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Potential method applied to steady-state heat conduction problems
MFXH’.
RES.
COMM.
Vo1.3, 285-290, 1976.
Pergamop Press.
Printed in USA.
POTENTIAL METHOD APPLIED TO STEADY-STATE HEAT CONDUCTION PROBLEMS Yu.A. Melnikov" Department of Mechanics and Mathematics, Dniepropetrovsk State University, Dniepropetrovsk, USSR (Received and accepted as ready for print 31 March 1976)
Introduction
A variant of the Green's function method is used here to solve steadystate heat conduction problems in multiply-connected bodies involving complicated geometries. An algorithm for constructing the Green's matrix of a layered strip is described and numerical results for two-and three dimensional particular cases are presented. This approach is general for systems of elliptic partial differential equations in mechanics and it has been employed in the past (see, for example, [l]-[7]) to obtain numerical solutions of various mechanical problems for bodies with intricate shapes. The solution of individual problems can be obtained by employing some numerical technique, for instance, the finite element method. This particular method is suitable for creating a general program (as, for example, NASTRAN) for solving a large class of such problems. However, many prob; lems can be solved even more effectively by means of potential theoretic approaches, such as the Green's function method which will be discussed in the sequel. The particular problems considered here will be used merely as examples to demonstrate the application of the method.
Statement of the Problem
Consider a two-dimensional, n-layered strip with an arbitrary hole L, and assume that the thermal conductivities Ak(k= 1,2,...,n) in each of the n layers are known continuous functions of the coordinate x (see Fig. l). The temperature u in the strip will satisfy the following relations: q
ax’ A,(x) -h(
au $) + hkW -a”%=
ay2
fk(x,y)
(k = 1,2 ,...,
4
*Presently visiting Professor of the Technological Institute, Northwestern University, Evanston, Illinois. Scientific Communication
285
(1)
286
Vo1.3, 'No.4
YU.A. MELNIKOV
a
0
FIG. 1
Uklx=a
= uk+lIx=ak,
lk
2 I x=ak
k
(3)
= 'k+l
(4) (5)
= Q(x,y)
BUklL
= u,(x,y) is the unknown temperature of the k-th layer, Al, A2 k and B are linear differential operators that depend on the boundary con-
where u
ditions, and Q(x,y) and fk(x,y) are given continuous functions. In the next section the Green's matrix for the relations (l)-(4) will be constructed in order to obtain the solution of the problems (1) - (5) by means of an integral representation in which the Green's matrix is the kernel. Construction of the Green's Matrix It will be assumed that problem.. (1) - (5) has
symmetry about the x-axis
and that the functions u,(x,y) and fk(x,y) can be represented in terms of Fourier integrals U,(%Y>
=1
U,(x,w)
TI
cos wy dw; fk(x,y) = ;.
0
r
Fk(x,w) cos wy dw
0
(6) Substitution of (6) into (1) - (4) leads to the following system of ordinary differential equations.
Vo1.3.,
No.4
HEAT CONDUCTION
d'Uk(x,w)
dhk
'k
dUk(x,w) dx
+dx
dx2
287
PROBLEMS
2 - w hkUk(x,w) = Fk(x,w)
(7)
= a
(8)
with the boundary conditions A
lw '1 x=a
= 0,
0
A&lx
= 0, n
(9) A set of linearly independent particular solutions of the homogeneous system corresponding to equations (7) can be obtained numerically by any suitable method (as, for instance, Euler or Runge-Kutta).
The Green's
matrix for the relations (7) - (9) n
g(x,5;w)
= (gij(x,P;wi
j
(10)
=1
,
may be obtained from this set of linearly independent particular solutions by employing a known procedure as, for example, that described and used in references Cl], [7]. Green's matrix is also n. be expressed as U(x,w)
a =
On this basis the solution of (7) - (9) may
n
s a
For n layers in the strip the order of the
g(x,S;w)F(5,w)d5
0
where U(x,w) and F(
cos wq dv
0
and substituting the resulting expression into the first of (6) and then reversing the order of integrations yields
g(x,S;w) oa
0
cos
WY
cos Nl dw1 f(l,‘Tl> 3
d’F
(11)
0
where u(x,y) and f(s,T) are vectors with components u,(x,y> and f,({,v), respectively. Assuming that the solutions of (1) = (4) is unique, the kernel of the double integral in (11) is the Green's matrix G(x,y;<,Ij) of the given problem.
Thus
288
YU.A.
G(x,y;S,V=
i
g(x,S;w)
j
.cos
M?XNIKOV
wy cos
WV
Vo1.3,
dw
(1.2)
0 Algorithm of Computation and Numerical Examples. Consider for example, the case in which the operator B is unity, meaning that the temperature is specified on the contour L.
