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FINITE ELEMENT ANALYSIS PROGRAM (FEAP)
FOR CONDUCTION HEAT TRANSFER
Jorge Martins Bettencourt
NAVAL POSTGRADUATE SCHOOLMonterey, California
THESISFINITE ELEMENT ANALYSIS PROGRAM (FEAP)
FOR CONDUCTION HEAT TRANSFER
by
Jorge Martins Bettencourt
December 1979
Thesis Advisor: G. Cant in
Approved for public release; distribution unlimited
T196177
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REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM
2. GOVT ACCESSION NO. S. RECIPIENT'S CATALOG NUMBER
4. TITLE r«n<< Subitum)
Finite Element Analysis Program (FEAP)
for Conduction Heat Transfer
5. TYPE OF REPORT a PERIOO COVERED
Master's & Engineer's ThesisDecember 1979
• PERFORMING ORG. REPORT NUMBER
7. AUTHORfaJ
Jorge Martins BETTENCOURT
S. CONTRACT OR GRANT NUMBERf*J
>. PERFORMING ORGANIZATION NAME AND ADDRESS
Naval Postgraduate SchoolMonterey, California 93940
10. PROGRAM ELEMENT. PROJECT, TASKAREA ft WORK UNIT NUMBERS
II. CONTROLLING OFFICE NAME AND ADDRESS
Naval Postgraduate SchoolMonterey, California 93940
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IS. SUPPLEMENTARY NOTES
IS. KEY WORDS (Contlmio an tavaraa atom II naeaaaafr •"<* lomntltf my Mae* numbor)
Conduction Heat Transfer, Finite Element, Numerical Analysis,Time Integration Operator, Computer Programming
20. ABSTRACT (Continue on ravaraa mid* II nooooomnj mnd Idmnlllr *T Moefe mmmbt)
The Finite Element Analysis Program (FEAP) was expanded to solve linearand nonlinear, two and three dimensional heat conduction problems. Theusual types of boundary conditions, including radiation, may be specified.A wide range of two- and three-time level schemes for the solution of timedependent problems is available in the program and a discussion of thosemost commonly used is presented. The algorithms for the solution of typical
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practical problems are described and several numerical examples arepresented. The results are compared with the available analyticalsolutions. A listing of this expanded version of FEAP and thecorresponding user's instructions are provided.
DD Form 1473. 1 Jan 73
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Approved for public release; distribution unlimited
Finite Element Analysis Program (FEAP)for Conduction Heat Transfer
by
Jorge Martins ^ettencourtLieutenant/ Portuguese Navy
B.S., Naval Postgraduate School, 1979
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
and
MECHANICAL ENGINEER
from the
NAVAL POSTGRADUATE SCHOOLDecember 1979
ABSTRACT
The Finite Element Analysis Program (FEAP) was expanded
to solve linear and nonlinear, two and three dimensional
heat conduction problems. The usual types of . boundary
conditions, including radiation, may be specified. A wide
range of two- and three-time level schemes for the solution
of time dependent problems is available in the program and a
discussion of those most commonly used is presented. The
algorithms for the solution of typical practical problems are
described and several numerical examples are presented. The
results are compared with the available analytical
solutions. A listing of this expanded version of FEAP and
the corresponding user's instructions are provided.
TABLE OF CONTENTS
I. INTRODUCTION - 7
II. STATEMENT OF OBJECTIVES — 9
A. EQUATION OF CONDUCTION. INITIAL AND BOUNDARY
CONDITIONS 9
B. NUMERICAL SOLUTION. DISCRETIZATION BY THE
GALERKIN METHOD. MATRIX FORMULATION 10
III. FINITE ELEMENT ANALYSIS PROGRAM 12
A. MAIN PROGRAM — - 12
B. FINITE ELEMENT SOLUTION MODULES - 18
1. Two Dimensional Heat Transfer Element 19
2. Three Dimensional Heat Transfer Element 23
C. TIME INTEGRATION MODULE — 26
IV. APPLICATION OF FEAP TO HEAT TRANSFER PROBLEMS 29
A. STEADY STATE PROBLEMS 29
1. Algorithms for Linear Problems with Macro
Program 29
2. Algorithms for Nonlinear Problems with
Macro Program 30
B. TIME DEPENDENT PROBLEMS - 31
1, Two and Three Point Recurrence Schemes
for First Order Equation 31
2, Algorithms for Time Dependent Problems
with Macro Program 38
V. NUMERICAL EXAMPLES - kl
A. RADIAL TEMPERATURE Dl STRI BU I Tl ON IN A HOLLOW/
CYLINDER kl
B. NONLINEAR STEADY STATE HEAT CONDUCTION IN A
SLAB WITH TEMPERATURE DEPENDENT CONDUCTIVITY — k2
C. TRANSIENT TEMPERATURE D I STRI BUI Tl ON IN A
HOLLOW CYLINDER U3
D. SLAB WITH RADIATION-CONVECTION BOUNDARY
CONDITIONS — — kk
VI. CONCLUSIONS AND RECOMMENDATIONS U6
APPENDIX A USER INSTRUCTIONS FOR FEAP 80
APPENDIX B SAMPLE EXAMPLE — - 93
APPENDIX C FEAP LISTING — - 101
LIST OF REFERENCES -— — 155
INITIAL DISTRIBUTION LIST 157
ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation to
Professor Gilles Cantln for is invaluable guidance, help and
friendship as instructor and thesis advisor. In addition,
the author wishes to thank Professor Olglerd Cecil
Zienkiewlcz for his interest, comments and thoughtful
advice.
The author is obligated to Professor Matthew Kelleher
and Professor David Salinas for their assistance during the
first stages of this work.
Finally the author wishes to thank his wife, Maria Joao,
and his daughters, Joana and Catarina, for their
understanding and encouragement throughout the course of
thi s work.
I . INTRODUCTION
This thesis describes an application of the Finite
Element method to Heat Transfer analysis through the use of
a computer program. This program was designed to solve
linear and nonlinear, steady and unsteady, two and three
dimensional heat conduction problems involving temperature
dependent thermophysi cal properties and complicated
radiation/convection boundary conditions.
The Finite Element Analysis Program (FEAP), programmed
by Professor R. L. Taylor in the Department of Civil
Engineering of the University of California/ Berkeley,
served as the point of departure for the present code. It
has now been expanded with two additional modules for Heat
Transfer analysis and time integration of first order
equations, respectively, but the original characteristics of
a research and educational tool in which the various modules
can be changed or added to as desired, were maintained.
Although all equations are retained in core, the program
can handle realistic engineering problems with several
hundred unknowns in most computer systems.
While the algorithms to solve steady heat transfer
problems are well studied and defined, more research has to
be done on the solution of unsteady problems. The
Zienkiewlcz two- and three-time level schemes were
Integrated In the program In such a way that a wide range of
choices of time integration algorithms is available to the
user. A limited study was performed in order to determine
the characteristics of the most used members of those
families of numerical schemes.
The FEAP code is written in FORTRAN IV language and this
version was constructed and tested on an IBM 360/67 computer
system with OS/67 release 18. Other systems and
installations will require modifications.
I I . STATEMENT OF OBJECTIVES
A. EQUATION OF CONDUCTION. INITIAL AND BOUNDARY
CONDITIONS
The problem considered for solution Is thoroughly
developed by Arpaci in Ref. 1 and is mathemat i cal
y
described in the region Q by the equation
p c |± = V . ( E VT ) + Q (1)
subjected to boundary conditions
T = T on r x(2)
and
( 1c VT ) n + q + q + q = on T 2 <3)
and initial condition
lim T = T ujt+0 °
V is the gradient operator, Ti and r 2 are mutually
exclusive parts of the boundary of the region SI , T is the
temperature and t is the time. The thermal capacity pc, the
thermal conductivity 1< and the rate of internal heat
generation Q are thermophysi cal properties dependent on
temperature.
In equation (3) n is an outward unit vector normal to
the boundary surface while q, q , q represent respectively
the Imposed heat flux and the rates of heat flow per unit
area due to convection and radiation defined as
q = h ( T - T ) (5)x c ^ ac
and
qr= Fa C T
4- T^
r ) = hr ( T - T
ar ) (6)
In (5) h is the temperature dependent convectfve heat
transfer coefficient. In (6) the parameter h Is defined byr
the expression
hr
= fa ( T2
+ T2
ar ) ( T + Tar ) , where F is
the radiant exchange factor and a the Stefan-Bol tzman
constant. Tac and Tar are the equilibrium temperatures for
which/ respect! vel y, no convection and radiation occurs.
B. NUMERICAL SOLUTION. DISCRETIZATION BY THE GALERKIN
METHOD. MATRIX FORMULATION
The spacewlse discretization of equation (1) using
cartesian coordinates can be acomplished by Galerkln's
principle as shown by Zienklewicz in Ref. 2 and Lew in Ref.
3.
Let the unknown function T be approximated, throughout
the solution domain at any time t, by the relationshipn
T = z N.fx.v.zl T.ftl = <N> {T} (7)Z N , (x,y,z) T (t) = <N> {T}1=1 x 1
where N. are the usual shape functions defined piecewise
element by element/ T. being the nodal parameters.
The result is
[K] {T> + [C] (T) + {F} = {0} (8)
where [K] and [C] are symmetric matrices defined/ on the
element level, as
T
f f h + h ) <N> <N> drJr e
* c r
10
(9)
[C]6
= / A PC<N^ <N> dfi
JT
(10)
The load vector F is
{F}e
= -/ <n5 q an +
n
/ <N5 ( q - h T1 2
c ach T ) drr ar J
(11)
where the surface integrals are performed only over the
surfaces where the prescribed boundary condition applies.
It must be noted that the set of equations (8) is
nonlinear since the matrices [K] , [C] and vector {F}are
dependent on T .
The system of equations (8) may now be solved by any
automated numerical technique. Code FEAP was written for
this purpose and Is described in the rest of this thesis.
11
III. FINITE ELEMENT ANALYSIS PROGRAM
The Finite Element Analysis Program (FEAP) was
programmed and published by Taylor in Ref. k where it is
well explained. The reader should consult this reference for
detai Is about i t.
This thesis is an expansion of the original program
towards the solution of Heat Conduction problems and as such
only the main features of the basic program will be referred
in this section. A more detailed explanation is reserved for
the new subroutines added to the program. The complete
user's instructions for FEAP are given in Appendix A and
they complement this section.
*
A. MAIN PROGRAM
FEAP can be separated Into two basic parts:
a) Data input module and preprocessor
b) Solution and output modules
The data input module must transmit sufficient
information to the other modules so that each problem can be
solved. All the input data is stored in a single array
(integer array M) which is partitioned to store all the data
arrays, as well as the global arrays, e.g., stiffness, mass,
load, etc.
The total capacity of the program Is controlled by the
dimension of the array M In the blank common of the main
12
program and the corresponding value of the variable MAX.
The partition of M Is performed In the control routine
(subroutine PCONTR) and the data used for It Is supplied by
the user in the title and control Information cards (first
two cards for every run of the program).
Then the nodal coordinates, element connections/
material properties, nodal loading and boundary conditions
may be input using the macro control statements. They are
interpreted by subroutine PMESH and each control is a
function independent of the others.
