Jan 2, 2010 - [21] S. Steinberg, P. J. Roache,"Symbolicmanipulation and Computational fluid dynamics" ...... [35] Isaac Fried, "Shear in C O and C 1 bending.
NASA
Contractor
Report
4544
/_ P
Application of Symbolic and Algebraic Manipulation Software in Solving Applied Mechanics Problems Wen-Lang
Tsai and Noboru
Kikuchi
(NASA-CR-4544) APPLICATION OF SYMBOLIC AND ALGEBRAIC MANIPULATION SOFTWARE IN SOLVING APPLIED MECHANICS Univ.)
PROBLEMS 164 p
N94-13798
(Michigan
Unclas
HI/31
CONTRACT
NCA2-136
August1993
N/ A National Aeronautics and Space Administration
0187791
NASA Contractor
Report 4544
Application of Symbolic and Algebraic Manipulation Software in Solving Applied Mechanics Problems Wen-Lang University
Tsai and Noboru
Kikuchi
of Michigan
Department Ann Arbor,
of Mechanical
Engineering
Michigan
Prepared for Ames Research Center CONTRACT NCA2-136 August
1993
National Aeronautics and Space Administration Ames Research Center Moffett Field, California 94035-1000
TABLE
OF
CONTENTS
SUMMARY CHAPTER I.
HISTORY
OF SYMBOLIC
1.1 Introduction 1.2 History
ALGEBRAIC
MANIPULATION
......
.........................
of SAM
systems
2 ....................
3
1.3 Conclusion
.........................
11
1.4 References
.........................
13
II. SURVEY
OF THE LITERATURE
MANIPULATION
2.2 Review
ON SYMBOLIC
AND
ALGEBRAIC
........................
2.1 Introduction
15
.........................
of SAM
applications
15 in engineering
..............
16
2.3 Conclusion
.........................
24
2.4 References
.........................
26
III. CAPABILITIES
OF THE SYMBOLIC
3.1 Introduction 3.2 What
can the symbolic
MANIPULATORS
29 29
and algebraic
OF SYMBOLIC
AUTOMATIC
PROBLEM
4.1 Introduction
manipulators
do?
.........
29 51
AND ALGEBRAIC
FORMULATION
MANIPULATION
TO
..............
52
.........................
4.2 Derivation
of equation
4.3 Autornatic
tensor
4.4 Approximation
4.6 Triangular
ALGEBRAIC
.........................
APPLICATION
4.5 Template
AND
........................
3.3 References IV.
AND
of motion
formulation of a function
for nonlinear stiffness
52 by SAM
..............
for shell problem by Fourier
numerical
analysis
and mass matrix
series
53
............
57
............
65
..............
construction
6_ ...........
72
iii PreCEDING
PAGE
BLANK
NOT
FILMED
4.7 Closed
form solution
4.8 Significance
and conclusion
4.9 References
.........................
V. APPLICATION
method
analysis
evaluation
5.7 References
94 problern
...........
98
....................
100
manipulation and result
...............
treatment
1(13
..............
107 .........
PROBLEMS
6.1 Introduction
AND ALGEBRAIC
MANIPULATION
115
6.5 References
115 ....................
for solving
and algebraic
116
shell problem manipulation
by FEM
...........
application
to plate problems
.........................
CONCLUSIONS
7.5 Contributions
and algebraic
and continuations
rnanipulation
limitations
manipulation
of this study
119
145
and mathematical
and algebraic
.....
145
........................ of symbolic
118
14(I
.......................
swelling
7.4 Symbolic
TO
....................
formulation
6.3 Methodology
7.2 Advantages
113
........................
6.2 Preliminary
7.6 Prospects
93
OF SYMBOLIC
PLATE
7.3 Internal
RING 93
................
APPLICATION
7.1 Introduction
TO A
....................
of symbolic
5.6 Numerical
VII.
MANIPULATION
--- RIGID-PLASTIC
for tile ling compression
element
6.4 Symbolic
80 89
AND ALGEBRAIC PROBLEM
Ik)rmulation
5.5 Application
THE
.......
.........................
5.2 Preliminary
5.4 Finite
element
........................
5.1 Introdt, ction
5.3 Matrix
of four-node
91
NONLINEAR
COMPRESSION
matrix
...................
OF SYMBOLIC
MATERIALLY
VI.
of stiffness
............
and education
................... of this research
iv
..........
..........
145 146 148 149
.............
15(1
7.7 References
.........................
