Fictitious Medium Research Articles

Viscous flow past a porous sphere within a nonconcentric fictitious spherical cell

The quasi steady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous sphere situated at an arbitrary position within a virtual spherical cell along the line connecting their centers is analyzed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman model governs the flow inside the porous particle and the flow in the fictitious envelope medium is governed by Stokes equations. A general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on both the porous particle and virtual spherical cell. Boundary conditions on the particle’s surface and the fictitious spherical envelope for four known cell models are satisfied by a collocation method for truncated series. Numerical solutions for the drag force exerted on the porous sphere in the presence of the cell are obtained for various cases of the effective distance between the center of the porous particle and the fictitious envelope, the volume ratio of the porous particle over the surrounding sphere, the viscosity ratio, the stress jump coefficient, and a coefficient that is proportional to the permeability. Streamlines in and around porous sphere are presented for the unit cell models at different values of relevant physical parameters. In the limits of the motions of the porous particle in the concentric position with the cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature.

Spectral and decomposition tracking for rendering heterogeneous volumes

We present two novel unbiased techniques for sampling free paths in heterogeneous participating media. Our decomposition tracking accelerates free-path construction by splitting the medium into a control component and a residual component and sampling each of them separately. To minimize expensive evaluations of spatially varying collision coefficients, we define the control component to allow constructing free paths in closed form. The residual heterogeneous component is then homogenized by adding a fictitious medium and handled using weighted delta tracking, which removes the need for computing strict bounds of the extinction function. Our second contribution, spectral tracking , enables efficient light transport simulation in chromatic media. We modify free-path distributions to minimize the fluctuation of path throughputs and thereby reduce the estimation variance. To demonstrate the correctness of our algorithms, we derive them directly from the radiative transfer equation by extending the integral formulation of null-collision algorithms recently developed in reactor physics. This mathematical framework, which we thoroughly review, encompasses existing trackers and postulates an entire family of new estimators for solving transport problems; our algorithms are examples of such. We analyze the proposed methods in canonical settings and on production scenes, and compare to the current state of the art in simulating light transport in heterogeneous participating media.

A BEM based methodology to solve inverse problems considering fictitious background media

This paper proposes a new BEM based inversion methodology named Fictitious Background Media Inverse Formulation (FBMIF) to solve inverse problems described in terms of the Helmholtz equation for inhomogeneous media. The BEM based approach considers a Green’s function related to a homogeneous background medium, a fictitious medium, which works as a transfer function. Thereafter, considering a nonrestrictive linearization scheme, the FBMIF is completely defined. Three examples are used to test the efficacy of the proposed model of inversion, which can be extended to other mathematical models different from those governed by the Helmholtz equation.

Nonuniform networks with restrictions on the residence time in maintenance mode

An open nonuniform network of queuing with the simplest incoming flow, exponential service at the nodes, and Markov routing was studied. In each of the nodes there is only one device, which can operate in several modes. For isolated nodes in a fictitious medium, conditions of reversibility, under which the stationary probability distribution of states of the network has a multiplicative form, were obtained. It is proved that customers flows coming out of the network are independent and Poisson.

Nonlinear Path Following Method

This paper develops a new guidance logic for a vehicle to converge to a specified, desired path of general shape in three dimensions. Conditions are found that ensure exponential convergence to the path for any initial vehicle position and velocity vector, excluding only the velocity vector normal to the desired path. The guidance logic includes a ghost vehicle that follows the desired path. The command accelerations are derived from fictitious forces that act in a frame of reference that moves with the ghost. The forces comprise a springlike force between vehicle and ghost and a drag force created by a fictitious medium that moves with the ghost. The motion of the ghost is determined indirectly by a nonlinear constraint that controls the real vehicle speed and leads to a differential-algebraic system of equations. The path equations are formulated in terms of differential geometry, in which time is replaced by distance along the vehicle path. The geometric equations are transformed into standard form differential equations. These are transformed again into kinetic equations for both a fixed medium and a moving medium. The transformed equations provide more explicit formulas for command accelerations and are used to ensure that commands do not exceed the capabilities of the vehicle. Often, the vehicle limitation is more constraining than the convergence conditions. Simulations of a miniature aircraft, using these equations, illustrate the effectiveness of the method. Other methods are compared.

An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems

An adaptive perfectly matched layer (PML) technique for solving the time harmonic electromagnetic scattering problems is developed. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the adaptive PML technique provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorbing layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.