Represent the
solution of (1) - (5) by the following sum [7] u(x,y) = v(x,y) + W(X,Y) Here
a .CO n v(x,y) = j G(x,y;S,Vf(S,'V 0 I
(13)
(14)
dS dTj
0
and W(X,Y)
= s G(x,y;5,Jl)h(l,VdL(5,'Il) L
(15)
The kernel G(x,y;s,v) of the representations above is the Green's matrix described in the previous section.
Function h({,T) is the unknown density
of the potential (15), which can be determined from the requirement that (13) satisfies the boundary conditions (5). Q(x,y) - %x,y)
=
On this basis we obtain (16)
s G(x,y;S,Vh(l,Ij)dL(%,V
which is a Fredholm integral equation for h({,T(). The term y(x,y) is equal to the value of the integral (14) on the line L.
After the integral
equation is solved for the potential density h(l,l), the solution (5) is immediately obtained from (13).
of (l)-
The described algorithm was ap-
plied to a two-layered strip with a tunnel-shaped hole for the following data: f(x,y)
q
0, Q(x,y) = 1, B = Al = 1 and A2 s a/a, + 1.
Fig. 2a gives
the variation of the thermal conductivities of the given materials transversely to the layers.
The previously mentioned set of the linearly independent particular solutions of the homogeneous system (7) can in this case be obtained in closed form because of the exponential representations of the Ak coefficients.
The integral equation (16) was solved by expressing the
integrals as finite sums by means of the trapezoidal rule.
The loca-
tions and spacing of the grid points must be generally determined by taking into account the size and shape of the hole L. we used a set of 24 uniformly spaced grid points.
In this example
The improper in-
tegral in (12) converges rather fast, and it could be calculated with
No.4
Vo1.3, No.4
t
HEAT CONDUCTION PROBLEMS
a
X
289
b FIG. 2
sufficient accuracy without any difficulties.
The resulting temperature
field for the case considered is shown in Fig. 2b. The three-dimensional case of a homogeneous layer with the tunnel S of an elliptical cross-section is shown in Fig. 3.
The boundary conditions
here are specified as
UI
,‘& x=0
+u’\,/x=1=o,
iax
III
S
=
cos32z
FIG. 3
290
YU.A. MELNIKOV
Vo1.3;No.4
Instead of the artificial boundary conditions on the surface S one could specify any condition that is more realistic from a physical point of view.
The artificial condition was selected here only for the purpose
of making the example simpler.
The three-dimensional problem considered
was reduced to a two-dimensional problem by means of Fourier series in the z direction, and subsequently the algorithm described above was employed.
From the standpoint of calculations such an approach to solve
three-dimensional problems does not require much additional work except for computer programming.
Acknowledgments
The author wishes to thank Professor B. A. Boley, Dean of the Technological Institute of Northwestern University for his critical coxunents and kind help.
It is also a pleasure to acknowledge the
help of Professor J. Dundurs of Northwestern University, who edited the manuscript, and the assistance of I. M. Dolgova, graduate student of Dniepropetrovsk State University, for developing the computer program.
Referneces 1.
2.
3.
4.
5.
6.
7.
S.P. Gavelya, "On a Method of Constructing Green's Matrices of Joined Shells," (in Ukrainian), Dep. An URSR, Ser. A, No. 12, 1969. S.P. Gavelya, Yu. A. Melnikov, I.A. Davidov, "Solution of Some Boundary Value Pr'oblems in Shell Theory," (in Russian), Dniepropetrovsk State University, USSR, 1971. S.P. Gavelya, Yu.A. Melnikov, "Calculating the Thermoelastic Equilibrium of a Toroidal Shell," (Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 6, pp. 191-197, NovemberDecember, 1971). S.P. Gavelya, Yu.A. Melnikov, "On Local Heating of a Toroidal Shell," (in Russian), Proceedings of VIII National Conference on Theory Plates and Shells, Moscow, 1973. Yu.A. Melnikov, I.M. Dolgova, " On the Elastic Equilibrium of a Nonuniformly Heated Strip with Arbitrary Holes," (in Russian), Abstracts of XIII Conference on Thermal Stresses, Kiev, USSR, 1974. B.G. Azanov, I.M. Dolgova, Yu.A. Melnikov, "Construction of Effective Algorithms for Boundary Value Problems in Thermoelasticity for Domains of a Complicated Shape," (in Russian), Mekhanika Tverdogo Tela, No. 5, 1975. Yu.A. Melnikov, I.M. Dolgova, "On Some Effective Algorithms for Solving Steady-State Heat Conduction Problems for Bodies of Complicated Shapes," (in Russian), Journal of Engineering Physics, Vol. 30, No. 1, 1976.
