Approximation Solution of Mixed Boundary-Contact Problems and
their Applications
MANANA CHUMBURIDZE, DAVID LEKVEISHVILI
Department of Computer Technology
Akaki Tsereteli State University
Kutaisi, Tamar Mephe St #59 postal code 4600
GEORGIA
Abstract: - In this article non-classical diffusion models of theory coupled-elasticity of static systems for isotropic
inhomogeneous elastic materials with thermal and diffusion variables has been investigated. Approximate
solutions for boundary-contact problems for theory coupled-elasticity in Cauchy hypothesis conditions has been
constructed. The tools applied in this development are based on the boundary integral methods and Greens
functions applications.
Key-Words: - Approximation solution; boundary-contact problems.
Received: January 11, 2023. Revised: October 19, 2023. Accepted: November 21, 2023. Published: December 31, 2023.
1 Introduction
This paper is devoted to the development of a new
method of approximate solutions of partial
differential equations (PDEs). In particular, a non-
classical diffusion models of theory coupled-
elasticity of static systems for isotropic
inhomogeneous elastic materials with thermal and
diffusion for two dimensional (2-D) areas variables
have been provided [1-4]. For consideration basic
boundary-contact problems for theory coupled-
elasticity (BCPTCE) in Cauchy hypothesis (CH)
conditions in infinite and finite domains of isotropic
inhomogeneous elastic media with inclusion of
several elastic materials and mixed contact
conditions have been investigated. Decomposition
Methods to build a numerical solution of partial
differential equations has been used [6].
Throughout of paper we introduce the following
notations:
two-dimensional Euclidean
space,-points of this
space, (Hilbert space,
is infinite
domain with inclusion another elastic material
(
,
bounded by the
close surface
with outward
positive normal vector.
2 Problem Formulation
The generalized model static systems of partial
differential equations of theory of coupled thermo-
diffusion model for 2-D isotropic inhomogeneous
elastic materials has the form [1]:
.
(1)
Where is the displacement vector, is
a characteristic of rotation, is the temperature
variation, is the chemical potential
,
Constants of
elasticity of
is a two-
dimensional Laplacian operator [3],
is a given vectors. Allow
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DOI: 10.37394/232021.2023.3.17
Manana Chumburidze, David Lekveishvili
E-ISSN: 2732-9976
140
Volume 3, 2023
us to introduce the matrices of differential operators
of static systems:
Where
,
,
Where, is of Levi-
Chamita’s symbol. Therefore (1) can be written in the
form:
)2(1,0,
)( rxfxUxL rr
r
The generalized stress operators of coupled thermo-
elasticity in
domains have the form [3]:
Where
-matrices of
stress operators on the plain [4]:
;
3 Problem Solution
In this work basic BCPTE in the case that couple-
stresses components, displacement components,
rotation, heat flux and temperature, concentration and
chemical potential are represented on the surface of
Holder class has been formulated.
The Cauchy-hypothesis condition for isotropic
inhomogeneous elastic materials with a center of
symmetry is implemented [3]:
It is assumed that surfaces are sufficiently smooth.
Problem. It is required to find regular solutions
of boundary contact
problems:
,(x) )()()( xfUxL rrr
(P)
In the radiation conditions for an infinite domain.
In the first iteration the Neiman’s [1-7] boundary
value problem is considered:
xfUxL )0(0
1
)0( (x)
The solution of (3) is presented in the following form:
0
)0(00
12
1
(x)
D
dyyfxyGU
Where
xyG
0
is Green’s tensor [2] defined for
problem (3).
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Further the Dirichlet’s boundary value problems [1-
5] is considered:
xfUxL )1(1
1
)1( (x)
The solution of (4) is presented in the following form:
S
y
T
D
sdyUyxGnyR
dyyfyxGxU
)0(
1
11
)1(1)1(
1
,,
2
1
),(
2
1
1
Where
xyG
1
is Green’s tensor defined for
problem (4).
