An approach for developing exact one dimensional conduction-convection finite elements is presented. Exact interpolation functions are derived based on solutions to the governing differential equations by employing a nodeless parameter. Exact interpolation functions are presented for combined heat transfer in several solids of different shapes, and for combined heat transfer in a flow passage. Numerical results demonstrate that exact one dimensional elements offer advantages over elements based on approximate interpolation functions

Dechaumphai, P.

Tamma, K. K.

Thornton, E. A.

English

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EXACT FINITE ELEMENTS FOR CONDUCTION AND CONVECTION
by
Earl A. Thornton
Associate Professor
Pramote Dechaumphai
Research Assistant
Kama r K. Tamma
Research Assistant
I Rf
Mechanical Enginecring and Mechanics Department
Old Dominion University
Norfolk, Virginia 23508
Presented at the
Second International Conference on
Numerical Methods in Thermal Problems
July 7-10, 1981
Venice, Italy
F- I
I
EXACT FINITE ELEMENTS FOR CONDUCTION AND CONVECTION
Earl A. Thornton l , Pramote Dechaumphai ii , Kumar K. Tama II
Mechanical Engineering and Mechanics Department
Old Dominion University, Norfolk, Virginia USA
SUMMARY
An approach for developing exact one-dimensional
conduction-convection finite elements is presented. Exact
interpolation functions are derived based on solutions to the
governing differential equations by employing a nodeless
parameter. Exact interpolation functions are presented for
combined heat transfer in several solids of different shapes,
and for combined heat transfer in a flow passage. Numerical
results demonstrate the exact one dimensional elements offer
significant advantages over elements based on approximate
interpolation functions.
1. INTRODUCTION
A significant development in finite element methodology
for forced convection analysis is the concept of upwind
weighting functions [1]. For the one-dimensional convective
diffusion equation upwind weighting functions are expressed
in terms of an upwind parameter which can be varied to adjust
the accuracy of the finite element solution. Optimal values
of the upwind parameter lead to exact values o` the tempera-
ture at nodal points. Other elements in one-dimensional
thermal and structural analysis also predict exact nodal
values. For example, one dimensional solutions [2] for heat
conduction with internal heat generation by linear ii-terpola-
tion functions yield exact temperatures at the nodes. The
ability of finite elements to predict exact nodal temperatures
has previously-been regarded as a property of a few particular
equations and to be of limited generality. There is, however,
( Associate Profes:.or
II ResearchAssistant
a broad class of-one-dimensional problems governed by linear
ordinary differential equations for which finite elements can
be developed to yield exact solutions. The one-dimensional
convective-diffusion equation and the one-dimensional conduc-
tion equation are special cases of this class of problems. In
a previous paper [3], the authors have presented two examples
of one-dimensional thermal-structural finite elements which
yield exact solutions for linear problems.
_s
	