To clarify what was said till this point, the discussion
of a simple example may be useful. Consider the mesh
presented in Figure 1 and assume that quadratic rectangular
elements as the one defined In Figure 2 are used. This
example is completly solved In Appendix B and will be used
throughout this text.
The sector of a circular ring was divided in six
elements and the 33 nodes were numbered. The order of
numbering is not crucial but In order to Improve the profile
of non-zero coefficients the following general rule should
be used.
The numbering should be such as to minimize the nodal
difference for each element (maximum node nummber minus
minimum node number).
The user can now proceed to the preparation of data for
the program. The first step consists of specifying the
problem title and the control information. The latter Is
x Figures are grouped at the end pp 47-79
13
clearly 33 nodes, six elements, one material set, two
spatial dimensions and eight nodes per element. If this is a
Heat Transfer problem we have one unknown per node
(temperature).
The program expects now the data cards for mesh
description. Any analysis require at least
a) coordinate data which follows the macro command COOR
b) connectivity table which follows the macro command
ELEM
c) material data which follows the macro command MATE
In addition, most analysis will require specification of
nodal boundary restraint conditions, macro BOUN, and the
corresponding nodal force or displacement values, macro
FORC. The term displacement refers to the value of the
unknown for each specific problem. In a Heat Transfer
problem the unknown Is temperature while in an Elasticity
analysis it is a geometrical displacement.
The TEMP macro command (temperature data) shall not be
used in a Heat Transfer analysis to specify nodal
temperature. It may be used to specify auxilary nodal
quanti ties.
If in our problem we assume the line defined by nodes
31, 32 and 33 at the temperature of 20, the line defined by
nodes 1, 2 and 3 at 820 and Insulated elsewhere, the macro
control statements and correspondent data cards will be
COOR
1 5.1
Ik
31 .3
k 5.116666667
29 .283333333
2 5 .1 3.
32 .3 3.
3 5 .1 6.
33 .3 6.
5 5.116666667 6.
30 .283333333 6.
(blank card)
POLA
1 33 1
(blank card)
ELEM1116(blank card)
30UN
11-13 -1
31 1 -1
33 -1
(blank card)
FORC
1 1 320.
3 820,
31 1 20.
33 20.
15
(blank card)
MATE
1 2
10. 10. 8000. 250. 3
END
In spite of the fact that the MATE command must be
Included In this part of the data preparation/ its
discussion is left to the section where the element modules
are treated.
After completing the mesh data input, we are ready to
initiate the problem solution.
FEAP has modules for variable algorithm capabilities and
which. If necessary, can be modified or expanded. The basic
aspect of the variable algorithm program is a macro
instruction language which can be used to construct modules
for specific algorithms as needed. This language is
Interpreted by subroutine PMACR and the complete list of
macro instructions available in this version of FEAP is
given in Appendix A. 3.
Here, as an example, we will discuss in detail the
algorithm for the solution of linear steady state problems
using the original explanation given in Ref. k.
To use the macro programming commands/ the user only
needs to learn the mnemonics of the language. If one wishes
to form the global stiffness matrix the program instruction
TANG Is used (TANG is the mnemonic for a symmetric tangent
stiffness matrix and for nonlinear elements would form and
16
assemble into the global stiffness the element tangent
stiffness computed about the current displacement state; for
linear elements this is just the linear stiffness matrix).
For a problem with an unsymmetric tangent stiffness the
macro command UTAN is used.
If one wishes to form the right hand side of the
equations modified for specified displacements one uses the
program Instruction FORM. The resulting equations are
solved using the Instruction SOLV.
Printed output can be obtained using the macro command
DISP,
The above instructions are sufficient to solve linear
steady state problems, that is, the macro Instructions
TANG ( or UTAN )
FORM
SOLV
DISP
are precisely the required Instructions to solve any linear
steady state problem.
If the same block of instructions is repeated twice or
more, considerable effort is wasted in preparing the macro
Instruction data. To rectify this, looping commands are
introduced as the Instruction pair
LOOP n
*
NEXT
which indicates that looping over all instructions between
17
LOOP and NEXT will occur n times.
Many other classes of problems can be solved using the
macro instruction list given in Appendix A. With the above
short explanation and the discussion presented in section IV
for Heat Transfer problems/ the reader will be able to
construct his own algorithms using the macro programming
language.
B. FINITE ELEMENT SOLUTION MODULES
Most of FEAP is common to a wide range of problems, but
some of the computations are linked to the type of problem
to be solved. It is clear that if one wishes to solve an
Elasticity problem, the calculation of the stiffness matrix
will be different from that used for a Heat Transfer
problem. It is also clear that differences also arise if one
is using triangular elements instead of quadratic
isoparametric elements as a way of describing the continuum.
The program was designed such that all computations
associated with any type of element are contained In an
element subroutine called ELMTnn where nn is between 01 and
05 In this version of FEAP. Each element type to be used Is
specified as part of the material property data, following
macro control MATE.
Since the objective of this work is the solution of Heat
Transfer problems, the element modules discussed here are
the two and three dimensional heat tranfer modules named
ELMT02 and ELMT03.
18
1. Two Dimensional Heat Transfer Element
Subroutine ELMT02 is accessed using the number two
in column ten of each material number card.
According to the value of the variable I SW, defined
in subroutine PMACR, a specific function is performed. If
ISW^l it reads and prints the material property data and
line boundary conditions. The stiffnes and mass matrices are
computed if I SW=3 and I SW=5 respectively. The load vector
due to internal forces and boundary conditions is calculated
when ISW=6.
The material properties may be input as constants or
as a table of temperature and property values. If any other
way of defining the property values is required by a
specific problem/ subroutines PKX , PKY , PROC or PQ which
calculate respectively the conductivity in x and y
directions, the thermal capacity and the heat generation per
unit volume, can be easily modified or substituted.
The shape functions used by this heat tranfer module
are calculated by subroutines SHAPE and SHAP2. These
routines are capable of constructing shape functions for a
three-node triangle, a four-node linear quadrilateral, an
eight-node quadratic serendipity quadrilateral, a nine-node
quadratic Lagrangian quadrilateral or any combination
in-between.
A four- to nine-node two dimensional element is
shown in Figure 3. As shown by Bathe and Wilson in Ref. 5
the temperature within the element is expressed at any time
19
in the local coordinate system s, t in terms of the nodal
temperatures T^ by
where
Nl
a
N2
-
N3
=
N4
"
N5
S
N6
'
N7
=
N8
=
Ng
T(s,t) = Z Ni(s / t) T
i
l-s)(l-t)M - (N5+N
8)/2 - N
gM
l+s)(l-t)/U - (N5+N
6)/2 - N
gM
l + s)(l + t)/U - (N_ +N,)/2 - N.M/ o y
l-s)(l+t)/U - (N7+N
g)/2 - H
g/k
l-s2)(l-t)/2
l+s)(l-t2)/2
l-s2)(l+t)/2
l-s)(l-t 2 )/2
l-s 2 )(l-t 2)
If any of the nodes from five to nine are omitted
the corresponding value of N is zero.
The connectivity table must follow the numbering
convention shown in Figure 3, otherwise wrong results will
be obtained.
The maximum number of nodes per element will depend
on the type of elements to be used and may be from three to
nine. The eight-node serendipity shape functions occur if
the ninth node number is omitted. The four-node
quadrilateral shape functions are computed if only the first
four nodal connections are non zero and the three-node
triangle if the first three nodal connections are non zero.
Finally, if a mid-side nodal connection Is omitted the edge
is 1 i near.
20
The line boundary conditions, closely related to the
type of element used, are defined in the cards following the
macro MATE. The line subjected to a specified boundary
condition is identified using the local coordinates. A line
is numbered 1 or 2 if it is perpendicular to the axis s or
t, respectively. Those numbers are positive or negative
according to which direction of the axis it is
perpendicular. Thus, as an example, the line defined by the
nodes 2, 6 an 3 is numbered 1 while the line defined by
nodes 1, 5 and 2 is numbered -2.
The boundary conditions are treated in subroutine
BC0ND2 and four types are considered:
a) specified flux
b) convection with constant heat tranfer coefficient
c) convection with temperature dependent heat
transfer coefficient
d) radiation
If more complex boundary conditions are required
subroutine BC0ND2 may be easily modified.
The numerical Integration necessary to evaluate the
matrices and vectors in equation (6) are performed using the
Gauss quadrature formula; the abscissas and weight
coefficients are calculated in subroutine PGAUSS. The user
may choose the number of points per direction used in the
integration from one to six, but the default value is four
points ,
To illustrate the above, we assume that our sector
21
used as an example Is part of the cross section of a hollow
cylinder made of a material with conductivity in x and y
directions of 10, specific heat 250 and density 8000. If the
outside surface is subjected to convection to an ambient
atmosphere at 20 degrees with a constant heat tranfer
coefficient of 200, the cards following the macro card MATE
are
MATE
1 2
10. 10. 8000. 250. 3
6 2 1 200. 20.
It must be noted that a plane analysis Is specified
and the integration will be performed using three points per
direction. The reader must also note that this version of
the problem is slightly different from the one given when
the mesh data preparation was discussed. In the previous
problem the line now subjected to a convection boundary
condition was at a specified constant temperature of 20. If
this second problem is to be solved the cards corresponding
to nodes 31-33 in the BOUN and FORC sets must be omitted.
Following is a list of subroutines included in the
two dimensional heat tranfer module:
ELMT02 : main routine; forms the necessary arrays
for the solution,
SHAPE : calculates the shape functions for the
triangle and the linear quadrilateral elements.
SHAP2 : adds the quadratic terms and the center node
22
to the shape functions.
PGAUSS : determine the abscissae and weight
coefficients for the numerical integration.
BC0ND2 : adds to the arrays calculated in ELMT02 the
contr ibui tion from the specified boundary conditions.
PKX : determine the temperature dependent
conductivity in the x direction.
PKY : determine the temperature dependent
conductivity in the y direction.
PROC : determine the temperature dependent thermal
capaci ty.
PQ : determine the temperature dependent heat
generation per unit volume.
CONV : determine the temperature dependent heat
transfer coefficient.
TABLE : called by PKX, PKY, PQ, PROC and CONV;
calculates the correspondent coefficient to a given
temperature by linear interpolation between two consecutive
entries in a table.
JACBB2 : calculates the jacoblan determinant when
an integration over a line is necessary.
2. Three Dimensional Heat Transfer Element
This module is called ELMT03 and Is accessed using
the number three in column ten of each material number card.
This element is the generalization of ELMT02 for
three dimensional space and little remains to be said about
its use.
23
The material properties are read in a similar way as
the two dimensional case, but the conductivity in the z
direction must be added. This module uses the same
subroutines as ELMT02 to read and calculate the temperature
dependent properties.
Subroutines SHAP3D and SHAP3 may construct
interpolation functions for any combination between a
eight-node linear brick and a 21-node Lagrangian brick.
