arXiv subjects

Vladimir Mityushev

Publications and source records attributed to Vladimir Mityushev.

12 recordsLinked to original sources

arXiv2021

Simulating local fields in carbon nanotube reinforced composites for infinite strip with voids

We consider the steady heat conduction problem within a thermal isotropic and homogeneous infinite strip composite reinforced by uniformly and randomly distributed non-overlapping carbon nanotubes (CNTs) and containing voids. We treat the CNTs as thin perfectly conducting elliptic inclusions and assume the voids to be of circular shape and act as barriers to heat flow. We also impose isothermal conditions on the external boundaries by assuming the lower infinite wall to be a heater under a given temperature, and the upper wall to be a cooler that can be held at a lower fixed temperature. The equations for the temperature distribution are governed by the two-dimensional Laplace equation with mixed Dirichlet-Neumann boundary conditions. The resulting boundary value problem is solved using the boundary integral equation with the generalized Neumann kernel. We illustrate the performance of the proposed method through several numerical examples including the case of the presence a large number of CNTs and voids.

math.NA↗

arXiv2019

Effective conductivity of a random suspension of highly conducting spherical particles

Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions $f$. Jeffrey (1973) suggested an analytical formula valid up to $O(f^3)$ for macroscopically isotropic random composites. A conditionally convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey's formula. The main revision concerns the proper investigation of the conditionally convergent sum and correction the $f^2$-term. A new model of symbolic computations is developed in order to compute the effective conductivity tensor. The corresponding algorithm is realized up to $O(f^{\frac{10}3})$. The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the $f^2$-term, and in the $f^3$-term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.

math-ph↗

arXiv2018

Lattice sums for polyanalytic functions

In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, by means of the Eisenstein summation, that the lattice sum $S_2$ is equal to $π$ for the square array. Further, it became clear that such an equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions including the conditionally convergent Eisenstein series. This paper is devoted to extension of the lattice sums to double periodic polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums where the number $π$ persists for macroscopically isotropic composites by our suggestion.

math.NT↗

arXiv2017

A relation on the effective conductivity of composites

Consider a 2D composites with non-overlapping equal inclusions imbedded in a host material of the normalized unit conductivity. The conductivity of inclusions takes two values $σ_1$ and $σ_2$ with the probabilities $p$ and $1-p$, respectively. We prove that the effective conductivity tensor of the considered three-phase random composite is equal to the effective conductivity tensor of the two-phase deterministic composite with the same inclusions of the conductivity $σ=[p(σ_1~-~σ_2)+~σ_2+σ_1σ_2] [1+σ_1-p(σ_1-σ_2)]^{-1}$.

math-ph↗

arXiv2017

Exact and "exact" formulae in the theory of composites

The effective properties of composites and review literature on the methods of Rayleigh, Natanzon--Filshtinsky, functional equations and asymptotic approaches are outlined. In connection with the above methods and new recent publications devoted to composites, we discuss the terms "analytical formula, approximate solution, closed form solution, asymptotic formula" etc... frequently used in applied mathematics and engineering in various contexts. Though mathematicians give rigorous definitions of exact form solution the term "exact solution" continues to be used too loosely and its attributes are lost. In the present paper, we give examples of misleading usage of such a term.

math-ph↗

arXiv2016

Closed-form evaluation of 2D static lattice sums

In the present paper, employing properties of the complete elliptic integrals of the first and second kind, we deduce closed-form formulae for the lattice sums and other new formulae. Applications to the effective properties of regular and random composites are discussed. We also discuss the Eisenstein summation method and the Rayleigh method used in computations.