152
APPENDIX
A
..........................
153
APPENDIX
B
..........................
154
LIST
OF
FIGURES
3.1
Error
between
3.2
Beam
under
4.1
Dynamic
4.2
Pictorially
4.3
Convergence
4.4
Behavior
5.1
Configuration
5.2
Plot of frictional
5.3
Arctangent
5.4
Physical
5.5
Discretization
5.6
Deformed
configuration
after
the 8th stage
for m=().5
.........
108
5.7
Deformed
configuration
after the 7th stage
for re=l).0
.........
108
5.8
Velocity
distribution
of m=0.5
case
...............
109
5.9
Velocity
distribution
of m=O.I) case
...............
110
5.10
Effective
stress
distribution
..................
111
5.11
Effective
stress
distribution
..................
112
6.1
Plate with hole under
6.2
Stress
6.3
Physical
6.4
Deformed
6.5
Stress
distribution
6.6
Three
different
6.7
Convergence
6.8
Deformed
the solutions uniform
system
of REDUCE
load
notations
of Fourier
of optimal
of domain
vs. variable
vs. relative
shape
of plate
of plate
under
...........
problem
tension
problem
bending
load
..........
load
97
..........
130
...........
134
............
for testillg
the convergence
of plate bending
solution under
125
...........
sizes of mesh
vi
97
1()1
tension
bending
plate
95
.............
..............
of clamped
94
.............
fiber of plate
shape
of outer
71
................
uniaxial
.53
68
.............
stress
uniaxial
of plate under
configuration
x3
for ring compression
uniform
manipulation
............
velocity
section
36
58
conditions
of fi'ictional
of ring cross
and algebraic
................
and boundary
approximation
distribution
of syrnbolic
approximation
location
configuration
......
41
of shell
series
stress
elimination
...................
for demonstration
vector
and Gauss
136 137 of plate
bending
.............. unilbl'm
.138 138
transverse
load
.....
139
LIST
OF
TABLES
Table 1.1
List of symbolic
and algebraic
systems
3.1
Comparison
of solutions
for 5*5 Hilbert
matrix
............
35
3.2
Comparison
of solutions
for 6*6 Hilbert
mauix
............
35
3.3
Comparison
of solutions
for 7*7 Hilbert
matrix
...........
_35
vii
...............
12
NOMENCLATURE
SAM
symbolic
V
strain
f_
domain
N
stress
resultant;
Na_
stress
tensor
M
moment
resultant;
M_a
moment
tensor
K
stiffness
matrix
and 'algebraic
energy;
manipulation
potential
energy;
velocity
of integration shape
function
mass matrix
strain £
smoothing
parameter
curvature
of curve coordinates
X,
y,
Z
Cartesian
U,
v,
W
displacement
components
R
radius
L
linear operator;
t
time
I
imagimu'y
q
load per unit; rotation
P
position
k
spring
0
angle
T
kinetic
energy
at_
metric
tensor
Da
operator
of covariant
D
material
property
da_
curvature
tensor
Ea_
strain
Fa
force components
Pa_
displacement
K_ 0
bending
V
poisson's
ratio
IJl
Jacobian
matrix
of curve Lagrangian;
length
symbol tensor
vector constant
tensor;
derivative
matrix;
Young's
flow matrix
modules
in surface
of shell
gradient
tensor
determinant
of displacement
gradient
...
VIII
wi
weight
si
area coordinates
_,_
natural
_ij _ 0",
.f U
C
parameter of triangular
coordinates
Cauchy
element
of quadrilateral
stress
effective
su'ess and strain
friction
rate
stress
velocity
vector
Lagrange
multiplier
d u
die velocity
A
alea
AU
difference
d
Generalized
h
thickness
m
friction
g
yield shear
between
U and U I
displacement
j
vector
of shell factor stress
ix
element
SUMMARY
Compared people
to numerical
in the research
simply
field.
interpreted
computations
analysis,
As its name
results
b, and c in the quadratic
manipulation algebraic
the example
above,
can be observed.
symbols
For example,
-4b
can be
the coefficients
and the results
x:.-
to
throughout
do not need to be known
, (-b
a,
in order
to
will be
z-4ac) 2a
three coefficients
can be specified
at least two unique
First unlike
are exact
are the same as those
derived
In the first chapter
of this report,
of symbolic and algebraic
chapter
and algebraic
and
solutions
chronologically
through
six demonstrate
four
describes
the automatic
the materially
nonlinear,
plate problems.