Let us introduce the following notations:
,..2,1
,
,
1)1(
1
0)0(
1
1
k
DxU
DxU
U
k
k
k
On the k-iteration the approximation solution for the
Neiman’s boundary value problem will be
constructed as follow:
xfUxL k)0(0
1
)0( (x)
Then corresponding solutions in Green’s tensor will
be presented in the form:
S
yk
D
k
sdyUnyyxG
dyyfyxGxU
)(,R),(
2
1
),(
2
1
)(
)1(00
)0(0)0(
1
1
Accordingly previous iteration the solution
for Dirichlet,s boundary value problems will be
constructed as follow:
xfUxL k)1(1
1
)1( (x)
(6)
The solution will be presented in Green’s tensor:
S
yk
T
D
k
sdyUyxGny
dyyfyxGxU
)0(
1
11
)1(1)1(
1
,,R
2
1
),(
2
1
1
Let us introduce the following notation:
....2,1,
1 kxUxUxV kkk
Then with respect to
)(xVk
the following conditions
is satisfied:
1,0
1,
(x)
kk
kxf
VxL
=
,
0
1
zU
,
Where
Accordingly of CH condition the following
equations is received:
(6)
Take in account the (6) in expressions of solutions in
D0 and D1 domains the following results has been
obtained
S
ykk sdyVnyyxGxV )(,R,
2
1
1
10)0(
xVsdyUyxK
sdyVyxGnyxV
k
S
y
k
S
yk
T
k
)0(
k
0
1
0
1
k
0
1
1
11)1(
,
)(,,R
2
1
Where
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s
TsdyGnyxGnyxK
,,R,,R),( 0011
(
)7
Take in account the symmetry properties of Greens
tensors for inner and outer problems:
xyGyxG T,,
We can rewrite (7) in the form:
),(
,,R,,R),(
1
0
1
0000
0
1
yxK
sdyGnyxGnyyxK
s
Obviously Vk(x) will be represented in the following
form:
)8(,..2
,)( )(, 11
0
1
11
0
1
k
yVKsdyVxyKxV k
S
ykk
Where
)9()(),(
11 sdyyxKK y
s
φφ
According to the following property of Green’s
tensor [3]:
3,2,1
,; 2
00
i
yx
c
yxG
xyx
c
yxG
i
Then
DLyxK 21 ),(
4 Conclusion
Thus, an approximate method solution of basic
boundary contact problems has been developed. The
static system of theory coupled-elasticity of thermo-
diffusion models for two-dimensional areas has been
investigated. The Greens functions applications in
the CH conditions to construct the numerical
solutions for materials with sufficiently smooth
surfaces has been provided.
The result is provided on the fundamental level and
useful tools for implementation in other theories to
solve mixed boundary-contact problems for statics
and oscillation systems.
References:
[1] Michael Paulus, Olivier JF Martin,A Green's
tensor approach to the modeling of
nanostructure replication and
characterizationChumburidze, Radio
Science 38(2),2003,DOI: 10.1029/2001RS0
02563
[2] A.S. Kravchuk , P. Neittaanmäki The
solution of contact problems using boundary
element method, Journal of Applied
Mathematics and Mechanics, Volume 71,
Issue 2, 2007, Pages 295-304
[3] Lekveishvili, David, and Menana
Chumburidze. "About Iterative Solution
Method of Boundary-Contact
Problems." Moambe-Bulletin of Akaki
Tsereteli State University 2 (2022).
[4] Chumburidze, Manana. "Approximate
Solution of Some Boundary Value Problems
of Coupled Thermo-
Elasticity." Mathematical and
Computational Approaches in Advancing
Modern Science and Engineering. Springer,
Cham, 2016. 71-80.
[5] Xiaoning Li, Daqing Jiang, Multiple
positive solutions of Dirichlet boundary
value problems for second order impulsive
differential equations, Journal of
Mathematical Analysis and Applications,
Pages 501-514,2006,
[6] Tarek Poonithara, Abraham Mathew,
Domain Decomposition Methods for the
Numerical Solution of Partial Differential
Equations Springer, © 2008
[7] Cheng, A. H.-D.; Cheng, D. T. (2005). "Heritage
and early history of the boundary element
method". Engineering Analysis with Boundary
Elements. 29 (3):
268. doi:10.1016/j.enganabound.2004.12.001
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DOI: 10.37394/232021.2023.3.17
Manana Chumburidze, David Lekveishvili
E-ISSN: 2732-9976
143
Volume 3, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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