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Approximation Methods for the Solution of Heat Conduction Problems Using Gyarmati's Principle.

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Bnnalen der Physik. 7. Folge, Band 31, Heft 1, 1974, S. 63-76
J. A. Berth, Leipzig
Approximation Methods for the Solution
of Heat Conduction Problems Using OYARMATI’S
Principle
By A. STARK
Abstract
After a brief description of GYARMATI’SGoverning Principle of Dissipative Processes,
general approximation methods &redeveloped for heat conduction phenomena on two scales.
On the first scale the temperature and heat current density fields are approximated by
two different sets of functions in such a way that the internal energy balance and the
imposed boundary conditions be satisfied. On the second scale we introduce a new temperatnre field related to the heat current density through the constitutive equation. Since the
principle are connected with each other by LIDOENDRE
dissipation potentials in GYARMATI’S
dual transformations we call ‘‘dual field methods” the approximation prpcedure based on
the two temperature fields. It is shown that the equations of the GALERKINmethod, the
method, when applied t o the appropriate
method of orthogonal projections, and the TREFJFPZ
problems, are included in the approximation schemes. Due t p their great importance a
special paragraph is devoted to the treatment of eigenvalue problems. Finally. the approximation methods are illustrated by a simple example and the results &recompared with the
exact solution.
1. On GYARMATI’B.PriIlCiplf3
I n 1965 a genuine variational piinciple was proposed by GYARMATI[l, 21
for the theory of non-equilibrium thermodynamics [3-61, by means of which
the evolution of dissipative transport processes in space-time can be described.
Later i t was proved in several papers that the principle is applicable to the phenomena of macroscopic continuum physics involving transport of mass, charge,
momentum, moment of momentum, energy etc., - phenomena which are
described in particular cases, within the framework of linear approximation, by
the classical FOURIER,
RICK,
NAVIER-STOKES
etc. equations. (For a brief summary see [7].) The GAussian (or general) form of the principle is especially
important, due t o its validity in quasi-linaer and in certain types of non-linear
cases [8-141.
GYARMATI’S
“Governing Principle of Dissipative Processes” [briefly GPDP)
demands the maximum of the functional
L ( F ~.., .,r,,J ~ . ., . , J,)
=
f
V
~ I V= maximum
(1.1)
for any instant of time under the constraints that the following balance equations
be satisfied :
A. STAKK
64
The LAQRANQiandensity 3 is defined in linear and quasi-linear cases as
or equivalently in the GAnssian least squares form
g =- 2
I
f
’
2 R i b (J ! - 2r =Li , , V r , ) . ( J k - - L1’=1
k~Vr,).
i,k=l
(1.3‘)
I n the formulae above vectorial symbolism is used for implicity’s sake, and p
means the mass density of the continuum, aiis the substantial time derivative
of the ith specific transport quantity (specific masses and charges of different
chemical components, specific moment, moment of momentum, internal energy
etc.) in question, Jiis the substantial current density, and oi is the source of ai
per unit volume and unit time. I n (1.3), CY is the entropy production, while y
and @ are the local dissipation potentials. According to OMSAQER’S
theory, CI
is always a bilinear expression of the independent currents Ji and conjugated
thermodynamical forces Xi defined as the gradients of certain Tiparameters
[I, 7,8, 91 ; y and @ are homogeneous quadratic forms of V r i and Ji, respectively.
The coefficients L i k and Rik are the generalized conductivities and resistances in
the ONSAQERiansense, the matrices of which are reciprocal to one another, i.e.,
f
2 LimRmk
m-1
I
2 R i m L m k = dik
m=l
=
(i, k
=
1, * .
9
f)
(1.4)
(where do is the KRONECKER
symbol) and they obey the famous ONSAGERsymmetries :
and R i k = R k i
(i, k = 1, . . . , f ) .
(1.5)
The importance of the local dissipation potentials y and @ is that they include
the total ONSAQERconstitutive theory, i.e., the linear (or quasi-linear) constitutive laws
Lik
= Lki
together with the reciprocal relations (1.5). For convenience comments on terminology may be useful [9-141: if the ONSAGERcoefficients are state independent ( L i k = ‘ L : k , R i k = R$) we call the theory linear (in dynamical sense),
while if they are depending on the variables Ti a quasi-linear type of theory is
cited. Evidently, in these two different cases we have the following general
transport equations for homogeneous continua [91 :
respectively.