The final chapter
The report
applications.
proposes
rigid-plastic
will
be
to avoid
Due to this breakthrough,
available. for improving
In addition,
reviews
the execution
three
efficiency
the
capability
to
conclusions
difficuilties
existing
the difficulties
found
of numerical
Chapters
manipulation.
Chapter
chapter
six discusses
of this report.
in the symbolic
previously
and algebraic
accomplishes
insolvable
in this research
programs.
five covers
and chapter
and successfully
of some
of symbolic
examples.
problems,
problem,
is
the application
The capabilities
by selected
mechanics
manipulation
regarding
and algebraic
the overall
one of the advantages
and
Second,
of human
and algebraic
field.
in chapter
the solution
from symbolic
is introduced.
the literature
ring compression
that there are some
and algebraic
by computer.
of symbolic
of applied
summarizes
a remedy
feasible
of symbolic
formulation
error
the extension
in the engineering
applications
of symbolic
the solutions
no round-off
the history
are demonstrated
four
It is well known
becomes
manipulation
manipulators
analysis,
by hand. Therefore
fomulations
The sencond
characteristics
numerical
and therefore
more sophisticated
introduced.
crucial
manipulation
retain
input to a computer
and
are required,
manipulation
solutions
become
forms.
ax2+bx+c=O
2a
values
can
is unfamiliar
as numbers. From
field.
equation
manipulation
and algebraic
which
+ 4bZ-4ac)
x_=
handle
symbolic
of symbolic
itself can be directly
(-b
and algebraic
operation
in terms
polynomial
find its roots. The equation
expressed
implies,
as a computerized
and express
If the numerical
symbolic
the
problems
is believed
to be
CHAPTER
HISTORY
1.1
OF
products might
and Algebraic
of modem say that SAM
ALGEBRAIC
MANIPULATION
severely
limits
addition
to numbers,
it think
more
on a variety
is a machine
intelligence
Can we ask a computer
which
finally
computers.
brain.
such as science,
industry
For instance,
mathematical
derivations
before
1953,
Therefore,
in
relations
for us? This is a to make
has had an impact although
has been so large
is
however,
"soul" to the computer,
originated
its impact
This viewpoint,
appropriate
and education.
sciences,
Traditionalists
of computers.
to do these analytical
This idea, which
computers.
numbers.
define
is one of the new
This is true if the viewpoint
capabilities
which
software
digital
led to the birth of SAM and added
like a human of fields,
as SAM)
only counts
symbols
is not as long as that of the classical history
of modern
which
artificial
there are many
book.
question
is a misuse
the emerging
(abbreviated
for use with highly developed
software
that a computer
in a calculus
Manipulation
technology
adopted
its history
that a record
of its
is deserving.
At the initiation systems
available
FORMAC,
on
of this dissertation
REDUCE
the
systems
Ironically,
or just skim
[1], includes
rewrite,
will get a complete
making important
them
details,
used today
and update
the history
over
historical
systems
market.
picture
briefly.
Only
This
more
documents
one book,
than
either
ignore in 1969
recent
since then. Therefore,
systems.
It is hoped
of SAM
the direction
purpose
systems.
a number years
the history by Jean
that the interested
of the chapter.
such
as
such as muMATH
it is necessary
Through
of SAM
old,
developments.
of the field and devote
is the ma.jor
how to run the SAM
systems.
ten
developed,
written
it is too old to cover
of the SAM
to grasp
were
were just being
most relevant
of the development
contribution.
than just knowing
Others
however,
the history
in 1986, there were already
of them
have been produced
they will be able
a further
effort
Some
and MACSYMA.
and MATHEMATICA.
major
AND
Introduction
Symbolic
great
SYMBOLIC
I
of these
E. Sammet Most of the to collect, researchers
an understanding themselves
of
towards
It is much
more
The first ideafor usingcomputersto do SAM can betracedbackto two mastertheses publishedin 1953([2],[3]). Threeyearslater,whatis believedto bethe earliestSAM system, called PM, wasdevelopedat IBM [4]. Now thereare manySAM systemson themarket for variouscomputers.Someof themaredesignedfor generalpurposeusage,while othershave beendevelopedfor particularapplications.Generallyspeaking,theevolutionof symbolicand algebraicmanipulationcanbeclassifiedintothreestages.Theyare: 1. The first generation(1953-1965)---softwareto appearin this generation was PM, ALGY, FORMAC,MATHLAB andALTRAN. Becauseof thelimitation of hardware capacity, the systemsin this stagewere small in size and immature in content. Therefore,mostof thembecameobsoleteor wererevised. 2. The 2nd generation(1966-1975)---software to appearin this generationwasREDUCE and MACSYMA. Thesesystemstook advantageof the improvementof hardware memorycapacity.They containmanybuilt-in functions,arelargeandarefor general purposeusage.All of themrunon themainframe. 3. The 3rd generation(1976-present)---somerepresentatives of more recentsystemsare muMATH, MATHEMATICA, and DERIVE. Unlike the systemsof the second generation,thesystemsin this generationaredesignedto run on microcomputers.This has
been
possible
microcomputers, or moderate
Details
of the histories
only
of
SAM
to
the
improvement
to the requirement
of
of most
users
memory
capacity
who just
need
in quick
manipulations.
