Maxwell solutions in media with multiple random interfaces

International Journal for Numerical Methods in Engineering, 1995

We present a new first-principle framework for the prediction of effective properties and statistical correlation lengths for multicomponent random media. The methodology is based upon a variational hierarchical decomposition procedure which recasts the original multiscale problem as a sequence of three scale-decoupled subproblems. The focus of the current paper is the computationally intensive mesoscale subproblem, which comprises: Monte-Carlo acceptance–rejection sampling; domain generation and parallel partition based on Voronoi tesselation; parallel Delaunay mesh generation; homogenization-theory formulation of the governing equations; finite-element discretization; parallel iterative solution procedures; and implementation on message-passing multicomputers, here the Intel iPSC/860 hypercube. Two (two-dimensional) problems of practical importance are addressed: heat conduction in random fibrous composites, and creeping flow through random fibrous porous media.

In this paper we present a methodology for the numerical computation of noisy electromagnetic fields excited by spatially distributed noise sources with arbitrary correlation. The method uses the field transfer function computed for deterministic fields and can be combined with available electromagnetic modeling tools.

SIAM Journal on Scientific Computing, 2018

Solving stochastic partial differential equations (SPDEs) can be a computationally intensive task, particularly when the underlying parametrization of the stochastic input field involves a large number of random variables. Direct Monte Carlo (MC) sampling methods are well suited for this type of situation since their cost is independent of the input complexity. Unfortunately, MC sampling methods suffer from slow convergence. In this manuscript, we propose an acceleration framework for elliptic SPDEs that relies on domain decomposition techniques and polynomial chaos (PC) expansions of local operators to reduce the cost of solving a SPDE via MC sampling. The approach exploits the fact that, at the subdomain level, the number of variables required to accurately parametrize the input stochastic field can be significantly reduced, as covered in detail in the prequel (Part A) to this paper. This makes it feasible to construct PC expansions of the local contributions to the condensed problem (i.e., the Schur complement of the discretized operator). The approach basically consists of two main stages: (1) a preprocessing stage in which PC expansions of the condensed problem are computed and (2) a Monte Carlo sampling stage where random samples of the solution are computed. The proposed method its naturally parallelizable. Extensive numerical tests are used to validate the methodology and assess its serial and parallel performance.

We discuss a mixed vector finite-element time-domain (FETD) algorithm for simulating Maxwell's equations in inhomogeneous doubly-dispersive media in the time-domain. The proposed FETD algorithm is based on first-order coupled Maxwell's equations and, edge and face element expansions for the electric and magnetic fields respectively. When compared to standard FETD scheme based on the second order wave equation, the mixed FETD scheme possesses several advantages without any significant computational drawback. In this paper, we specifically explore the fact that frequency dispersion in permittivity and permeability tensors can easily be incorporated in the mixed scheme.

2018

This paper presents an approach of one-and two-dimensional random field simulation methods using a correlated random vector and the Karhunen-Loève expansion. Comparison of the authors' analytical solution of the Fredholm integral equation of the second kind with the numerical solution using the finite element method and the inverse vector iteration technique is presented. Numerical approach and sample realizations of one-and two-dimensional random fields are presented using described techniques as well as generated probability distribution functions for chosen point of the analysed domain.

IEEE Transactions on Antennas and Propagation, 2000

A split-step finite-difference time-domain (FDTD) method is presented for 3-D Maxwell's equations in general anisotropic media. The general anisotropic media can be characterized by full permittivity and permeability tensors. Stability analysis of the proposed split-step FDTD method using the Fourier method is presented. The eigenvalues of the Fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the Courant limit time step, thereby illustrating the stable and non-dissipative nature of the split-step FDTD method in general anisotropic media. Numerical results are presented to further validate the accuracy and stability of the proposed split-step FDTD method in general anisotropic media.

International Journal for Numerical Methods in Engineering, 2002

We present a new approach to time domain hybrid schemes for the Maxwell equations. By combining the classical FD-TD scheme with two unstructured solvers, one explicit ÿnite volume solver and one implicit ÿnite element solver, we achieve a very e cient and exible second-order scheme. The secondorder accuracy of the hybrid scheme is veriÿed through convergence studies on perfectly conducting as well as dielectric and diamagnetic circular cylinders. The numerical results also show its superiority to the FD-TD scheme.

IEEE Transactions on Antennas and Propagation, 2013

In this paper a three-dimensional (3-D) numerical technique based on the finite difference frequency domain (FDFD) is implemented to calculate the scattering from arbitrary shaped objects embedded in a continuous random medium. The total field/scattered field (TF/SF) algorithm is integrated with the FDFD to minimize the memory consumption and speed up the calculations. For validation purposes, the radar cross-section of a 2-D conducting cylinder in random medium is calculated using the FDFD technique and compared to previously published data based on the current generator method. An upgrade to a 3-D solver was then inspired once the idea of using the multi-grid technique was introduced to accelerate the convergence rate of the BICGSTAB iterative solver. Thus, allowing the FDFD technique to become a robust method to solve the scattering problem from large targets embedded in random medium. Therefore, using the introduced simulating scheme, one can easily elucidate any scattering information out of real life targets surrounded by random environmental effects.

