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Two-dimensional heat transfer problem using the boundary integral equation
Two-dimensional Heat Transfer Problem Using the Boundary Integral Equation by
YOUNG
Department PA 19085,
W.
CHUN
of Mechanical
Engineering,
Villanova
University,
Villanova,
U.S.A.
ABSTRACT : Mathematical properties of the variational solution and solution of the boundary integral equation of a two-dimensional heat transfer problem are studied. It is first reviewed that a boundary integral expression is valid for the classical solution, and then it is shown that a unique solution of the boundary integral equation is identical to the vuriational solution in Sobolev space H’(Q) even when the classical solution does not exist. To represent the boundary integral equation ,for the two-dimensional problem, Green’s ,formula in Sobolev space is utilized on the solution domain excluding a circle with a small radius p centered at the singular point. By letting p tend to zero it is shown that for the heat transfer problem, a boundary integral expression is valid for the variational solution. From this fact, one can obtain a numerical approximation of the variational solution by the boundary element method even when the classical solution does not exist.
I. Zntvoduction The boundary element method (BEM) has recently become a popular and powerful numerical method, next to the finite element method (FEM). One advantage of the method is that the equations apply only on the boundary of the solution domain. An immediate consequence of this is that the order of the system algebraic equations generated by the method is considerably smaller than those generated by FEM or the finite difference method (FDM). Another advantage is that, on the boundary of the domain, the solution obtained by the method is more accurate than from either the FEM or FDM (1,2), so it is preferable to use the BEM when one needs a solution only on the boundary of the domain. When using the method, one approximates the solution of the boundary integral equation. However, it has not been well investigated whether a unique solution of the boundary integral equation exists and, if it does exist, what its mathematical properties are. In this paper, it is first reviewed that a boundary integral expression for the twodimensional heat transfer problem is valid for the classical solution, then it is shown that a unique solution of the boundary integral equation is identical to the variational solution in the Sobolev space of order one, H’(R), even when the classical solution does not exist. For this, Green’s formula in the space can be utilized. However, a singularity of the fundamental solution prevents one from applying the formula. In this paper, this difficulty is removed by replacing the
The Franklin Institute001&0032,‘92 $5 OOf0.00
1147
Y. W. Chun integral domain Q by R-B, and letting p tend to zero, where B,, is a circle with sufficiently small radius p centered at the singular point.
II. Statement
of the Steady-state
Heat Transfer Problem
Consider a homogeneous isotropic heat conducting body R which is a simply connected and bounded domain in R’ with a Lipschitz boundary f = r, u Tz u r3 u r4, where I-,, Tz, rj and r4 are disjoint parts of r. It is supposed that convection into the ambient medium occurs on boundaries r, and Tz, temperature is kept constant at the prescribed value of T3 on r3, and r4 is insulated. The state equation for the system is described by the mixed boundary value problem as V’O = 0
o=o
in
on
Q
(1)
r3
(3) (4)
where 8 = T- T,, 8, = T, - Tir T is the temperature in the domain, T, is the ambient temperature, n is the outerward unit normal vector, h is the convection coefficient and k is the conduction coefficient. Since the classical solution 8 E C’(a) n C2(sZ) to this problem does not exist (3), where !Z is the closure of the domain fl, one is concerned with the variational solution in H’(Q).
III. Boundavy Integral Expression for the Classical Solution For the heat transfer problem, the boundary integral equation is formulated on the basis of Green’s formula with a fundamental solution (4). The fundamental solution for the Laplace operator is given by the following theorem (57). Theorem I Let EE D’(R’)
such that V’E=
is called the fundamental problem, it is given by
solution
-8(x--);
s,t~R’
(5)
of V* at x = <. Then for the two-dimensional
E=
-Clogr,
(6)
where D’(R’) is the distribution space in R*, 6 is the Dirac Delta function, arbitrary constant, and r is the distance from t to x. Let u and z: be C’(Q) n C’(0) functions. Then Green’s formula 1148
C is an
Journal of the Frankhn lnstitutc Pcrgamon Press Lid
Boundary Integral Equation
s
J-&g +g)ds
(vV%-uv22r)dX =
R
holds (7) where ds is a boundary element. If the classical solution 0 E C’(n) n C’(Q) exists, one can substitute u by 0 in Eq. (7). However, the singularity of E prevents one from substituting z)by E in the equation. One way of overcoming this difficulty is to replace Sz by a-- B,,(c), where B,>(t) is a circle with small radius p centered at the singular point 5. One can then conclude from Eq. (7) that
s
(EV20-6V’E)dx
= I
(Eg
-e!$dJ
Q-B,,
+iB,, (Eg where aB, is the boundary on Q-B,, one has
-Og)ds,
foreEC’(li)
n C2(Q),
of B,. In Eq. (8), since V2B = 0 and V2E = -6(x
EV28dx
= 0
8V2Edx
= 0.