An eight- to twenty one-node three dimensional solid
element is shown in Figure k. The natural coordinates (r, s
and t) of the eight corner nodes are (±1/ ±1, ±1) , of the
twelve mid-edge nodes are (0, +1, ±1)/ (±1/ 0, +1) and (+1,
+ 1, 0) and of the center node are (0/ 0, 0).
The temperature within the element is defined in
terms of the nodal temperatures T^ at any time by [Ref. 5]
T(r,s,t) = Z Ni(r / s,t)Ti
If we define
GCB,^) = .5(l + eiB) , for &
±= +1
= 1-B2
for Bt=
and
g±
G(r, r.)G(s,s- )G(t / ti
)
where r., s- and t- are the natural coordinates of the
element nodal points, the interpolation functions N. are
Nl
= gl " (N
9+N
12+N
17)/2 " N
21/8
N2
= g2 " (N
9+N
10+N
18)/2 - N
21/8
N3
= g3
- (N 10+ Nn+ N19 )/2
- Nn /8
N4 " g 4 " (N
11+N
12+N 20 ) / 2 " N 2l/ 8
Ik
N5
= g 5- (N 13+ N 16+ N
17)/2 - N
21/8
N6
- g 6- (N 13+ N 14+ N 18
)/2 - N21
/8
N7
=g7 " (N
14+N
15+N
19)/2 " N
21/8
N8
= g8
- (N 15+ N 16+ N20
)/2 - N21
/8
N. = g. - N-,M for j=9,...,20
N 21= g 21
If any of the nodes from nine to twenty one are
omitted the corresponding value of g is zero.
The numbering convention used in the definition of
the shape functions and shown in Figure k must be followed
by the user when the connectivity table Is but It and the
surface boundary conditions are specified. The surfaces are
numbered +1, +2, +3 as they are perpendicular to the
positive or negative directions of the axis r, s, t,
respectively.
As the numerical integration routine is the same as
for ELMT02, the user may choose from one to six points per
direction, the default value being four points.
Following is a list of subroutines included in the
three dimensional heat transfer module:
ELMT03 : main routine; forms the matrices necessary
for the solution.
SHAP3D : calculates the shape functions for the
eight node linear brick.
SHAP3 : adds the quadratic terms to the shape
functions and the center node for the Lagrangian brick.
BC0ND3 : adds to the matrices the contr
I
bui 1 1 on from
25
the specified boundary conditions.
PKZ : determine the conductivity in the z direction
JACBB3 : calculates the jacobian determinant when an
integration over a surface is necessary.
C. TIME INTEGRATION MODULE
The first order ordinary differential equation solver is
accessed when the macro command 0DE1 is utilized and uses
the Zienkiewlcz two- and three-time level schemes. The
details of these algorithms are treated in section IV.
This module was designed for the solution of first order
equations but can, with little programming effort, be
expanded in order to include higher order algorithms and
solve higher order differential equations.
In order to economize computer memory space at some cost
of execution time, this module only uses the space
previously reserved for the mesh construction. As the mesh
data Is not needed during the execution of a time step
integration, all the corresponding arrays are kept in file 9
during that period. Once the next displacement vector is
calculated and the current displacement vector is saved in
file 10, all the Information in file 9 Is retrieved and the
mesh data restored.
File 10 is used as a working space during the time
integration and the current displacement vector is saved in
it for possible future use. This vector Is retrieved if the
algorithm used In the next time step integration Is a three
point scheme,
26
Besides the time step size change using the appropriate
macro instructions (macro TOD, an optional automatic time
step adjustment was incorporated in the program. The norm of
the difference between the displacements vectors at two
consecutive times is computed at each step. If the norm is
less than a predetermined value ATmax / the time step size
is doubled before going to the next step, whereas if the
norm is greater than a prespecified value AT./ the time
step size is halved and the calculation for that time step
is repeated until the norm is acceptable. The magnitudes of
the maximum and minimum values of the norm are problem
dependent and must be choosen by the user. If they are not
specified no time step adjustment will be performed. The
recalculation of the displacement vector is allways done
using the two point scheme/ even when a three point scheme
was utilized for the first calculation.
The macro command 0DE1 used to access this module
must be followed by a second macro command in order to
determine the function to be performed. The secondary macro
instructions are INIT, LINE or QUAD.
The couple 0DE1 INIT reads the integration constants
theta , beta and gamma, the parameters for the automatic
time step adjustment and the Initial displacement vector.
This data must follow the macro program (see user's
instructions). No time integration is performed by this
instruction.
The couple 0DE1 LINE performs the two point scheme
27
and the current displacement vector is substituted by the
new calculated displacement vector.
The couple 0DE1 QUAD has a function similar to 0DE1
LINE but uses the three point scheme.
Once the execution of an 0DE1 macro command is
complete, the mesh data is restored and the displacement at
time t replaced by the displacement at time t+ At, when the
secondary macro command is LINE or QUAD. If INIT is used,
the displacement vector becomes the given initial
displacement vector.
The subroutines included in this module are:
P0DE1 : control routine; reserves working space and
calls the appropriate subroutines.
INIT : reads the constants theta, beta and gamma and
the maximum and minimum values for the norm used in the time
step adjustment.
PLINE : performs the two point integration scheme.
PQUAD : performs the three point integration scheme.
TERM : performs the automatic time step adjustment.
28
IV. APPLICATION OF FEAP TO HEAT TRANFER PROBLEMS
The program FEAP requires from the user the knowledge of
the algorithm to be used In the solution of the problem to
be treated.
In this section the algorithms used In the solution of
heat conduction problems are discussed.
A. STEADY STATE PROBLEMS
1. Algorithms for Linear Problems with Macro Program
In this case equation (8) becomes
[K] {T} + {F} = {0} (12)
The algorithm used to solve (12) is described by
[K] {v} = -{F} -[K] {T } (13)
{T} = {T } + {v} (HO
where {T°} are the specified nodal temperatures.
The macro instruction TANG builds the left hand side
of (13) while FORM forms the vector in the right hand side.
The instruction SOLV solves the system (13) and performs the
step defined by (1U).
Consequently the simplest macro program used to
solve this problem Is
TANG
kFORMSOLV
DISP
29
2 . Algorithms for Nonlinear Problems with Macro
Program
Equation (12) still applies to this case but now [K]
and {F} may be dependent on the nodal temperatures {T}. The
well known Newton-Raphson iteration may be used and the
algorithm is described as
[KfT1)] {v
1} = -{FCT
1)} - [KCT
1)] {T
1} = {R
1} (15)
{Ti+1
} = {T1
} + {v1
} (16)
where {T 1} are the nodal temperatures at the ith iteration.
The algorithm defined by (15) and (16) is the one used for
the linear problem repeated several times. Then the macro
program may be
LOOP n
TANG
FORM
SOLV
DISP
NEXT
DISP
where n are the user's guess of the number of Iterations
necessary to obtain equilibrium. However, the program has an
internal check on the value of the vector norm||R
I I ,
where
MRU = (? |r£i2)1/2
Wheneverk-1
k
|IR
1! |
< TOL x max|\9?
| |(j=l,...,t )
J -9where TOL Is a predefined tolerance ( 10 is the default
30
value), the iteration ceases and a skip to the macro command
immediately following the first NEXT occurs. Usually one
begins with zero as the initial guess of the displacement
vector; however, any other vector may be used.
B. TIME DEPENDENT PROBLEMS
1. Two and Three Point Recurrence Schemes for First
Order Equation
The set of differential equations
[K]{T> + [C]{T} + {F} = {0} (8)
may be solved using one of the many recursive schemes
described in the literature. From them, the two and three
time level schemes presented by Zienklewicz [Ref.4] offer a
wide range of choices for the solution of linear and
nonlinear problems.
The recurrence relation for the two-time level
scheme can be wri tten as
( ^-[C] + e[K]){Tn+1 > + C-^[c] + (i-e)[K]){T
n ) +
{F} - {0} , < G < 1 (17)
where the subscripts n and n+1 denote evaluation at time t
and t+At. The vector {F} Is defined as
{F} = {Fn+1
> + (l-0){Fn } (18)
The choice of the parameter defines the particular
scheme to be used and the reader will recognize a well known
series of finite difference formulas with a modification of
using a weighted loading term {~F} .
Consider the system of decoupled equations in terms
of the modal participation variables T^
31
c T. + k. T. + £. =11 11 l(19)
for free response/ i.e., f^=0, expression (17) becomes
(i c. k.0) CT-) n+1+ C-^r C. + k.(l-G)) (T
i)n=0 (20)
Substituting the relation
CTiW = ^Vn (21)
in (20)/ the resulting characteristic equation of the
recurrent scheme solved for X gives
X = [l-ki(l-0)At/c
i]/[l+k
iAt/ Ci ]
The scheme will be unconditionally stable if
|X| < 1
for any value of p. where
p i=
(ki/c
j.)At (22)
This condition is satisfied for
> 1/2
On the other hand if
< < 1/2
stability is conditional requiring
Pi < 2/(1-20)
Figure 5 shows how X varies with p- for the schemes
discussed later and how it compares with the exact value of
X = expC-p^
The schemes considered are
a) Crank-Nicolson with 0=1/2
b) 0=2/3 proposed by Zienkiewicz [Ref. h]
c) 0=3M proposed by the author
d) 0=0.873 proposed by Liniger In Ref. 6,
The Zienkiewicz three-time level scheme Is defined
32
as
CY [C] +6 At[K]){Tn+1 > + ((1-2y)[C] + (l/2-26 +Y)At[K]){Tn
}+
C-(1-Y)[C] + (l/2+B-Y)At[K]){Tn _ 1
} + At{F} = {0} (23)
where the subscripts n, n+1 and n-1 denote evaluation at
time t, t+At and t-At respectively.
The vector { F} is defined as
{F} = 6{Fn+1 > + (l/2-20+Y ){Fn } + (1/2+B-y) i?n^ (2U)
and the parameters y anc' 6 define the particular scheme to
be used.
Considering again the system of decoupled equations
(19) and using the relations (21) and (22) one finds that
the characteristic equation for the three-time level scheme
i s
(Y+BP^ A2
+ [(1-2Y ) + (l/2-2B+Y)Pi]X +
[-(1-Y) + (l/2+$-Y)p1
] = (25)
Wri ting
g = [l+Cl/2+Y)Pil/[Y+ 3p i ]
h = [-l+(l/2- Y)p i]/[Y + Bp
i ]
the roots of (25) may be written as
Xl,2
=C 2 "g^ 2 * [C2-g)
2-4(l-h)] 1/2
/2
These roots will be complex if the quantity under
the square root is negative. Then the modulus of A is
|X| = (l-h)1/2
The scheme is stable if IM<1 for any p^and Wood in
Ref. 7 shows that this condition is satisfied if
Y>l/2 and 3>y/2
33
Equation (25) has two roots: X- / the principal root
which is the aproximation to the exact
X = expOp^
and the spurious root X~. For relative stability one must
have
I
X
x I< |x
2 |
Also a negative root or complex roots can produce
osci
1
lation.
In Figure 6a-f the real roots X, and X- and the
modulus |X| for the complex roots are plotted against p.
for the schemes:
a) 6=1/3, y s 1/2 proposed by Lees in Ref. 8
b) 6=3A, y = 1 proposed by Hogge in Ref.
