Mapping physical problems on fractals onto boundary value problems within continuum framework

In this Letter, we emphasize that methods of fractal homogenization should take into account a loop structure of the fractal, as well as its connectivity and geodesic metric. The fractal attributes can be quantified by a set of dimension numbers. Accordingly, physical problems on fractals can be mapped onto the boundary values problems in the fractional-dimensional space with metric induced by the fractal topology. The solutions of these problems represent analytical envelopes of non-analytical functions defined on the fractal. Some examples are briefly discussed. The interplay between effects of fractal connectivity, loop structure, and mass distributions on electromagnetic fields in fractal media is highlighted. The effects of fractal connectivity, geodesic metric, and loop structure are outlined.     

分形多孔介质中流体流动的LBM模拟

基于分形理论构造了规则Sierpinski地毯结构来模拟多孔介质孔隙结构,采用Lattice Boltzmann Method(LBM)分别对不同孔隙率与进出口压差下的分形多孔介质中的流体流动进行了数值模拟.结果表明,分形多孔介质的某些流场结构也具有分形特征;体积流量和流场进出口压差呈线性关系,符合Darcy定律;流体穿过分形多孔介质的体积流量与孔隙率成指数函数关系.     

Fractal transition radiation

Phenomena related to fractal properties of moving emitting objects and electrodynamic structures are of considerable interest now. Electromagnetic radiation accompanying fractal transition processes is considered.     

Analysis and design of electromagnetic band gap structures with stratified and inhomogeneous media

In this study, a generalized and easy to use method based on Hill's equation and chain matrix concept is used to analyze the electromagnetic wave propagation in stratified dielectric and inhomogeneous media with arbitrary profiles.Numerical simulations are performed to compute the reflection and transmission of several Electromagnetic Bandgap (EBG) structures for various permittivity profiles and given polarization as well as incidence angles of driving fields.Multilayer structures are analyzed and optimized to enhance their selectivity performances. Obtained results agree excellently well with other published data.     

Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane

Electromagnetic scattering by buried objects may involve a plane-wave expansion of the related fields, which depends on the objects' geometry. Furthermore, involved media in realistic cases are lossy, which requires the analytic continuation of formulae known for the lossless cases, due to the complex nature of the wave vectors. This problem has been covered in a previous paper, but the expression found still does not converge in some areas of space. In this paper, a new, convergent, expression of the spectrum of cylindrical functions in lossy media is analytically computed and its convergence limits are discussed.     

分形多孔介质的粒子扩散特点(Ⅰ)

本文讨论了粒子在分形结构中的扩散,随机Sierpinski地毯上的扩散模拟表明:分形结构中的扩散已不再满足 Fick扩散定律。扩散速度较欧氏空间减慢了,即“扩散慢化”效应。而且可以看到由于结构的自相似特征,扩散过程也表 现出某种自相似性(标度性),分形结构和欧氏空间中的扩散特点是有本质的差异的。     

A fractal streaming current model for charged microscale porous media

An analytical expression for the streaming current in fractal porous media is developed based on the capillary model and the fractal theory for porous media. The proposed fractal model is expressed as a function of the space charge density at the solid–liquid interface, the fluid flow rate, the Debye–Huckel parameter, the minimum and maximum pore/capillary radii and fractal dimensions for porous media. The results are compared with available experimental data and good agreement is found between them. In addition, factors influencing the streaming current in porous media are also analyzed.     

变形双重介质分形油藏非达西流动分析

考虑与实际生产相符的介质的双重特性和分形特征,并考虑介质的变形,引入双重分形介质渗透率模数,建立应力敏感地层双重分形介质系统的流动方程.采用Douglas-Jones预估-校正法获得无限大地层定产量生产时变形双重分形介质模型的数值解和无限大地层定压生产时分形介质双孔模型的数值解,作出了典型的压力曲线图版.     

Monte Carlo methods for turbulent tracers with long range and fractal random velocity fields

Monte Carlo methods for computing various statistical aspects of turbulent diffusion with long range correlated and even fractal random velocity fields are described here. A simple explicit exactly solvable model with complex regimes of scaling behavior including trapping, subdiffusion, and superdiffusion is utilized to compare and contrast the capabilities of conventional Monte Carlo procedures such as the Fourier method and the moving average method; explicit numerical examples are presented which demonstrate the poor convergence of these conventional methods in various regimes with long range velocity correlations. A new method for computing fractal random fields involving wavelets and random plane waves developed recently by two of the authors [J. Comput. Phys. 117, 146 (1995)] is applied to compute pair dispersion over many decades for systematic families of anisotropic fractal velocity fields with the Kolmogorov spectrum. The important associated preconstant for pair dispersion in the Richardson law in these anisotropic settings is compared with the one obtained over many decades recently by two of the authors [Phys. Fluids 8, 1052 (1996)] for an isotropic fractal field with the Kolmogorov spectrum. (c) 1997 American Institute of Physics.     