	 The purpose of the paper is to present an approach for
developing one-dimensional conduction-convection finite ele-
ments which yield exact nodal temperatures and an exact
variatior, of temperature within an element for steady-state
linear analysis. The one-dimensional conduction-convection
problems considered are first discussed. A nodeless variable
approach for deriving exact interpolation functions is then
presented and illustrated for several cases. Next, the gener-
al form of the element equations is presented and discussed.
Finally, the benefits of the exact finite element formulation
are illustrated for two problems by comparing numerical solu-
tions from finite elements with exact and linear interpolation
functions.
2. EXACT ELEMENT FORMULATION
2.1 Governing Equations and Boundary Conditions
The geometry and terminology for eight one-dimensional
conduction-convection cases are shown in Fig. 1. Cases 1-7
are solids of various shapes where heat transfer consists of
conduction which may be combined with: (a) surface convection,
(b) internal heat generation, and (c) surface heating. Case
(8) is a one-dimensional flow where the heat.transfer consists
of fluid conduction and mass-transport convection which may be
combined with: (a) surface convection, and (c) surface heat-
ing. In the figure, Q is the volumetric heat generation rate,
q is the surface heating rate, h is the convective heat
transfer coefficient and T is the environmental temperature
for the convection heat exchange. Heat transfer in Cases 1-8
is described by differential equations of the form
ao (x) _7+a1(x)rx + a2 (x)T = a3 (x)	 (1)
dx
where a i , i - 0,1,2,3, are functions which depend on the
geometry and thermal parameters of each case. Finite element
equations for eq. (1) may be formulated by the method of
weighted residuals, but due to the presence of the odd-order
derivative an unsymmetrical coefficient matrix will result.
However, if eq. (1) is first cast into self-adjoint form, the
differential operator will be symmetric. Then, finite element
matrices can be derived by the method of weighted residuals or
a variational method, and coefficient matrices will be
0 q	 h'T"'
x
L
Case 1, ROD
q WX hTfth, L	q
Case 2, SLAB
TI.
To
-I-
q	
h Tr
the
l	 '
L
L
Q
9
Case 7, SPHERICAL SHELL Case 8. FLOW PASSAGE
Figure 1
ONE DIMENSIONAL CONDUCTION AND CONVECTION CASES
Case 3. HOLLOW CYLINDER Case 4, HOLLOW SPHERE
Case 5, CYLINDRICAL SHELL
° = bi
4
14
4	 hAlm
Case 6. CONICAL SHELL
- ds cosy TV --	 •-
- d[PdT]
	 h PT
	
h PT
Tx_ dx
	
kA	 0
where	 P - exp(-mcx/kA)
A cos a kt cos a
--	 kA P
e
symmetrical, [2]. Eq. (1) is written in self-adjoint form by
multiplying it by the factor
P - exp[t(a 1/ao) dx]
such that
az [P(x^] + Q(x)T - R (x)	 (2)
where Q(x) - a 2 P/ao and R(x) - a3 P/ao.
The governing self-adjoint differential equations for
Cases 1-8 are shown in Table 1. Additional terms not previous-
• ly defined are k the thermal conductivity, p the perimeter of
the rod or flow passage, A the cross-sectional area of the rod
or flow pass.;ge, m the mass flow, rate and c the specific heat.
Table 1 shows the .xefficie. cs of eq. (2) for each case. Note
that for each case a diffQ,ent differential equation may
TABLE 1
r	 GOVERNING SELF-ADJOINT DIFFERENTIAL EQUATIONS
dx [P(x 'dx] + Q(x)T - R(x)
Heat Loads
Case Conduction Convection	 Convection Source Surface
Flux
(a)	 (a)	 (b)	 (c)
	
1 - dx[t]	 T	 T	 k
2 - dx -	 --	 --	 --	 .
d [ dT]3	 - dr rdr	 --	 --	 kr	 --
4	 ^[r2^]	 --	 --	 kr•2	 --
	d dT	 h	 h
5	
- ds [ds ] 	kt T	 kt T^	 k	 kt
5	 - ts Csas ]	 kt sT	 kt sT-	 i S	 kts
I	 7
8
e
• ^Y•	 V ll^i
t^ dT
i
- .
	
•	 An .
result by combining conduction with each of the three heat
ioads or combinations of these heat loads.
The boundary conditions for &4. (2) consist of specifying
the temperature or the temperature gradient at the endpoints
of the solution domain, a<x<b. For cases 1-8, the boundary
conditions considered herein are:
T = constant, or
-k 7 = h(T-T.)
2.2 I nterpolation Functions
Exact interpolation functions for finite element formula-
tions of eq. (2) can be derived from the general solution to
the differential equation. The general solution has the form,
	
T(x) - c 1 f1 (x) + c2 f2 (x) + G(x)
	 (4)
where f (x) and f (x) are linearly independent solutions of
the homogeneous 4uation,c and c are constants of integra-
tion, and G(x) is a particular solution. A finite element
with two nodes is formulated by imposing the conditions
	