Returning to the extremum principle (1.1)we emphasize that the balance
equations (1.2) are a priori given for any thermodynamical theory, therefore
Solution of Heat Conduction Problems
55
they must be considered as constraints (subsidiary .conditions) t o GYARMATI’S
principle (1.1).Taking this into account and varying with respect t o Ti and Ji
simultaneously, the explicit form of (1.1)
f
+ 2’ (
v r i
-
i=l
+ ($&
Rik Jk)
k=l
kz
. dJ,)
dV
f
(Ji
I?
-
Lik v r k
(L9)
is obtained, where is the surface of volume V [9]. It is clear from this explicit
form of the GPDP that the quasi-linear transport equations (1.Qthe constitutive
equations (1.6), and the natural boundary conditions
(1.10)
(where n is the outward normal t o Q) are all included in it. The validity of the
GPDP for quasi-linear cases is due to the following theorem of GYARMATI:
+ @) = 0 (i = 1, ..., f )
ar,
(1.11)
i.e., the partial derivative of the sum of dissipation potentials with respect t o
any of the parameters Ti vanishes [8-141. Obvisouly (1.11)is a consequence
of the fact that the dissipation potentials y and CD are connected with each other
by L E G E N D R E dual transformations where the parameters ri play the role of
passive variables [7, 151. We wish to mention already here that the general
approximation methods, developed on the basis of GPDP in the following
sections, are closely related t o the L E G E N D R E transformation properties of y
and @.
It is important t o remark in connection with the varied form (1.9) of the
GPDP that two partial forms, viz. the “force representation”
and the “flux representation”
(1.13)
can be obtained as special cases from i t [7]. Referring once again to (l.ll),we
note t h a t this formula is valid for the real physical process only and therefore
in case of approximate I’, and Ji fields its contribution to the total variation (1.9)
and force representation (1.12) is different from zero.
A. STARK
56
2. General approximation methods for heat conduction phenomena
For simplicity's sake we shall treat here only heat conduction phenomena
in a rigid body [16]. We can write GYARMATI'S
principle in this actual case in
FOURIER
picture [7, 171, commonly used in older literature, as
s
s
v
(a** - y** - @**) dV
(2.1)
where the relations
a** = Tab;
v** = T % + J@**
;
TZ@
are valid. Here V denotes a bounded region in the 3-dimensional EucLIDian spaqe
with a smooth boundary Q, T is the temperature, 1 is the ordinary heat conduction coefficient. The heat current density J satisfies the following internal
energy balance equation
au
aT
(2.2)
or e c - + V . J = a , ,
e,,+V.J=a,,
=
at
where e is the mass density of the continuum, u is the specific internal energy,
c is the heat capacity, and a, is the source of internal energy per unit volume and
unit time. All functions occuring in the text are considered continuously differentiable as many times as needed. We shall accept two simple types of boundary
conditions :
and
TI, = 0
(2.3)
+
(J * n
H T ) In = f ( w )
(2.4)
where H is the surface conductivity [16], and f ( w ) is a given function of the surface coordinates w e Q. It is assumed that the variational problem formulated
by (2.1), (2.2),and the appropriate boundary condition (2.3) or (2.4) and, in case
of nonstationary problems, by the initial temperature field
q t = o = TO(4,
(2.5)
has a unique solution.
We seek for the sequence of approximate temperature fields {T(n)}(A =
1, 2, . . .) in the form:
where {pli}&l are members of a suitable complete set of linearly independent
functions, which in case of boundary condition (2.3) satisfy ailR= 0. The
coefficients ai are variational parameters to be determined. Substituting (2.6)
into the internal energy balance (2.2), we find that the approximate heat current
density J(") must obey the equation
Solution of Heat Conduotion Problems
67
Here &") = o(T(")),dn) = c(.Tcn))etc. Let J&z, t , al,...,a,,, ixl, ..., a,) be a
vector fulfilling (2.7) and also (2.4) if this boundary condition is imposed. Then
J(n)may be represented, in general, approximately as follows :
j=1
where (tpdEl is an appropriate complete system of linearly independent divergence-free vector functions V lpi = 0, j F 1, 2, . . . which in case of boundary
condition (2.4) satisfy y j . nln = 0. The coefficients
are variational parameters.
