Thermal simulation of heterogeneous structural components in microelectronic devices
The steady state nonlinear problem of thermal conduction for isotropic strip with foreign rectangular inclusion that heats as an internal thermal source with heat dissipation has been considered. The methodology to solve this problem and its application for the specific dependence of the thermal-...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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| Zitieren: | Thermal simulation of heterogeneous structural components in microelectronic devices / V.I. Gavrysh, D.V. Fedasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 439-443. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1187382017-06-01T03:04:45Z Thermal simulation of heterogeneous structural components in microelectronic devices Gavrysh, V.I. Fedasyuk, D.V. The steady state nonlinear problem of thermal conduction for isotropic strip with foreign rectangular inclusion that heats as an internal thermal source with heat dissipation has been considered. The methodology to solve this problem and its application for the specific dependence of the thermal-conductivity coefficients on temperature has been offered. 2010 Article Thermal simulation of heterogeneous structural components in microelectronic devices / V.I. Gavrysh, D.V. Fedasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 439-443. — Бібліогр.: 7 назв. — англ. 1560-8034 PACS 74.25.fc http://dspace.nbuv.gov.ua/handle/123456789/118738 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The steady state nonlinear problem of thermal conduction for isotropic strip
with foreign rectangular inclusion that heats as an internal thermal source with heat
dissipation has been considered. The methodology to solve this problem and its
application for the specific dependence of the thermal-conductivity coefficients on
temperature has been offered. |
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Article |
| author |
Gavrysh, V.I. Fedasyuk, D.V. |
| spellingShingle |
Gavrysh, V.I. Fedasyuk, D.V. Thermal simulation of heterogeneous structural components in microelectronic devices Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Gavrysh, V.I. Fedasyuk, D.V. |
| author_sort |
Gavrysh, V.I. |
| title |
Thermal simulation of heterogeneous structural components in microelectronic devices |
| title_short |
Thermal simulation of heterogeneous structural components in microelectronic devices |
| title_full |
Thermal simulation of heterogeneous structural components in microelectronic devices |
| title_fullStr |
Thermal simulation of heterogeneous structural components in microelectronic devices |
| title_full_unstemmed |
Thermal simulation of heterogeneous structural components in microelectronic devices |
| title_sort |
thermal simulation of heterogeneous structural components in microelectronic devices |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/118738 |
| citation_txt |
Thermal simulation of heterogeneous structural components
in microelectronic devices / V.I. Gavrysh, D.V. Fedasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 439-443. — Бібліогр.: 7 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT gavryshvi thermalsimulationofheterogeneousstructuralcomponentsinmicroelectronicdevices AT fedasyukdv thermalsimulationofheterogeneousstructuralcomponentsinmicroelectronicdevices |
| first_indexed |
2025-07-08T14:33:44Z |
| last_indexed |
2025-07-08T14:33:44Z |
| _version_ |
1837089666077032448 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 439-443.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
439
PACS 74.25.fc
Thermal simulation of heterogeneous structural components
in microelectronic devices
V.I. Gavrysh, D.V. Fedasyuk
Lviv Polytechnic National University
E-mail: fedasyuk@lp.edu.ua
Abstract. The steady state nonlinear problem of thermal conduction for isotropic strip
with foreign rectangular inclusion that heats as an internal thermal source with heat
dissipation has been considered. The methodology to solve this problem and its
application for the specific dependence of the thermal-conductivity coefficients on
temperature has been offered.
Keywords: isotropic, thermal conduction, foreign inclusion, thermosensitive, heat
dissipation.
Manuscript received 03.06.10; accepted for publication 02.12.10; published online 30.12.10.
1. Introduction
In the design process of individual junctions and
structural components for microelectronic devices, there
appears the necessity to mathematically model thermal
processes in thermosensitive structures with foreign
inclusions that are one of the important parts in modern
engineering researches. The construction and study of
mathematical models for thermoconductivity processes
in thermosensitive structures with foreign inclusions
requires the development of new efficient methods to
solve boundary mathematical physics problems of both
theoretical and practical significance.
The thermal process is nonlinear, and the presence of
foreign inclusions complicates these mathematical models
but increases the research accuracy, which is topical as to
their heat resistance. This requires new algorithms for
construction and appropriate software development for the
temperature condition analysis of individual junctions and
structural elements of electronic devices.
In the process of solving the boundary thermal
conductivity problems, it is important to consider the
dependence of thermal-and-physical characteristics on
temperature, which allows describing the distribution of
temperature fields and their gradients in the systems
under consideration to be more accurate and precise.
The approximate analytical solution for half-space
with a small foreign heat dissipating cylindrical
inclusion was obtained in the paper [1]. The analytical
solution for the isotropic strip with the rectangular
inclusion of an arbitrary size is produced in the paper
[2]. General thermal conductivity equations for the
thermosensitive piecewise homogeneous solids are
presented in the works [3, 4].
2. Problem statement
Due to the complex structure of separate junctions and
construction elements in modern electronic devices the
necessity in modeling of thermosensitive systems with
foreign inclusions has arisen. Therefore we consider the
isotropic thermosensitive strip with rectangular inclusion
with square hl4 referred to the Cartesian coordinate
system Oxy with its origin in the center of inclusion
(Figure). At the coupling intervals
lyyhLx :, , hxlxLy :, , the
conditions of ideal thermal contact are fulfilled. At the
strip edges xdlxK :, 1 ,
hxlxK :, , the conditions of convective heat
exchange with the constant environment temperature ct
are given. In the inclusion area
lyhxyx ,:,0 with uniformly distributed
internal heat sources, the power is q0.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 439-443.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
440
3. Partially linearized initial boundary problem
The distribution of steady state temperature field ),( yxt
in the thermosensitive strip is obtained by solving the
nonlinear thermal conductivity equation [3, 4]
,),(),(
),,(),,(
0 lyNhxNq
y
t
tyx
yx
t
tyx
x
(1)
taking into account the following boundary conditions:
cdly
dly
tt
y
t
t
1
1
)(1 ,
cdly
dly
tt
y
t
t
2
2
)(1 ,
0
x
t , 0
x
x
t
, (2)
where ),(),()()()(),,( 101 lyNhxNttttyx
is the thermal conductivity coefficient of the
heterogeneous strip; 10 , − heat conduction
coefficients of the inclusion and strip; − coefficient
of heat elimination from the strip edges K ;
)()(),( SSN ; )(S − asymmetric
unit functions.