T(x 1 ) = T 1	T(x2) - T2	(5)
where x
the nodal temperature values To accommodate Ae' particular
solution G(x), the element temperature interpolation is
written as
T(x) = N0 (x) To + N 1 (x) T 1 + N2 (x) T2	(6)
where T is a nodeless parameter. The nodeless parameter To
is similar to a nodeless variahle [2] except the nodeless
parameter is uniquely determined for each differential equa-
tion. Note that N (x) is zero at the nodes to satisfy (5).
The interpolation ?unctions N , i - 0,1,2 are determined by
imposing the boundary conditions, eq. (5), on eq. (4) and then
writing the result in the form of eq. (6). The nodeless para-
meter is identified by writing the particular solution
G(x) - To g(x). Thus, in general
f (x )g(x )-f (x )g(x )
	
No (x) - g(x) +[ 2 1	 2	
2 2	 1 J 
fl(x) 
+
fl(x2)g(x1)0	
J f2(x)fl(xl)g(x2)	
(7a)
[ 
el
N
1 (x) - f2 (x2 )f l (x) - fl(x2)f2(x) 	
(7b)
N (x) s fl (xl )f2 (x) - f2(xl)fl(x)	
(7c)2
where
	 0 - f1 (X1 )f2 (x2 ) - fl(x2)f2(xl).
Element interpolation functions derived in the form of eq. (2)
are not generally the same as used in conventional finite
elements since they result from the differential equation
solution. Interpolation functions, in fact, may range from
simple polynomials to higher transcendental functions depend-
ing on the equation. The exact interpolation functions, eqs.
(7), are derived for specified temperature boundary conditions,
but they also yield exact solutions for the gradient boundary
conditions, eq. (3), when these boundary conditions are con-
sistently incorporated in the finite element equations. Since
the interpolation functions are exact, temperature gradients
and fluxes computed from these functions are also exact.
Nodeless parameters and element interpolation functions
have been derived for Cases 1-8 with heat loads (a) - (c), but
for brevity only typical results are presented herein. Typical
nodeless parameters appear in Table 2 and typical interpolation
functions appear in Table 3.
TABLE 2
NODELESS PARAMETERS
Case	 To	 Case	 To
1(a) TW	 1(b)	
^a2/k
1(b) OL2/2k	 8(b)	 qpL/mc
3(b)	 Ob2/4kw	 8(a)	 T
4(b)	 0/6k
where w - ln(b/a)
I
..
TABLE 3
ELEMENT INTERPOLATION FUNCTIONS
Case
No
N1 N2
1(a) 1-N1-N2 Binh m(L-x) sinh mx
sink mL sinh mL
1(b) N1N2
1 L
3(b)
2	 N	
!-2
	 2
- w + 
	 - - w w 
In (
r) w In (a)
4(b) (r-a)(b-rM r+a+b)
r
a b_r
r	 a
b r_a
r	 a
l+sin a
7(b) slog(cosa)-N2 log(cos
log
L
a)	 1-N2
sin
a
L
+sni 	
allog
1•sin a J
[l
8(b) L - N2 1-N2 1 - e2axZT.
8(a) 1-N -n ,
1	 2
eax sinh e(L-x)
 sinh SL
ea(x-L) sinh Bx
si—n a
where
	
m = (hp/kA)}
w s ln(b/a)
CL 
= me/2kA
a - (1112 + a2 ) 
}
i
2.3 Element Matrices
For general interpolation functions, element matrices for
eq. (2) are
x	 dN dN	 x
Kid = f 2 P 
ax	
dx + f 2 Q Ni N^dx	 (8a)
x l 	 xl
F i	 f 
x2 
R N, dx
x1
where K is a typi(.al term in the symmetric conductance
matrix,'And F  is P. typical term in the heat load vector. For
a node on the boundary additional terms are required to repre-
sent surface heating or a convective heat exchange. For the
exact interpolation functions, element matrices have the
-general form
KOO	 KO 1
	