We pointed out in the previous section that thei volume integral (1.2) is
maximum a t any instant of time for the real physical process, i.e., for the exact
values of the parameters ri and current densities Ji. It is fundamentally important that the maximum is zero for any time [9]. However, in the course of the
approximation procedure the volume integral generally becomes a function of
time and therefore i t appears natural t o integrate it over the time interval
0 < t < 00, during which the process is considered [12]. Taking this into account,
by virtue of the expressions (2.6) and (2.8) for Ten) and J ( " Bwe
~ ) can rewrite
GYARMATI'Sprinciple aa
-
a) Approximation method in general representation
Performing the independent variations in (2.9) with respect to the parameters ai and pi we obtain in the usual way the two sets of EULER-LAGRANQE
equations
(2.10)
and
(2.11)
A. STARK
58
together with the transversality conditions belonging t o (2.10)
(2.12)
I n (2.12) the subscript denotes that the values of the parameters [xi are evaluated a t the moment t = 00. I n case of nonstationary problems the values of
the parameters [xi are precribed a t the moment t = 0 by the initial condition
(2.5)
(2.13)
are the FOURIER
coefficients of To(z).
By using the expression (2.9)
where ai(0)
for 9,
we easily derive the detailed forms of eqs. (2.10)-(2.12):
(2.14)
(i = 1, ..., n ) ;
(2.15)
By determining the parameters
pj
=
p,([xl, ..
. )
[x,
pi from the linear system of equations (2.15)
oil, . . *, oi*, t )
(j = 1, .. k)
. )
(2.17)
and substituting them into (2.14) and (2.16) we obtain a set of second order
ordinary differential equations (2.14) together with the transversality conditions
(2.16) and initial condition (2.13) for the evaluation of the coefficients [xi.
Integrating these differential equations we find the dependence on time of the
parameters mi E a p k ):
mi
= ol?(t)
(i = 1, . . ., n ) .
(2.18)
Solution of Heat Conduction Problems
59
Using these values of the coefficients ai we have the approximate temperature
field T("):
(2.19)
Furthermore, we evaluate the parameters
(2.17), i.e., we can write
pi = /3pk)by
means of relation
pj = pj(a;, . . . , an*,A;, .. ., .,*, t ) = p*(t)
3
( j = 1, . . . , k)
and hence, according to (2.8), the approximate heat current density is
J(n)
Jhk)
=
(2.20)
n
J(?i)
0
(X,t,@,
...,a,*,Lq,..., L
a
+z:Pj*(t)y,(x).
i=l
(2.21)
After the general representation of GYARMATI'Sprinciple we consider partial
variations.
b) Force representation
I n t,his case (see section 1)we vary with respect t o
i.e.,
= 0. Therefore, from (2.14) i t follows that
T("),
keeping J('sk) fixed,
&7["jk)
aT(")
+ $ (Jcm+ A(") VFW) -.
dQ=O
I>
aor,
Suppose the heat conduction coefficient is state independent;
(i=1, ...,n).
a;l
= 0.
aT
Then
clearly the equations of the GALERKINmethod are obtained, generalized for
natural boundary conditions [18]. Since the RITZmethod (or energetic method)
and the method of minimal surface integrals [MI,when applied to the appropriate heat conduction problem, lead to equations identical to those of the
GALERKINmethod, they are also included in eqs. (2.22). In the other hand, if A
an
is a function of temperature, i.e. - # 0, then, as mentioned already a t the end
aT
of the previous section in connection with GYARIATI'S theorem ( l . l l ) , the
contribution arising from the variation of the phenomenological coefficients
differs from zero. I n our case this means that generally
Therefore, we see that in the quasi-linear case the system of equations (2.22)
contains-due to the presence of J("sk) in (2.23)-the unknown coefficients pj
A. STARE
60
which must be determined by means of eq. (2.15)
which yields for Fj eq. (2.17). Using these expressions for the parameters
in
(2.22) and solving the resulting system of first order in time differential equations for ai,taking into account the initial condition (2.13), we can represent
the hpproxirnatk temperature,field T(")in the form:
Pi
(2.24)
i=l
It is clear that the approximate heat current density field is:
E
where
Pj"(t) = g,S,(a:, ..., a,*,
q,..., &*,, t )
(j= 1, ... , k).
Notice that in general the values of the two sets of parameters
are different from those of the general representation.
{/?:}F=l
(2.24)
and
c) Flux representation
This variational procedure is charaterised by conditions dT(") = 0, dJ(niL)#O.
Taking this into account, the'system of equations (2.14)-(2.16) becomes
(i = 1, ..., n ) ;
(2.27)
(2.29)
We distinguish two important subcases.
i. If the approximate heat current field J(n,k)is not independent of the parameters ai, and maybe its time derivatives ;Xi, then, proceeding as above, we can
determine the approximate temperature field
T(n)
T ( n , k )=
c a$@)V i ( 4
n
(2.30)
i=l
and the approximate heat current density field
J(n)
J(n,k)= J (0n ) (2, t ,
a:,
.. . , a t , Oi, .. . , Oi)
L
+ z@(t)
Y~(X).
j- 1
(2.31)
61.