(8)
- <) = 0
s Q-B,, and
s
(10)
Q-8,)
The first term in the integration
s s
E;;ds
= E(p)
Since
IG-’
iiB ;; s
3B,>
over aB, in Eq. (8) becomes
(11)
i’
[e(x) - e(r)] ds I d Cp ’ s ?Bp
“B”
Id(x)-e(t)1
ds
d Cw2 max l&x> -e(t) XdB,,
the second term in the integration
s
BEds
?B,,
an
= Cp-’
(12)
over dB, in Eq. (8) becomes
e(x) ds
s c2B,,
I +O asp + 0,
+ c,-
’
e(t) ds = c0,eg), sPB,,
asp --,
0, (13)
where o2 = 21r is the boundary length of the unit circle in R2 and co2p is the boundary length of the circle with radius p. If one chooses C = l/m2 and substitutes Eqs (9), (IO), (11) and (13) into Eq. (8) then Vol. 329, No. 6, pp. 1147-l 152, 1992 Pnnted m Great Britain
1149
Y. W. Chun
holds as p goes to zero. If 5 is on I, Eq. (14) has singularity. Then one can divide the boundary I by I, and I- I,:, where I, is a half circle with small radius E centered at the singular point 4. Then Eq. (14) becomes
(15) The first term of the boundary
over I,: in Eq. (15) becomes
integration
ae ,ds s r, on
d
E--t0
(16)
while the second term becomes -j-
r,
BEds dn
= Cc-’
Q(x) ds.
(17)
s r,
Since ICE-’
{e(x)-U([)}dsl
< Cc~-’
s r,
s r,
I‘J(x) - Q(t) Id.s
d ICw2~~~x lQ(x)-H(5)I Eq. (17) becomes,
as
e-+0,
(18)
with C = l/cuz, 13(x) ds + Ce ’
C&K’ s r, Substituting
+O
O(c) ds = ;Co,o([)
= {8(5).
s r,
(19)
Eqs (16) and (19) into Eq. (15) and letting e go to zero, one obtains
When one uses the BEM for the problem with 5 E I, 0(r) is obtained numerically from Eq. (20), while it is obtained from Eq. (14) when 5 EQ. From the above results, one can see that the classical solution, if it exists, can be approximated numerically from the boundary integral equations (14) and (20) by dividing the boundary into small segments. However, since the classical solution does not exist in the mixed boundary value problem described in Section II, one cannot use Eqs (14) and (20) directly. IV. Boundary Integral Expression for the Weak Solution The variational
form of the state equation
of Eqs (l)-(4)
is given as
(21) 11.50
Journal
of the Franklin lnsutute Pergamon Press Lfd
Boundary Integral Equation where K is the admissible set given by K = {vlv E N’(R), u = 0 on I,}. It is known that the weak solution of Eq. (21) is unique in H’(R) by the Lax-Milgram theorem (7,s). To represent the boundary integral equation for the variational weak solution 0 E H’(R), one needs the following theorem (9). Theorem II Green’s formula
in Sobolev
space :
s
(DV2U--V2U)dx
= jr (z$
-u;)di:
(22)
n
holds for the domain R with Lipschitz boundary I if U, VE H’(Q V2), where H’(Q, V’) = {UIUE H’(0) such that V2u E L2(R)}. The variational solution 8 is in H’(Q, V’), but the fundamental solution is not. In fact, it is in Cm(R2- (4)) (10). Then u in Eq. (22) can be substituted by 0 but v cannot be by E. This difficulty is removed by replacing R by Q-B,, since E is in H’(Q, VI) in R-B,. Then one can conclude from Eq. (22) that
s
(EV28-0V’E)dx
= jr (Eg
-t$)ds
R-B,
+ j.,
(EE
-0g)ds.
fordEH’(fi,V’).
(23)
By a similar way to that used in Section III, Eq. (23) becomes Q(5) = jr(Eii as p goes to zero. Likewise,
If one inserts the boundary respectively, one can get
-B~~ds,forall~~R,0~H1(R.V2).