Input the function [5]
xhSlySd
lySd
lyNdd
lxt
lht
lxt
lht
yxt
yht
yxt
,
,
10
,
,
10
,
,
10
,
0
1 ,
(3)
and obtain by differentiating with respect to х and у:
.,
,,
,
,,
10
10
10
lyNxhS
y
t
tt
x
t
yxt
y
xhSlyS
x
t
tt
lyS
x
t
tt
x
t
yxt
x
hx
ly
ly
(4)
Considering the expressions (4) and equation (1),
we can write:
.,
,
10
0
10
10
lyNxhS
y
t
tt
y
lyNxhSq
xhSlyS
x
t
tt
lyS
x
t
tt
x
hx
ly
ly
(5)
Here, is the Laplace operator.
cdly
dly
tt
y
1
1
, 0
x
,
0
xx
, cdly
dly
tt
y
2
2
. (6)
Hence, the nonlinear boundary equations (1), (2)
with application of the input function (3) can be reduced
to a partially linearized problem (5), (6).
4. Fully linearized boundary problem
Let us approximate ),,(),,(),,( 1dlxtlxtyht
),( 2dlxt functions with the following expressions
,)()(),(
,)()(),(
),()(),(
),()(),(
1
1
)()(
1
)(
12
1
1
)()(
1
)(
11
)(
1
1
)(
1
)(
1
)(
1
1
)(
1
)(
1
m
k
kkk
m
k
kkk
j
l
j
n
j
l
j
l
j
h
j
n
j
h
j
h
xxStttdlxt
xxStttdlxt
xxStttlxt
yyStttyht
(7)
where )()()()( ,,,
kk
l
j
h
j tttt are unknown approximating
temperature values ;, lly j ;... 121 nyyy
;,0,0, hhx j ;... 121 nxxx ;,0 *xxk
;... 121 mxxx *x is the abscissa value for which
the temperature is practically equal to ct (is obtained
from the relevant linear problem); n – number of
partitions in the intervals ll, , 0,h and h,0 , m –
number of partitions in the interval *,0 x .
Substituting expressions (7) in the equation (5) and
boundary conditions (6) at the edges of strip K , a
linear boundary problem for finding the function is
obtained
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 439-443.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
441
),,(|)|(
)(|)|()()()(
)()()()()(
)()()()()(
0
1
1
)(
11
)(
10
)()(
1
)(
11
)(
10
1
1
)()(
1
1
1
)(
11
)(
10
)()(
1
lyNxhSq
yyxhStttt
lySxxtttt
lySxxtttt
n
j
j
h
j
h
j
h
j
h
j
j
l
j
l
j
n
j
l
j
l
j
j
n
j
l
j
l
j
l
j
l
j
(8)
,)()(
,)()(
1
1
)()(
1
)(
1
1
1
)()(
1
)(
1
2
1
c
m
k
kkk
dly
c
m
k
kkk
dly
txxSttt
y
txxSttt
y
(9)
,0
||
x
0
||
xx
,
where )( is the asymmetric Dirac delta function .
5. Construction of analytical solution to the linear
boundary problem (8), (9)
Applying the Fourier integral transform by the x
coordinate for the boundary problem (8), (9), we obtain
the ordinary differential equations with constant
coefficients:
)(
,
sh2
)(
)(
2
1
)()(
1
1
1
)(
11
)(
100
1
1
)()(
1
)(
11
)(
10
)()(
1
1
1
)(
11
)(
10
2
2
2
j
h
j
h
j
n
j
h
j
h
j
n
j
l
j
l
j
l
j
l
j
xi
l
j
l
j
n
j
l
j
l
j
xi
yytt
ttlyNq
hi
lyStttte
lyStt
ttei
y
j
j
(10)
and the following boundary conditions:
,)(
2
,)(
2
1
1
)()(
1
1
1
)()(
1
2
1
m
k
xi
kk
dly
m
k
xi
kk
dly
k
k
ett
i
y
ett
i
y
(11)
where
dxe xi
2
1
− function transformant;
1i .
The general solution to the equation (10) is:
.)()(ch)()(ch),(
)()(ch
)()(sh2
)()(ch1
)()(
)()(ch1
)()(
2
1
2
0)()(
1
1
1
)(
11
)(
10
)()(
1
1
1
)(
11
)(
10
)()(
1
1
1
)(
11
)(
10
21
lySlylySlylyN
q
yySyytt
tthi
lySlytt
tte
lySlytt
tte
i
eCeC
jj
h
j
h
j
n
j
h
j
h
j
l
j
l
j
n
j
l
j
l
j
xi
l
j
l
j
n
j
l
j
l
j
xi
yy
j
j
Here C1 and C2 are integrating constants.
