Electromagnetism on anisotropic fractal media

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green–Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.     

Closed fractal interpolation surfaces

Based on the construction of bivariate fractal interpolation surfaces, we introduce closed spherical fractal interpolation surfaces. The interpolation takes place in spherical coordinates and with the transformation to Cartesian coordinates a closed surface arises. We give conditions for this construction to be valid and state some useful relations about the Hausdorff and the Box counting dimension of the closed surface.     

Construction of fractal surfaces by recurrent fractal interpolation curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system (RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.     

Spatial distributions of islands in fractal surfaces and natural surfaces

The size and spatial distributions of islands in random fractal surfaces and natural islands are investigated. Multi-scale island distributions from a fractal surface are analyzed to determine the size distribution of islands, such as the number of islands greater than a particular size, and this was found to exhibit a Korcak-type empirical relation. In the same multi-scale analysis, the spatial distribution of islands, which is the number of islands less than or equal to a particular distance between a pair of islands, is also investigated to determine the effects of differences in scale. The spatial distribution of distance between uniformly distributed islands is analytically calculated and compared with the island distributions of the fractal surface. We seek to determine the degree of resemblance between simulated spatial distributions and the natural island distribution of French Polynesia. We also estimate the size and spatial distributions of islands according to different sea levels.     

Multiple fractal aggregation and simulation

Condensation of metallic vapors has provided novel evidence of simultaneous production of multiple fractal systems through the aggregation of monomers. Simulations have been effected successfully with stochastic procedures at several levels. The growth of monomers is also discussed.     

Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions

A construction method of Fractal Interpolation Surfaces on a rectangular domain with arbitrary interpolation nodes is introduced. The variation properties of the binary functions corresponding to this type of fractal interpolation surfaces are discussed. Based on the relationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.     

Modeling anisotropic conducting surfaces by strip structures

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostika, pp. 193–199, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Yield surfaces for non-homogeneous anisotropic shells of revolution

The state of stress in a rotationally symmetric shell is characterized by the direct stresses and moments in the circumferential and longitudinal directions. It is assumed that the material of the shell is rigid perfectly plastic and that the yield stress of the material varies over the thickness of the shell. The material has different yield stresses in tension and compression and the yield stresses in the principal directions have different values. The yield condition for the shell is obtained in terms of the stress resultants assuming that the material of the shell obeys the maximum shear stress criterion.     

The Minkowski dimension of the bivariate fractal interpolation surfaces

We present a new construction of continuous bivariate fractal interpolation surface for every set of data. Furthermore, we generalize this construction to higher dimensions. Exact values for the Minkowski dimension of the bivariate fractal interpolation surfaces are obtained.     

Stability of anisotropic capillary surfaces between two parallel planes

We study the stability of capillary surfaces without gravity for anisotropic free surface energies. For a large class of rotationally symmetric energy functionals, it is shown that the only stable equilibria supported on parallel planes are either cylinders or a part of the Wulff shape. The first author is partially supported by Grant-in-Aid for Scientific Research (C) No. 16540195 of the Japan Society for the Promotion of Science.     

Dimension analysis on a class of Bush type fractal surfaces

Abstract. In this paper,the authors construct a class of fractal surfaces,Bush type surfaces,based on the Bush type functions. The Box dimension,Packing dimension and Hausdorff dimen-sion of such surfaces are investigated.     

Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension

Recurrent bivariate fractal interpolation surfaces (RBFISs) generalise the notion of affine fractal interpolation surfaces (FISs) in that the iterated system of transformations used to construct such a surface is non-affine. The resulting limit surface is therefore no longer self-affine nor self-similar. Exact values for the box-counting dimension of the RBFISs are obtained. Finally, a methodology to approximate any natural surface using RBFISs is outlined.     

An alternative version of constructing limit surfaces for anisotropic solids

This paper deals with constructing closed everywhere smooth surfaces in 6D space of symmetric second-rank tensors by means of a conoidal anisotropic transformation of the unit sphere S⁵. A special case of such transformations is proposed and the surface convexity conditions are pointed out. These surfaces can be utilized in the strength and plasticity theories of isotropic and anisotropic solids.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 488–493, July–August, 1995.     