Solution of Heat Conduction Problems
ii. I f J(nik)does not depend on mi and hi, then eqs. (2.28) give, in general, only
the functional dependencemof the parameters pj on mi and ;Xi:
(2.32)
pi= &(a1,..., a*,hi, oln, t ) (j = 1, ..., k).
. . . I
The unknown values of mi may be evaluated then by applying the general or
force representations. However, it may occur that the coefficients pi will be
independent of ai and hi.This means that the approximate value of the heat
current density is not influenced by the assumed form of the temperature field.
To clarify this important point, consider the following linear ( A = & = const)
and stationqry variational problem :
1
(2.33)
6 J (&VT 5)'- G d V = 0;
+
V
VJ
TI,
(2.34)
(2.35)
= C~(U)Z
;
=
0.
We seek for the approximate heat current density in the form :
(2.36)
where
V . J ~ ( T=
) au(z)
a.nd
7 .tpj = 0 ( j
= 1, ..., k).
(2.37)
Substituting (2.36) into (2.33) and varying with respect to pj we get:
k
/$(.loVT + J,, + ,zpjvj) j=l
r;
Suppose the functions {
<Vi,Yk)
lpi
dV = 0 (i = 1, ..., k).
(2.38)
~ ~ }t ot be= orthonormal,
~
i.e.
-J~im'y)k.dv=8ikV
Since, due to (2.35) and (2.37) (VT, yi) = 0 (i = 1, ..., k) we can conclude
that
- ( J o , v j ) ( j ' = l , ..., k)
(2.39)
and hence the approximate heat current density field is
J
k
M J ( k ) = .Io@)
- <Jo,yj)
2
j= 1
Vj(Z).
(2.40)
The above outlined approximation procedure for the evaluation of the heat
current density J is equivalent t o the application of the method of orthogonal
projections [18].
3. The dual field methods
In the previous section we developed three approximation methods based
principle. These three methods, as i t will be clear from the followon GYARMATI'S
A. STARK
62
ing, should be considered the first scale of approximation methods related t o the
Governing Principle of Dissipative Processes. However, we have additional
possibility to double the set of approximation methods on a second scale. The
root of this possibility is very deep, and originates in the fact that the entropy
production CI is a symmetric (bilinear) expression in terms of current densities
Ji and conjugated forces V r i , furthermore I+J and @ are connected with each
dual transformations with respect to these variables. On
other by LEQENDRE
the other and the structure of the varied form (1.9) of the GPDP (1.2) also
possesses an analogous duality property with respect to the transport equations
and linear constitutive laws. Indeed, we can read from the volume integral in
(1.9) that the transport equations and the linear constitutive laws are following
simultaneously from it. This fact ensures the possibility to develop the second
scale of approximation methods, for which the name dual field methods seems
t o be most appropriate.
We emphasize once more that the dual field methods are based on the
property of GYARMATI'S
principle that its variations with respect to current
densities lead t o the corresponding constitutive laws
which may be considered as starting points for approximation procedures. Due
to the existence of these equations one of the different sets of the dual independent variables {Vri, . . . , Vr,} and { Ji, . . ., Jt} may be considered as fundamental
for any approximation method. It is dictated by the very nature of irreversible
transport phenomena to select the variables Tias fundamental ones, since their
gradients Vri are the driving forces of dissipative transport processes. In the
spirit of the ideas developed above, we introduce, by definition, a second set of
parameters
rendered t o the current densities as
r!
I
Needless t o say that the duality property of the GPDP is preserved, since the
is a natural consequence of it. It is
possibility of introducing the dual fields
obvious that in case of exact solutions the two sets of variables ri and T!
coincide, i.e., Ti= for all i . By using (1.3)and (3.2) we can rewrite the Governing Principle of Dissipative Processes in the following alternative form :
which, together with the balance equations
t
ecii+CV.(LikVI'j)=ai
k=l
(i= I,...,/)
(3.4)
serves as a basis for the dual field methods. It is remarkable that the perfect
symmetry of (3.3) in the dual forces V r i and Vl'i(O) is a direct consequence of the
GAussian form of the GPDP, the theoretical and practical importance of which
was many times emphasized [l, 2, 7, 91.