Having applied the boundary conditions (11), a
partial solution to the problem (10), (11) is obtained
.)(ch
)(ch
2sh
1
)(ch
2sh
sh2sh
)()(ch)()(ch),(
)(ch
2sh
sh
)()(ch
)()(sh2
)(ch
2sh
sh
)()(ch1
)()(
)(ch
2sh
2sh
)()(ch1
)()(
2
1
1
1
21
1
1
21
21
2
21
21
2
0
2
21
1
1
1
)(
11
)(
10
)()(
1
2
21
2
1
1
)()(
1
)(
11
)(
10
2
21
1
)()(
1
1
1
)(
11
)(
10
m
k
xi
kk
m
k
xi
kk
j
jj
n
j
h
j
h
j
h
j
h
j
n
j
l
j
l
j
l
j
l
j
xi
l
j
l
j
n
j
l
j
l
j
xi
dlyett
dlyett
ddl
dly
ddl
ddl
lySlylySlylyN
q
dly
ddl
ydl
yySyy
tttthi
dly
ddl
d
lySly
tttte
dly
ddl
dl
lySly
tttte
i
k
k
j
j
(12)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 439-443.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
442
Applying the inverse Fourier transform to the
relation (12), the following expression for the function
could be found:
.)(ch
2sh
sin
)(ch
2sh
sin
)(ch
2sh
sh2sh
)()(ch
)()(ch,
sinsin
)(ch
2sh
sh
)()(ch
sinsin
)()(
)(ch
2sh
sh
)()(ch1
sin
)()(
)(ch
2sh
2sh
)()(ch1
sin
)()(
1
1
1
1
0 21
21
2
0 21
2
1
1
1
2
21
21
0
0
2
21
1
1
1 0
)(
11
)(
10
)()(
1
2
21
2
1
1 0
)()(
1
)(
11
)(
10
2
21
1
0
1
1
)(
11
)(
10
)()(
1
m
k
k
kk
k
m
k
kk
j
jj
n
j
h
j
h
j
h
j
h
j
n
j
jl
j
l
j
l
j
l
j
j
n
j
l
j
l
j
l
j
l
j
ddly
ddl
xx
tt
ddly
ddl
xx
tt
ddly
ddl
ddl
lySly
lySlylyN
hxhx
q
ddly
ddl
ydl
yySyy
hxhx
tttt
ddly
ddl
d
lySly
xx
tttt
ddly
ddl
dl
lySly
xx
tttt
(13)
Substituting the specific subjection coefficients of
inclusion materials and strip thermal conduction in the
relation (3), (13) and equalizing the expressions of -
function on the edges K and on contact intervals
xL ,
yL the system of nonlinear algebraic equations to
determine unknown values of temperature )( h
jt ,
),1()( njt l
j , )(
kt , ),1()( mktk is obtained.
The desired temperature field for the nonlinear
boundary heat conduction problem (1), (2) is determined
from the nonlinear algebraic equation obtained after the
application of relations (3), (13) and substitution of the
specific expressions for coefficient dependences of
thermal conductivity inherent to inclusion and strip
materials.
6. Partial examples and analysis of the results
There is such a dependency between the coefficient of
thermal conductivity and temperature in many practical
cases [6, 7]:
)1(0 kt ,
where k,0 are reference and temperature coefficients
for the thermal conductivity.
Then, applying expressions (3), (13), the formula to
determine the temperature t in areas is obtained
lyhxyx ,:,0 ,
0
10
0
0211
k
k
t
,
121 ,:, dlydlhxyx ,
lydlhxyx 22 ,:, ,
1
0
1
12
11
k
k
t
,
13 ,:, dlylhxyx ,
1
10
1
1211
k
k
t
,
where
;
2
1
1
2
1
1
2
1
1
10
0
0
1
00
0
0
1
010
0
0
1
0
0
0
1
010
0
0
1
0
0
0
1*
1
ly
hx
ly
hx
ttkk
ttkk
ttkk
;
2
11
;
2
11
1
0
1
1
1
0
1
1
k
k
t
k
k
t
ly
ly
hx
hx
;
2
11
1
0
1
1
k
k
t
ly
hx
ly
hx
0
30
0
0211
k
k
t
ly
ly
;
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 439-443.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
443
y
d
l2
d 2h
O
x
Fig. Thermosensitive strip with rectangular inclusion.
.
2
1
1
2
1
1
2
1
1
;
2
1
1
2
1
1
2
1
1
2
1
1
||
0
0
0
1
100
0
0
1
||
00
0
0
1
10
0
0
1
00
0
0
1
10
0
0
1*
3
||
10
1
0
0
00
1
0
0
||
0
1
0
0
010
1
0
0
10
1
0
0
00
1
0
0
0
1
0
0
010
1
0
0*
2
lz
hx
lz
hx
lz
lz
hx
lz
hx
lz
lz
ttkk
ttkk
ttkk
ttkk
ttkk
ttkk
ttkk
Being based on numerical analysis, it has been
determined that it is enough to select the partition
number n of the intervals ll; , 0;h and h;0 that
equals seven and the partition number m of the interval
*;0 x that equals eleven. Numerical calculations have
been performed for the following materials: strip
material – ceramics VK-94-1, the inclusion material –
tungsten, which show that the consideration of
dependency of the thermal conductivity coefficients on
temperature leads to reduction of the temperature field
compared with non-thermosensitive system (thermal-
and-physical parameters are independent on
temperature) by 3.8% for selected materials.
7. Conclusions
The original nonlinear heat conductivity equation (1) has
been partially linearized using the relation (3) that
describes a new input function . The proposed
piecewise linear approximation of temperature on the
boundaries
xL ,
yL of an inclusion and at the edges K
of the strip with the expressions (7) that allowed to
totally linearize the nonlinear boundary problem of
thermal conductivity (5), (6) for a thermosensitive
system with the foreign inclusion. An analytical solution
to the input function has been produced as a
formula (3), which enabled to construct new algorithms
and developed software to evaluate the temperature
gradients in the area of foreign inclusion and to forecast
the operation mode of electronic devices with increased
thermal resistance and prolonged operating period.
References
1. Yu.M. Kolyano, Yu.M. Krychevets, E.H. Ivanyk,
V.I. Gavrysh, Temperature field in half-space with
foreign inclusion // Inzhenerno-fizich. zhurnal
(Minsk) 55 (6), p. 1006-1011 (1988) (in Russian).
2. V.I. Gavrysh, V.A. Volos, The problem of the strip
with rectangular inclusions heat conductivity // The
Lviv Polytechnic State University Bulletin: Applied
Mathematics. No. 320, p. 28-33 (1997) (in
Ukrainian).
3. Ya.M. Podstryhach, V.A. Lomakin, Yu.M. Kolya-
no, Thermoelasticity of Heterogeneous Body
Structure. Nauka. Moscow, 1984 (in Russian).