Central European Journal of Mathematics, 2012

For the Maxwell equations in time-dependent media only finite difference schemes with time-dependent conductivity are known. In this paper we present a numerical scheme based on the Magnus expansion and operator splitting that can handle time-dependent permeability and permittivity too. We demonstrate our results with numerical tests.

2020

Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. In this paper, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green's kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal $\epsilon^{-2}$ asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.

2010

Abstract: In this paper, two different methods are studied to solve numerically time dependent Maxwell's equations in two dimensions. The methods are, the Discontinuous Galerkin Method (DGM) and the Variational Iteration Method (VIM). Discontinuous Galerkin method is tested for different number of triangulation elements, different time intervals and different orders of basis functions. Comparisons between the solutions of Maxwell's equations using DGM and VIM are presented.

IEEE Transactions on Antennas and Propagation, 2015

The electromagnetic (EM) scattering problem from three-dimensional (3-D) arbitrary composite objects is proposed using the random auxiliary sources (RAS) method. Based on direct application of the boundary conditions with the uniqueness theorem and the use of random equivalent problems concept, more degrees-of-freedom to the sources' positions are added resulting in significantly efficient solutions with lower memory requirements. The technique does not require any singularity treatment due to placing the equivalent sources away from the boundaries. While boundary conditions are not enforced exactly, an iterative framework is introduced that can achieve an acceptable level of error in their satisfaction for an arbitrary, randomly generated set of equivalent sources. The presented technique promises a significant reduction in the execution time and memory requirements compared to the surface-equivalent-based method of moments (MoM). The solution stability, repeatability, and numerical noise susceptibility are investigated thoroughly through this work. Also, a novel edge correction scheme has been implemented to extend the capabilities of this procedure to structures with sharp edges. The results of the presented technique are compared to series solutions for conducting spheres and a commercially available MoM code for arbitrarily shaped objects and combinations of different materials. Index Terms-Electromagnetic (EM) scattering, fast methods, frequency domain analysis, random auxiliary source (RAS) method. I. INTRODUCTION E LECTROMAGNETIC (EM) solutions for open-boundary problems are required to be rapid, especially for recent demanding applications [1] involving electrically large objects. Many research groups have devoted decades of extensive work toward accelerating the solution speed and improving the accuracy and range of applications [2]. The method of moments (MoM) is typically the preferred method for open problems due to its systematic approach and reasonably fast response. This is a result of directly implementing the surface equivalence principle on the boundaries of the given object(s) [3] without a need to have volumetric meshing. Recently, large antennas and scattering problems still challenge the capabilities

Physical Review B, 2009

Localization of electromagnetic waves in two-dimensional ͑2D͒ and three-dimensional ͑3D͒ media with random permittivities is studied by numerical simulations of the Maxwell's equations. Using the transfermatrix method, the minimum positive Lyapunov exponent ␥ m of the model is computed, the inverse of which is the localization length. Finite-size scaling analysis of ␥ m is carried out in order to check the localizationdelocalization transition in 2D and 3D. We show that in 3D disordered media ␥ m exhibits two distinct types of frequency dependence over two frequency ranges, hence indicating the existence of a localizationdelocalization transition at a critical frequency c . The critical exponent of the localization length in 3D is estimated to be, Ӎ 1.57Ϯ 0.07. At the transition point in the 3D media, the distribution function of the level spacings is independent of the system size, and is represented well by the semi-Poisson distribution. The 2D model can be mapped onto the 2D Anderson model and, hence, there is no localization-delocalization transition.

Electrical, Control and Communication Engineering, 2014

In electromagnetic problems, the problem geometry may not always be exactly known. One example of such a case is a rotating machine with random-wound windings. While spectral stochastic finite element methods have been used to solve statistical electromagnetic problems such as this, their use has been mainly limited to problems with uncertainties in material parameters only. This paper presents a simple method to solve both static and time-harmonic magnetic field problems with source currents in random positions. By using an indicator function, the geometric uncertainties are effectively reduced to material uncertainties, and the problem can be solved using the established spectral stochastic procedures. The proposed method is used to solve a demonstrative single-conductor problem, and the results are compared to the Monte Carlo method. Based on these simulations, the method appears to yield accurate mean values and variances both for the vector potential and current, converging clo...

IEEE Transactions on Antennas and Propagation, 2018

A 3D general purpose implementation of the Random Auxiliary Sources (RAS) method is proposed, benefiting from the Rao-Wilton-Glisson (RWG) testing functions. The testing procedure is presented in terms of the RWG function parameters. The performance of proposed RAS with the RWG testing function is compared with point matching. Also, a simple treatment for mixed boundary problems is introduced to enable solving complex electromagnetic problems. Furthermore, a waveport formulation is proposed that is better suited for the iterative approach of the RAS method to facilitate the modular analysis of components. A gradient-based testing formulation for the transmitting port is exploited to enforce the matching to the forward wave only on a waveport. Nevertheless, several test cases are presented to evaluate the accuracy and performance of the proposed method in comparison to commercial full-wave solvers showing a reasonable agreement.