Search results for: Vladimir Mityushev

Items from 1 to 16 out of 16 results

article

Cluster method in composites and its convergence

Vladimir Mityushev

Applied Mathematics Letters > 2018 > 77 > C > 44-48

Extensions of Maxwell’s self-consistent approach from single- to n - inclusions problems lead to cluster methods applied to computation of the effective properties of composites. We describe applications of Maxwell’s formalism to finite clusters and explain the uncertainty arising when n tends to infinity by study of the corresponding conditionally convergent series.

article

An analytical formula for the effective conductivity of 2D domains with cracks of high density

Ekaterina Pesetskaya, Roman Czapla, Vladimir Mityushev

Applied Mathematical Modelling > 2018 > 53 > C > 214-222

We construct the conformal mapping of the square with circular non-overlapping holes onto the square with non-overlapping slits. It is constructed as a solution of the Riemann–Hilbert problem for a multiply connected domain in a class of double periodic functions. The Riemann–Hilbert problem is reduced to a system of functional equations which is solved with an arbitrary order of approximations. On...

article

Effective elastic constants of hexagonal array of soft fibers

Piotr Drygaś, Simon Gluzman, Vladimir Mityushev, Wojciech Nawalaniec

Computational Materials Science > 2017 > 139 > C > 395-405

Analytical formulae for the effective elastic constants of 2D composites with soft unidirectional fibers arranged in the hexagonal array are obtained for arbitrary concentration of fibers, from dilute case to percolation. It is supposed that every section of composites perpendicular to fibers is the hexagonal array with circular holes or soft inclusions. First, a polynomial approximation in concentration...

article

O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej

Agata Matczyk, Vladimir Mityushev, Zbigniew Powązka

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam... > 2009 > 2 > 153-157

The lattice sum S_2 for the square array conditionally converges. Having used physical arguments, Rayleigh chose an order of summation in such a way that S_2 = π. The Eisenstein summation method applied to S2 yields the same result. This paper is devoted to a rigorous proof of S_2 = π for the Eisenstein summation method. The study can be used in class for students as an interesting example which illustrates...

article

Wprowadzenie funkcji trygonometrycznych przez szeregi Eisensteina

Maciej Klakla, Renata Liszka, Vladimir Mityushev

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam... > 2009 > 2 > 97-103

In the paper, we describe how to introduce the trigonometric functions using their functional characteristics and the Eisenstein series.

article

Optimal and stable random pattern formations

Vladimir Mityushev

Journal of Theoretical Biology > 2017 > 422 > C > 12-17

We develop a graph model to describe the vast range of patterns observed in biological structures. For any given number of spotty patterns, a finite number of structures (optimal graphs) is precisely described. The construction of the optimal graphs is based on the minimization of the diffusion dissipation energy. The notion of geometrical stability of structures is introduced. It is demonstrated...

article

Effective elastic properties of random two-dimensional composites

Piotr Drygaś, Vladimir Mityushev

International Journal of Solids and Structures > 2016 > 97-98 > C > 543-553

Consider 2D two-phase random composites with circular inclusions of concentration f. New analytical formulae for the effective constants are deduced up to O(f⁴) for macroscopically isotropic composites. It is shown that the second order terms O(f²) do not depend on the location of inclusions whilst the third order terms do. This implies that the previous analytical methods (effective medium approximation,...

article

Pattern formations and optimal packing

Vladimir Mityushev

Mathematical Biosciences > 2016 > 274 > C > 12-16

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...

article

Basic sums and their random dynamic changes in description of microstructure of 2D composites

Vladimir Mityushev, Wojciech Nawalaniec

Computational Materials Science > 2015 > 97 > C > 64-74

We discuss microstructures which can be associated to 2D random two-phase composites with circular non-overlapping inclusions. A set of geometrical parameters em is introduced and called the basic sums. These basic sums perform the same role in description of microstructure as the n-point correlation functions. The main advantage of em is their fast computation including high order terms. Besides...

article

A fast algorithm for computing the flux around non-overlapping disks on the plane

Vladimir Mityushev, Natalia Rylko

Mathematical and Computer Modelling > 2013 > 57 > 5-6 > 1350-1359

Conductivity of n non-overlapping disks embedded in a two-dimensional background can be investigated by the method of images which is based on the successive application of the inversions with respect to circles. For closely placed disks, the classical method of images yields slowly convergent series. The method of images can be treated as an application of successive approximations to a system of...

article

Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions

Roman Czapla, Wojciech Nawalaniec, Vladimir Mityushev

Computational Materials Science > 2012 > 63 > Complete > 118-126

We study the effective conductivity of equal unidirectional infinite circular cylinders randomly distributed in a uniform host (disks on the plane). The problem is reduced to a boundary value problem for the two-dimensional Laplace equation. A symbolic–numerical algorithm was proposed in the previous papers to solve the boundary value problem with arbitrary deterministic locations of disks. Application...

article

Riemann-Hilbert Problems for Multiply Connected Domains and Circular Slit Maps

Vladimir Mityushev

Computational Methods and Function Theory > 2012 > 11 > 2 > 575-590

A conformal mapping of a multiply connected circular domain onto a complex plane with circular slits is obtained. No restriction on the location of the boundary circles is assumed. The mapping is derived in terms of the uniformly convergent Poincaré series by solution to a Riemann-Hilbert boundary value problem.

article

Schwarz-Christoffel Formula for Multiply Connected Domains

Vladimir Mityushev

Computational Methods and Function Theory > 2012 > 12 > 2 > 449-463

We derive a Schwarz-Christoffel formula for the conformal mapping of an arbitrary n-connected domain D bounded by mutually disjoint circles ¦z − a k¦ = rk, k = 1,2,…,n, onto the exterior of mutually disjoint polygons. The derivation is based on the exact solution to a Riemann-Hilbert problem for D without any geometric restriction imposed upon the location of the non-overlapping disks ¦z − ak¦ ≤ r...

article

O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej

Agata Matczyk, Vladimir Mityushev, Zbigniew Powązka

Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam... > 2009 > 2 > 153-157

article

Wprowadzenie funkcji trygonometrycznych przez szeregi Eisensteina

Maciej Klakla, Renata Liszka, Vladimir Mityushev

Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam... > 2009 > 2 > 97-103

In the paper, we describe how to introduce the trigonometric functions using their functional characteristics and the Eisenstein series.

article

Conformal mapping of the domain bounded by a circular polygon with zero angles onto the unit disc

Vladimir Mityushev

Annales Polonici Mathematici > 1998 > 68 > 3 > 227-236

The conformal mapping ω(z) of a domain D onto the unit disc must satisfy the condition |ω(t)| = 1 on ∂D, the boundary of D. The last condition can be considered as a Dirichlet problem for the domain D. In the present paper this problem is reduced to a system of functional equations when ∂D is a circular polygon with zero angles. The mapping is given in terms of a Poincaré series.

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