Modeling of Nomex® Honeycomb Cores, Linear and Nonlinear Behaviors

The purpose of this study is to develop tools dedicated to the design of sandwich panels involving Nomex® honeycomb cores. Special attention is paid to the ability to perform full three dimensional calculations up to failure of such structures. In the first part, the determination of effective elastic properties of Nomex® honeycomb cores is carried out thanks to strain based periodic homogenization technique. Using an equivalence in energy between a real honeycomb and a fictitious continuous medium, it becomes possible to evaluate the elastic behavior of Nomex® cores, starting from the knowledge of the behavior of the constitutive paper. Next, relying on experimental observations, the strengths of Nomex® honeycomb cores are evaluated with a linear Euler's buckling analysis. Results are compared with data coming from manufacturers, and give satisfaction. In order to carry out these two first studies, the NidaCore software has been developed using the finite element code Cast3M from CEA. The last part deals with the modeling of the nonlinear compressive response of Nomex® cores. A model based on the thermodynamics of irreversible process is proposed and the identification technique detailed. Good agreement between experimental data and computed values is obtained.

Calculation of the Earth Return Impedance of a Conductor Inside a Tunnel by Using the Fictitious Medium Method

This paper describes an efficient numerical approach for the calculation of the per-unit length earth-return impedance of a conductor inside a tunnel. The influence of the tunnel on the per-unit length impedance is replaced by a set of equivalent current sources via the fictitious medium method. The main advantage of the proposed approach is that it can solve the boundary value problem within the tunnel region. The proposed method can reduce the calculation region and, thus, leads to time savings. A comparison of the numerical results with published data shows very good agreement. The proposed method is a good candidate for impedance calculation of a conductor inside a tunnel.

On Determining the Percolation Reynolds Number and the Characteristic Linear Dimension for Ideal and Fictitious Porous Media

Various versions of representations of the percolation Reynolds number for porous media with isotropic and anisotropic flow properties are considered. The formulas are derived and the variants are analyzed with reference to model porous media with a periodic microstructure formed by systems of capillaries and packings consisting of spheres of constant diameter (ideal and fictitious porous media, respectively). A generalization of the Kozeny formula is given for determining the capillary diameter in an ideal porous medium equivalent to a fictitious medium with respect to permeability and porosity and it is shown that the capillary diameter is nonuniquely determined. Relations for recalculating values of the Reynolds number determined by means of formulas proposed earlier are given and it is shown that taking the microstructure of porous media into account, as proposed in [1, 2], makes it possible to explain the large scatter of the numerical values of the Reynolds number in processing the experimental data.

An Adaptive Perfectly Matched Layer Technique for Time-harmonic Scattering Problems

We develop an adaptive perfectly matched layer (PML) technique for solving the time-harmonic scattering problems. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. The derived finite element a posteriori estimate for adapting meshes has the nice feature that it decays exponentially away from the boundary of the fixed domain where the PML layer is placed. This property makes the total computational costs insensitive to the thickness of the PML absorbing layers. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.

An Adaptive Perfectly Matched Layer Technique for Time-harmonic Scattering Problems

We develop an adaptive perfectly matched layer (PML) technique for solving the time-harmonic scattering problems. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. The derived finite element a posteriori estimate for adapting meshes has the nice feature that it decays exponentially away from the boundary of the fixed domain where the PML layer is placed. This property makes the total computational costs insensitive to the thickness of the PML absorbing layers. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.

A fast quasi-multiple medium method for 3-D bem calculation of parasitic capacitance

A quasi-multiple medium (QMM) method is proposed to accelerate the boundary element method (BEM) for the 3-D parasitic capacitance calculation. In the QMM method, a homogeneous dielectric is decomposed into a number of fictitious medium blocks, each with the same permittivity of original medium. By the localization character of BEM, the QMM method makes great sparsity to the coefficient matrix of the overall discretized BEM equations. Then, using storing technique of sparse matrix and iterative equation solvers, the sparsity is explored to greatly reduce CPU time and memory usage of BEM computation. The computational complexity of the QMM accelerated BEM for a single-medium model problem is analyzed, and it is concluded as O( N), if the number of iterations is bounded. Numerical results verify the theoretical analysis and show the accelerating efficiency of the QMM method for calculation of 3-D parasitic capacitance.

Fast capacitance extraction of actual 3-D VLSI interconnects using quasi-multiple medium accelerated BEM

A quasi-multiple medium (QMM) method based on the direct boundary element method (BEM) is presented to extract the capacitance of three-dimensional (3-D) very large scale integration interconnects with multiple dielectrics. QMM decomposes each dielectric layer into a few fictitious medium blocks, and generates an overall coefficient matrix with high sparsity. With the storage technique of a sparse blocked matrix and iterative equation solver generalized minimal residual, the QMM can greatly reduce the CPU time and memory usage of large-scale direct BEM computation. Numerical examples of 3-D multilayered and multiconductor structures cut from actual layout show the efficiency of the QMM method for capacitance extraction. We also compared the QMM accelerated BEM with geometry independent measured equation of invariance (GIMEI) and Zhu's overlapping domain decomposition method (ODDM).