4. Yu.M. Kolyano, Methods of Heat Conductivity and
Thermoelasticity of Heterogeneous Bodies.
Naukova Dumka, Kyiv (1992) (in Russian).
5. D. Fedasyuk, V. Gavrysh, A. Kuzmin, Non-linear
heat exchange problem for the piecewise homo-
geneous strip with foreign inclusion // Proc. Intern.
Conference “Modern Problems of Mechanics and
Mathematics”, vol. 1, Lviv, 2008 (in Ukrainian).
6. V.A. Lomakin, The Theory of Thermoelasticity of
Heterogeneous Bodies. Moscow University
Publishing House, Moscow, 1976 (in Russian).
7. R. Berman, Heat Conductivity of Solids. Mir,
Moscow, 1979 (in Russian).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 439-443.
PACS 74.25.fc
Thermal simulation of heterogeneous structural components
in microelectronic devices
V.I. Gavrysh, D.V. Fedasyuk
Lviv Polytechnic National University
E-mail: fedasyuk@lp.edu.ua
Abstract. The steady state nonlinear problem of thermal conduction for isotropic strip with foreign rectangular inclusion that heats as an internal thermal source with heat dissipation has been considered. The methodology to solve this problem and its application for the specific dependence of the thermal-conductivity coefficients on temperature has been offered.
Keywords: isotropic, thermal conduction, foreign inclusion, thermosensitive, heat dissipation.
Manuscript received 03.06.10; accepted for publication 02.12.10; published online 30.12.10.
1. Introduction
In the design process of individual junctions and structural components for microelectronic devices, there appears the necessity to mathematically model thermal processes in thermosensitive structures with foreign inclusions that are one of the important parts in modern engineering researches. The construction and study of mathematical models for thermoconductivity processes in thermosensitive structures with foreign inclusions requires the development of new efficient methods to solve boundary mathematical physics problems of both theoretical and practical significance.
The thermal process is nonlinear, and the presence of foreign inclusions complicates these mathematical models but increases the research accuracy, which is topical as to their heat resistance. This requires new algorithms for construction and appropriate software development for the temperature condition analysis of individual junctions and structural elements of electronic devices.
In the process of solving the boundary thermal conductivity problems, it is important to consider the dependence of thermal-and-physical characteristics on temperature, which allows describing the distribution of temperature fields and their gradients in the systems under consideration to be more accurate and precise.
The approximate analytical solution for half-space with a small foreign heat dissipating cylindrical inclusion was obtained in the paper [1]. The analytical solution for the isotropic strip with the rectangular inclusion of an arbitrary size is produced in the paper [2]. General thermal conductivity equations for the thermosensitive piecewise homogeneous solids are presented in the works [3, 4].
2. Problem statement
Due to the complex structure of separate junctions and construction elements in modern electronic devices the necessity in modeling of thermosensitive systems with foreign inclusions has arisen. Therefore we consider the isotropic thermosensitive strip with rectangular inclusion with square
hl
4
referred to the Cartesian coordinate system
Oxy
with its origin in the center of inclusion (Figure). At the coupling intervals
(
)
{
}
l
y
y
h
L
x
£
±
=
±
:
,
,
(
)
{
}
h
x
l
x
L
y
£
±
=
±
:
,
, the conditions of ideal thermal contact are fulfilled. At the strip edges
(
)
{
}
¥
<
+
=
+
x
d
l
x
K
:
,
1
,
(
)
{
}
h
x
l
x
K
£
±
=
-
:
,
, the conditions of convective heat exchange with the constant environment temperature
c
t
are given. In the inclusion area
(
)
{
}
l
y
h
x
y
x
£
£
=
W
,
:
,
0
with uniformly distributed internal heat sources, the power is q0.
3. Partially linearized initial boundary problem
The distribution of steady state temperature field
)
,
(
y
x
t
in the thermosensitive strip is obtained by solving the nonlinear thermal conductivity equation [3, 4]
,
)
,
(
)
,
(
)
,
,
(
)
,
,
(
0
l
y
N
h
x
N
q
y
t
t
y
x
y
x
t
t
y
x
x
×
×
-
=
=
ú
û
ù
ê
ë
é
¶
¶
×
l
¶
¶
+
ú
û
ù
ê
ë
é
¶
¶
×
l
¶
¶
(1)
taking into account the following boundary conditions:
(
)
c
d
l
y
d
l
y
t
t
y
t
t
-
×
a
-
=
¶
¶
×
l
+
=
+
+
=
1
1
)
(
1
,
(
)
c
d
l
y
d
l
y
t
t
y
t
t
-
×
a
=
¶
¶
×
l
-
-
=
-
-
-
=
2
2
)
(
1
,
0
=
¥
®
x
t
,
0
=
¶
¶
¥
®
x
x
t
,
(2)
where
[
]
)
,
(
)
,
(
)
(
)
(
)
(
)
,
,
(
1
0
1
l
y
N
h
x
N
t
t
t
t
y
x
×
×
l
-
l
+
l
=
l
is the thermal conductivity coefficient of the heterogeneous strip;
1
0
,
l
l
− heat conduction coefficients of the inclusion and strip;
±
a
− coefficient of heat elimination from the strip edges
±
K
;
)
(
)
(
)
,
(
h
-
z
-
h
+
z
=
h
z
+
-
S
S
N
;
)
(
z
±
S
− asymmetric unit functions.