Solution of Heat Conduction Problems
63
a) Applieatlon of dual field methods in general form
Keeping in mind the aim to develop approximation methods, we introduco
again the sequence of approximate temperature fields :
By virtue of (3.5) we can write the approximate form of the internal energy
balance as
aT(n)- v . (p)
pC(n)-V T W ) = ,p)
(3.6)
at
which serves for the determination of the To(n)
field As in the previous section,
we consider two simple types of boundary conditions
Therefore, we see that
TO(n)= TO(,)(z, t ,
an, A,,
*,
&n)-
(3.9)
I n case of necessity the functions T @ )may also be subjected to the appropriate
boundary condition.
Obviously, the total variation of GYARMATI'Sprinciple
(3.10)
o v
may be written as:
dV
while
(3.12)
represent the corresponding transversality conditions. Using the expression
(3.10) for 2 the EULER-LAGRANQE
equations (3.11) and the transversality condi-
A. STABX
64
tions (3.12) belonging t o i t may be represented in a detailed form:
]
V . (A(") VT("))- up) __
v
(i =1,
...)n)
and
6!
+n
(-;Z(n)
aTO(n)
VTO(n)+ A(n)VT(n))
.an = 0
ahi L = m
(i = 1, ..., n ) .
(3.14)
Solvjng the set of second order ordinary differential equations in time (3.12)
and taking the transversialty condition (3.14) and initial condition (2.6) into
account we determine the parameters ai = a?)
a( = a f ( t ) (i = 1, ...)n ) .
(3.15)
Therefore the approximate temperature field T(%)is given b y
n
(3.16)
while the corresponding To(")field :
To(")= TO(")(z,t , a:, ..., a,*,
dr:,
..., &,*).
(3.17)
b) Dual force representation
I n this e m we vary with respect to Ten), keeping To(")
fixed,i.e. 6To(")= 0.
Taking this into account we obtain from (3.11) and (3.13) the equations
Solution of Heat Conduction Problems
66
or more kxplicitly the equations
(i = 1, ...)n )
for the determination of the parameters ai. Now we show that the equations of
the TREFBVTZ
method [18] are included in (3.18) or (3.19). Indeed, let us consider
the following linear ( A = & = const) and stationary variational problem:
-
\$.(VT - VT(o))2dV
= maximum;
(3.20)
L
-1odTo
=
0;
T oIn = f ( ~ ) ( W E Q).
(3.21)
(3.22)
Obviously, TO is the exact solution of the problem. We represent T(%)
in the form
(3.5):
where the functions (VJ:=~ are harmonic
dVi= 0 (i = 1, . . ., n ) .
(3.23)
Varying the actual form of GYARMATI'Sprinciple with respect to Ten), we get
(3.24)
( i = 1, ..., n ) .
As a consequence of (3.23), the volume integral is zero. Since the value of TO on
SZ is given [see (3.22)], we can represent (3.24) in the form:
(3.25)
where
-+i= V ~ ~ . r z ( i = 1, ... , n ) .
an
Thus, the equations of the TREFFTZ
method [IS] are obtained.
5
Ann. Physik. 7. Folgr, Bd.31
A. STARK
66
c) Dual flux representation
This representation is characterized by the condition BT@)= 0 and can be
used only if To(")
contains variational parameters, i.e., if it is not independent of
0 1 and
~
Ori. Then equations (3.11)-(3.14) for the evaluation of the parameters ai
become
(3.26)
(i = 1, ..., n )
and
(3.27)
or written out in detail
(i = 1, ..., n )
and
(i = 1, ...)n ) .
(3.29)
Solving the second order differential equations (3.26) and taking into account
the transversality conditions (3.27) and the initial condition (2.6) we determine
in the usual way the dual fields T(")and To@).
Solution of Hest Conduction Problems
67
As a simple application of the dual flux representation consider the following
eigenvalue problem :
-J*(VT
-VT0)2dV
= maximum;
(3.30)
-&AT" = A T ;
(3.31)
TIn = TO], = 0
(3.32)
where A is the eigenvalue. Now, we take the approximate T(n)field in the form
(3.33)
In
where qli = 0 (i = 1, . . . n). Using the balance equations (3.31) and the boundary condition (3.32) we determine the corresponding
field
To(")= T0(")(z,
t , O C ~. ., . an, A ) .
(3.34)
(3.35)
and hence by applying the GAUSS theorem we obtain the result
-10
[ (T("1
- p ( n ) ) ~ v T o ( "d~
) + 1, $ ( p n ) - ~ o ( n )~) v p ( n. )d B = 0,
J
R
(3.36)
As a consequence of the boundary conditions (3.32) the surface integral is zero.