of the systems
For the sake of clarification,
History
1.2.1
not
but also due
checks
individually. 1.2
due
will be described
a summary
in the following
is also included
in Table
subsections
1.1.
systems
PM
PM is believed developed
by George
Although
the first
Written
E. Collins
document
in assembly
addition,
to be the earliest
subtraction
performing
the same
coefficients.
Between
at the IBM research
was
language
published
1956 and
in 1966
algebraic center
of multiple-precision
on multivariate 1966
system
in Yorktown
[4] its beginning
for the IBM 701 computer,
and multiplication operations
computerized
polynomials
PM was reprogrammed
PM contained integers, with
in the world. Heights, dates
It was
New York.
back
to 1956.
the subroutines and
subroutines
multiple-precision
for the newer
for for
integer
IBM computers
(e.g.
709,
7090
with
various
and
7094),
and augmented
improvements
(e.g.
allocation).
In 1966 Dr. Collins
Wisconsin.
With
system
The
Alto,
ALGY
For example,
the PM system
replacement,
written
expressed
• OPEN
science
dynamic
god)
storage
at the University
was converted
PM, as the first SAM
Callender
: expanding : making
• FCTR
: factoring
• TRGA
: expanding
Which
of
to the SAC-I system,
was still
for example,
1961
to FORTRAN
SAM Lab-Philco
in the
Co. in Paio
[5]. It was interactive
as input,
with some
exponentiation,
0 and 1 were denoted
systems
and
deviations.
and all natural
integers
by 0/1 and 1/1, respectively.
It
such as :
in the parenthesis
substitution
a given
expression
the sin(a+b)
the authors
didn't
system,
similar
Development around
of ** to represent
the expression
is why
ALGY
at Western
and Sanford,
in a notation
as fractions,
it was one of the earliest
was started
a few commands,
• SBST
1.2.3
functions,
the $ was used instead
only contained
FORMAC
work
by Bernick,
expressions
Although
has few
developmental
California
allowed
come
which
into sin(a)cos(b)+sin(b)cos(a)
said that only
into extensive
is still popular
two hours
use, some
instruction
of its ideas
was were
enough
to use it.
succeeded
by the
today.
FORMAC
FORMAC
is an acronym
E. Sammet
and
Robert
July,
[1].
Five
1962
specifications
computer
improvement
themselves
(not
of FORmula
G. Tobey
months
was prepared
of extensive
IBM 7090/94 further
of computer
and
as integer
ALGY
world.
months
(such
of list processing
a professor
students,
new operations
in the field.
Although
were
became
In spite of its eventual
very significant 1.2.2
the incorporations
the aid of graduate
in 1973.
to include
at IBM's
later
Boston
(December,
and implementation
experiments, in April,
Compiler.
Advanced
1962), design
the first complete
the first started
version
FORMAC
in November,
1964.
4
It was developed
Programming
draft
from
was released
for public
This
of FORMAC
version
use
in
of language
thereafter.
was successfully feedback
by J.
Department
complete
immediately
1964. For the sake of obtaining
to the system,
by IBM)
MAnipulation
After
running
18
on the
users to make by the authors was
written
in
FORTRAN
IV. Three
PIM was released The
manipulations.
and COEFF
obtains
control
DIST
4. Expression
analysis
and LOWPOW
8. User defined
FORMAC general
purpose
history
and
system
larger In 1977,
in
added
notations
to allow
instead
PL/I FORMAC
of
it to do
of numbers
in
in expressions capabilities
can be
the parentheses
of sums,
expression,
etc.
by b.
differentiation.
returns
the coefficient
and denominator,
and lowest
for storing
of exprl
respectively.
in expr2.