Input the function [5]
(
)
(
)
(
)
(
)
[
]
(
)
(
)
(
)
(
)
(
)
[
]
(
)
(
)
(
)
(
)
(
)
[
]
(
)
(
)
(
)
(
)
x
h
S
l
y
S
d
l
y
S
d
l
y
N
d
d
l
x
t
l
h
t
l
x
t
l
h
t
y
x
t
y
h
t
y
x
t
-
×
ï
þ
ï
ý
ü
-
×
z
z
l
-
z
l
+
+
+
×
z
z
l
-
z
l
-
ï
î
ï
í
ì
-
×
z
z
l
-
z
l
+
z
z
l
=
J
-
±
+
-
-
±
-
±
ò
ò
ò
ò
,
,
1
0
,
,
1
0
,
,
1
0
,
0
1
,
(3)
and obtain by differentiating with respect to х and у:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
,
,
,
,
,
,
1
0
1
0
1
0
l
y
N
x
h
S
y
t
t
t
x
t
y
x
t
y
x
h
S
l
y
S
x
t
t
t
l
y
S
x
t
t
t
x
t
y
x
t
x
h
x
l
y
l
y
×
-
´
´
þ
ý
ü
î
í
ì
¶
¶
l
-
l
-
¶
¶
l
=
¶
J
¶
-
×
ï
þ
ï
ý
ü
+
×
ú
û
ù
ê
ë
é
¶
¶
l
-
l
-
ï
î
ï
í
ì
-
-
×
ú
û
ù
ê
ë
é
¶
¶
l
-
l
+
+
¶
¶
l
=
¶
J
¶
-
=
-
-
-
=
+
=
(4)
Considering the expressions (4) and equation (1), we can write:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
,
,
1
0
0
1
0
1
0
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
-
×
ú
û
ù
ê
ë
é
¶
¶
×
l
-
l
¶
¶
-
-
×
-
×
-
-
ï
þ
ï
ý
ü
-
×
ú
ú
û
ù
+
×
÷
ø
ö
ç
è
æ
¶
¶
×
l
-
l
×
-
ï
î
ï
í
ì
ê
ê
ë
é
-
-
×
÷
ø
ö
ç
è
æ
¶
¶
×
l
-
l
¶
¶
=
J
D
-
=
-
-
-
-
=
+
=
l
y
N
x
h
S
y
t
t
t
y
l
y
N
x
h
S
q
x
h
S
l
y
S
x
t
t
t
l
y
S
x
t
t
t
x
h
x
l
y
l
y
(5)
Here,
D
is the Laplace operator.
(
)
c
d
l
y
d
l
y
t
t
y
-
×
a
-
=
¶
J
¶
+
=
+
+
=
1
1
,
0
=
J
¥
®
x
,
0
=
¶
J
¶
¥
®
x
x
,
(
)
c
d
l
y
d
l
y
t
t
y
-
a
-
=
¶
J
¶
-
-
=
-
-
-
=
2
2
.
(6)
Hence, the nonlinear boundary equations (1), (2) with application of the input function (3) can be reduced to a partially linearized problem (5), (6).
4. Fully linearized boundary problem
Let us approximate
),
,
(
),
,
(
),
,
(
1
d
l
x
t
l
x
t
y
h
t
+
±
±
)
,
(
2
d
l
x
t
-
-
functions with the following expressions
,
)
(
)
(
)
,
(
,
)
(
)
(
)
,
(
),
(
)
(
)
,
(
),
(
)
(
)
,
(
1
1
)
(
)
(
1
)
(
1
2
1
1
)
(
)
(
1
)
(
1
1
)
(
1
1
)
(
1
)
(
1
)
(
1
1
)
(
1
)
(
1
å
å
å
å
-
=
-
-
-
+
-
-
=
-
+
+
+
+
-
±
-
=
±
+
±
-
±
-
=
±
+
±
-
-
+
=
-
-
-
-
+
=
+
-
×
-
+
=
±
-
×
-
+
=
±
m
k
k
k
k
m
k
k
k
k
j
l
j
n
j
l
j
l
j
h
j
n
j
h
j
h
x
x
S
t
t
t
d
l
x
t
x
x
S
t
t
t
d
l
x
t
x
x
S
t
t
t
l
x
t
y
y
S
t
t
t
y
h
t
(7)
where
)
(
)
(
)
(
)
(
,
,
,
-
+
±
±
k
k
l
j
h
j
t
t
t
t
are unknown approximating temperature values
]
[
;
,
l
l
y
j
-
Î
;
...
1
2
1
-
<
<
<
n
y
y
y
]
[
]
[
;
,
0
,
0
,
h
h
x
j
-
Î
;
...
1
2
1
-
<
<
<
n
x
x
x
]
[
;
,
0
*
x
x
k
Î
;
...
1
2
1
-
<
<
<
m
x
x
x
*
x
is the abscissa value for which the temperature is practically equal to
c
t
(is obtained from the relevant linear problem); n – number of partitions in the intervals
]
[
l
l
,
-
,
]
[
0
,
h
-
and
]
[
h
,
0
, m – number of partitions in the interval
]
[
*
,
0
x
.
Substituting expressions (7) in the equation (5) and boundary conditions (6) at the edges of strip
±
K
, a linear boundary problem for finding the function
J
is obtained
[
]
[
]
[
]
),
,
(
|)
|
(
)
(
|)
|
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
0
1
1
)
(
1
1
)
(
1
0
)
(
)
(
1
)
(
1
1
)
(
1
0
1
1
)
(
)
(
1
1
1
)
(
1
1
)
(
1
0
)
(
)
(
1
l
y
N
x
h
S
q
y
y
x
h
S
t
t
t
t
l
y
S
x
x
t
t
t
t
l
y
S
x
x
t
t
t
t
n
j
j
h
j
h
j
h
j
h
j
j
l
j
l
j
n
j
l
j
l
j
j
n
j
l
j
l
j
l
j
l
j
×
-
×
-
-
-
d
¢
×
-
×
l
-
l
×
-
-
-
+
-
d
¢
×
l
-
l
×
-
-
-
-
×
-
d
¢
×
l
-
l
×
-
=
J
D
-
-
=
-
-
±
+
±
+
±
±
+
-
-
-
+
-
+
-
=
-
-
+
+
-
-
=
+
+
+
å
å
å
(8)
,
)
(
)
(
,
)
(
)
(
1
1
)
(
)
(
1
)
(
1
1
1
)
(
)
(
1
)
(
1
2
1
ú
ú
û
ù
ê
ê
ë
é
-
-
×
-
+
×
a
=
¶
J
¶
ú
ú
û
ù
ê
ê
ë
é
-
-
×
-
+
×
a
-
=
¶
J
¶
å
å
-
=
-
-
-
+
-
-
-
-
=
-
=
-
+
+
+
+
+
+
=
c
m
k
k
k
k
d
l
y
c
m
k
k
k
k
d
l
y
t
x
x
S
t
t
t
y
t
x
x
S
t
t
t
y
(9)
,
0
|
|
=
J
¥
®
x
0
|
|
=
¶
J
¶
¥
®
x
x
,
where
)
(
z
d
-
is the asymmetric Dirac delta function .