分形介质的传热与传质分析(综述)

本文论述了分形介质的分形理论和数学基础,并简要综述了用分形理论和方法研究分形介质的传热与传质特性(如多孔介质的渗透率、热导率以及池核态沸腾换热)方面目前所取得的研究进展,最后扼要展望了用分形理论和方法进一步研究分形介质的传热与传质的可能的若干课题和方向。     

The scattering of electromagnetic waves in fractal media

Abstract

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

静电放电火花产生的电磁场树枝模型

 静电放电火花产生的电磁脉冲会对电子系统的正常工作造成严重的干扰，甚至造成系统的损伤。利用时域有限差分法建立了静电放电火花产生的电磁场的数值模型，模型中充分考虑了放电电极上的静电荷对电场的影响。把由此模型放电计算的电磁场值与由此解析方法得到的场值进行了比较，结果吻合良好。由此可以用此模型来研究静电放电火花产生的电磁场与电子系统的能量耦合问题。     

随机多孔介质的蒙特卡罗重构与渗流特性模拟

根据多孔介质微观结构的分形尺度标度特征,采用蒙特卡罗方法分别重构随机多孔介质的微观颗粒和孔隙结构,并基于分形毛管束模型研究多尺度多孔介质的气体渗流特性,建立多孔介质微观结构和宏观渗流特性的定量关系。结果表明：分形蒙特卡罗重构的多孔介质微细结构接近真实介质结构,气体渗流特性的计算结果与格子玻尔兹曼模拟数据较为吻合; 多孔介质气体渗透率随着克努森数的增加而增大,孔隙分形维数对于气体渗流的微尺度效应具有显著影响,而迂曲度分形维数对于表观渗透率和固有渗透率的比值影响可以忽略。提出的分形蒙特卡罗方法具有收敛速度快且计算误差与维数无关的优点,有利于深入理解多尺度多孔介质的渗流机理。     

The scattering of electromagnetic waves in fractal media

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

Nonlinear electrodynamic lag of electromagnetic signals in a magnetic dipole field

We investigate the propagation of electromagnetic waves in the magnetic dipole and gravitational fields of a neutron star, which follows the laws of nonlinear electrodynamics in a vacuum. Electromagnetic signals in these fields are shown to propagate along different rays and at different velocities, depending on their polarization. We found the law of motion for these signals along rays. We calculate the difference between the propagation times of electromagnetic signals with different polarizations from the same source to the detector. This difference is shown to reach a measurable value of 1 μs in some cases.     

分形分层粗糙面电磁波透射特性的混合算法

运用基于矩量法(method of moment,MoM)结合基尔霍夫近似(Kirchhoff approximation,KA)的混合算法研究一维带限Weierstrass分形分层介质粗糙面的电磁波透射问题.在混合算法中将上层粗糙面和下层粗糙面分别划分到MoM区域和KA区域,数值计算得到透射系数随透射波的散射角的变化,讨论粗糙面分维数、高度起伏均方根、底层介质介电常数、中间层介质介电常数和厚度、入射波频率对透射系数的影响,结果表明上层粗糙面参数对透射系数有显著影响,而下层粗糙面参数对透射系数影响较小.     

Fractal structures in turbulence

We present a qualitative overview of our work on the issue of fractal structures in turbulence. We explain why fully developed turbulence is not space filling and describe how its fractal dimension can be estimated theoretically. The implications of the fractal nature of turbulence on transport processes like turbulent diffusion and on fluctuations in passive scalars are discussed. The latter affect wave propagation in turbulent media and these effects are examined. In addition we consider clouds in the atmosphere which are claimed to have fractal perimeters (or surfaces) and outline the physical reasons for this phenomenon. The fractal dimension of clouds is tied to the theory of turbulent diffusion and is computed theoretically. Indications of the road ahead are given.     

Lagrangian and Hamiltonian Mechanics on Fractals Subset of Real-Line

A discontinuous media can be described by fractal dimensions. Fractal objects has special geometric properties, which are discrete and discontinuous structure. A fractal-time diffusion equation is a model for subdiffusive. In this work, we have generalized the Hamiltonian and Lagrangian dynamics on fractal using the fractional local derivative, so one can use as a new mathematical model for the motion in the fractal media. More, Poisson bracket on fractal subset of real line is suggested.     

Acoustic-elastodynamic interaction in isotropic fractal media

This research explores the acoustic-elastodynamic interaction in isotropic fractal media. The analysis discusses the direct coupling of two constitutive models under dynamic loading: a continuous solid and an isotropic fractal medium. We consider two situations where in the first, the fractal medium is enclosed within a thin spherical shell (interior problem), while in the second, the fractal medium extends infinitely outside the shell (exterior problem). The two problems are simulated analytically, and the exact solution for the shell displacement is expressed in closed form in the Laplace domain. The formulation of the radiation condition for infinite fractal media is essential to derive the exterior problem’s solution. This study represents a meaningful idealization of real-application problems involving the interaction of multi-constitutive media, e.g. the human brain, whereby fractal features affect the response of this body under various excitations.     

Waveguide devices with homogeneous complementary media

We report the design of microwave waveguide devices based on complementary media. Various kinds of waveguide devices that possess nearly 100% transmission efficiency are proposed, such as waveguide bends, splitters, connectors, and shifters. Compared with previous work on waveguide devices of low reflection and minimized distortion based on transformation optics, our transform media are homogeneous metamaterials. Electromagnetic simulations by a finite-element method on detailed examples have been performed to validate the designs and these functionalities can be close to the practical.