HIGHPOW
power.
the var to secondary
to pnnt out the required trigonometric,
storage.
expressions.
logarithm,
exponentiation,
square
root,
etc.
function
is now
a in expr
partial
: these include
function,
and
all parentheses
for expanding
the numerator
: SAVE(var)
functions
removes
law to all products
replaces
the highest
: PRINT_OUT(expr)
hyperbolic
remarkable. one.
return
allocation
7. Built-in
: EXPAND
performs
return
commands
to variables
The major
: COEFF(exprl,expr2)
and DENOM
written
:
the distributive
: DERIV
most
symbols
EXPAND
of variables.
: EVAL(expr,a,b)
3. Differentiation
6. Output
LET assigns
categories
for applying
5. Storage
of new commands
of simplification
2. Substitution
NUM
were
a couple
of FORMAC
systems. kept
substitution,
the coefficients
into the following
1. User
there
makes
the new version
of FORMAC
For example,
SUBST
1967),
for use on IBM/360
version
IV. In addition,
FORTRAN,
divided
later (November,
by the authors
FORTRAN
FORTRAN algebraic
years
: the user can define
one
of the most
to receive
numbers
popular
extensive
of users,
the new version,
functions
usage
SAM
5
systems.
worldwide.
its accumulated called
as needed.
FORMAC
It is the first
With
the advantages
contributions 73, was
reasonably
released
to SAM to replace
of longer field
are
the old
1.2.4
MATHLAB
MATHLAB in 1964
l system
[6]. Its source
example,
language
is LISP,
PLEASESIMPLIFY(x,y)
of 1967, the first version which
was developed
operated
through
a teletype-like
second
version
1.2.5
is the command
keyboard
were more ALGOL-like.
character
MATHLAB
display
and
The
was the first complete
For
MATHLAB
input
scope.
words.
it as y. In the fall
version,
The
Co.
68,
output
were
notations
on-line
in the
system.
ALTRAN
ALTRAN New
Jersey
running
is a system
1964
adapted
on
at the BELL TELEPHONE
M. D. Mcllroy,
the IBM
were a mixture
use in the BELL Lab., 1.2.6
developed
by W. S. Brown,
in late
ALTRAN
D. C. Leagues
7090/7094,
7040/44,
of FORTRAN
its contributions
Laboratory and
etc.
G. S. Stoller
The
II and FORTRAN
in Murray
basic
Hill,
[1]. It was
languages
IV. Since
which
it was limited
to
to the SAM field were small.
REDUCE
REDUCE
was
developed
by A. C. Hearn
[7]. At that time, he met Dr. John McCarthy, the use of LISP
for the problems
theoretical
physicist,
issued
This
[8].
elementary system, name
paper
particle
only
physics.
intended
expressions
problems,
ALGOL-like
in 1970. The dialect
was written.
the beginnings
later (1968),
[9]. The name
(call
REDUCE
big improvement
RLISP),
rather
than
At this time, the REDUCE
of a user community.
Thereafter,
from
its name
to produce
very
manageable
system notation
was also released 2 was implemented
6
to
this paper.
himself,
to a more
was
algebra Its was large form".
REDUCE
was written of LISP to users,
2,
in an
in which making
successfully
is not to be confused with MATLAB. MATLAB is the numerical software for matrix operations, while MATHLAB is another symbolic and algebraic manipulator.
1 MA'rHLAB
as a
techniques
from the new version,
the parenthesized
REDUCE
Dr. Hearn,
a general
originated
was that the whole
2 system
then,
describing
the author
the results
for distinction
who suggested
of SAM
then as now_tended
then reduce
in 1963
1966, the first publication
of REDUCE
system
Since
the first paper
from
California,
language,
application
to the description
rather
at this time was called
appeared
REDUCE
According
physics.
area. In August
the specific
as a wit. He said "algebra
for many
The system
Two years
was published
is not an acronym.
about
Corporation,
of the LISP
particle
in the SAM
talked
of Rand
an inventor
of elementary
has worked
"REDUCE",
actually
which
memory.
at MITRE
as English
x and name
by the second
with 256 K core
with a fixed
are defined
to simplify
was replaced
machine
and his employees
but the commands
of MATHLAB
on a PDP-6
by C. Engeiman
on
theMichiganTerminal a long silence, added real
REDUCE
in this version, arithmetic
released.
in the REDUCE
equation
15, 1987.
Each
on MTS,
Computer
design
confirmed
makes
new packages
were
arbitrary versions
were
also
1985, REDUCE
capabilities.