9
c) 3=0.61*6, y=1.218U proposed by Wood [Ref. 7]
d) 6 =l*/5, y=3/2 proposed by Zienklewicz [Ref. k]
e) 6=9/10, y = 3/2 proposed by the author
f) Fully implicit algorithm, 6=1, Y=3/2 [Ref. 7].
From Figure 6a one may predict a very strong
oscillatory behaviour for the Lees algorithm a).
In order to study the various schemes, the physical
problem represented by the nondimensional ized linear heat
conduction equationST .
52 T
3T 6x 2
was solved in a bar of length k and width 1 subjected to the
boundary conditions
T=l at x=0
|I=0 at x=46x
31*
and initial condition
T=0 at t=0
i.e., consider a step change in the surface temperature.
This problem was also studied by Wood and Lewis in Ref. 10.
The problem was discretlzed in space using ten equal
length two dimensional quadratic elements through the length
of the bar. This type of elements was chosen because it was
reported by Wood and Lewis as giving the worst numerical
resul ts.
In order to avoid the discontinuity in the loading
term caused by the step change of the surface temperature at
the initial time, the problem was first solved starting from
time t=At, where At Is the time step size. The value of the
temperature distribution at t=At is provided by the exact
analytical solution. This procedure also provides the
necessary two starting vectors for the three-time level
schemes.
The temperature of the nodes at x=l is used as
reference value to compare the various schemes.
In Figure 7a-d the results obtained from the
various two-time level schemes with At=2 for the temperature
at x=l are compared with the analytical solution. For this
Ideal starting conditions the Crank-Nicol son, 0=1/2, proves
ttobe the most accurate algorithm. When the step size was
Increased to 10, all the schemes performed well as shown in
Figure Sa-d.
The corresponding results for the three-time level
35
schemes are presented in Figure 9a-f and Figure lOa-f for
At=2 and At=10 / respectively.
Strong oscllations were observed with the Lees
algori thm a)
.
For At=2 the schemes c), d) and e) performed well.
For the larger step size of 10 the safest schemes are e) and
f).
The step change in the loading term has been
reported [Ref. 10] as producing relative instability v/hen
some of the schemes are used, particularly with the
Crank-Nicolson algorithm. The noise may be supressed
reducing the magnitude of the time step but this is
impraticable in most problems. Other techniques may be used
to reduce the amplitude of the oscillations as those
discussed by Wood and Lewi s [Ref. 10] and Gresho and Lee in
Ref. 11.
The step change In the loading term in the problem
in study Is due to the variation of the surface temperature
at the initial time t=0. That discontinuity in the force
term is illustrated In Figure 11a. When the time dimension
is discretized It has been common practice to calculate the
numerical solution at the nodes t=0, t=At, etc. When a
three-time level scheme is used the node at t=- At is
considered assuming all conditions steady before t=0. The
force values used by this method are indicated by the
circles in Figure 11a, This procedure transfers to the
numerical solution the uncertainty in the value of the force
36
term at time t=0 Introduced by the mathematical
discontinuity assumed in the analytical solution. This
problem is avoided if one uses the procedure illustrated in
Figure lib. The nodes considered for the discretization are
those at t=-At/2, t=At/2, t=3At/2 / etc. The discontinuity at
t=0 is smoothed by the Interpolation of the loading term
inherent to the particular scheme being used. The values of
the force term for every node are well defined and no
uncertainty is associated with any of them. For the
three-time level schemes the node at t=-3At/2 is used
assuming all conditions steady before t=0.
The problem studied was solved using both starting
procedures. The results obtained wi th the first procedure,
start at t=0 / using the two- and three-time level schemes
are shown in Figures 12a-d and 13a-f, respectively. The
second procedure, start at t»- At/2, gives the results shown
in Figures l^a-d and 15a-f for the two- and three-time level
schemes, respectively.
One may conclude that, in this simple problem, when
the procedure Illustrated in Figure lib is followed, the
step change in the loading term do not deteriorate the
accuracy of the numerical results obtained with the schemes
tested, with the exception of the Crank-Nicol son algorithm.
This scheme shows an oscillatory behaviour not observed when
the step change in the force term Is not present.
The starting procedure of Figure lib also proves to
be an efficient way of starting the time Integration with
37
the three-time level schemes.
2. Algorithms for Time Dependent Problems with Macro
Program
A simplified form of the two-time level scheme is
available in FEAP and is described by
^V + WJHW C-^[cn ] (i + e)[Kn]){T
n } +
At{F} = {0} (26)
where the subscripts n and n+1 denote evaluation at time t
and t+At, respectively.
The three-time level scheme programmed in FEAP is
described by the expression
(y[Cn ] +B At[K
n]){T
n+1 } + ((1-2y) [CJ + (1/2-2S+Y) At [KJHV+ C-d-Y)[Cn] + Cl/2+e-Y)At[Kn
]){Tn . 1
}
+ {Fn
> = {0} (27)
where the subscripts n, n+1 and n-1 denote evaluation at
times t,At+ t and t-At respectively.
It must be noted that in (26) and (27) the vector
{F }is an aproximation of the interpolated force vector {F}.n
If the force vector is strongly dependent on time/
this aproximation may produce wrong results.
In problems where the stiffness matrix is constant
and the force term results from the specified boundary
displacements or where the specified boundary forces are
only time dependent/ the interpolation of the force term may
be easily acompllshed since the force/displacement boudary
values can be changed at any time using the macro commands
MESH or PROP. This procedure was found partcularly useful
38
when a step change In boundary displacements or forces Is
applied at the initial time.
The first order ordinary differential equation
solver module is acessed using the macro command 0DE1
followed by one of the following macro commands:
I N I T to specify the Initial displacement vector
and the integration constants
LINE to perform the two-time level algorithm
QUAD to perform the three-time level algorithm
Since the matrices [K] and [C] and the vector {F}
must be formed before the time integration/ the macro
commands TANG, CMAS or LMAS and FORM are closely associated
with the use of 0DE1. It should be noted that the macro FORM
forms the vector
{R(Tn)}. -{F
n}- [K
n ] {Tn }
which Is allways dependent on the current displacement.
Therefore the macro command FORM must be used every time
step and preceed 0DE1,
To solve a fully nonlinear problem the following
macro program may be used:
DT At
0DE1 INIT
LOOP n
TANG
CMAS ( or LMAS )
FORM
0DE1 LINE ( or QUAD )
39
TIME
DISP
NEXT
The macro command program must be followed by the
data cards necessary to specify the integration constants
6/ y and 8 and the Initial displacement vector.
If the matrices [K] and/or [C] are not temperature
or time dependent then the instructions TANG and/or CMAS or
LMAS must be placed outside the loop.
For the simplest case of a linear problem, the macro
program may be
DT At
0DE1 I NIT
TANG
CMAS ( or LMAS )
LOOP n
FORM
0DE1 LINE ( or QUAD )
TIME
DISP
NEXT
kO
V. NUMERICAL EXAMPLES
A. RADIAL TEMPERATURE DISTRIBUTION IN A HOLLOW
CYLINDER
Consider the hollow cylinder of infinite length already
discussed in section III. Consider the case in which the
inside wall is maintained at a constant temperature and heat
is lost by convection through the outer wall. This problem
was considered for solution by Lew [Ref. 3],
The geometrical and thermophysl cal characteristics are:
Inside radius (i"^n ) 0.1 m
Outside radius ( r out ) 0.3 m
Thermal conductivity (k) 10 W/m °C
Outside ambient temperature 20 °C
Heat transfer coefficient 200 W/m 2°C
Inside wall temperature (T, ) 820 °C
The finite element model of this problem was completly
discussed before and is shown in Figure 1. The complete
computer output is given in Appendix B. It must be noted
that the element arc width used is arbitrary since the
problem Is symmetric and no heat is conducted perpendicular
to the radius.
The comparlsion between the analytical and FEAP
solutions Is shov/n in Figure 15. In this case the values
provided by the finite element model used are very accurate
and no further mesh refinement is necessary.
kl
B. NONLINEAR STEADY STATE HEAT CONDUCTION IN A SLAB
WITH TEMPERATURE DEPENDENT CONDUCTIVITY
A liquid is boiled by a flat electric heater plate of
thickness 2L. The internal energy Q generated electrically
may be assumed to be uniform. The boiling temperature of the
liquid/ corresponding to a specific pressure, is Tw .
In the finite element analysis for the temperature
di str i bui tlon in the heater, ten equal length quadratic
elements were used to represent the unit cross section
through the half plate thickness L, since the problem is
symmetric. The extremity of the resulting slab is assumed at
the temperature T while the other is insulated.w
The thermal conductivity is assumed temperature
dependent according to the expression
k = a(l+bT)
where a and b are constants.
Assuming the following values for the parameters in a
consistent system of units
T = 0.0w
a = 0.5
L = 1.0
Q = 1.0
the problem was solved for b equal 0.0, 1.0 and 2.0.
The numerical results are compared with the analytical
solutions in Figure 16 and one may conclude that FEAP
provides exact solutions in this problem.
hi
.
C. TRANSIENT TEMPERATURE D I STRI BU IT I ON IN A HOLLOW
CYLINDER
The same hollow cylinder of infinite length treated
before is considered again but the case solved is the one
with both outside and inside walls maintained at constant
temperatures. Thermal diffusivity/ a=k/ pc is specified as
20.00005 m /sec. Initially the cylinder is at an ambient
temperature TQ
of 20°C; suddenly the temperature of the
Inside wall is raised to 820°C while the outside wall is
maintained at 20°C. All other conditions are the same as for
the steady state problem.
In order to obtain the unsteady solution the number of
elements in the finite element model was doubled, thus
twelve radial quadratic elements were used. The element arc
width chosen is the same 6° as before.
The two time-level scheme with 0=2/3 was used for the
time integration. In order to compare the results the
dimensi onless temperature T*, radius r* and Fourier number
Fo were used. They are defined as
T* = CT - TQ)/(T
in- T
Q)
r* = r/r.in
and
Fo = ott/r.
A time step size At of 2 seconds was chosen initially.
After 10 sec, At was Increased to 10 sec and to 50 sec after
200sec. After 1000 sec, At was increased to 100 sec.
Figure 17 shows the evolution of the temperature of the
U3
center interior points of the cylinder and Figure 18
compares the analytical values with the FEAP solution for
the radial temperature di stri bui tion for two different
times.
D. SLAB WITH RADIATION-CONVECTION BOUNDARY CONDITIONS
Consider a plane slab of thickness L, with constant
thermophysical properties (k= p 0=1)/ subjected to
simultaneous radiation and convection at x=0 and insulated
at x = L (0<x<L). The slab is initially at a uniform
temperature T and it was decided to consider zero ambient
temperatures for convection and radiation.
It was also decided to consider the Biot number
Bi = hcL/k = 1
where h is the convection heat tranfer coefficient andc
R = eaTgL/k = 4
where e is the emissivity and a the Stefan-Bol tzman
constant.
The same mesh as in Example B was employed for the
finite element solution. The time integration was performed
by the three-time level scheme with y =1/2 and 6=1/3. A
variable time step size was chosen. A value of At=0.001 was
used initially. After t=0.02, At was increased to 0.0025.
