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Discrete & Continuous Dynamical Systems - S

2015 , Volume 8 , Issue 3

Issue on the workshop “Electromagnetics-Modelling, Simulation, Control and Industrial Applications"

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Dietmar Hömberg and  Guanghui Hu
2015, 8(3): i-i doi: 10.3934/dcdss.2015.8.3i +[Abstract](35) +[PDF](96.9KB)
The workshop Electromagnetics-Modelling, Simulation, Control and Industrial Applications was held at Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin during May 13-17, 2013. Organizers of this workshop were Dietmar Hömberg (WIAS), Ronald H. W. Hoppe (University of Augsburg/University of Houston), Olaf Klein (WIAS), Jürgen Sprekels (WIAS) and Fredi Tröltzsch (Technical University of Berlin). More than sixty researchers from mathematical, physical, engineering and industrial communities participated in this scientific meeting. This special issue of DCDS-S, which contains eleven research-level articles, is based on the talks presented during the workshop. Electromagnetism plays an important role in many modern high-technological applications. Our workshop brought together prominent worldwide experts from academia and industry to discuss recent achievements and future trends of modelling, computations and analysis in electromagnetics. The contributions to this volume cover the following topics: finite and boundary element discretization methods for the electromagnetic field equations in frequency and time domain, optimal control and model reduction for multi-physics problems involving electromagnetics, mathematical analysis of Maxwell's equations as well as direct and inverse scattering problems.
    We particularly emphasize that most articles are devoted to mathematically challenging issues in applied sciences and industrial applications. The optimal control and model reduction presented by S. Nicaise et al. arise from electromagnetic flow measurement in the real world. The contribution by G. Beck et al. focuses on a generalized telegrapher's model which describes the propagation of electromagnetic waves in non-homogeneous conductor cables with multi-wires. The numerical analysis of boundary integral formulations carried out by K. Schmidt et al. is motivated by asymptotic models for thin conducting sheets. The integral equation system established by B. Bugert et al. and the analysis and experiments performed by H. Gross et al. make new contributions to direct and inverse electromagnetic scattering from diffraction gratings, respectively. The locating and inversion schemes for detecting unknown configurations proposed by H. Ammari, G. Bao, J. Li and X. Liu et al. could be important and useful in radar and medical imaging, non-destructive testing and geophysical exploration. Last but not least, one can also find important mathematical applications regarding the estimate of the second Maxwell eigenvalues obtained by D. Pauly and the regularity of solutions to Maxwell's system at low frequencies due to P-E. Druet.
    We hope the presented papers will find a large audience and they may stimulate novel studies on electromagnetism. Finally we would like to express our gratitude to the Research Center MATHEON and the Weierstrass Institute, whose financial support made the workshop possible.
Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging
Habib Ammari , Josselin Garnier and  Vincent Jugnon
2015, 8(3): 389-417 doi: 10.3934/dcdss.2015.8.389 +[Abstract](35) +[PDF](536.4KB)
The detection, localization, and characterization of a collection of targets embedded in a medium is an important problem in multistatic wave imaging. The responses between each pair of source and receiver are collected and assembled in the form of a response matrix, known as the multi-static response matrix. When the data are corrupted by measurement or instrument noise, the structure of the response matrix is studied by using random matrix theory. It is shown how the targets can be efficiently detected, localized and characterized. Both the case of a collection of point reflectors in which the singular vectors have all the same form and the case of small-volume electromagnetic inclusions in which the singular vectors may have different forms depending on their magnetic or dielectric type are addressed.
Radar cross section reduction of a cavity in the ground plane: TE polarization
Gang Bao and  Jun Lai
2015, 8(3): 419-434 doi: 10.3934/dcdss.2015.8.419 +[Abstract](25) +[PDF](410.7KB)
The reduction of backscatter radar cross section(RCS) in TE polarization for a rectangular cavity embedded in the ground plane is investigated in this paper. It is established by placing a thin, multilayered radar absorbing material(RAM) with possibly different permittivities at the bottom of the cavity. A minimization problem with respect to the backscatter RCS is formulated to determine the synthesis of RAM. The underlying scattered field is governed by a generalized Helmholtz equation with transparent boundary condition. The gradient with respect to the material permittivity is derived by the adjoint state method. A fast solver for the Helmholtz equation is presented for the optimization scheme. Numerical examples are presented to show the efficiency of the algorithm for RCS reduction.
Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures
Beatrice Bugert and  Gunther Schmidt
2015, 8(3): 435-473 doi: 10.3934/dcdss.2015.8.435 +[Abstract](40) +[PDF](606.7KB)
This paper is concerned with the study of a new integral equation formulation for electromagnetic scattering by a $2\pi$-biperiodic polyhedral Lipschitz profile. Using a combined potential ansatz, we derive a singular integral equation with Fredholm operator of index zero from time-harmonic Maxwell's equations and prove its equivalence to the electromagnetic scattering problem. Moreover, under certain assumptions on the electric permittivity and the magnetic permeability, we obtain existence and uniqueness results in the special case that the grating is smooth and, under more restrictive assumptions, in the case that the grating is of polyhedral Lipschitz regularity.
Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations
Pierre-Étienne Druet
2015, 8(3): 475-496 doi: 10.3934/dcdss.2015.8.475 +[Abstract](36) +[PDF](532.5KB)
We show that $L^p$ vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We use these tools to investigate the regularity of the solution to the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation `in H' of the problem, in a reference geometrical setting allowing for material heterogeneities and nonsmooth interfaces.