math.CA↗

arXiv2015

Pattern formations and optimal packing

Patterns of different symmetries may arise after solution to reaction-diffusion equations. Hexagonal arrays, layers and their perturbations are observed in different models after numerical solution to the corresponding initial-boundary value problems. We demonstrate an intimate connection between pattern formations and optimal random packing on the plane. The main study is based on the following two points. First, the diffusive flux in reaction-diffusion systems is approximated by piecewise linear functions in the framework of structural approximations. This leads to a discrete network approximation of the considered continuous problem. Second, the discrete energy minimization yields optimal random packing of the domains (disks) in the representative cell. Therefore, the general problem of pattern formations based on the reaction-diffusion equations is reduced to the geometric problem of random packing. It is demonstrated that all random packings can be divided onto classes associated with classes of isomorphic graphs obtained form the Delaunay triangulation. The unique optimal solution is constructed in each class of the random packings. If the number of disks per representative cell is finite, the number of classes of isomorphic graphs, hence, the number of optimal packings is also finite.

cond-mat.soft↗

arXiv2015

Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders

Effective conductivity of a 2D composite corresponding to the regular hexagonal arrangement of superconducting disks is expressed in the form of a long series in the volume fraction of ideally conducting disks. According to our calculations based on various re-summation techniques, both the threshold and critical index are obtained in good agreement with expected values. The critical amplitude is in the interval $(5.14,5.24)$ that is close to the theoretical estimation $5.18$. The next order (constant) term in the high concentration regime is calculated for the first time, and the best estimate is equal to $(-6.229)$. Final formula is derived for the effective conductivity for arbitrary volume fraction.

cond-mat.stat-mech↗

arXiv2015

Composites with invisible inclusions: eigenvalues of R-linear problem

An new eigenvalue $\mathbb R$-linear problem arisen in the theory of metamaterials is stated and constructively investigated for circular non-overlapping inclusions. An asymptotic formula for eigenvalues is deduced when the radii of inclusions tend to zero. The nodal domains conjecture related to univalent eigenfunctions is posed. Demonstration of the conjecture allows to justify that a set of inclusions can be made neutral by surrounding it with an appropriate coating.

math-ph↗

arXiv2015

Random composites and the generalized method of Schwarz I. Conductivity problems

Two-phase composites with non-overlapping inclusions randomly embedded in matrix are investigated. A straight forward approach is applied to estimate the effective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, i.e., for all events of the considered probabilistic space $\mathcal C$ by the generalized method of Schwarz. Second, the effective properties are calculated in analytical form and averaged over $\mathcal C$. This method is related to the classic method based on the average probabilistic values involving the $n$-point correlation functions. However, we avoid computation of the correlation functions and compute their weighted moments of high orders by an indirect method which does not address to the correlation functions. The effective properties are exactly expressed through these moments. It is proved that the generalized method of Schwarz converges for an arbitrary multiply connected doubly periodic domain and for an arbitrary contrast parameter. The proposed method yields effective in symbolic-numeric computations formulae of high order in concentration. In particular, the Torquato--Milton parameter $ζ_1$ is exactly written for circular inclusions.

math-ph↗

arXiv2014

Optimal packing of spheres in $\mathbb R^d$ and extremal effective conductivity

Optimal packing of spheres in $\mathbb R^d$ is studied by optimization of the energy $E$ (effective conductivity) of composites with ideally conducting spherical inclusions. It is demonstrated that the minimum of $E$ over locations of spheres is attained at the optimal packing. The energy is estimated in the framework of structural approximations. This method yields upper bounds and sometimes exact values for the maximal concentrations of spheres in $\mathbb R^d$. A constructive algorithm for the optimal locations of spheres associated to the classes of the Delaunay graphs is constructed.

math.MG↗

arXiv2014

Series, Index and Threshold for Random 2D Composite

Effective conductivity of a 2D random composite is expressed in the form of long series in the volume fraction of ideally conducting disks. The problem of a {\it direct} reconstruction of the critical index for superconductivity from the series is solved with good accuracy, for the first time. General analytical expressions for conductivity in the whole range of concentrations are derived and compared with the regular composite and existing models.

cond-mat.stat-mech↗