Perfectly matched anisotropic layer for the numerical analysis of unbounded eddy-current problems

An extension of the perfectly matched layer (PML) technique in quasi-static fields is developed. The new low frequency PML is based on a fictitious medium with diagonal tensor anisotropy. On the basis of a theoretical investigation, the material properties of the anisotropic layer are specified, so that it will be reflectionless for an arbitrary eddy-current field that may exist in free space. Furthermore, the PML is designed in such a way that outgoing eddy-current fields are sufficiently absorbed. The effectiveness of the low-frequency PML is validated by the implementation of the finite-element solution of a simple two-dimensional eddy-current problem as well as a more complicated three-dimensional one.

A constitutive model for soil reinforced by continuous threads

The aim of this paper is the formulation of a constitutive law for soil reinforced by continuous threads. Such a law is necessary to analyse the deformation pattern of structures, or earthworks, where this recently developed technique is employed. It is assumed that the reinforced soil can be seen as a continuum obtained by the superposition of two continua: unreinforced soil and a fictitious medium which is able to sustain only tensile stresses up to a certain failure level. It is also assumed that the two continua act in parallel so that strains are the same, while stresses are summed. It is shown that starting from the constitutive law of the two continua it is possible to derive the constitutive law for the reinforced soil. It is further shown that it is possible to reproduce reasonably well the deviatoric stress-strain law observed in triaxial compression tests, even by assuming very simple constitutive laws for sand and the threads. It is shown however that the model is neither able to describe the recorded peak nor to reproduce correctly the trend of volumetric strains.

Calculation of the alternating magnetic field near a metal surface

A medium-addition method and its recent development are recommended for calculating the 2-D alternating magnetic field in the vicinity of a defect. A fictitious medium with negative conductivity is suggested to simulate the defect. The prominent advantages of this method are that it requires much less computer capacity and is time saving. It is concluded that a medium-addition method can be used to solve certain electromagnetic field problems in nondestructive testing if the media are linear and the defect is a tiny crack or a small hole.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The effect of the angular dispersion term on energy deposition of fast α-particles slowing down in a D-T plasma

The linear Fokker-Planck equation for external charged particles slowing down in a fully ionized D-T plasma has been solved in one-dimensional plane geometry by the technique of Legendre polynomial and Fourier series expansion. The vacuum boundary condition at the plasma surface has been treated by appending to it a fictitious medium in which angular dispersion is absent. For alpha particles slowing down in a fully ionized D-T plasma the effect of the angular dispersion term on the energy deposition profiles has been estimated quantitatively and found to be small. It is found that it may be masked by errors introduced by the discretisation procedures employed in the numerical solutions of the Fokker-Planck equation.

Bend radiation in optical fibers

Abstract A new, relatively simple way of calculating bend radiation is developed. As is often done in propagation over gently curved surfaces, the fiber curvature is accounted for by placing a straight fiber in a fictitious medium that is inhomogeneous in the plane of the bend. For a gently curved fiber, the medium is slowly varying and the WKB method is used to approximate the radial field dependence. This solution is related to the unperturbed straight fiber solution through an ansatz consistent with the WKB solution. The propagating modes in this approximation remain orthogonal, allowing an immediate generalization to the multimoded case. The radiation loss per unit length is calculated two ways and is consistent with results in the literature.

Micromagnetics and Magnetostatics of an Iron Single Crystal Whisker

The magnetization process of an oriented iron single crystal is investigated. A solution of the micromagnetic equations for a multidomain configuration in an applied field demonstrates the dominance of magnetostatics in determining the susceptibility. The magnetic charge is found to be almost entirely on the surface, justifying the model of a fictitious medium of constant intrinsic susceptibility. This model is then used to explain the observed small coil susceptibility. Analytic expressions for the susceptibility and for an upper limit on the volume charge density (div M) are also developed. The major results of the magnetostatics calculation can be summarized by a useful local approximation for the demagnetizing field.

An interpretation of Choudhary and Felsen's perturbation method for tracing the phase paths of evanescent fields

A method for constructing the complex eikonal of an evanescent optical field, given its values on some surface, is described. It proceeds by successive approximations whose steps consist of tracing real rays in appropriate fictitious media. The first step, valid for weakly evanescent fields, is essentially a method previously described by Choudhary and Felsen.