well-posedness of the problem, based on error estimates in mesh-dependent norms. We consider a space-time polynomial basis to implement numerically the method and to perform some numerical experiments showing promising results.

Enriched finite elements for time-harmonic Webster’s equation

This work presents an enriched finite element method (FEM) dedicated to the numerical resolution of Webster’s equation in the time-harmonic regime, which models many physical configurations, e.g. wave propagation in acoustic waveguides or vibration of bars with varying cross-section. Building on the wave-based methods existing in the literature, we present new enriched finite element bases that account for both the frequency of the problem and the heterogeneity of the coefficients of the equation. The enriched method is compared to the classical fifth-order polynomial FEM, and we show they share the same asymptotic convergence order, present the same easiness of implementation and have similar computational costs. The main improvement brought by these enriched bases relies on the convergence threshold (the mesh size at which the convergence regime begins) and the convergence multiplicative constant, which are observed to be (i) better than the ones associated with polynomial bases and (ii) not only dependent on the resolution (the number of elements per wavelength), but also on the frequency for a fixed resolution, making the method we propose well adapted to high-frequency regimes. Moreover, taking into account the heterogeneity of the coefficient in Webster’s equation by using an element-dependent enrichment leads to a significant decrease of the approximation error on the considered examples compared to a uniform enrichment, again with almost no additional cost. Several possible extensions of this work are finally discussed.

The halfspace matching method: A new method to solve scattering problems in infinite media

We are interested in acoustic wave propagation in time harmonic regime in a two-dimensional medium which is a local perturbation of an infinite isotropic or anisotropic homogeneous medium. We investigate the question of finding artificial boundary conditions to reduce the numerical computations to a neighborhood of this perturbation. Our objective is to derive a method which can extend to the anisotropic elastic problem for which classical approaches fail. The idea consists in coupling several semi-analytical representations of the solution in halfspaces surrounding the defect with a Finite Element computation of the solution around the defect. As representations of the same function, they have to match in the infinite intersections of the halfspaces. It leads to a formulation which couples, via integral operators, the solution in a bounded domain including the defect and its traces on the edge of the halfspaces. A stability property is shown for this new formulation.

An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell’s equations

In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell’s equations in waveguide configurations. The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell’s equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.