Due t o the balance equation (3.31) the volume integral may be rewritten as
(T(n)- To(?'))ST(")d V
v
=
0
(3.37)
or
(3.38)
Solving the secular equation for A , we determine the eigenvalues AP), .. . A:).
It is remarkable that the above described approximation procedure is equivaIent
t o a version of the RITZmethod, used by FICHERA
in obtaining lower bounds for
eigenvalues [IS].
4. On eigenvalue problems
Because of the great importance of eigenvalue problems, i t seems desirable
to show how the Governing Principle of Dissipative Processes may be applied in
such cases. Suppose the eigenvalue equation t o be solved under appropriate
boundary conditions is
+
-V . (A(z)V T ) f ( z )T = AT
(4.1)
where A(x) > 0, and A is the eigenvalue t o be determined. It is assumed that the
eigenvalue problem is non-degenerate. I n terms of GYARMATI'SPrinciple the
5*
68
equivalent variational problem has the form
/+(vT
-
+ f ) z d ~= maximum
V
and
V .J
+ f(z)T= A T
(4.3)
where J and T satisfy the imposed boundary conditions. There exist several
possibilities to treat this problem from the viewpoint of approximation procedures.
1. We take the approximate T(")
field as a linear combination of independent
coordinate functions {pli}y==l
=
5
i=l
"(rp&)
(4.4)
and apply the force representation. The solution of the resulting secular equation for A gives the different eigenvalues LIP), .. ., Ak).Thus, in the variational
procedure the eigenvaluc A is considered as constant, not subject t o variation.
However this formal process does not lead to success in the general representation and, in general, neither in flux representation. This is the consequence
of the fact that the secular equation yields an algebraic equation of power 2n
for the evaluation of A and, generally, we get n complex conjugate pairs
AP),A?)*, .. . , A?), Ae)*for the eigenvalues, which must be real however. Therefore we accept the point of view that A is a variational parameter and take T(fl)
normed, which reduces the number of independent parameters aci t o n - 1.
Evaluating in the usual way the quantity J("sk)from the balance equation (4.3)
and performing the spatial integration in GYARMATI'S
principle (4.2)we obtain
an expression of the form:
G(ori, . . . ,
pi, . .., pk,A ) = maximum.
(4.5)
Solving simultaneously the system of equations
-aG
_
aori - 0
(i = 3 , ..., n - 1);
-aG_ - 0
( j = 1, ..., k);
V i
(4.7)
we determine the real eigenva.lues A?", ..., AEk)and the corresponding set of
eigen've ctors.
2. The eigenvalue problem may be reformulated as a non-stationary one in
the following way
m
Solution of Heat Conduction Problems
and
Zi + V
J
69
+ f ( z ) T = 0.
(4.10)
The exact solution can be expressed in principle in terms of the eigenfunctions
(gi}C1of (4.1) and the eigenvalues belonging t o them as
*
m
T
=
C ci(0)
exp (-At)gi(x).
(4.11)
d.~
=1
Therefore, if we take T(n)in the form (4.4) and determine
from the valance
equation (4.3), the application of the general, force, and flux representations
lead t o an approximate solution of the form
J(nik'
n
TW)M T W )= C ai(o) exp (--d("&)t)
pli(s)
(4.12)
i-1
where the exponents contain the approximate eigenvalues APk)',. ..,@k).
It is obvious that the above considerations may be easily extended to the
dual field methods as well.
6. Applications
Now we want t o show the application of approximation methods developed
in the previous sections t o a purely illustrative example, We consider transient
heat conduction in a rigid bar, the ends of which are held a t the same constant
temperatures. By choosing appropriate nondimensional variables, the problem
may be formulated in terms of GYARMATI'S
principle as follows:
m l
where the temperature T and heat current J mtisfy the balance equation
and the boundary conditiom
Tls,=-i = TI,,, = 0.
We accept the initial condition
= T o @ )= sin nx.
T
The exact solution of the problem is well known [16] :
T = exp ( -nat) sin nx.
(5.5)
We shall take only one coordinate function pll satisfying the boundary conditions (5.3) and seek for the approximate temperature field T ( l )in the form
T(1)= q ( t ) pli(5) = a 1 ( t ) (5 - 2)
(54
where a1 is a variational parameter to be determined. Taking into account the
initial condition (5.4), we easily evaluate its FOURIER
coefficient
315
#(O) = a, = (5.7)
4n3
-
6 Ann. Physig. 7. Folfle, Bd. 31
A. STARK
70
Substituting (5.6) into the balance equations (5.2)
a~cu
ajci)
-+-=o
at
ax
where we have taken p1 E 1, since the only solution of equation
aw = 0 is
ax
y = const; p, is a variationa1 parameter. By using (6.6) and (6.9),GYARMATI'S
principle (5.1) may be written as
=I1
+ J('))e dz dt = maximuin.
