Papers in an open access journal “Mathematical Problems in Engineering”
Asymptotic-group analysis of algebraic equations, A. D. Shamrovskii, I. V. Andrianov, and J. Awrejcewicz
Volume 2004 (2004), Issue 5, Pages 411-451.
Artificial small parameter method—solving mixed boundary value problems, I. V. Andrianov, J. Awrejcewicz, and A. Ivankov
Volume 2005 (2005), Issue 3, Pages 325-340.
Analysis of natural in-plane vibration of rectangular plates using homotopy perturbation approach, Igor V. Andrianov, Jan Awrejcewicz, and Vladimir Chernetskyy
Volume 2006 (2006), Article ID 20598, 8 pages.
On an elastic dissipation model as continuous approximation for discrete media, I.V. Andrianov, J. Awrejcewicz, and A. O. Ivankov
Volume 2006 (2006), Article ID 27373, 8 pages.
Dynamics of a reinforced viscoelastic plate, Igor V. Andrianov, Jan Awrejcewicz, and Irina V. Pasichnik
Volume 2006 (2006), Article ID 89675, 8 pages.
Asymptotic Solution of the Theory of Shells Boundary Value Problem, I.V. Andrianov and J. Awrejcewicz
Volume 2007 (2007), Article ID 82348, 25 pages.
Love and Rayleigh Correction Terms and Padé Approximants, I. Andrianov and J. Awrejcewicz
Volume 2007 (2007), Article ID 94035, 8 pages.
Improved continuous models for discrete media, Andrianov, I.V., Awrejcewicz, J., Weichert, D.
Volume 2010 (2010), Article ID 986242, 35 pages.
- Andrianov I.V., Awrejcewicz J. and Barantsev R.G. (2003)
Asymptotic Approaches in Mechanics: New Parameters and Procedures. Appl. Mech. Rev., vol.56, No 1, pp.87-110
This survey is devoted to recent achievements in the field of asymptotic approaches.
Here we consider the asymptotics in relation to completely new and sometimes unexpected parameters.
Some procedures leading to improvement and isolation of the essential analytical structure of the perturbation
series are presented. It has been also shown that a lot of relatively simple at first glance problems of the
perturbation theory is still far from a complete solution. Different asymptotic techniques to solve the same
problem and their influence on the results are briefly illustrated and discussed.
This review paper contains 310 references.
- Andrianov I.V., Danishevs’kiy V.V. and Weichert D. (2002)
Asymptotic Determination of Effective Elastic Properties of Composite Materials
with Fibrous Square-Shaped Inclusions. Eur. J. Mech. A/Solids, vol. 21, N 6, pp. 1019-1036
We propose an asymptotic approach for evaluating effective elastic properties of two-components periodic
composite materials with fibrous inclusions. We start with a nontrivial expansion of the input elastic
boundary value problem by ratios of elastic constants. This allows to simplify the governing equations
to forms analogous to the transport problem. Then we apply an asymptotic homogenization method, coming
from the original problem on a multi-connected domain to a so called cell problem, defined on a characterizing
unit cell of the composite. If the inclusions’ volume fraction tends to zero, the cell problem is solved by means o
f a boundary perturbation approach. When on the contrary the inclusions tend to touch each other we use an asymptotic
expansion by non-dimensional distance between two neighbouring inclusions. Finally, the obtained “limiting” solutions
are matched via two-point Padé approximants. As the results, we derive uniform analytical representations for effective
elastic properties. Also local distributions of physical fields may be calculated. In some partial cases the proposed
approach gives a possibility to establish a direct analogy between evaluations of effective elastic moduli and transport
coefficients. As illustrative examples we consider transversally-orthotropic composite materials with fibres of square
cross section and with square checkerboard structure. The obtained results are in good agreement with data of other authors.
- Andrianov I.V. and Awrejcewicz J. (2001)
New Trends in Asymptotic Approaches: Summation and Interpolation Methods. Applied Mechanics Review, V.54, No.1, pp.69-92.
In this paper, we present in some detail new trends in application of asymptotic techniques to mechanical problems.
First we consider the various methods which give a possibility to extend a space of application of perturbation series and hence to omit their local character. [Download]
- Tokarzewski S., Andrianov I. (2001)
Effective coefficients for real non-linear and fictitious linear
temperature-dependent periodic composites. International Journal of
Non-Linear Mechanics, V.36, 187-195.
It has been proved that the effective conductivities for non-linear,
temperature-dependent composites and the so-called linear, fictitious
ones coincides. Due to the fact all homogeneous methods, exact or
approximate, developed previously for linear composites apply
immediately to non-linear, temperature-dependent ones. Numerical example
illustrating the results obtained is provided. [Download]
- Andrianov I., Awrejcewicz J. (2000)
Method of small and large delta in nonlinear dynamics – a comparative
analysis. Nonlinear Dynamics, V.23, 57-66.
New asymptotic approaches for dynamical systems containing a power
nonlinear term xn
are proposed and analysed. Two natural limiting cases are studied: n tends to the unit and n tends to infinity. In the first
case, the “small delta method” (SDM) is used and its applicability for
problems is outlined. For the second case a new asymptotic
approach is proposed (conditionally we call it “large delta method” -
LDM). Error estimations lead to the following conclusion: the LDM may be
used, even for small n, whereas
the SDM has a narrow application area. Both of the discussed approaches
overlap all values of the parameter n. [Download]
- Andrianov I., Danishevs’kyy V.,
Tokarzewski S. (2000) The method of quasifractional approximants in
application to mechanical problems. Theoretical Foundations of Civil
Engineering, V.8, 371-376.
Practically any physical or mechanical problem, which includes a
variable parameter epsilon, can be approximately solved as epsilon
approaches zero or infinity. How can this “limiting” information be used
in the study of the system at intermediate values of epsilon ? In some
instances the answer may be given by two-point Padé approximants (PA).
However, one of the main shortages of PAs is related to the presence of
logarithmic or other complicated components in numerous asymptotic
expansions, which do not allow rational approximations. Such
difficulties are essential for the most of real mechanical problems. In
order to overcome them so called method of quasifractional approximants
(QA) may be used. It allows one to obtain approximate solutions in
closed analytical forms, valid for all values of governing parameters.
Here we introduce QAs for the Thomas-Fermi boundary value problem and
for effective transport properties of regular arrays of spheres. [Download]
- Andrianov I., Danishevs’kyy V.,
Tokarzewski S. (2000) Quasifractional approximants for effective
conductivity of regular arrays of spheres. Archives of Mechanics,
We study the effective heat conductivity of regular arrays of perfectly
conducting spheres embedded in a matrix with the unit conductivity.
Quasifractional approximants allow us to derive an approximate
analytical solution, valid for all values of the spheres volume
fraction. As the bases we use a perturbation approach for small spheres
and an asymptotic solution for large ones. Three different types of the
spheres space arrangement (simple, body and face centred cubic arrays)
are considered. Obtained results give a good agreement with numerical
- Andrianov I., Starushenko G.,
Danishevs’kyy V., Tokarzewski S. (1999) Homogenization procedure and Padé
approximants for effective heat conductivity of composite materials with
cylindrical inclusions having square cross section. Proc. R. Soc. Lond. A, V.455, 3401-3413.
An analytical solution, describing the effective heat conductivity of
composite materials with a periodic array of cylindrical inclusions with
square cross section, has been obtained by means of asymptotic methods
and Padé approximants for any values of inclusions concentration and
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