After t=0.2 / At was increased to 0.01. After 1=0.4, At was
Increased to 0.025. After t=1.0 / At was increased to 0.05.
The history of the ratios T /T and T /TQ
are plotted in
Figure 19 against the Fourier number
Fo = at/L2
T is the surface tmperature at x=0 and T fs the surfacew m
temperature at x=L.
The finite element results are compared with those given
by Haji-Sheick and Sparrow in Ref. 12 and obtained from a
Monte Carlo method.
kS
VI. CONCLUSIONS AND RECOMMENDATIONS
The computer program In this thesis provides an accurate
and reliable means for solving a variety of Heat Transfer
problems. The use of this program and efforts to increase
its versatility are highly encouraged.
The capabilities of the program, while quite
significant, can still be improved. The present version is
designed for an "in-core" solution technique, which
restricts the problem size to within the computer core-size.
The capacity to handle large problems may be increased
through the use of some "out of core" technique for solving
the system of equations and even for building the mesh data.
In that case the size of the problems treated would be
restricted only by the availability of external storage
devices.
In order to check the mesh Input and to easily interpret
the output, a graphics module must be included in the
program.
Although the time integration module seems to provide
reliable results in linear and some nonlinear problems, it
should be subjected to further research in order to test it
with different types of nonlinear analysis.
46
0.1
«3" \0 CTl i—
I
et \0 0\ r-i -rt <£> CTi
i—I iH H N N N N0.2
to
Figure 1. Mesh for Hollow Cylinder Problem
Figure 2. Eight -Node Serendipity Quadrilateral
47
Figure 3. Local Node Number Sequence for a QuadraticLagrangian Element
48
Figure 4. Local Node Number Sequence for a 21 -NodeThree Dimensional Element
49
-,001
CO
Ji
O
s•H
1H
.1
50
(a)
Figure 6. Zienkiewicz Three-Time Level Schemes
51
Figure 6. (continued)
52
(e)
Figure 6. (continued)
53
H *-
Tuo-Tlnm Laval SchemeTurn exact start ing point*Stap atze - 2.0Theta - 8.S
— exact solution+ FEAP solution
W Al
Time
(a)
Two-Tlma Level SchemeTurn exact starting pointsStep size - 2.0Theta - 0.BB7
i i i ucj w -«• <o a>
Time
cuCU N at
018CO
(b)
Figure 7.
54
Turn-Time Laval SohameTwo oxnot starting pointsStep alza - 2.0Thete - 8.75
Tim©
sOJ
OJ01 OJ OJ
CD S«»
CO
Two-Ttme Level SchemaTwo exaot starting pointsStep alza - 2.0Theta - 0.878
to
Time
<9M MM 01
CD(U
s
Figure 7. (continued)
55
Tao-Tim Lava) SohemeTwo exaot etarttng. pointsStep elze - 10.0That* -0.5
exact solution+ FEAP solution
_i_s
Tim©
s s s a0> 8
(a)
Tan-Tina Laval SchemaTwo exaot starting pointsStap size - 10.0Thata - 0.667
s sto
Time
s s s am
00
Figure 8.
56
Tee-Tlee Laval SoheaeTwo axset atartlno. point*Stop size - 10.0That* - 0.79
—i
8S sto
9to
sis.
Time
(c)
Two-Time Level SchemeTwo exeat starting pointsStep size •* 10.0That a - 0.878
8sIO
sis.
aa
Time
(d)
Figure 8. (continued)
57
1.1
1
.9 A
.8 . / \° 7 • y3•
C.
a * 3. f\
.• .4 1
.31
.2 11
.1
^~s N/
Three-Time Lsval SchemeTun exact start Ing pointsStep size - 2.0Bets - 0.333Gamma m 0.5
exact solutionFEAP solution
7^f
-i-Al Al CD
Time
C9Al
Al to s
(a)
Three-Time Level SchemeTwo exsot starting pointsStep size - 2.0Beta - 0.75Gamma - 1.0
CO
Time
BAl
m <rAl Al
COAl
COAl
a
00
Figure 9.
58
Thrae-Tlme Laval SchemaTwo exact starting pointsStep etze - 8.8Bet* - 0.646Gemma - 1.2184
CO
Time
so»
01Ol oi
toni
S
(c)
Three-Time Level SchemaTwo exact starting pointsStep size - 2.0Beta - 0.8Gamma -1.5
Time
Figure 9. (continued)
MM toM CO
01iso»
59
1.1
1
.9
.8
o .7
3+»id
c
.6
-/Three-Time Leva)Turn axact start
1
Schemeng points
n Step size - 2.0Q.g
-//Beta - 0.9
.4
.3
Gamma -1.5
.2 11
.1
°J1 Al <e <0 cd 9 w * (D a s Alai
V CDAl Al
09Al
SMTime
(e)
Three—Time Level SchemeTwo exaot starting pointsStep slzs - 2.0BETR - 1.0Gamma ** 1.9
Time
Figure 9. (continued)
AlAl
(0Al Al
s
60
\// \ /
V\/
Thr*e-Tl«e Level SchemeTwo exsot etsrtlng point*Step elze - 18.0Bete 8.333Gem* - 8.9
exact solutionFEAP solution
S
Time
(a)
I.3r
Three-Tine Level SchemeTwo exsot at»rting point*Step elze - 18.8Bet* - 8.73<S«nm* -1.0
—i
Ss s sn
Time
(b)
Figure 10.
61
Three-Tlaa Laval SohaaaTwo axaot atartino. polntaStap alza - 10.8Bat* - 0.646Gamma - 1.2184
—i
8s sID
sto
s s•
Time
(c)
Thraa-Tlma Laval SohemaTwo axaot atartlng potntaStap alza - 18.8Beta - 8.8Gamma -1.5
s (9 S s«D
s 8
Time
(d)
Figure 10. (continued)
62
Thraa-T1«a Laval SonomaTwo ox*ot starting pointsStop atza - 18.0Bata -0.9Gaana - l.S
sto s s
IN.sa
Time
(e)
Thrae-Tlma Laval SoheaaTwo axaot atartlng potntaStap atza - 10.0Bata - 1.0Gamma -1.5
s» s«D
sIV
Time
(£)
Figure 10. (continued)
63
o e-
e--At -At/2
1 1-
At/2 At
time
(a)
J*/
/
-At -At/2
(b)
—I HAt/2 At
time
Figure 11. Step Change in the Loading Term
64
Two-Tlme Level SohemeStep elze - 2.0Start at time - 0.0That* -0.5
exact solutionFEAP solution
W T 05
Time
aw s SB S
1.1
I
.8
.8
° 71-
34>
C.8 -
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C APFEM3IX ACC LSEP INSTRUCTIONS FOR FEAPCC A.l TITLE ANC CONTROL INFORMATIONCCC... TITLE CAFC-FCRMAT(2QA4)C THE TITLE CARC ALSO SERVES AS A START CF FRCELEM CARC,C THE FIRST FOUR (4) COLUMNS MUST CONTAIN THE STARTC WCRD FEAFCC COLUMNS DESCRIPTION VARIABLEC 1 TC 4 MUST C^TAIN FEAP TITL(l)C 5 TO EC ALPHANUVERIC INFORMA- TITL ( I ) ,1 = 2 ,20C TION TO EE PRINTEDC WITH OUTFLT AS PAGEC HEADER,CCC... CCNTFCL CARC-F0RMAT(7IS)CC COLUMNS DESCRIPTION VARIAELEC 1 TC 5 NUMBER CF NCCES NUMNPC 6 TC 1C NLMBER CF ELEMENTS NUMELC 11 TC 15 NUMBER CF MATERIAL SETS NUMMATC 16 TC 20 SPATIAL CIMEf^SION NDMC (MAXIMLM, LP TO 3)C 21 TC 25 NUMBER CF UNKNOWNS NDFC PER NOCE (MAXIMUM,C LP TO 6)C 26 TC 20 NUMBER CF NOCES PER NENC ELEMENT (MAXIMUM)C 31 TC 25 NOT USEDC
80
cccc.ccccccccccccccccccccccccccccccccc.cccccccccccccccccccccccc.ccccccccccc
A. 2 CATA INPUT: MACRO CONTROL STATEMENTS
INPUT MACFC CCNTROL CARCS-FORMAT ( A4)THE INFLT CF EACH OAT* SEGfENT IS CONTROLLED EY THEVALUE ASS1GNEC TO CC. THE FOLLOWING VALUES ARE ADMI-SSIBLE ANC EACH CC CAPC MUST BE IMMEDIATELY FGLLOWECEY THE APPROPRIATE DATA.
CC VALLECOORFCLA
SPHE
ELEMPATEBOLNFORC
TEMPPRIN
NOPRPAGEENC
DATA TC BE INPUTCOORDINATE DATACONVERT PCLAR TO CARTESIANCOORDINATESCONVERT SPHERICAL TO CARTESIANCOORCINATESELEMENT DATAMATERIAL DATANODAL ECUNCARY CONDITICN CATAPRESCRIBED NODAL DISPLACEMENT/FORCEDATA. FCR HEAT TRANSFER PROBLEMSTHIS IS PRESCRIBED NODAL TEMPERA-TURE/FLUX CATATEMPERATURE DATAPRINT SLBSEQUENT MESH DATA
(DEFAULT MODE)DO NCT FP^T SUBSEQUENT MESH CATAPRINTED OUTPUT CONTROLMUST EE LAST CARD IN MESH DATA,TERMINATES MESH INPUT
EXCEPT FCR THE ENC CAPC THE DATA SEGMENTS CAN BEIN ANY CRDER. IF THE VALLES CF BOUN,FCRC, CF TEMPARE ZEFC, NC INPUT CATA IS REQUIRED.
COORDINATE CATANUST IMMECIATELTHE CCCFCINATEAND THE VALLETHE VALUES CF (
THE VALLE INFLTNOCAL COGRCINATLINE DESCFIEECCARDS. THE VALUUSING THE N ANCTHE SECLEhCE N,AS A NEGATIVE NSIGN WILL EE CH
- FORMY FOLLCCATA CAF THE CXL (I), I
ON THEES CANBY THEE OF THNG ONN+NGtN
UMBER,ANGED.
AT(2I5,7F10.0)W A COOR MACRO CARD.PC CCNTAINS THE NODE NUMBER NCCRDINATES FOR THE fsCCE. CNLY=1,NCM ) ARE USED, WHERE NDM ISCCNTROL CARD.
EE GENERATED ALONG A STRAIGHTVALUES ON TWO SUCCESSIVEE NCCE NUMBER IS COMPUTEDTHE FIRST CARD TO COMPLLTE+2*NG, ETC. NG MAY BE INPUTIF IT HAS INCORRECT SIGN THENODES NEED NOT BE IN ORCER.