5. Construction of analytical solution to the linear boundary problem (8), (9)
Applying the Fourier integral transform by the x coordinate for the boundary problem (8), (9), we obtain the ordinary differential equations with constant coefficients:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
]
}
)
(
,
sh
2
)
(
)
(
2
1
)
(
)
(
1
1
1
)
(
1
1
)
(
1
0
0
1
1
)
(
)
(
1
)
(
1
1
)
(
1
0
)
(
)
(
1
1
1
)
(
1
1
)
(
1
0
2
2
2
j
h
j
h
j
n
j
h
j
h
j
n
j
l
j
l
j
l
j
l
j
x
i
l
j
l
j
n
j
l
j
l
j
x
i
y
y
t
t
t
t
l
y
N
q
h
i
l
y
S
t
t
t
t
e
l
y
S
t
t
t
t
e
i
y
j
j
-
d
¢
×
-
´
ê
ê
ë
é
´
l
-
l
+
×
×
x
x
+
+
ú
ú
û
ù
-
×
-
×
l
-
l
-
-
+
×
-
´
ï
î
ï
í
ì
ê
ê
ë
é
´
l
-
l
×
x
×
p
=
J
x
-
¶
J
¶
-
±
±
+
-
=
±
+
±
+
-
=
+
+
+
+
+
+
+
+
x
-
-
-
+
-
=
-
+
-
+
x
å
å
å
×
×
(10)
and the following boundary conditions:
,
)
(
2
,
)
(
2
1
1
)
(
)
(
1
1
1
)
(
)
(
1
2
1
å
å
-
=
x
-
-
+
-
-
-
=
-
=
x
+
+
+
+
+
=
×
+
×
px
a
-
=
¶
J
¶
×
+
×
px
a
-
=
¶
J
¶
m
k
x
i
k
k
d
l
y
m
k
x
i
k
k
d
l
y
k
k
e
t
t
i
y
e
t
t
i
y
(11)
where
ò
¥
¥
-
x
J
×
p
=
J
dx
e
x
i
2
1
− function
J
transformant;
1
-
=
i
.
The general solution to the equation (10) is:
[
]
(
)
(
)
[
]
(
)
(
)
[
]
(
)
(
)
]
}
.
)
(
)
(
ch
)
(
)
(
ch
)
,
(
)
(
)
(
ch
)
(
)
(
sh
2
)
(
)
(
ch
1
)
(
)
(
)
(
)
(
ch
1
)
(
)
(
2
1
2
0
)
(
)
(
1
1
1
)
(
1
1
)
(
1
0
)
(
)
(
1
1
1
)
(
1
1
)
(
1
0
)
(
)
(
1
1
1
)
(
1
1
)
(
1
0
2
1
l
y
S
l
y
l
y
S
l
y
l
y
N
q
y
y
S
y
y
t
t
t
t
h
i
l
y
S
l
y
t
t
t
t
e
l
y
S
l
y
t
t
t
t
e
i
e
C
e
C
j
j
h
j
h
j
n
j
h
j
h
j
l
j
l
j
n
j
l
j
l
j
x
i
l
j
l
j
n
j
l
j
l
j
x
i
y
y
j
j
-
×
-
x
+
+
×
+
x
-
´
´
x
-
-
-
×
-
x
-
´
ê
ê
ë
é
´
l
-
l
×
x
-
-
-
×
-
x
-
×
-
´
´
l
-
l
×
-
-
+
×
+
x
-
×
-
´
ï
î
ï
í
ì
´
l
-
l
×
´
´
x
×
p
-
×
+
×
=
J
+
-
-
±
±
+
-
=
±
+
±
+
+
+
-
=
+
+
x
-
-
-
+
-
=
-
+
-
+
x
x
-
x
å
å
å
Here C1 and C2 are integrating constants.