(24)
if 5 is on I, Eq (24) becomes
conditions
of Eqs (2))(4)
into Eqs (24) and (25),
From the above results it is evident that if one finds the solution of Eqs (26) and (27) in H’(Q), it is identical to the variational solution since it is unique in H’(Q). For the solution of the problem, Eqs (26) and (27) can be approximated numerically by dividing the boundary into small segments. Vol 329, No. 6. pp. ,147-l Printed in Great Britain
152, 1992
1151
Y. W. Chun V. Discussion and Conclusions In this paper it is shown that for the problem governed by the I,aplace operator, a unique solution of the boundary integral equation exists and is identical to the variational solution in H’(Q), which can be served as the basic mathematical theory of the boundary element method for the problem. Therefore, it is possible to get a numerical approximation of the variational solution by the boundary element method even when the classical solution does not exist. The procedure in this paper can also be applied to the three-dimensional heat transfer problem, and it is expected that a similar procedure can be applied to other elliptic boundary value problems.
References (1) C. A. Mota Soares and K. K. Choi, “Boundary element in shape optimal design of structures”, Advanced Study Institute on Computer Aided Optimal Design ; Structural and Mechanical Systems, Troia, Portugal, 1986. (2) C. S. Lee and Y. M. Yoo, “Investigation of the boundary element method for engineering application”, Boundary Element VII : Proc. 7th Int. Conf. (Edited by C. A. Brebbia), 1985. (3) W. L. Wendland, E. Stephan, Darmstadt and G. C. Hsiao, “On the integral equation method for the plane mixed boundary value problem of the Laplacian”, Math. Met/z. Appl.. Vol. 1, pp. 2655321, 1979. Element Method for Engineers”, Pentech Press, (4) C. A. Brebbia, “The Boundary London, 1980. (5) F. John, “Partial Differential Equations”, 3rd Edn, Springer, New York, 1978. (6) J. Barros-Neto, “An Introduction to the Theory of Distributions”, Marcel Dekker, New York, 1973. (7) D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer, Berlin, 1977. (8) K. Rektorys, “Variational Methods in Mathematics, Science and Engineering”, D. Reidel, Boston, MA, 1980. (9) J.-P. Aubin, “Applied Functional Analysis”, Wiley, New York, 1979. (10) L. Hormander, “The Analysis of Linear Partial Differential Operators II”, Springer, Berlin, 1983.
Jaurnal
1152
of the Frankhn lnstitutc Pergamon Press Ltd
YOUNG
Department PA 19085,
W.
CHUN
of Mechanical
Engineering,
Villanova
University,
Villanova,
U.S.A.
ABSTRACT : Mathematical properties of the variational solution and solution of the boundary integral equation of a two-dimensional heat transfer problem are studied. It is first reviewed that a boundary integral expression is valid for the classical solution, and then it is shown that a unique solution of the boundary integral equation is identical to the vuriational solution in Sobolev space H’(Q) even when the classical solution does not exist. To represent the boundary integral equation ,for the two-dimensional problem, Green’s ,formula in Sobolev space is utilized on the solution domain excluding a circle with a small radius p centered at the singular point. By letting p tend to zero it is shown that for the heat transfer problem, a boundary integral expression is valid for the variational solution. From this fact, one can obtain a numerical approximation of the variational solution by the boundary element method even when the classical solution does not exist.
I. Zntvoduction The boundary element method (BEM) has recently become a popular and powerful numerical method, next to the finite element method (FEM). One advantage of the method is that the equations apply only on the boundary of the solution domain. An immediate consequence of this is that the order of the system algebraic equations generated by the method is considerably smaller than those generated by FEM or the finite difference method (FDM). Another advantage is that, on the boundary of the domain, the solution obtained by the method is more accurate than from either the FEM or FDM (1,2), so it is preferable to use the BEM when one needs a solution only on the boundary of the domain. When using the method, one approximates the solution of the boundary integral equation. However, it has not been well investigated whether a unique solution of the boundary integral equation exists and, if it does exist, what its mathematical properties are. In this paper, it is first reviewed that a boundary integral expression for the twodimensional heat transfer problem is valid for the classical solution, then it is shown that a unique solution of the boundary integral equation is identical to the variational solution in the Sobolev space of order one, H’(R), even when the classical solution does not exist. For this, Green’s formula in the space can be utilized. However, a singularity of the fundamental solution prevents one from applying the formula. In this paper, this difficulty is removed by replacing the
The Franklin Institute001&0032,‘92 $5 OOf0.00
1147
Y. W. Chun integral domain Q by R-B, and letting p tend to zero, where B,, is a circle with sufficiently small radius p centered at the singular point.