COLUMNS DESCRIPTION1 TC 5 NCDE NUMEER
TC 10 GENERATCR INCREMENTTC 2C XI COORCINATETC 3C X2 COORCINATETC 40 >3 COORDINATETERMINATE WITH ELANK CARD(S)
61121il
VARIABLENNGXL(1)=X(1,N)XL(2)=X(2,N)XL(3I=X(3,N)
FCLAR TC CARTESIAN CONVERSION CATA-FORMAT (3 15 ,2F10.0
)
NUST II-MECIATELY FOLLCW A FOLA MACRO CARD.THE FCLAF CCCRCINATES (P,THETA) MUST BE PREVIOUSLY IN-PUT AS CCCRCINATE DATA (MACRC CARD COOP) WHERE R IS XIANC THETA IS X2. THE ISOCAL POLAR COORDINATES CF NODESM, NI + INC, M + 2*INC,..., NE ARE CONVERTED TCCARTESIAN COORDINATES.
COLU*NS1 TC 5
6 TC 10
DESCRIPTIONFIRST NOCE TC BECCNVEPTECLAST NODE TO BE
VARIABLENI
NE
81
CCNVERTEC11 TC 15 NODAL INCREMENT16 TC 25 X COORCINATE OF THE
CRIGIN26 TC 25 Y COORDINATE OF THE
CRIGINTEPMNATE WITH ELANK CARCC5!
CC 11 TC 15 NODAL INCREMENT INCC 16 TC 25 X COORCINATE OF THE XOCC 26 TC 25 Y COORDINATE OF THE YOCCCC
C... SPHERICAL TC CART. CCNVERSION OATA-FORMATOI 5 ,2F10.0)C fLST IMMEDIATELY FCLLCW A SPHE MACRO CARC.C THE SPHERICAL COORDINATES (R,THETA,PHI ) MUST BE PRE-C VICUSLY INFLT AS COORCINATE DATA (X1,X2,X3) FCLLQWINGC PACRC CAFC CCCR. THETA IS THE ANGLE FORMED BY R ANC XC AND PH IS THE ANGLE FCRMED BY R AND Z.C THE NCCAL SPHERICAL CCCPCINATES OF NODES NI, M + INC,C M42*INC,...,NE ARE CONVERTED TO CARTESIAN CCCFDINATESCC COLUMNS DESCPIFTICN VARIABLEC 1 TC 5 FIRST NCCE TC BE NIC 6 TC 10 LAST NCCE TC BE NEC CCNVERTECC 11 TC 15 NOPAL INCREMENT INCC 16 TC 2C X COORDINATE OF THE XOC ORIGINC 26 TC 25 Y COORCINATE OF THE YOC CRIGINC 36 TC 45 Z COORDINATE OF THE ZOC CRIGINC TEPMNATE WITH ELANK CARC(S).CCCC... ELEMENT C4T* - F0RMAT(16I5)C MLST INMECIATELY c OLLCW AN ELEM MACRO CARD.C THE ELEMENT CATA CARD CCNTAINS THE ELEMENT NUMBER,C MATERIAL SET NUMBER (WHICH ALSO SELECTS THE ELEMENTC TYPE, SEE MATERIAL PRCFEPTY CATA), AND THE SEQUENCEC CF NCCES CCNNECTEO TO THE ELEMENT. IF THERE ARE LESSC THAN NEN NCCES EITHER LEAVE THE APPROPRIATE FIELDSC ELANK CR PUNCH ZEROS.C ELEMENTS MUST BE IN ORDER. IF ELEMENT CARDS AREC CMITTEC THE ELEMENT DMA WILL BE GENERATED FRCMC THE PREVICLS ELEMENT WITH THE SAME MATERIAL NUMBERC ANC THE NCCES AUL TNCFEMENTED BY LX ON THE PREVIOUSC ELEMENT. GENERATION TC THE MAXIMUM ELEMENT NUMEERC CCCUPS WHEN t ELANK CARD IS ENCOUNTERED.CCC COLUMNS DESCRIPTION VARIABLEC 1 TC 5 ELEMENT NUMEER LC 6 TC 10 MATERIAL SET NUMBER IX(NEN1,L)C 11 TC 15 NODE 1 NUMSEF IX(1,U)C 16 TC 20 NCDE 2 NLMeEF IX(2,LJC ETC.C ETC. NCDE NEN NUMEER IX(NEN,L)C ETC. GENERATION INCREMENT LXC TEFMINATE WITH ELANK CARD(S).CCcC... MATERIAL FFCFERTY DATA - FCRMAT ( 15 ,4X , II , 17A4 )
C MUST IMMECIATELY FOLLCW A MATE MACRO CARD.CC COLUMNS DESCRIPTION VARIABLEC 1 TC 5 MATERIAL SET NUMBER MAc e tc «
C 10 ELEMENT TYPE NUMBER IELC 11 TC IS ALPHANUMERIC INFORMATION XHECC TC BE CUTPUTC EACH MATERIAU CARD MUST BE FOLLOWED IMMEDIATELY
82
C BY THE fATERIAL PRCFEPTY DATA REQUIRED FCR THEC ELEMENT TYPE IEL BEING USED, E. G. , SEE SECTION A.
4
C ANC A. 5 FCP HEAT TRANSFER ELEMENTS.CCCC... eCUNCARY RESTRAINT DAT* - FORMAT(16I5)C *UST IhNECIATELY FCLLCW A EOUN MACRO CARD.C FOR EACH NCDE WHICH HAS AT LEAST ONE DEGREE CF FREEDCMC WITH A SPECIFIED DISPLACEMENT, A BOUNDARY CONDITIONC CARD l"LST EE INPUT. THE CONVENTION USEC FCR BCUNDARYC RESTRAINTS ISC .ECO NO RESTRAIN, FORCE SPECIFIEDC .NE.C RESTRAINED. CISPLACENENT SPECIFIEDC VALUES OF FORCE OR DISPLACEMENT INPUT IN FORCCC COLUMNS DESCRIPTION VARIABLEC 1 TC 3 NODE NUMBER NC 6 TC 10 GENERATION INCREMENT NXC 11 TO 15 DOF 1 BCLNDARY CODE I DL (1 ) =ID(1 ,N)C 16 TC 20 DOF 2 ECUNCARY CODE IDL ( 2 )=I0( 2,N)C ETC.C ETC. COF MDF EGUNCARY CODE IDL(NDF)=C ID(NDF,N)C WHEN GENERATING BCUNCARY CONDITION CODES FCPC SUESECUENT NODES ICL IS SET TO ZERO IF IT WASC INPUT .GE.C, AND IS SET TO -1 IF INPUT NEGA-C TIVE. ALL DEGREES OF FREEDOM WITH NON-ZERCC COOES APE ASSUMED FIXEC.C TERMINATE WITH ELANK CARD(S).CCCC... NOCAL FCFCEC BOUNDARY VALUE DATA - FORMAT ( 21 5 ,7F10. 0)C NUST INNECIMELY FOLLOW A FORC ^ACRO CARC.C FOR EACH NCDE WHICH HAS A NON-ZERO NODAL FCRCE CRC CISPLACENENT A FORCE CARC NUST BE INPUT OR GENERATED.C GENERATION IS THE SANE AS FCR CO-ORDINATE CAT*. THEC VALUE SPECIFIED IS A FORCE IF THE CORRESPODING RES-C TRAINT CCCE IS ZERO ANC A CISPLACEMENT IF THE CORRES-C FGNDING RESTRAINT CODE IS NON-ZEPO.CC CCLUNNS DESCRIPTION VARIABLEC 1 TO 5 NODE NUMBER NC 6 TC 10 GENERATICN INCREMENT NGC 11 TO 2C DCF 1 FCRCE (CISP.) XL(1)=F(1,N)C 21 TO 20 CCF 2 FORCE (CISP.) XL(2I=F(2,N)C ETC.C ETC. DCF NDF FCRCE (DISP.) XL(NOF) =C F(NDF,N)C TEPMNATE WITH A ELANK CARC.CCCC... TEMPERATURE CATA CARD - FORMAT( 21 5.F10.0)C KST IfKECIATELY FCLLCW A TEMP MACRO CARC.C NOT USEC IN HEAT TRANSFER PROBLEMS.C FCR EACH NCCE WHICH HAS A NON-ZERO TEMPERATURE THEC VALUE KST EE INPUT. GENERATION OF VALUES CAN BEC FERFGRNEC AS DESCRIBEC FOR CO-ORDINATES.CC COLUMNS DESCRIPTION VARIABLEC 1 TC 5 NODE NUN8ER NC 6 TC 10 C-ENERATICN INCREMENT NGC 11 TC 2C NODAL TENFEPATURE XL(1I=T(N)C TERMINATE WITH ELANK CARD(S).CccC... PAGE CCNTFCL DATA - FCFNAT(Al)C NUST INMEEIATELY FOLLOW A PAGE MACRO CARDC THE VALUE IN CCLUMN 1 CCNTFOLS THE PRINTED OUTPUT
83
C ACCORDING TC THE FOLLCWING CONVENTION:C 1 TITLE IS WRITEN fil THE TOP OF A NEW PAGEC "THE FRINT5D OUTFIT IS CONTINUOUS WITHOUTC SKIFFING PAGES (CEFAUT MODS).