0 -1
(5.10)
a) General representation
The general expressions (2.14)-(2.16) lead in the present case t o the following EULEE-LAGRANGE
equations
107 ..
16
7
(5.11)
- -%
p
=0
630
and
7
(6.12)
&I - 481 = 0
16
+
+ %PI
together with the transversality condition
(6.13)
Evaluating from (5.12) the parameter
v
81 = g
I
as
.
1
and substituting i t into (5.11) and (6.12) we obtain
- 105a1 = 0;
(dri
1OaI) m = 0.
Hence the solution is
316
*(t) = a1
exp [-( 105)1'2t ] .
+
4763
Consequently, the approximate temperature field :
315
!Z(') = -exp [-(105)1/2 t ] (z - 23)
4723
(5.14)
(5.16)
(6.16)
(5.17)
(6.18)
and the corresponding heat current field :
(5.19)
Solution of Heat Conduction Problems
71
b) Force representation
Applying (2.22) we obtain the differential equation
iW,
+ 10,50c, = 0
(5.20)
and hence the solution is
315
a: ( t ) = - exp (- 10,5t).
4n3
Thus the approximate temperature field T(') may be written as
315
T(1) = -exp
(-10,St) (z- 9)
43z3
while the corresponding heat current field :
$1)
=
3307,5
--
(5.21)
(5.22)
(5.23)
4 9
c) Flux representation
By using eqs. (2.27)-(2.30) we have the differential equations
(5.24)
(5.25)
and transversality condition
(5.26)
Simple calculation yields the result :
315
a: ( t ) =
exp (-lot).
(5.27)
The approximate temperature field:
(5.28)
and the corresponding heat current field:
(5.29)
Dual field methods
Obviously, the alternative formulation of the variational problem (5.1)- (5.4)
in terms of approximate dual fields reads
001
0 -1
6*
aTq'ax ))e dx
dt = maximum
(5.30)
A. STARK
72
with the approximate balance equation
(6.31)
boundary conditions
T O(1) Iz=-l = Toci)[z=l= 0
(5.32)
and initial condition
= T,(x)= sin nx.
T
(5.33)
We take again
(5.34)
T(1)= 01 i(t) (5 - x8)
and, by using (5.31) and (5.32) we get
(5.35)
a) Dual general representation
Applying (3.13)-(3.14) we obtain the EULER-LAGRANGE
equation
& - 10501, = 0
(5.36)
and transversality condition
(dll
+ 10ori)
It-
03
(6.37)
= 0.
Hence we can represent the solution as
315
m?(t) = G e x p [-(105)1/2t].
(5.38)
Therefore, the approximate temperature field :
(5.39)
and the corresponding dual field :
315
exp [-(105)1/2 t ] -x - (6'0
6
4n3
and finally the approximate heat current:
= (105)1/2-
+20
(5.40)
(5.41)
b) Dual force representation
By using (3.19) we drive the differential equation
6 , s 10,601, = 0
(5.42)
Solution of Heat Conduction Problems
73
and hence the solution is
315
a f ( t )= -exp
(-10’5t).
479
The approximate temperature field :
315
T(’) = -exp
(-10,5t) (z - 9)
4n3
and the corresponding dual field :
3307’5
7 x 3
TMi) = -exp (-10,5t) (wx 43t3
(5.43)
(5.44)
+f )
(5.46)
while the heat current field
(5.46)
c) Dual flux representation
The application of eq.(3.28)-(3.29) yields the differential equation
d
(kl + IOa,) = 0
at
(6.47)
and transversality condition
((xi
+ lOa,)
rn
= 0.
(6.48)
Therefore the approximate temperature field :
(6.49)
while the corresponding dual temperature field :
3150
T~l)=-exp(-IOt)
479
and finally the approximate heat current field :
3150
$1)
=
-4763
(5.60)
(5.61)
Now we summa,rizethe results in Table 1.
From Table 1we see that the exact eigenvalue n2 = 9,8696 is best approximated by the flux representation, where it is equal to 10. Further on, in the
mean the temperature field is approximated best by the flux representation,
while the heat current field by the force representation. Finally, we mention
that for exact solutions the LAaRANaian density in GYARMATI’Sprinciple is
equal to zero, otherwise it is negative. Therefore the integral of L over the spatial
coordinates and time defines a natural accuracy measure for the approximation
procedure. I n this respect the best results are achieved by the general form of
the Governing Principle of Dissipative Processes.