RES.
COMM.
Vo1.3, 285-290, 1976.
Pergamop Press.
Printed in USA.
POTENTIAL METHOD APPLIED TO STEADY-STATE HEAT CONDUCTION PROBLEMS Yu.A. Melnikov" Department of Mechanics and Mathematics, Dniepropetrovsk State University, Dniepropetrovsk, USSR (Received and accepted as ready for print 31 March 1976)
Introduction
A variant of the Green's function method is used here to solve steadystate heat conduction problems in multiply-connected bodies involving complicated geometries. An algorithm for constructing the Green's matrix of a layered strip is described and numerical results for two-and three dimensional particular cases are presented. This approach is general for systems of elliptic partial differential equations in mechanics and it has been employed in the past (see, for example, [l]-[7]) to obtain numerical solutions of various mechanical problems for bodies with intricate shapes. The solution of individual problems can be obtained by employing some numerical technique, for instance, the finite element method. This particular method is suitable for creating a general program (as, for example, NASTRAN) for solving a large class of such problems. However, many prob; lems can be solved even more effectively by means of potential theoretic approaches, such as the Green's function method which will be discussed in the sequel. The particular problems considered here will be used merely as examples to demonstrate the application of the method.
Statement of the Problem
Consider a two-dimensional, n-layered strip with an arbitrary hole L, and assume that the thermal conductivities Ak(k= 1,2,...,n) in each of the n layers are known continuous functions of the coordinate x (see Fig. l). The temperature u in the strip will satisfy the following relations: q
ax’ A,(x) -h(
au $) + hkW -a”%=
ay2
fk(x,y)
(k = 1,2 ,...,
4
*Presently visiting Professor of the Technological Institute, Northwestern University, Evanston, Illinois. Scientific Communication
285
(1)
286
Vo1.3, 'No.4
YU.A. MELNIKOV
a
0
FIG. 1
Uklx=a
= uk+lIx=ak,
lk
2 I x=ak
k
(3)
= 'k+l
(4) (5)
= Q(x,y)
BUklL
= u,(x,y) is the unknown temperature of the k-th layer, Al, A2 k and B are linear differential operators that depend on the boundary con-
where u
ditions, and Q(x,y) and fk(x,y) are given continuous functions. In the next section the Green's matrix for the relations (l)-(4) will be constructed in order to obtain the solution of the problems (1) - (5) by means of an integral representation in which the Green's matrix is the kernel. Construction of the Green's Matrix It will be assumed that problem.. (1) - (5) has
symmetry about the x-axis
and that the functions u,(x,y) and fk(x,y) can be represented in terms of Fourier integrals U,(%Y>
=1
U,(x,w)
TI
cos wy dw; fk(x,y) = ;.
0
r
Fk(x,w) cos wy dw
0
(6) Substitution of (6) into (1) - (4) leads to the following system of ordinary differential equations.
Vo1.3.,
No.4
HEAT CONDUCTION
d'Uk(x,w)
dhk
'k
dUk(x,w) dx
+dx
dx2
287
PROBLEMS
2 - w hkUk(x,w) = Fk(x,w)
(7)
= a
(8)
with the boundary conditions A
lw '1 x=a
= 0,
0
A&lx
= 0, n
(9) A set of linearly independent particular solutions of the homogeneous system corresponding to equations (7) can be obtained numerically by any suitable method (as, for instance, Euler or Runge-Kutta).
The Green's
matrix for the relations (7) - (9) n
g(x,5;w)
= (gij(x,P;wi
j
(10)
=1
,
may be obtained from this set of linearly independent particular solutions by employing a known procedure as, for example, that described and used in references Cl], [7]. Green's matrix is also n. be expressed as U(x,w)
a =
On this basis the solution of (7) - (9) may
n
s a
For n layers in the strip the order of the
g(x,S;w)F(5,w)d5
0
where U(x,w) and F(
cos wq dv
0
and substituting the resulting expression into the first of (6) and then reversing the order of integrations yields
g(x,S;w) oa
0
cos
WY
cos Nl dw1 f(l,‘Tl> 3
d’F
(11)
0
where u(x,y) and f(s,T) are vectors with components u,(x,y> and f,({,v), respectively. Assuming that the solutions of (1) = (4) is unique, the kernel of the double integral in (11) is the Green's matrix G(x,y;<,Ij) of the given problem.
Thus
288
YU.A.
G(x,y;S,V=
i
g(x,S;w)
j
.cos
M?XNIKOV
wy cos
WV
Vo1.3,
dw
(1.2)
0 Algorithm of Computation and Numerical Examples. Consider for example, the case in which the operator B is unity, meaning that the temperature is specified on the contour L.