84
C A. 3 PROBLEM SCLUTION: MACRC PROGRAMMING CCMANCSCCC... MACRO FRCGRAMMING COMMANDS - FORMAT( 2( A4 »1X) , F10.0
)
C FGLLCW1NG IS A LIST OF MACFO INSTRUCTIONS WHICHC CAN BE LSEC TO CONSTRICT SCLUTION ALGORITHMS. THEC FIRST INSTRLCTION MUST eE A CARD WITH MACR INC COLUMNS 1 TC 4. THE INDICATED ISW VALUE IS USED BYC EACH ELEMENT ROUTINE TC FEFFCRM THE APPROPRIATEC CCMPUTATIGNS.CC COLUMNS COLUMNS COLUMNS DESCRIPTIONC 1 TC 4 6 TC 9 11 TO 20C CHEC FEPFCRM CHECK OF MESH (ISW=2)C CMAS CONSISTENT MASS FORMULATIONC (ISW=5)C CCNV CISPLACEMENT CONVERGENCE TESTC DATA REAC DATA **MACRO COMMANDS. AC **MACRC CARD IS INSERTEC AS CATAC FOLLOWING THE MACRC PROGRAMC CISP N PRINT NOCAL DISPLACEMENTS EVERYC N STEPS IN LOOPC **CT V SET TIME INCREMENT TC VALUE VC EIGE CCMPLTE DOMINANT EIGENVALUE ANDC VECTCR OF CURRENT MASS ANDC SYMMETRIC STIFFNESSC EXCD EXPLICIT CENTEREC CIFFEPENCEC INTEGRATION OF ECUATICNS CFC NCTICN USING LUMPEC MASS. FIRSTC CALL RESERVES MEMORY ONLY.C FORM FORM RIGHT HAND SIDE OF ECUA-C TICNS (ISW=6)C LMAS LLMPED MASS FORMULATION (ISW=5)C LCCP N LOOP N TIMES ALL INSTRUCTIONSC EETWEEN MATCHING NEXTC INSTRUCTION.C MESH INPUT MESH CHANGES (MUST NOTC CHANGE BCUNDARY RESTRA INTS ) .CATAC FOLLCWS MACRO PROGRAM. SEEC SECTION A. 2.C NEXT END CF LOOP INSTRUCTIONC CCE1 INIT INPUT INITIAL CONDITIONS ANDC CCNSTANTS OF INTEGRATION FORC FIRST ORDER ORDINARY DIFFEREN-C TIAL EQUATION SOLUTION (CATAC FCLLCWS MACRO PROGRAM)C 0CE1 LINE TWO-FOINT INTEGRATION SCHEMEC CCE1 CUAC THREE-POINT INTEGRATION SCHEMEC PRCP 1 INPUT PROPORTIONAL LCAD TABLEC (CATA FOLLOWS MACRO PROGRAM)C REAC COMPUTE NODAL RE ACTIONS ( ISW = 6
)
C SOLV SOLVE TANGENT EQUATI CNS. UPDATEC NCCAL DISPLACEMENTS.C STRE N FPINT ELEMENT VAPIAELES (E.G.,C STRESSES) EVERY N STEPS IN LOOPC ( ISW=4
)
C TANG SYMMETRIC TANGENT STIFFNESSC FORMLLATION (ISW=3)C TIME ADVANCE TIME BY CT VALUEC *TIM INDICATE EXECUTION TIME FCRC EVERY MACRO INSTRUCTIONC **TCL V SET SOLUTICN CONVERGENCE TOLE-C PANCE TO VALUE VC UTAN UNSYMMETRIC TANGENT STIFFNESSC FORMLLATION (ISW=3)C ENC END CF MACRO PROGRAM COMMANDS.C CATA FCR PROGRAM FOLLOWS INC CRDER OF USECcCC... FIRST CRCEF C.C.E. SCLVFR CATA
85
C MUST FCLLCW THE MACRC CARD END AND CORRESPONDS TOC THE MACFC INSTFUCTICN CCE1 INIT. THE FIRST CARDC INPUTS THE CCNTROL INFCPNATION AND THE NODAL INITIALC CISPLACE^ENTS ARE INPLT ON THE FOLLOWING CARDS.C NC ALTCMATIC TIME STSF ACJLSTMENT IS PERFORMEC IFC CUMAX CR CLMIN ARE LEFT BLANK OR ZERO.CC... CCNTRCL C/FD - FORMAT (4F10.0)CCC COLUMNS DESCRIPTION VARIABLEC 1 TC 1C INTEGRATION PARAMETER C5C THETA FCP TWC PCINTC SCHEME (DEFALLT VALUEC THETA=2/2)C 11 TC 2C INTEGRATICN FARAMETER CIC GAMMA FCR THREE POINTC SCHEMEC (DEFAULT GAMMA=1.5)C 21 TC 20 INTEGRATION PARAMETER C2C EETA FCF THREE PCINTC SCHEMEC (CEFAULT EETA=.8)C 21 TC «G MAXIMUM DISPLACEMENT DUMAXC ALLOWEDC 41 TC 50 MNIMUM CISPLACEMENT DUMINC ALLOWEDCC... INITIAL CCNCITICN DAT* - FCRMAT ( 2I5.7F10 .0 )
C FOR EACH NCDE fcHICH HAS A NON-ZERO INITIAL DISFLACE-C MENT A C*FC MUST BE INPUT CR GENERATED. THE GENERATIONC IS THE SAME AS FOR CCCFCIN/TE CATA.CC COLUr*NS DESCRIPTION VARIABLEC 1 TC 5 NCDE NUMEEP NC 6 TO 1C GENERATICN INCREMENT NGC 11 TG 10 CCF 1 DISPLACEMENT XL( 1 )=U ( 1+N-l
)
C ETC.C ETC. COF NDF CISPLACEMENT XL(NCF)=CC TERMINATE WITH BLANK CARD(S).CCCC... FFCFCRTICNAL LCAC CAPC - FCRMAT t2 I5.6F10 .0 )
C fi SIMPLIFIED PROPORTIONAL LOADING IS PERMITTEC WITHC PROF = Al + A2*TIME + A2*( SIN ( A4*TIME + A5))**LC VHERE THE COEFFICIENTS APE INPUT ON A CATA CAPCC FOLLOWING ThE END MACRC CARC ACCORDING TO THEC FCLLCWING TAeLE.CC COLUMN CESCRIFTIONC 6 TC 10 LC 11 TC 2C MINIMUM TIME FOR WHICH PRCP ISC COMPUTECC 21 TC 2C MAXIMLf TIME FCR WHICH FRCF ISC COMPUTEDC 21 TC 40 AlC 41 TC SO A2C 51 TC iC A3C 61 TC 1C A4C 71 TC EC A5CC
86
cccccccccccccccccccccccccccc.cccccccccccccccccccccccccccccccccccccccccccc
A. 4 TWC CIMENSIONAL FEAT TRANSFER ELEMENT DATA
NLST FCLLCW THE MATE MACPC CARC.THE TWC Cir<ENSIONAL HEAT TRANSFER ELEMENT IS CALLEDELPT02 ANC THUS THE ELEMENT TYPE NUMBER IN CCLUMN 10CF EACH NATEFIAL SET NLMeEF CARD MUST BE 2 WHEN THISELEMENT IS REQUESTED. THE SECOND CARD GIVES GENERALINFORMATICN ANC IS PREPAREC AS FOLLOWS:
CENEFAL INFCFMATION CATA - FORMAT< 5F10 .0,415
)
COLUMNS1 TC 1C
11 TC 2C
21 TC -C
21 TC *C
41 TC 5C
51 TC 55
56 TC 60
61 TC 65
66 TC 70
DESCRIPCONDUCTIVITY(IGNORED IFCEPENDENT)CONDUCTIVITY(IGNOREC IFCEPENDENT)SPECIFIC FEAIF TEMPEPATUCENT)DENSITY (IGNTEMPERATUREHEAT GENEFATUN!"!" VOLUMEIF TEMFEFATUDENT)NUMBER CF INFCINTS FEP C(DEFAULT 4)
GEOMETRY TYP.SO. 2 FCR flX
.NE.2 FCR PLTOTAL NUNEERWITH SPECIFICARY CCNCITIELEMENTS WITMATERIAL S C TCODE TO INCICF THE M7ERTIES IS TEMPCEPENDENT..EO.O IF ALLPROPERTIES.NE.O IF ANYRIAL PRCPERPERATURE CE
TIONIN X DIR.
TEMPERATURE
IN Y DIR.TEMPERATURE
T (IGNOREDRE DEPEN-
CRED IFCEPENDENT)ION PER(IGNOREDFE CEPEN-
TEGRATIONIRECTION
VARIABLED(l)
C(2)
ISYMMETRYANE GEOMETRYOF LINES
ED BGUN-CNS INF THE SAMENUMBER
CATE IF ANYIAL PRCPER-ERATURE DE-
MATERIALARE CONSTANTOF THE MATE-TIES IS TEM-PENDENT
D(3)
D<4)
C(5)
NGP
KAT
NLBC
INOL
THE FCLLCWING CATA ARE FECUIRED IF ANY OF THE MATERIALPROPERTIES IS TEMPERATURE CEPENDENT. THEIR CRCER ISCRUCIAL ANC TFCSE CARCS WHICH CORRESPOND TO CCNSTANTMATERIAL FFCFERTIES MUST BE OMITTED.
87
C... ^ATEFIAL FFCFEPTY COCE -F0FMAT(4I5)C THIS CARC IS ALLWAYS REQUIRED IF INOL.NE.O ANC MUSTC IMMEDIATELY FCLLOW THE GENERAL INFORMATION DATA CARD.CC COLUMNS DESCRIPTION VARIABLEC 1 TC 5 .EQ.O CONSTANT CONDUCTI- IKXC VITY IN X DIR.C .NE.O TEMP. CEP. KXC 6 TC 1C .ECO CCNSTANT CONDUCTI- IKYC VITY IN Y DIR.C .NE.O TEMP. CEP. KYC 11 TC 15 .EQ.O CCSTANT HEAT CA IRCCC PACITY (SPECIFIC HEAT *C DENSITY)C .NE.O TEfF. CEP. HEATC CAPACITYC 16 TC 2C .EQ.O CCNSTANT HEAT GE- IQC NERATICN PER UNIT VOL.C .NE.O TEMP. CEP. QCCC... CCNDUCTIVITY IN X DIRECTION TABLEC THE TEMPERATURE DEPENCENT CONDUCTIVITY IN THE XC CIPECTICN (KX) IS CALCLLATED BY LINEAR INTERPOLATIONC BETWEEN TfcC CCNSECUTIVE ENTRIES IN THIS TAeLE. THEC FIRST CARC OF THIS SET CF DATA INDICATES THE NUMBERC CF ENTFIES IN THE TABLE. IT MUST BE FOLLOWED BY THEC SAME NLMBEP CF CARDS VITH A PAIR OF VALUES (TEMPE-C PATURE ANC CORRESPONDENT PROPERTY) IN EACH ONE. THEIRC CRCEP IS CFLCIAL. THE FAIRS MUST BE ORDERED ACCORDINGC TO THE INCREASING VALUE CF TEMPERATURE.C CMIT IF IKX. ECOCC FIRST CARC -FOPMAT(I5)CC COLUMNS DESCRIPTION VAPIAELEC 1 TC 5 NUMBER CF ENTRIES OF NC THE TABLE TC BE INPUTCC TABLE CATA - FORMAT
(
2F1G.C)C FCR EACH FAIR CF VALUES A CARC MUST BE INPUT. THEC TOTAL IS ThE NUMBER SPECIFIED IN THE PRECEDING CARD.C LOWEST TENFERATURE IN THE FIRST CARD, SECOND LOWESTC IN THE SECCND CARD, ETC.CC COLUMNS DESCRIPTION VARIABLEC 1 TC 1C TEMPERATURE XX(I)C 11 TC 20 CONDUCTIVITY YYCI)CCC... CONDUCTIVITY IN Y DIRECTION TABLEC PREPARED IN THE SAME V»AY AS THE CODUCTIVITY IN THEC X CIRECTICN TABLE.C CMIT IF IKY. ECOCCC... HEAT CAPACITY TABLEC PREPARED IN THE SAME feAY AS THE CONDUCTIVITY IN THEC X CIRECTICN TAELE.C CMIT IF IRCC.ECOCCC... HEAT GENERATION PER UNIT VOLUME TABLEC PREPARED IN THE SAME WAY AS THE CONDUCTIVITY IN THEC > CIPECTICN TAELE.C CMIT IF IQ.EQ.CCcC... LINE BOUNDARY CONDITION DATA - FORMAT( 31 5 , 2F1C.0)C CMIT IF NLEC.ECO.C A CARC MLST INPUT FCR EACH LINE BOUNDARY CONDITION.C IF THE SAME LINE IS SUBJECTED TO MORE THAN ONE TYPE
88
C CF BCUNCAPY CCNDITTCN, A CARD MUST BE USED FCP EACHC CNE CF THESE TYPES.C 1J-E TCTAl KNEER OF CAPCS NUST BE NLBC.C T»-E PRCPER7> VALUE IS CEFINED:C FCR CCfvVECTICN - CONSTANT HEAT TRANSFER COEFFICIENTC UGNCREC IF KBCOND ( I ) .EC .4)C FOR FLLX - FLUX PER UNIT AREAC FCR RADIATI^ - PRODUCT CF EMISSIVITY BY STEFAN-C BOLTZMAN CCNSTANTC ThE AMEIENT TEMPERATURE IS IGNORED FOR FLUX BCUNDARYC CGr^DITlCNCC COLUMNS DESCRIPTION VARIABLEC 1 TC f ELEMENT ^"PEER KEL(I)C 6 TC 1C CODE TO INDICATE BOUN- KBCONDdlC CARY CCr^CITKN TYPEC .EO.l FLLXC .EQ.2 CONVECTION (CONS-C TANT CCEFFICIENT)C .5Q.3 RADIATIONC .5Q.4 CONVECTION (TEMP.C DEP. CCEFF.)C 11 TC 15 CODE TO INDICATE LINE KLINE(I)C .EQ. 1 S=+l LINEC .=Q.-1 S=-l LINEC .=0. 2 T=+l LINEC .EQ.-2 T=-l LINEC 16 TC 2f PROPERTY VALLE PRCFB(I,1)C 26 TC 25 AMBIENT TEMPERATURE PR0PB(I,2)CC
89
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A. 5 THREE CIMENSIONAL HEAT TRANSFER ELEMENT DATA
fliST FCLLCfc THE MATE MCRC CARD.THE THREE CIMENSIONAL HEAT TRANSFER ELEMENT IS CALLEDELNTQ3 ANC THUS THE ELEMENT TYPE NUMBER IN CCLLMN 10CF EACH MATERIAL SET NLMEEP CARD MUST BE 3 WHEN THISELEMENT IS REQUESTED. THE SECOND CARD GIVES GENERAL1NFCRMTICN ANC IS PREPAREC AS FCLLOWS:
GENERAL INFCRMATION DATA - FORMAT (6F10 .0,415 )
COLUMNS1 TC 1C
11 TC 2C
21 TC 2C
21 TC AC
41 TC 50
51 TC 6C
61 TC 65
66 TC 7C
71 TC 75
76 TC EC
DESCONOUCTI(IGNORECCEPENDENCONDUCT!(IGNORECCEPENDENCDNDUCTI(IGNORECCEPENDENSPECIFICIF TEMFECENT)CEMSITYTEMPERATHEAT GENUNIT VCLIF TEMPECENT)NUMBER CPOINTS F(DEFAUL
GEOMETRY•EQ.2 FC.NE.2 FCTCTAL NLWITH SPECARY CCNELEMENTSMATERIALCODE TCCF THE MTIES ISDEPENDEN.EO.O IFPROPER!