Instead
on the APOLLO
of the
precision
University
of
workstation of Michigan.
15, 1988.
one of the most
published
After
3.2 in April,
and additional
well-known
to be used in a wide
of papers
variety
in different
SAM
of areas.
systems.
Its general
Its contributions
are
fields.
SCHOONSCHIP
SCHOONSCHIP Its major
was designed
applications
used for other
by M. Veltman
are in the field of high energy
calculations.
computer.
1.2.8
It can deal easily
It was limited
to use within
at CERN,
physics,
Switzerland
but it is sufficiently
with expressions
in 1964
[10].
general
to be
of 104 to 105 terms
on the CDC
CERN.
ANALITIK
ANALITIK direction first
once in January
it possible
3, upgraded
(CAEN)
Alexander.
factorization,
in 1984, REDUCE bug fixes
by Mike
significantly
REDUCE
Network
has become
Several
3.3 was first implemented
Engineering
by the number
1.2.7
contains
of Michigan
multivariate
Following
3.1 released
3.3 was also updated
purpose
6000
solving.
the REDUCE
system
in 1983.
integration,
of them
Aided
REDUCE
of the University
3 was distributed
were REDUCE
implementation
(MTS)
such as analytic
and
They
3.3 on July
System
was developed
of the well known
paper
discussing
ALGOL-like possessed
the system
and close interactive
dependent, 1.2.9
Soviet
at the Institute
of Cybernetics
cybernetician
and academician
features
to that of traditional
and batch processing
ANALITIK
was published
in 1964.
mathematical modes.
has only run on the MIR-2
Since
in Kiev,
Soviet
Union,
V. M. Glushkov The
notation
[11]. The
language
and
natural
its implementation
by the
it used
was
language.
is highly
machine
computer.
FLAP
FLAP Laboratory
was
written
in Dahlgren,
in LISP VA. prior
1.5 by A. H. Morris, to 1967 [11. Obviously,
to the public.
7
Jr. at the
U.S.
the FLAP
system
Naval wasn't
Weapons released
It
1.2.10
SAC
The Wisconsin,
SAC
system
Madison.
portable
general
system,
PM in IBM
was
The first version,
purpose
system,
[see
1.2.1].
was programmed
in ALDES
in 1973
The
to 1974.
FORTRAN
designed
was distributed
developed
to replace
SAC-1
was replaced which
system
also
E. Collins
at the
provided
of
in 1967 [4]. This was a highly
one of the very earliest by SAC-2
was designed
University
computer
in July,
by Rudiger
the translator
1980.
Loos
from
algebra
The
SAC-2
and G.E. Collins
ALDES
to standard
its portability.
is an acronym
by C. Engelman,
implementation describing
ARPA
networks
graphic
facilities
in May,
Unfortunately, implemented
the
MAC's
J. Moses
1969. The
in 1971
([12]
have
made
University
SYmbolic
system
MAC
[14]
[15]).
didn't
powerful
have
in CAEN
in 1968.
in size
It was made
has a lot of built-in
of Michigan
It was originally
at M.I.T.
has quintupled
it one of the most
workstation
MAnipulator.
for project
[13]
1972. MACSYMA
on the APOLLO
have a graphics 1.2.12
in July,
it appeared
which
of project
W. Matin,
of it began
paper
since
available
The
the first over
the
mathematical
functions
SAM
in the world.
it until
Systems
September,
of the University
1988.
of Michigan
and
The
one
still doesn't
package.
SCRATCHPAD
Although be traced
the name
back
to 1965.
Griesmer,
Richard
J. Watson
Research
the name later,
(allowing
Center,
in March
When
David
Yorktown
Heights,
by Dick After
to organize
and
published
Jenks
Jenks,
in 1970, the initial
was designed
New
York
called
commands,
in 1975. After due to personnel
returned
the "mode-base"
and
Dick Jenks
8
new
went
Heights
118]).
such
Unfortunately, in 1970. One
demonstrated
at
as history
file
SCRATCHPAD/1
a stagnation
Jim Griesmer
in the progress
left the group
to the University
to be a
of Utah
in the fall of 1977,
by Dick Jenks
H.
at the IBM Thomas
was
features,
on it can
by James
in Bonn
the first completed
changes.
originated
[17]
presented
this, there seemed
to Yorktown ideas
([16]
SCRATHPAD/1, some
work
principally
Yun, and their colleagues,
combining
system
at IBM research
Dick
system
Fred Blair,
1971.
system
of education
sabbatical.