Represent the
solution of (1) - (5) by the following sum [7] u(x,y) = v(x,y) + W(X,Y) Here
a .CO n v(x,y) = j G(x,y;S,Vf(S,'V 0 I
(13)
(14)
dS dTj
0
and W(X,Y)
= s G(x,y;5,Jl)h(l,VdL(5,'Il) L
(15)
The kernel G(x,y;s,v) of the representations above is the Green's matrix described in the previous section.
Function h({,T) is the unknown density
of the potential (15), which can be determined from the requirement that (13) satisfies the boundary conditions (5). Q(x,y) - %x,y)
=
On this basis we obtain (16)
s G(x,y;S,Vh(l,Ij)dL(%,V
which is a Fredholm integral equation for h({,T(). The term y(x,y) is equal to the value of the integral (14) on the line L.
After the integral
equation is solved for the potential density h(l,l), the solution (5) is immediately obtained from (13).
of (l)-
The described algorithm was ap-
plied to a two-layered strip with a tunnel-shaped hole for the following data: f(x,y)
q
0, Q(x,y) = 1, B = Al = 1 and A2 s a/a, + 1.
Fig. 2a gives
the variation of the thermal conductivities of the given materials transversely to the layers.
The previously mentioned set of the linearly independent particular solutions of the homogeneous system (7) can in this case be obtained in closed form because of the exponential representations of the Ak coefficients.
The integral equation (16) was solved by expressing the
integrals as finite sums by means of the trapezoidal rule.
The loca-
tions and spacing of the grid points must be generally determined by taking into account the size and shape of the hole L. we used a set of 24 uniformly spaced grid points.
In this example
The improper in-
tegral in (12) converges rather fast, and it could be calculated with
No.4
Vo1.3, No.4
t
HEAT CONDUCTION PROBLEMS
a
X
289
b FIG. 2
sufficient accuracy without any difficulties.
The resulting temperature
field for the case considered is shown in Fig. 2b. The three-dimensional case of a homogeneous layer with the tunnel S of an elliptical cross-section is shown in Fig. 3.
The boundary conditions
here are specified as
UI
,‘& x=0
+u’\,/x=1=o,
iax
III
S
=
cos32z
FIG. 3
290
YU.A. MELNIKOV
Vo1.3;No.4
Instead of the artificial boundary conditions on the surface S one could specify any condition that is more realistic from a physical point of view.
The artificial condition was selected here only for the purpose
of making the example simpler.
The three-dimensional problem considered
was reduced to a two-dimensional problem by means of Fourier series in the z direction, and subsequently the algorithm described above was employed.
From the standpoint of calculations such an approach to solve
three-dimensional problems does not require much additional work except for computer programming.
Acknowledgments
The author wishes to thank Professor B. A. Boley, Dean of the Technological Institute of Northwestern University for his critical coxunents and kind help.
It is also a pleasure to acknowledge the
help of Professor J. Dundurs of Northwestern University, who edited the manuscript, and the assistance of I. M. Dolgova, graduate student of Dniepropetrovsk State University, for developing the computer program.
Referneces 1.
2.
3.
4.
5.
6.
7.
S.P. Gavelya, "On a Method of Constructing Green's Matrices of Joined Shells," (in Ukrainian), Dep. An URSR, Ser. A, No. 12, 1969. S.P. Gavelya, Yu. A. Melnikov, I.A. Davidov, "Solution of Some Boundary Value Pr'oblems in Shell Theory," (in Russian), Dniepropetrovsk State University, USSR, 1971. S.P. Gavelya, Yu.A. Melnikov, "Calculating the Thermoelastic Equilibrium of a Toroidal Shell," (Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 6, pp. 191-197, NovemberDecember, 1971). S.P. Gavelya, Yu.A. Melnikov, "On Local Heating of a Toroidal Shell," (in Russian), Proceedings of VIII National Conference on Theory Plates and Shells, Moscow, 1973. Yu.A. Melnikov, I.M. Dolgova, " On the Elastic Equilibrium of a Nonuniformly Heated Strip with Arbitrary Holes," (in Russian), Abstracts of XIII Conference on Thermal Stresses, Kiev, USSR, 1974. B.G. Azanov, I.M. Dolgova, Yu.A. Melnikov, "Construction of Effective Algorithms for Boundary Value Problems in Thermoelasticity for Domains of a Complicated Shape," (in Russian), Mekhanika Tverdogo Tela, No. 5, 1975. Yu.A. Melnikov, I.M. Dolgova, "On Some Effective Algorithms for Solving Steady-State Heat Conduction Problems for Bodies of Complicated Shapes," (in Russian), Journal of Engineering Physics, Vol. 30, No. 1, 1976.