II. Statement
of the Steady-state
Heat Transfer Problem
Consider a homogeneous isotropic heat conducting body R which is a simply connected and bounded domain in R’ with a Lipschitz boundary f = r, u Tz u r3 u r4, where I-,, Tz, rj and r4 are disjoint parts of r. It is supposed that convection into the ambient medium occurs on boundaries r, and Tz, temperature is kept constant at the prescribed value of T3 on r3, and r4 is insulated. The state equation for the system is described by the mixed boundary value problem as V’O = 0
o=o
in
on
Q
(1)
r3
(3) (4)
where 8 = T- T,, 8, = T, - Tir T is the temperature in the domain, T, is the ambient temperature, n is the outerward unit normal vector, h is the convection coefficient and k is the conduction coefficient. Since the classical solution 8 E C’(a) n C2(sZ) to this problem does not exist (3), where !Z is the closure of the domain fl, one is concerned with the variational solution in H’(Q).
III. Boundavy Integral Expression for the Classical Solution For the heat transfer problem, the boundary integral equation is formulated on the basis of Green’s formula with a fundamental solution (4). The fundamental solution for the Laplace operator is given by the following theorem (57). Theorem I Let EE D’(R’)
such that V’E=
is called the fundamental problem, it is given by
solution
-8(x--);
s,t~R’
(5)
of V* at x = <. Then for the two-dimensional
E=
-Clogr,
(6)
where D’(R’) is the distribution space in R*, 6 is the Dirac Delta function, arbitrary constant, and r is the distance from t to x. Let u and z: be C’(Q) n C’(0) functions. Then Green’s formula 1148
C is an
Journal of the Frankhn lnstitutc Pcrgamon Press Lid
Boundary Integral Equation
s
J-&g +g)ds
(vV%-uv22r)dX =
R
holds (7) where ds is a boundary element. If the classical solution 0 E C’(n) n C’(Q) exists, one can substitute u by 0 in Eq. (7). However, the singularity of E prevents one from substituting z)by E in the equation. One way of overcoming this difficulty is to replace Sz by a-- B,,(c), where B,>(t) is a circle with small radius p centered at the singular point 5. One can then conclude from Eq. (7) that
s
(EV20-6V’E)dx
= I
(Eg
-e!$dJ
Q-B,,
+iB,, (Eg where aB, is the boundary on Q-B,, one has
-Og)ds,
foreEC’(li)
n C2(Q),
of B,. In Eq. (8), since V2B = 0 and V2E = -6(x
EV28dx
= 0
8V2Edx
= 0.
(8)
- <) = 0
s Q-B,, and
s
(10)
Q-8,)
The first term in the integration
s s
E;;ds
= E(p)
Since
IG-’
iiB ;; s
3B,>
over aB, in Eq. (8) becomes
(11)
i’
[e(x) - e(r)] ds I d Cp ’ s ?Bp
“B”
Id(x)-e(t)1
ds
d Cw2 max l&x> -e(t) XdB,,
the second term in the integration
s
BEds
?B,,
an
= Cp-’
(12)
over dB, in Eq. (8) becomes
e(x) ds
s c2B,,
I +O asp + 0,
+ c,-
’
e(t) ds = c0,eg), sPB,,
asp --,
0, (13)
where o2 = 21r is the boundary length of the unit circle in R2 and co2p is the boundary length of the circle with radius p. If one chooses C = l/m2 and substitutes Eqs (9), (IO), (11) and (13) into Eq. (8) then Vol. 329, No. 6, pp. 1147-l 152, 1992 Pnnted m Great Britain
1149
Y. W. Chun
holds as p goes to zero. If 5 is on I, Eq. (14) has singularity. Then one can divide the boundary I by I, and I- I,:, where I, is a half circle with small radius E centered at the singular point 4. Then Eq. (14) becomes
(15) The first term of the boundary
over I,: in Eq. (15) becomes
integration
ae ,ds s r, on
d
E--t0
(16)
while the second term becomes -j-
r,
BEds dn
= Cc-’
Q(x) ds.