K02 To Fo
K10	 K11	 K12 T 1 =
F1
K20	 K21	 K22 T2 F2
since i = 0,1,2. Because T
o
 is known the first equation may
be uncoupled from the nodal unknowns in the second and third
equations. Thus, the exact element matrices have the same
site as a conventional linear Element and can be written as
K11	 K12	 T 1	 F1	 K10
- To	
(9)
K21 K22	
T2	
F2	 K20
Eq. (9) can be simplified further by noting that K	 =0 for
self-adjoint equations. The last result can be shn$
observing that N1 =1,2 are solutions of the ordinary
differential equation, (2). Multiplying this equation by N
and integrating by parts shows that K =K, n 0. Thus, for °
self-adjoint equations, element matri12s h ged to be evaluated
only for 1=1,2.
As emphasized previously, the development of the element
matrices, eqs. (8), and element equations, eq. (9), are based
upon the differential equation being written in the self-
adjoint form, eq. (2). As an alternate ap proach, exact finite
element matrices may also be derived from the standard form of
the differential equation, eq. (1). However, element matrices
.
(8b)
will not be symmetrical due to the odd-order derivative, and
the exact element equations will have Koi # 0, 1-1,2.
The principal advantage of the exact one-dimensional
finite elements is the superior accuracy in --Comparison to
elements based on approximate interpolation functions. The
principal disadvantage of the exact elements is the additional
effort required to form the more complex interpolation
functions and to evaluate the element matrices. This disad-
vantage has been overcome, in part, by using a computer-based
symbolic manipulation language MACSYMA to perform the algebra
and calculus required for these derivations.
3. NUMERICAL EXAMPLES
Two numerical examples are presented to illustrate the
benefits of the exact elements; other examples for Cases 1 and
:I appear in [4]. The solution of problems with exact elements
typically requires only a few elements because mesh refinement
is not required in regions of large temperature gradlen,ts.
The number of elements used is determined by changes in
geometry, material properties or heating variations. After
nodal temperatures are computed, temperatures are computed at
several points within each element by using the element inter-
polation functions, eq. (6).
3.1 Coffee Spoon with Conduction and Convection
Rod elements with conduction and convection (Table 1,
Case 1(a)) are used to model one-dimensional heat transfer in
a coffee spoon (Fig. 2). The lower one-half of the spoon is
convectively heated by the coffee at a specified temperature
of 150°F, and the upper one-half of the spoon is convectively
cooled by the atmosphere at a specified temperature of 50°F.
The ends of the one-dimensional spoon model are a>sumed to
have negligible heat transfer. Element matrices for the e!oc t
finite elements appear in reference [4]. Temperatures for tht
spoon model are computed from: (1) two exact finite elements,
(2) two elements with linear temperature interpolation,(3) ten
elements with linear temperature interpolation.
The temperature distributions (Fig. 2) show the exact
temperature distribution computed by the two exact elements
with three nodal unknown temperatures compared with predictions
from the elements with linear temperature interpolation. Two
linear elements predict the temperatures at the three nodes
with fair accuracy, but the linear interpolation functions aie
not capable of representing the zero temperature gradient
boundary conditions. Temperatures computed from ten linear
elements, however, show excellent agreement with the exact
finite element solution. Solutions for similar problems have
also shown this trend; typically about five to ten elements
with linear interpolation per exact element are required for
acceptable accuracy.
^	
160	 h= 0.05
r Tan 50 ^`
140
	
h=010
	
L^2 x
^.=150
r
120
T, °F	 —exact fe.
100	
2 elements
--linear fa
2 elements
--- linear fa
80	 10 elements
0
600
	 0.2	 0.4	 0.6
	