Modeling aspects to improve the solution of the inverse problem in scatterometry
Hermann Gross , Sebastian Heidenreich , Mark-Alexander Henn , Markus Bär and  Andreas Rathsfeld
2015, 8(3): 497-519 doi: 10.3934/dcdss.2015.8.497 +[Abstract](31) +[PDF](4570.8KB)
The precise and accurate determination of critical dimensions (CDs) of photo masks and their uncertainties is relevant to the lithographic process. Scatterometry is a fast, non-destructive optical method for the indirect determination of geometry parameters of periodic surface structures from scattered light intensities. Shorter wavelengths like extreme ultraviolet (EUV) at 13.5 nm ensure that the measured light diffraction pattern has many higher diffraction orders and is sensitive to the structure details. We present a fast non-rigorous method for the analysis of stochastic line edge roughness with amplitudes in the range of a few nanometers, based on a 2D Fourier transform method. The mean scattering light efficiencies of rough line edges reveal an exponential decrease in terms of the diffraction orders and the standard deviation of the roughness amplitude. Former results obtained by rigorous finite element methods (FEM) are confirmed. The implicated extension of the mathematical model of scatterometry by an exponential damping factor is demonstrated by a maximum likelihood method used to reconstruct the geometrical parameters. Approximate uncertainties are determined by employing the Fisher information matrix and additionally a Monte Carlo method with a limited amount of samplings. It turns out that using incomplete mathematical models may lead to underestimated uncertainties calculated by the Fisher matrix approach and to substantially larger uncertainties for the Monte Carlo method.
Mathematical modelling of multi conductor cables
Geoffrey Beck , Sebastien Imperiale and  Patrick Joly
2015, 8(3): 521-546 doi: 10.3934/dcdss.2015.8.521 +[Abstract](34) +[PDF](1996.4KB)
This paper proposes a formal justification of simplified 1D models for the propagation of electromagnetic waves in thin non-homogeneous lossy conductor cables. Our approach consists in deriving these models from an asymptotic analysis of 3D Maxwell's equations. In essence, we extend and complete previous results to the multi-wires case.
Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method
Jingzhi Li , Hongyu Liu and  Qi Wang
2015, 8(3): 547-561 doi: 10.3934/dcdss.2015.8.547 +[Abstract](36) +[PDF](1073.7KB)
Some effective imaging schemes for inverse scattering problems were recently proposed in [13,14] for locating multiple multiscale electromagnetic (EM) scatterers, namely a combination of components of possible small size and regular size compared to the detecting EM wavelength. In this paper, instead of using a single far-field measurement, we relax the assumption of one fixed frequency to multiple ones, and develop efficient numerical techniques to speed up those imaging schemes by adopting multi-frequency and Multilevel ideas in a two-stage manner. Numerical tests are presented to demonstrate the efficiency and the salient features of the proposed fast imaging scheme.
The factorization method for scatterers with different physical properties
Xiaodong Liu
2015, 8(3): 563-577 doi: 10.3934/dcdss.2015.8.563 +[Abstract](31) +[PDF](616.8KB)
The scattering of time-harmonic acoustic plane waves by a mixed type scatterer is considered. Such a scatterer is given as the union of several components with different physical properties. Some of them are impenetrable obstacles with Dirichlet or impedance boundary conditions, while the others are penetrable inhomogeneous media with compact support. This paper is concerned with modifications of the factorization method for the following two basic cases, one is the scattering by a priori separated sound-soft and sound-hard obstacles, the other one is the scattering by a scatterer with impenetrable (Dirichlet) and penetrable components. The other cases can be dealt with similarly. Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of the modified factorization methods.
Optimal control of magnetic fields in flow measurement
Serge Nicaise , Simon Stingelin and  Fredi Tröltzsch
2015, 8(3): 579-605 doi: 10.3934/dcdss.2015.8.579 +[Abstract](28) +[PDF](4673.9KB)
Optimal control problems are considered for transient magnetization processes arising from electromagnetic flow measurement. The magnetic fields are generated by an induction coil and are defined in 3D spatial domains that include electrically conducting and nonconducting regions. Taking the electrical voltage in the coil as control, the state equation for the magnetic field and the electrical current generated in the induction coil is a system of integro-differential evolution Maxwell equations. The aim of the control is a fast transition of the magnetic field in the conduction region from an initial polarization to the opposite one. First-order necessary optimality condition and numerical methods of projected gradient type are discussed for associated optimal control problems. To deal with the extremely long computing times for this problem, model reduction by standard proper orthogonal decomposition is applied. Numerical tests are shown for a simplified geometry and for a 3D industrial application.
On Maxwell's and Poincaré's constants
Dirk Pauly
2015, 8(3): 607-618 doi: 10.3934/dcdss.2015.8.607 +[Abstract](30) +[PDF](402.7KB)
We prove that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincaré's constants. In other words, the second Maxwell eigenvalues lie between the square roots of the second Neumann-Laplace and the first Dirichlet-Laplace eigenvalue.
Asymptotic boundary element methods for thin conducting sheets
Kersten Schmidt and  Ralf Hiptmair
2015, 8(3): 619-647 doi: 10.3934/dcdss.2015.8.619 +[Abstract](40) +[PDF](1083.8KB)
Various asymptotic models for thin conducting sheets in computational electromagnetics describe them as closed hyper-surfaces equipped with linear local transmission conditions for the traces of electric and magnetic fields. The transmission conditions turn out to be singularly perturbed with respect to limit values of parameters depending on sheet thickness and conductivity. We consider the reformulation of the resulting transmission problems into boundary integral equations (BIE) and their Galerkin discretization by means of low-order boundary elements. We establish stability of the BIE and provide a priori $h$-convergence estimates.

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