Dispersive effective material parameters for Maxwell's equations

We study how effective material parameters can be defined for Maxwell's equations when taking dispersion into account. The reasoning is based on the concept of dispersion relations, and is consequently primarily concerned with lossless media. We essentially require that the effective material parameters should produce the same dispersion relations as the heterogeneous problem, which implies the effective material is primarily connected to the phase velocity of the waves. Material parameters which exhibit temporal dispersion only, can be defined if the propagation direction is fixed

A comparison of two numerical methods for homogenization of Maxwell's equations

When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. The classical way of determination of the homogenized coeffi- cients consists of solving an elliptic problem in a unit cell. This method and the Floquet-Bloch method, where an eigenvalue problem is solved, are numerically compared with respect to accuracy and contrast sensitivity. The Floquet-Bloch method is shown to be a good alternative to the classical homogenization method, when the contrast is modest

Time-Dependent Cauchy-Navier Splines and their Application to Seismic Wave Front Propagation

In this paper a known orthonormal system of time- and space-dependent functions, that were derived out of the Cauchy-Navier equation for elastodynamic phenomena, is used to construct reproducing kernel Hilbert spaces. After choosing one of the spaces the corresponding kernel is used to define a function system that serves as a basis for a spline space. We show that under certain conditions there exists a unique interpolating or approximating, respectively, spline in this space with respect to given samples of an unknown function. The name "spline" here refers to its property of minimising a norm among all interpolating functions. Moreover, a convergence theorem and an error estimate relative to the point grid density are derived. As numerical example we investigate the propagation of seismic waves

Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations

The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well known for the Helmholtz equation, and second the large scale systems obtained from the vector valued equations in three spatial dimensions need to be solved by iterative methods, since direct factorizations are not feasible any more at that scale. As for the Helmholtz equation, classical iterative methods applied to discretized Maxwell equations have severe convergence problems.We explain in this paper a family of domain decomposition methods based on well chosen transmission conditions. We show that all transmission conditions proposed so far in the literature, both for the first and second order formulation of Maxwell's equations, can be written and optimized in the common framework of optimized Schwarz methods, independently of the first or second order formulation one uses, and the performance of the corresponding algorithms is identical. We use a decomposition into transverse electric and transverse magnetic fields to describe these algorithms, which greatly simplifies the convergence analysis of the methods. We illustrate the performance of our algorithms with large scale numerical simulations

Software for evaluating probability-based integrity of reinforced concrete structures

In recent years, much research work has been carried out in order to obtain a more controlled durability and long-term performance of concrete structures in chloride containing environment. In particular, the development of new procedures for probability-based durability design has proved to give a more realistic basis for the analysis. Although there is still a lack of relevant data, this approach has been successfully applied to several new concrete structures, where requirements to a more controlled durability and service life have been specified. A probability-based durability analysis has also become an important and integral part of condition assessment of existing concrete structures in chloride containing environment. In order to facilitate the probability-based durability analysis, a software named DURACON has been developed, where the probabilistic approach is based on a Monte Carlo simulation. In the present paper, the software for the probability-based durability analysis is briefly described and used in order to demonstrate the importance of the various durability parameters affecting the durability of concrete structures in chloride containing environment

Discontinuous Galerkin Finite Element Methods for Maxwell\u27s Equations in Dispersive and Metamaterials Media

Discontinuous Galerkin Finite Element Method (DG-FEM) has been further developed in this dissertation. We give a complete proof of stability and error estimate for the DG-FEM combined with Runge Kutta which is commonly used in different fields. The proved error estimate matches those numerical results seen in technical papers. Numerical simulations of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. We propose a leap-frog discontinuous Galerkin Finite Element Method to solve the time-dependent Maxwell\u27s equations in metamaterials. The stability and error estimate are proved for this scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided. The wave propagation simulation in the double negative index metamaterials supplemented with perfectly matched layer (PML) boundary is given with one discontinuous Galerkin time difference method (DGTD), of which the stability and error estimate are proved as well in this dissertation. To illustrate the effectiveness of this DGTD, we present some numerical result tables which show the consistent convergence rate and the simulation of PML in metamaterials is tested in this dissertation as well. Also the wave propagation simulation in metamaterals by this DGTD scheme is consistent with those seen in other papers. Several techniques have appeared for solving the time-dependent Maxwell\u27s equations with periodically varying coefficients. For the first time, I apply the discontinuous Galerkin (DG) method to this homogenization problem in dispersive media. For simplicity, my focus is on obtaining a solution in two-dimensions (2D) using 2D corrector equations. my numerical results show the DG method to be both convergent and efficient. Furthermore, the solution is consistent with previous treatments and theoretical expectations

Transport modeling of simple fluids and nano-colloids : thermal conduction mechanisms and coarse projection

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Nuclear Science and Engineering, 2006.Includes bibliographical references (p. 159-166).In the first part of this thesis, the modes of microscopic energy fluctuations governing heat flow in nano-colloids are quantitatively assessed by combining linear response theory with molecular dynamics (MD) simulations. The intrinsic thermal conductivity is decomposed into self and cross correlations of the three modes that make up the microscopic heat flux vector, namely, the kinetic, the potential and the virial. By this decomposition analysis, the interplay between the molecular mechanisms that govern the variation of the thermal conductivity with volume fraction and solid-fluid interaction is examined. For a specific system of nanosized platinum clusters which interact strongly with host liquid xenon, a significant thermal conductivity enhancement is obtained as a result of self correlation in the potential energy flux. The effect saturates at higher volume fractions due to the cross-mode correlation between the potential and the virial flux. A strong solid-fluid coupling also introduces an amorphous-like structural transition and a pronounced cage effect that significantly reduces the self diffusion of the nano-clusters. These attendant structural and diffusive effects, unlike the self correlation of the potential flux, are amenable to experimental observations. The cluster-fluid interface is characterized by large fluctuations in the potential energy which is indicative of an unusual exchange of potential energy among the interfacial fluid atoms. For small nano-clusters, the interfacial layers interact with each other to form a percolating network. The research findings highlight the importance of surface interactions and show that the interfacial thermal resistance emanating from the self correlation of the collision flux is not the limiting mechanism for heat transfer in nano-colloids.(cont.) This thesis also addresses several theoretical concerns regarding the microscopic thermal transport in colloids by using non-equilibrium molecular dynamics simulations (NEMD). The time averaged microscopic heat flux which assumes spatial homogeneity is shown to be applicable to nano-colloidal systems. Further, it is demonstrated that the thermal conductivity from a NEMD simulation is statistically equivalent to that of an equilibrium linear response evaluation only under certain dynamic conditions at the cluster-fluid interface. The concept of interfacial dynamical similarity is developed to establish this equivalence. The proposed thermal conduction model is consistent with several experimental observations such as the anomalous enhancement at small volume fractions with very small nanoparticles (3-10nm), limiting behavior at higher volume fractions, and the lack of correlation of the enhancement to the intrinsic thermal conductivity of the nano-clusters. The model also suggests possible avenues for optimizing the colloids by developing nano-clusters that have functionalized surface layers to maximize the interactions with the fluid atoms. In the second part of this thesis, smooth field estimators based on statistical inference and smoothing kernels are developed to transfer molecular data to the continuum for hybrid and equation-free multiscale simulations. The field estimators are then employed to implement coarse projection, a multiscale integration scheme, for a shear driven flow in an enclosure. This thesis shows that the spatial continuity and smoothness of the microscopically generated coarse variables, geometrically similar initial conditions and the separation of timescales are essential for the correct coarse field evolution with coarse projection.by Jacob Eapen.Sc.D