Having applied the boundary conditions (11), a partial solution to the problem (10), (11) is obtained
[
]
(
)
(
)
(
)
(
)
[
]
(
)
(
)
(
)
(
)
(
)
[
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
)
(
ch
)
(
ch
2
sh
1
)
(
ch
2
sh
sh
2
sh
)
(
)
(
ch
)
(
)
(
ch
)
,
(
)
(
ch
2
sh
sh
)
(
)
(
ch
)
(
)
(
sh
2
)
(
ch
2
sh
sh
)
(
)
(
ch
1
)
(
)
(
)
(
ch
2
sh
2
sh
)
(
)
(
ch
1
)
(
)
(
2
1
1
1
2
1
1
1
2
1
2
1
2
2
1
2
1
2
0
2
2
1
1
1
1
)
(
1
1
)
(
1
0
)
(
)
(
1
2
2
1
2
1
1
)
(
)
(
1
)
(
1
1
)
(
1
0
2
2
1
1
)
(
)
(
1
1
1
)
(
1
1
)
(
1
0
ï
þ
ï
ý
ü
ú
ú
û
ù
-
-
x
-
a
+
ê
ê
ë
é
+
+
+
x
-
a
´
´
+
+
x
x
+
ú
ú
û
ù
÷
÷
ø
ö
+
+
x
+
+
x
x
-
+
x
+
+
-
×
-
x
+
+
×
+
x
-
×
x
-
-
÷
÷
ø
ö
ç
ç
è
æ
+
+
x
+
+
x
-
+
x
-
-
×
-
x
´
´
ê
ê
ë
é
l
-
l
×
-
×
x
-
-
ú
û
ù
ê
ë
é
+
+
x
+
+
x
x
+
-
-
x
-
´
´
-
l
-
l
×
-
-
ú
û
ù
ê
ë
é
+
+
x
+
+
x
+
x
+
+
+
x
-
´
´
ï
î
ï
í
ì
-
×
l
-
l
×
×
x
×
p
-
=
J
å
å
å
å
å
-
=
x
-
-
+
-
-
=
x
+
+
+
+
+
-
-
-
=
±
+
±
+
±
±
+
+
-
=
+
+
+
x
-
-
-
+
-
=
-
+
-
+
x
m
k
x
i
k
k
m
k
x
i
k
k
j
j
j
n
j
h
j
h
j
h
j
h
j
n
j
l
j
l
j
l
j
l
j
x
i
l
j
l
j
n
j
l
j
l
j
x
i
d
l
y
e
t
t
d
l
y
e
t
t
d
d
l
d
l
y
d
d
l
d
d
l
l
y
S
l
y
l
y
S
l
y
l
y
N
q
d
l
y
d
d
l
y
d
l
y
y
S
y
y
t
t
t
t
h
i
d
l
y
d
d
l
d
l
y
S
l
y
t
t
t
t
e
d
l
y
d
d
l
d
l
l
y
S
l
y
t
t
t
t
e
i
k
k
j
j
(12)
Applying the inverse Fourier transform to the relation (12), the following expression for the function
J
could be found:
(
)
[
]
(
)
(
)
(
)
(
)
[
]
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
[
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
)
(
ch
2
sh
sin
)
(
ch
2
sh
sin
)
(
ch
2
sh
sh
2
sh
)
(
)
(
ch
)
(
)
(
ch
,
sin
sin
)
(
ch
2
sh
sh
)
(
)
(
ch
sin
sin
)
(
)
(
)
(
ch
2
sh
sh
)
(
)
(
ch
1
sin
)
(
)
(
)
(
ch
2
sh
2
sh
)
(
)
(
ch
1
sin
)
(
)
(
1
1
1
1
0
2
1
2
1
2
0
2
1
2
1
1
1
2
2
1
2
1
0
0
2
2
1
1
1
1
0
)
(
1
1
)
(
1
0
)
(
)
(
1
2
2
1
2
1
1
0
)
(
)
(
1
)
(
1
1
)
(
1
0
2
2
1
1
0
1
1
)
(
1
1
)
(
1
0
)
(
)
(
1
ï
þ
ï
ý
ü
x
-
-
x
+
+
x
×
x
-
x
-
a
+
+
x
+
+
x
+
+
x
×
x
-
x
-
a
+
+
x
ú
û
ù
+
+
x
+
+
x
x
-
+
x
+
-
×
-
x
+
+
+
×
+
x
-
x
+
x
-
-
x
+
+
x
ú
û
ù
ê
ë
é
+
+
x
+
+
x
-
+
x
-
-
-
x
´
´
x
+
x
-
-
x
l
-
l
×
-
-
-
x
ú
û
ù
ê
ë
é
+
+
x
+
+
x
x
+
-
-
x
-
´
´
x
-
x
-
l
-
l
-
-
x
ú
û
ù
ê
ë
é
+
+
x
+
+
x
+
x
+
+
+
x
-
´
´
x
-
x
ï
î
ï
í
ì
l
-
l
×
-
×
p
-
=
J
å
ò
ò
å
ò
å
ò
å
ò
ò
å
-
=
¥
-
-
+
-
¥
-
=
+
+
+
+
+
-
¥
-
-
=
¥
±
+
±
+
±
±
+
+
-
=
¥
+
+
+
-
¥
-
=
-
+
-
+
-
-
+
m
k
k
k
k
k
m
k
k
k
j
j
j
n
j
h
j
h
j
h
j
h
j
n
j
j
l
j
l
j
l
j
l
j
j
n
j
l
j
l
j
l
j
l
j
d
d
l
y
d
d
l
x
x
t
t
d
d
l
y
d
d
l
x
x
t
t
d
d
l
y
d
d
l
d
d
l
l
y
S
l
y
l
y
S
l
y
l
y
N
h
x
h
x
q
d
d
l
y
d
d
l
y
d
l
y
y
S
y
y
h
x
h
x
t
t
t
t
d
d
l
y
d
d
l
d
l
y
S
l
y
x
x
t
t
t
t
d
d
l
y
d
d
l
d
l
l
y
S
l
y
x
x
t
t
t
t
(13)
Substituting the specific subjection coefficients of inclusion materials and strip thermal conduction in the relation (3), (13) and equalizing the expressions of
J
-function on the edges
±
K
and on contact intervals
±
x
L
,
±
y
L
the system of nonlinear algebraic equations to determine unknown values of temperature
)
(
h
j
t
±
,
)
,
1
(
)
(
n
j
t
l
j
=
±
,
)
(
-
k
t
,
)
,
1
(
)
(
m
k
t
k
=
+
is obtained.
The desired temperature field for the nonlinear boundary heat conduction problem (1), (2) is determined from the nonlinear algebraic equation obtained after the application of relations (3), (13) and substitution of the specific expressions for coefficient dependences of thermal conductivity inherent to inclusion and strip materials.
6. Partial examples and analysis of the results
There is such a dependency between the coefficient of thermal conductivity and temperature in many practical cases [6, 7]:
)
1
(
0
kt
-
×
l
=
l
,
where
k
,
0
l
are reference and temperature coefficients for the thermal conductivity.