(17)
s r,
Since ICE-’
{e(x)-U([)}dsl
< Cc~-’
s r,
s r,
I‘J(x) - Q(t) Id.s
d ICw2~~~x lQ(x)-H(5)I Eq. (17) becomes,
as
e-+0,
(18)
with C = l/cuz, 13(x) ds + Ce ’
C&K’ s r, Substituting
+O
O(c) ds = ;Co,o([)
= {8(5).
s r,
(19)
Eqs (16) and (19) into Eq. (15) and letting e go to zero, one obtains
When one uses the BEM for the problem with 5 E I, 0(r) is obtained numerically from Eq. (20), while it is obtained from Eq. (14) when 5 EQ. From the above results, one can see that the classical solution, if it exists, can be approximated numerically from the boundary integral equations (14) and (20) by dividing the boundary into small segments. However, since the classical solution does not exist in the mixed boundary value problem described in Section II, one cannot use Eqs (14) and (20) directly. IV. Boundary Integral Expression for the Weak Solution The variational
form of the state equation
of Eqs (l)-(4)
is given as
(21) 11.50
Journal
of the Franklin lnsutute Pergamon Press Lfd
Boundary Integral Equation where K is the admissible set given by K = {vlv E N’(R), u = 0 on I,}. It is known that the weak solution of Eq. (21) is unique in H’(R) by the Lax-Milgram theorem (7,s). To represent the boundary integral equation for the variational weak solution 0 E H’(R), one needs the following theorem (9). Theorem II Green’s formula
in Sobolev
space :
s
(DV2U--V2U)dx
= jr (z$
-u;)di:
(22)
n
holds for the domain R with Lipschitz boundary I if U, VE H’(Q V2), where H’(Q, V’) = {UIUE H’(0) such that V2u E L2(R)}. The variational solution 8 is in H’(Q, V’), but the fundamental solution is not. In fact, it is in Cm(R2- (4)) (10). Then u in Eq. (22) can be substituted by 0 but v cannot be by E. This difficulty is removed by replacing R by Q-B,, since E is in H’(Q, VI) in R-B,. Then one can conclude from Eq. (22) that
s
(EV28-0V’E)dx
= jr (Eg
-t$)ds
R-B,
+ j.,
(EE
-0g)ds.
fordEH’(fi,V’).
(23)
By a similar way to that used in Section III, Eq. (23) becomes Q(5) = jr(Eii as p goes to zero. Likewise,
If one inserts the boundary respectively, one can get
-B~~ds,forall~~R,0~H1(R.V2).
(24)
if 5 is on I, Eq (24) becomes
conditions
of Eqs (2))(4)
into Eqs (24) and (25),
From the above results it is evident that if one finds the solution of Eqs (26) and (27) in H’(Q), it is identical to the variational solution since it is unique in H’(Q). For the solution of the problem, Eqs (26) and (27) can be approximated numerically by dividing the boundary into small segments. Vol 329, No. 6. pp. ,147-l Printed in Great Britain
152, 1992
1151
Y. W. Chun V. Discussion and Conclusions In this paper it is shown that for the problem governed by the I,aplace operator, a unique solution of the boundary integral equation exists and is identical to the variational solution in H’(Q), which can be served as the basic mathematical theory of the boundary element method for the problem. Therefore, it is possible to get a numerical approximation of the variational solution by the boundary element method even when the classical solution does not exist. The procedure in this paper can also be applied to the three-dimensional heat transfer problem, and it is expected that a similar procedure can be applied to other elliptic boundary value problems.
References (1) C. A. Mota Soares and K. K. Choi, “Boundary element in shape optimal design of structures”, Advanced Study Institute on Computer Aided Optimal Design ; Structural and Mechanical Systems, Troia, Portugal, 1986. (2) C. S. Lee and Y. M. Yoo, “Investigation of the boundary element method for engineering application”, Boundary Element VII : Proc. 7th Int. Conf. (Edited by C. A. Brebbia), 1985. (3) W. L. Wendland, E. Stephan, Darmstadt and G. C. Hsiao, “On the integral equation method for the plane mixed boundary value problem of the Laplacian”, Math. Met/z. Appl.. Vol. 1, pp. 2655321, 1979. Element Method for Engineers”, Pentech Press, (4) C. A. Brebbia, “The Boundary London, 1980. (5) F. John, “Partial Differential Equations”, 3rd Edn, Springer, New York, 1978. (6) J. Barros-Neto, “An Introduction to the Theory of Distributions”, Marcel Dekker, New York, 1973. (7) D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer, Berlin, 1977. (8) K. Rektorys, “Variational Methods in Mathematics, Science and Engineering”, D. Reidel, Boston, MA, 1980. (9) J.-P. Aubin, “Applied Functional Analysis”, Wiley, New York, 1979. (10) L. Hormander, “The Analysis of Linear Partial Differential Operators II”, Springer, Berlin, 1983.
Jaurnal
1152
of the Frankhn lnstitutc Pergamon Press Ltd