Blood flow simulation through fractal models of circulatory system

The blood flow in human arteries has been analytically calculated according to Poiseuille’s equation. Geometry of the fractal arterial trees has been described in previous article [Gabryś E, Rybaczuk M, Kędzia A. Fractal model of circulatory system. Symmetrical and asymmetrical approach comparison. Chaos, Solitons & Fractals, in press]. Blood vessel trees are consisted of straight, rigid cylindrical tubes. In each bifurcation two new children segments appears according to Murray law.Blood flow in circulatory system is driven by the pressure differences at the two ends of the blood vessel. A mathematical analysis shows the continuous dependence of the solution on vessel tree parameters and boundary condition.     

A fully anisotropic mechanism for formation of trapped surfaces in vacuum

We present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum. More precisely we provide local conditions on null data, concentrated in a neighborhood of a short null geodesic segment (possibly flat in all other directions) whose future development contains a trapped surface. This extends considerably the previous result of Christodoulou (Monographs in Mathematics. European Mathematical Society, Switzerland, 2009) which required instead a uniform condition along all null geodesic generators. To obtain our result we combine Christodoulou’s mechanism for the formation of a trapped surface with a new deformation process which takes place along incoming null hypersurfaces.     

Verification of an anisotropic Coulomb adhesion-friction law for contact surfaces with periodic structure

Sliding friction forces and so-called adhesion forces are the main mechanical characteristics to describe contact interaction. Both together are representing 2D surface constitutive laws in analogy to e.g. elasto-plasticity for 3D continua. The classical model to generalize the Coulomb friction law into anisotropic domains is to introduce an anisotropic friction tensor. Michalowski and Mroz in [1] proposed the structure of the friction tensor considering the sliding of a rigid block on an inclined surface. Zmitrowicz in [2] developed the theoretical basis for the structure of the friction tensor on symmetry groups for the tensor. The current contribution is aimed at verification of this modeling process based on a homogenization procedure for a very fine discretization representing the exact structure of the surface. The validation issue with realistic experiments given in [4] is discussed as well. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An investigation of the behavior of simulation response surfaces

This paper is part of a research stream whose purpose is to study the effect of simulation response surface behavior on the choice of appropriate simulation optimization search technique. This paper's research lays some groundwork by examining the behavior of simulation response surfaces themselves. The point here is not to criticize existing simulation-optimization techniques (such as Response Surface Methodology (RSM). Rather, one point is to emphasize the care and precision that must be used to invoke extant procedures properly, while another is to demonstrate the need for additional methods such as nonparametric approaches. In particular, this paper examines a simple, inventory-simulation model under various experimental conditions, including some factors under a user's control, and some not. Both point and region estimates of surface characteristics are determined and graphed while such factors as number of replications, simulation run length, and demand and lead-time variances are varied. It is found, for example, that even for this simple surface such optimization techniques as first-order RSM can be inappropriate over 21–98% of the feasible region, depending on the case. Four implications are noted from the research: the care that should be exercised with existing simulation-optimization techniques; the need for a simulation-optimization starter; the importance of examining global, nonparametric-metamodeling approaches to simulation optimization; and the desirability of investigating a multi-strategy approach to optimization. The paper concludes with a call for further research investigating these suggestions.     

Numerical simulation of convective motion in an anisotropic porous medium and cosymmetry conservation

The onset of convection in a porous anisotropic rectangle occupied by a heat-conducting fluid heated from below is analyzed on the basis of the Darcy–Boussinesq model. It is shown that there are combinations of control parameters for which the system has a nontrivial cosymmetry and a one-parameter family of stationary convective regimes branches off from the mechanical equilibrium. For the two-dimensional convection equations in a porous medium, finite-difference approximations preserving the cosymmetry of the original system are developed. Numerical results are presented that demonstrate the formation of a family of convective regimes and its disappearance when the approximations do not inherit the cosymmetry property.