SCRATCHPAD
to backtrack),
of the SCRATCHPAD
was chosen
was not used for the first paper,
version
was eventually
manager
The
D. Jenks,
a revised
users
manual
of SCRATCHPAD
of SCRATCHPAD
SYMSAM/II
agreed
George
MACSYMA
MACSYMA
year
by Dr.
SAC-I,
language,
SAC-2
to maintain
1.2.11
developed
in 1973. This
David
for a Yun
led to the
NEWSPADwhich thereafterwasrenamedto SCRATCHPAD84 at theNew York conference in 1984.However,the nameof SCRATCHPAD84 wasnot quiteappropriatesinceit would takemorethanoneyearto finishthesystem.Thereforeit waschangedinto SCRATCHPADII, which is the nameusednow. It becameavailablein 1985for testandevaluationto a limited numberof usersfrom an IBM ownedmainframevia telenet,CSNETandARPANET. As yet, it is not commerciallyavailable. 1.2.13
CAMAC
The version
CAMAC
system
of it ran interactively
language.
When
transferred acronym
Vera
was
moved
IBM
of Combinatorial
1.2.14
and
Pless
to the Circle's
was for a specific
by Vera
written
Circle
by William
and Algebraic
in 1973 at M.I.T.
in FORTRAN
to Chicago
370-158
Pless
Machine
with
in 1975, Pattern.
Aided
[19].
sections
name
Computation.
first
in assembler
the CAMAC
The
The
system
of CAMAC
As the name
was is an
implies,
it
application.
SHEEP
SHEEP [(20].
[21])
was
called
SHEEP
version,
by I. Frick
at the University
for manipulating
It runs on DEC
1, was written
SHEEP
1.2.15
designed
It was specialized
uses is MACRO-10.
PDP
in assembler
2 is written
in standard
components
of Stockholm, of tensors.
10 and PDP 20. The code for the DEC-
Sweden
The source
first version
in 1975 language
of SHEEP,
10/20 computer.
Unlike
it
now
the first
LISP.
ORTOCARTAN
ORTOCARTAN 1977
was designed
[22].
Its name
Riemann,
Ricci,
set of CARTAN
is written is an acronym
Einstein forms.
uses, such as inverting 1.2.16
matrices
by Andrzej
and is due to the specific
and Weyl tensors Although
It was designed
from a given
the author of arbitrary
said it could
Krasinski
in Poland
application
to the calculation
tensor
using an ORTHonormal
metric
be relatively
rank, it did not come
easily
into wide
extended
in of
for other
use.
MAPLE
The Gentleman
MAPLE
system
and Gaston
[24]). The name Canadian
in LISP.
identity.
Gonnet
"MAPLE" There
was
designed
by
at University
Char,
of Waterloo,
is not an acronym were two goals
Bruce
but rather
which
9
oriented
Keith
Canada
in December
it was simply MAPLE's
Geddes,
chosen
design.
W.
Morven 1980 ([23]
as a name with a
The
first was to be
usedon a time sharingmainframecomputer.The secondwasto run it on a microprocessorbased workstation. This was the major difference betweenMAPLE and REDUCE (or MACSYMA). 1.2.17
muMATH
wntten David
R. Stoutmeyer
version Inc.,
in muSIMP
was called
was
to users
(a LISP-like
and
Albert
by David
for the CP/M-80
core memory personal
required.
computer
DERIVE
has
DERIVE
taken
product
called,
the place
the numerical,
functions
of muMATH.
II family
algebraic
in 1977
and graphical
The DERIVE
first
House,
was distributed
Z80 machine
with 64 K bytes
was not released RAM
significant functions
system
The
later, the Software
needs 256 K bytes The
[25].
by
muMATH-79,
muMATH-83,
of muMATH-83.
and developed
for the IBM
memory.
Recently,
improvement together,
rather
is that than just
requires
512 K memory
space
for
system.
It was originally
designed
execution.
1.2.18
MATHLIB
MATHLIB and developed
& SMP
is an interactive
under
general
the auspices
in 1978.