CRIPVI^YIF
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MATERIALRE CONSTANT
90
COLUNNS DESCRIPTION VARIABLE1 TC 5 .EO.O CCNSTANT CONDUCTI-
VITY IN X CIR..NE.O TEfP. CEP. KX
IKX
6 TC 1C •EQ.O CONSTANT CONDUCTI-VITY IN Y DIR..NE.O TENF. CEP. KY
IKY
11 TG 15 .EQ.O CONSTANT CONDUCTI-VITY IN Z CIR..NE.O TEMF. CEP. KZ
IKZ
16 TC 20 .EQ.O CCSTANT HEAT CA IROC
C .NE.O IP ANY OF THE MATE-C RIAL PPCPERTIES IS TEM-C PERATLRE DEPENDENTCCC THE FOLLOWING CATA ARE REQUIRED IF ANY OF THE MATERIALC PROPERTIES IS TEMPERATURE CEPENDENT. THEIR ORCER ISC CRUCIAL ANC THCSE CARCS WHICH CORRESPOND TC CCNSTANTC MATERIAL FFCPERTIES MUST BE OMITTED.CC... MATERIAL PROPERTY CODE -F0PMATC4I5)C THIS CARC IS ALLWAYS REQUIRED IF INOL.NE.O ANC MUSTC IMMEDIATELY FCLLOW THE GENERAL INFORMATION CAT* CARC.CCCCccccccccC PACITY (SPECIFIC HEATC DENSITY)C .NE.O T<=NP. CEP. HEATC CAPACITYC 21 TC 25 .EQ.O CCNSTANT HEAT GE- IQC NERATICN PEP UNIT VOL.C .NE.O TENF. CEP. QCCC... CONDUCTIVITY IN X DIRECTION TABLEC THE TEI-FEFATURE DEPENCENT CONDUCTIVITY IN THE X
C DIRECTICN (K>> IS CALCLLATEC BY LINEAR I NTEPPCLAT IONC EETWEEN TWC CONSECUTIVE ENTRIES IN THIS TABLE. THEC FIRST CAFC CF THIS SET CF CATA INDICATES THE NUMBERC CF ENTRIES IN THE TABLE. IT MUST BE FOLLOWED eY THEC SACS NLN6ER CF CARDS WITH A PAIR OF VALUES (TEMPE-C PATURE ANC CCFPSSPCNCENT FFOPERTY) IN EACH ONE. THEIRC CRCER IS CRLCIAL. THE PAIRS MUST BE ORDERED ACCCRDINGC TC THE INCREASING VALUE CF TEMPERATURE.C CMIT IF IKX. ECOCC FIRST CAFC -FCFMAT(I5)CC COLUMNS DESCRIPTION VARIABLEC 1 TC 5 NUMBER CF ENTRIES CF NC THE TABLE TC BE INPUTCC TABLE CATA - FORMAT (2 F10.0)C FCR EACH PAIR OF VALUES A CARD MUST BE INPUT. THEC TOTAL IS THE NUMBER SPECIFIED IN THE PRECEDING CARC.C LOWEST TEI^FERATURE IN THE FIRST CARD, SECCND LCWESTC IN THE SECCNC CARD, ETC.CC COLUMNS DESCRIPTICN VARIABLEC 1 TC 10 TEMPERATURE XX(I)C 11 TO 2C CONDUCTIVITY YYU)CcC... CONDUCTIVITY IN Y DIRECTION TABLEC PREPAREC IN THE SAME WAY AS THE CODUCTIVITY IN THEC X CIPECTICN TAELE.C CMIT IF IKY. ECOCCC... CONDUCTIVITY IN Z DIRECTION TABLEC FREPARED IN THE SAME WAY AS THE CODUCTIVITY IN THEC X CIRECTICN TABLE.
91
C CMIT IF IKZ.EC.OCcC... FEAT CAFACITY TABLEC PREPARED IN THE SAME WAY AS THE CONDUCTIVITY IN THEC X CIRECTICN TABLE.C CMIT IF IRCC.EC.OCCC... HEAT GENERATION PER UMT VCLUME TABLEC FREPAREC IN THE SAME WAY AS THE CONDUCTIVITY IN THEC > DIRECTION TAELE.C CMIT IF IC.EG.CCCC... SURFACE ECLNCARY CONDITION DATA - FORM AT (315 ,2F10.0 )
C CMIT IF NSEC.EC.O.C A CARD MLST INPUT FCR EACH SURFACE BOUNCARY CCNCITICN.C IF TFE SAME SURFACE IS SUBJECTED TO MORE THAN CNE TYPEC CF BCUNCAFY CCNDITICN', A CARD MUST BE USED FCR EACHC CNE CF THESE TYPES.C TFE TCTAL NLMEER OF CARCS MUST BE NSBC.C "THE PRCPERT> VALUE IS CEFINED:C FOR CONVECTICN - CONSTANT FEAT TRANSFER COEFFICIENTC (IC-NCPEC IF KBCCNCm.EC.4)C FOR FLLX - FLUX PER UMT AREAC FCR RACIATICN - PRODUCT OF EMISSIVITY BY STEFAN-C BOLTZMAN CCNSTANTC THE AMEIENT TEMPERATLRE IS IGNORED FOR FLLX BCLNDARYC CCNDITICNCC COLUMNS DESCRIPTION VARIABLEC 1 TC 5 ELEMENT NUMBER KEL(I)C 6 TC 1C CODE TC INDICATE BCUN- KBCCNC(I)C CARY CONCITICN TYPEC .EQ.l FLLXC .EQ.2 CCNVECTION (COMS-C TANT CCEFFICIENT)C .EQ.3 R/SCIATIONC .EQ.4 CCNVECTION (TEMP.C DEP. CCEFF.)C 11 TC 15 CODE TC INDICATE SURF. KSURFUIC .EQ. 1 R=+l SURFC .EQ.-l P=-l SURFC .EG. 2 S=+l SURFC .EO.-2 S=-l SURFC .EO. 3 T=+l SURFC .EQ.-3 T=-l SURFC 16 TC 25 PROPERTY VALUE PROPBUtl)C 26 TC 11 AMBIENT TEMPERATURE PRCFB(I,2)
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LIST OF REFERENCES
1. Arpaci, V. S., Conduction Heat Transfer Addi son-WesleyPublishing Company / 1966.
2. Zienklewicz, 0. C. and Parekh, C. J., "Transient FieldProblems: Two-Dimensional and Three-Dimensional Analysisby Isoparametric Finite Elements," International Journalfor Numerical Methods in Engl neeri nJT. \T. 2, p. 61-71,January-March 1970.
3. Lew, G. T., A Three Dimensional Solution of theTransient Field Problem Using Isoparametric FiniteElements , M.S.M.E. Thesis, Naval Postgraduate School,Monterey, 1972.
k. Zienklewicz, 0. C, The Finite Element Method , 3rd ed.,McGraw-Hill, 1977.
5. Bathe, K. J. and Wilson, E. L., Numerical Methods inFinite Element Analysis , Prentice-Hall, 1976.
6. Liniger, W., "Optimization of a numerical integrationmethod for stiff systems of ordinary differentialequations," IBM Research Report RC2198 , 1968.
7. ^ Wood, W. L., "On the Zienklewicz Three- andTwo-Ti me- Level Schemes Applied to the Integration ofParabolic Equations," I nternatlonal Journal forNumerical Methods In Engineering, v. 12, p. 1717-1726,1978.
8. Lees, M., "A Linear Three-Level Difference Scheme forQuasi linear Parabolic Equations," Mathematics ofComputation, v. 20, p. 516-622, 1966.
9. Hogge, M. A., A Survey of Direct Integration Proceduresfor Nonlinear Transient Heat Transfer, presented at theInternational Conference on Numerical Methods In ThermalProblems, University College, Swansea, U. K., 2-6 July,1979.
10. Wood, W. L. and Lewis R. W., "A Comparision of TimeMarching Schemes for the Transient Heat ConductionEquation," International Journal for Numerical Mpfhnd<;in Engineering, v, 9, p. 679-689, 1975.
11. Gresho, P. M. and Lee, R. L., Don't Supress the WigglesThey're Telling You Something!, presented at the
155
Symposium on Finite Element Methods for ConvectionDominated Flows, ASME Winter Annual Meeting, New York,2-7 December, 1979.
12. Haj i-Shei kh, A. and Sparrow, E. M., "The Solution ofHeat Conduction Problems by Probability Methods,"dovirhai Of" tteat- Tran s fe
r
, v. 89, p. 121-131, May 1967.
156
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