0.8
	
1.0
x fL
Figure 2
R00 ELEMENT ANALYSIS OF COFFEE SPOON
WITH CONDUCTION AND CONVECTION
3.2 Heat Transfer in Merging Flows
In merging flows, significant temperature gradients may
occur at the flow confluence; therefore, merging flows provide
i
	
	
a good test for finite element evaluations. In reference [4],
the authors compared conventional (Bubnov-Galerkin) finite
element solutions with upwind (Petrov-Galerkin) solut i ons for
steady-state and transient merging flows. Fig. 3 presents
the geometry and terminology of a merging flow where conduction 	 S
is combined with mass-transport and surface convection.	 p
a
Temperatures for the merging flow (Fig. 3) were computed 	 3
from: (1) three exact finite elements with four nodes, ar'
(2) 75 conventional elements with 76 nodes. There is good
agreement between the two solutions except in the vicinity
;.
	
	 of the flow confluence where the conventional element shows
osculations indicating a need for mesh refinement in the
region of large temperature gradients. The clear superiority
of the exact elements is demonstrated. Exact nodal tempe;-a-
tures are also predicted in problems of this type (without
surface convection) by optimum upwind elements, but a larger
s
.
i'
number of nodes are required since upwind elements do not
predict exact temperatures within an element.
	
2.0
	`.T=2
° ti .^
• m
1.5 -6- exact f e.
3 elements.4 nodes l • 1 ^,
	
T(x)
	 o conventional fe.
75 elements	 °ip
76 nodes
1.0
0
0	 0.4	 0.8 X/L 12	 1.6	 2.0
Figure 3
EXACT AND CONVENTIONAL FINITE ELEMENT ANALYSIS
OF MERGING FLOW WITH CONDUCTION, MASS TRANSPORT
CONVECTION AND SURFACE CONVECTION.
4. CONCLUDING REMARKS
An approach for developing one-dimensional, exact finite
elements for linear, steady conduction-convection analysis is
presented. The element interpolation functions employ the
solution from the governing second-order di fferenti al equati on by
utilizing a nodeless parameter approach. The elements require
two nodes and yield exact nodal temperatures and an exact
variation of the temperature within an element. Exact nodal
interpolation functions are presented for several cases of
conduction and convection. Numerical results are presented
for conductior. and surface convection in a rod element model,
and for conduction, mass - transport com<<=:.!ion and surface
convection in a merging, one-dimensional flow model. The
examples demonstrate that for one-dimensional analysis, the
exact elements offer significant advantages over conventional
elements based on approximate interpolation functions.
REFERENCES
s	 1. CHRISTIE, I., GRIFFITHS, 0.	 F., MITCHELL, A. R.,
ZIENKIEWICZ, 0. C. - Finite Element Methods for Second
Order Differential Equations with Significbn,	 First
Derivatives.
	
International Journal for Numerical
Methods in Engineering, Vol 10, pp. 1339-1396, 1976.
'	 2. ZIENKIEWICZ, 0. C. - The Finite Element Method, third
edition, McGraw-Hill, 1977.
3. THORNTON, E. A., DECHAUMPHAI, P., WIETING, A. R. and
TAMMA, K. K. - Integrated Transient Thermal-Structural
Finite Element Analysis.	 Proceedings of the AIAA/ASME/
ASCE/AHS 22nd Structures, Structural Dynamics and Materials
Conference, Atlanta, Georgia, AIAA Paper No. 81-0480,
Apri 1 6-8, 1981.
4. THORNTON, E. A. and DECHAUMPHAI, P. - i:onvective Heat
Transport in Merging Flows, Finite Elements in Wa or
Resources, Ed. Wang, S. Y., Brebbia, C
	
A.,
	
ons 
iGray, W. G. , and Pi nder, G. F., The Unversity of
Mississippi, 1980.
ACKNOWLEDGEMENT
This study was supported in part by the National Aero-
nautics and Space Administration under grant NSG 1321.
r
i


Exact finite elements for conduction and convection

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