I am very grateful t o Professor I. GYARMATI
for valuable suggestions and
particular interest in the present work.
315
Exact
Dualflux
479
T
t ) (x - 9)
-x--+-
(lo
-x--+-
(lo
(6'0
x -x--+-
4ns
10 t )
ifso)
: ifso)
3150
To(') = exp (-
x
439
t)
:3
[-(105)lla t ]
p')= 3307'5 exp (-10,5
x
= exp (-nst) sin nx = exp (-9,869 t ) sin nx
T(')= -exp (-103
Dual
force
315
(Z- 9)
4n8
= (105)1/a316 exp
t ) (z- 6')
exp (-10 t ) ( x - ZS)
[-(105)'/2 t ]
T(')= ens
315
T")= 4n9 exp (-10,6
=4 9 exp
315
Approximate temperature field
T(')= 4ns exp [-(106)1/* t ] ( x - 6')
315
2'")
~~
Dual
general
Flux
Force
General
Representation
Table 1
2
4n5
J=
--R
JI1)=
4
4)
exp
(lo
(-la2L)
2
COB
nx
4
[-(105)'I*t]
t)
[-(105)1/2 t ]
3307 6
erp (-10,5
479
2
x ---+-
-
(lo
x ---+-
315
2
5") = -(105)11a -exp
479
Go
d- z '
x ---+
4
- -316
Bx~
x ---+(6'0
J ( l )= -(106)"*
Approximate heat current
?4
F
?
IP
4
Solution of Heat Conduction Problems
75
References
El] GYARMATI,
I., Zsurn., Fiz. Himii (Moscow) T. 89, N. 6 (1966) 1489.
[2] GYARMATI,I., Acta Chim. Hung. 43 (1965) 363.
131 ONSAQER,
L., Phys. Rev. 97 (1931) 405; 58 (1931) 2265.
H. G., Thermodynamik der Irreversiblen Prozesse, Handb. der
[4] MEIXNER,J., REIIC,
Phys. 3 Bd. 2 T1. Berlin, Gottingen, Heidelberg: Springer 1959.
DE GROOT,S. R.,MAZUR,
P., Non-equilibrium Thermodynamics, Amsterdam: NorthHolland Publ. Co. 1962.
[S] HAASE,R., Thermodynamik der Irreversiblen Prozesse, Darmstadt: Dr. Dietrich
Steinlropff 1963.
[7] GYARMATI,
I., Non-equilibrium Thermodynamics, Field Theory and variational
Principles, Berlin, Heidelberg, New York: Springer 1970.
[a] GYARMATI.
I., Z. Phys. Chem 289 (1968) 133.
[9] GYARMATI,
I., Ann. Physik, Leipzig 7, 28 (1969) 363.
1101 SANDOR,J., Zsurn. Fiz. Himii (Moscow) T. 54 (1970), N. 11, 2727.
1111 SANDOR,
J.. Acta Chim. Hung. 67 (1971) 303.
[12] VERHAS,J.,Third Conference on Drying held in Budapest, 19-21. 10. 1971. Section A
Paper 4.
[13] VINCZE,GY., Ann. Physik, Leipzig 7, 27 (1971) 225.
Z., Ann. Physik, Leipzig 7, 27 (1971) 341.
[14] FARKAS, H., NO~ZTICZIUSZ,
[16] COURANT,
H.,
HILBERT,D., Methods of Mathematical Physics, Interscience, New York
1963.
[lS] CARLSLAW,
M. S., JAEQER,G. C., Conduction of Heat in Solids, 2nd .Ed., Clarendon
Press, Oxford 1969.
H., Phys. Chem. 289 (1968) 124.
[17] PARKAS,
[18] 1\IIgHLIN,S. G., Variational Methods in Mathematical Physics, New York: Tve Macmillan Co. 1964. See also the.recent edition in Russian: Variacionniije metodc v mathematicseszkoj fizike, Moscow: Nrtuka 1970.
BapHaqHOHHbIe MeTOrhI 6 MaTeMaTHYeCKOfi G H a H K e , Moc~Ba,HayKa, 1970.
151
Godollo (Hungary), University, Institute of Physics.
Bei der Redaktion eingegangen am 12. Juni 1973.
Anschr. d. Verf.: Dr. A. STARE
Institute of Physics, Univ. of Godo116
H-2103 Godo116 (Hungary)
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