Then, applying expressions (3), (13), the formula to determine the temperature t in areas is obtained
(
)
{
}
l
y
h
x
y
x
£
£
=
W
,
:
,
0
,
0
1
0
0
0
2
1
1
k
k
t
÷
÷
ø
ö
ç
ç
è
æ
J
+
l
J
-
-
=
*
,
(
)
{
}
1
2
1
,
:
,
d
l
y
d
l
h
x
y
x
+
£
£
-
-
>
=
W
,
(
)
{
}
l
y
d
l
h
x
y
x
-
<
£
-
-
£
=
W
2
2
,
:
,
,
1
0
1
1
2
1
1
k
k
t
l
J
×
-
-
=
,
(
)
{
}
1
3
,
:
,
d
l
y
l
h
x
y
x
+
£
<
£
=
W
,
1
1
0
1
1
2
1
1
k
k
t
÷
÷
ø
ö
ç
ç
è
æ
J
+
l
J
-
-
=
*
,
where
;
2
1
1
2
1
1
2
1
1
1
0
0
0
1
0
0
0
0
1
0
1
0
0
0
1
0
0
0
1
0
1
0
0
0
1
0
0
0
1
*
1
l
y
h
x
l
y
h
x
t
t
k
k
t
t
k
k
t
t
k
k
-
=
=
-
=
=
ï
þ
ï
ý
ü
×
ï
î
ï
í
ì
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
×
l
l
-
+
-
l
l
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
-
×
l
l
+
l
l
-
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
-
×
l
l
+
l
l
-
=
J
;
2
1
1
;
2
1
1
1
0
1
1
1
0
1
1
k
k
t
k
k
t
l
y
l
y
h
x
h
x
l
J
×
-
-
=
l
J
×
-
-
=
-
=
-
=
=
=
;
2
1
1
1
0
1
1
k
k
t
l
y
h
x
l
y
h
x
l
J
×
-
-
=
±
=
=
±
=
=
0
3
0
0
0
2
1
1
k
k
t
l
y
l
y
÷
÷
ø
ö
ç
ç
è
æ
J
+
l
J
-
-
=
*
=
=
;
y
d
1
l2
d
2
2h
O
x
Fig. Thermosensitive strip with rectangular inclusion.
.
2
1
1
2
1
1
2
1
1
;
2
1
1
2
1
1
2
1
1
2
1
1
|
|
0
0
0
1
1
0
0
0
0
1
|
|
0
0
0
0
1
1
0
0
0
1
0
0
0
0
1
1
0
0
0
1
*
3
|
|
1
0
1
0
0
0
0
1
0
0
|
|
0
1
0
0
0
1
0
1
0
0
1
0
1
0
0
0
0
1
0
0
0
1
0
0
0
1
0
1
0
0
*
2
l
z
h
x
l
z
h
x
l
z
l
z
h
x
l
z
h
x
l
z
l
z
t
t
k
k
t
t
k
k
t
t
k
k
t
t
k
k
t
t
k
k
t
t
k
k
t
t
k
k
-
=
=
=
=
-
=
-
=
=
=
=
=
-
=
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
l
l
×
-
+
-
l
l
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
-
l
l
×
+
l
l
-
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
-
l
l
×
+
l
l
-
=
J
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
-
l
l
×
+
l
l
-
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
l
l
×
-
+
-
l
l
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
-
l
l
×
+
l
l
-
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
×
ú
ú
û
ù
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
l
l
×
-
+
-
l
l
=
J
Being based on numerical analysis, it has been determined that it is enough to select the partition number n of the intervals
]
[
l
l
;
-
,
]
[
0
;
h
-
and
]
[
h
;
0
that equals seven and the partition number m of the interval
]
[
*
;
0
x
that equals eleven. Numerical calculations have been performed for the following materials: strip material – ceramics VK-94-1, the inclusion material –
tungsten, which show that the consideration of dependency of the thermal conductivity coefficients on temperature leads to reduction of the temperature field compared with non-thermosensitive system (thermal-and-physical parameters are independent on temperature) by 3.8% for selected materials.
7. Conclusions
The original nonlinear heat conductivity equation (1) has been partially linearized using the relation (3) that describes a new input function
J
. The proposed piecewise linear approximation of temperature on the boundaries
±
x
L
,
±
y
L
of an inclusion and at the edges
±
K
of the strip with the expressions (7) that allowed to totally linearize the nonlinear boundary problem of thermal conductivity (5), (6) for a thermosensitive system with the foreign inclusion. An analytical solution to the input function
J
has been produced as a formula (3), which enabled to construct new algorithms and developed software to evaluate the temperature gradients in the area of foreign inclusion and to forecast the operation mode of electronic devices with increased thermal resistance and prolonged operating period.
References
1.
Yu.M. Kolyano, Yu.M. Krychevets, E.H. Ivanyk, V.I. Gavrysh, Temperature field in half-space with foreign inclusion // Inzhenerno-fizich. zhurnal (Minsk) 55 (6), p. 1006-1011 (1988) (in Russian).
2.
V.I. Gavrysh, V.A. Volos, The problem of the strip with rectangular inclusions heat conductivity // The Lviv Polytechnic State University Bulletin: Applied Mathematics. No. 320, p. 28-33 (1997) (in Ukrainian).
3.
Ya.M. Podstryhach, V.A. Lomakin, Yu.M. Kolyano, Thermoelasticity of Heterogeneous Body Structure. Nauka. Moscow, 1984 (in Russian).
4.
Yu.M. Kolyano, Methods of Heat Conductivity and Thermoelasticity of Heterogeneous Bodies. Naukova Dumka, Kyiv (1992) (in Russian).
5.
D. Fedasyuk, V. Gavrysh, A. Kuzmin, Non-linear heat exchange problem for the piecewise homogeneous strip with foreign inclusion // Proc. Intern. Conference “Modern Problems of Mechanics and Mathematics”, vol. 1, Lviv, 2008 (in Ukrainian).
6.
V.A. Lomakin, The Theory of Thermoelasticity of Heterogeneous Bodies. Moscow University Publishing House, Moscow, 1976 (in Russian).
7.
R. Berman, Heat Conductivity of Solids. Mir, Moscow, 1979 (in Russian).
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
440
_1338974804.unknown
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|