California
and became
commercially
In addition,
its graphical
operations. by over
150
subsystem mathematical
It was one of the products
different of
VAX/780
devices.
has
more
of Mathematics
of Innosoft
available output
SAM
at Harvey
International
in 1983. It can perform
Mudd College,
Inc.,
of Claremont,
numerical
and symbolic
is device-independent
and allows
it to be processed
The
embodied
in the
than
PRS 250
subroutine built-in
functions
for
algebraic
manipulation
of
expressions. product
of Technology. under
to allow
of memory
graphics
MATHLIB
SMP is another Institute
purpose
of the Department
California,
SMP
or Apple
1983. The muMATH-83
over
Two years
and the first product,
system
The second
combines
algebraic normal
Stoutmeyer
was designed
of Hawaii
and was experimental.
operation
until
muMATH
Rich at the University
muMATH-77
founded
language),
of Innosoft
It's written
the UNIX
operating
for easy conversion
space
for typical
International
in C language system.
between
Inc.. It was designed and was originally
In addition, operating
usage.
10
there
systems.
developed
is a special It needs
at the California
design
to run on character
at least 2.5 megabytes
in
1.2.19
MATHEMATICA
MATHEMATICA versions VAX,
of MATHEMATICA IBM ,Cray, Roman
integrates
the algebraic
Its source
language
version
computation, in Ccode,
requirement
FORTRAN
for normal are expected
DEC
Wolfram,
of Illinois
and graphical
are several
Macintosh,
by Stephen
at the University
as well as DERIVE
Mathcad
is developed
appeared
on market
itself
machines.
driven.
This unique
minimum
disk. Unlike This allows
feature
sometimes
around
Inc.
Daniel
in 1988 [26].
functions
It
together
code, and text form.
operation
is about
3.7 mega
to be two dominant
at Cambridge,
1987. This system
the graphic
systems
space
version
can run in IBM, are
most of the SAM systems,
turn out a unmanageable,
two
in
by simply
in typing
latest
PCs and unix
RAM
and
seven
in this system
picking
but also reduces
hard-to-be-debugged
The
Macintosh
inputs
earlier
of MAPLE
in C language.
megabyte
the command
with machine
not only saves users lots of efforts
The
the core functions
It is written
requirements
users to communicate
Massachusetts.
adopted
capacity.
in 1992. This newest
The
hard
by Mathsoft,
by including
3.1 is available
megabyte
1.3
memory
such as for Apple
and implemented
to be outputed
Inc. There
decade.
and extented
which
is C. The
Research,
Mathcad
version
menu
of computers,
numerical
expressions
The MATHEMATICA
1.2.20
of Wolfram
and their colleagues
manipulation,
the resultant
based
for a variety
E. Maeder,
and allows
the coming
product
and so forth. It was designed
Grayson,
bytes.
is a recent
are
and clicking. human
errors
results.
Conclusion
From
• The
a history
SAM
function
systems
systems,
• At present, depends
evolved
from
to compact
there
on many
familiarization facility,
of the SAM system,
is no unique variables,
with software,
we can draw the following
small,
systems best such
immature
which
as computer
and so forth.
11
:
to well-designed,
multi-
can be used on microcomputers.
system.
the problem
systems
conclusions
The
definition availability,
to be solved,
software
of the
best
availability contents,
SAM
system
of software, circumference
system
remarks
year
PM
1956
IBM
ALGY
1961
WDLP
FORMAC
1962
MATHLAB
1964
ALTRAN
1964
BELL
RED UCE
1963
Rand
i,
SCHOONCHIP
IBM - Boston MITRE
Co. Lab.
Co.
CERN
1964
ANALITIK
1964
FLAP
1967
SAC
1967
MACSYMA
Co.
Soviet
U.S.
Unions
Navy
Uni. of Wisconsin
1968
M.I.T.
SCRATCHPAD 1965 CAMAC SHEEP ORTOCA
RTA N
MAPLE muMATH
& DERIVE
MATHLIB
1973
Vera Pless
1975
Sweden
1977
Poland
1980
Canada
Heights
Uni. of Hawaii
1977 1977
SMP
IBM-Yorktown
Harvey
Mudd
College
1977
Caltech
MATHEMATICA
1988
Uni. of Illinois
Mathcad
1992
3.1
Table 1.1 • List of symbolic
12
Mathsoft
and algebraic
systems
Inc.
1.4
References
[1] Jean. E. Sammet, Hall inc., 1969.
"Programming
Languages
"Analytic Philadelphia,
differentiation Pennsylvania,
[2] H. G. Kahrimanian, Temple University,
[3] J. Nolan, cambridge,
[4] George 8, Aug.
"Analytic differentiation Massachusettes, 1953.
E. Collins, 1966.
[5] M. D. Bernick,
WJCC,
[6] C. Engelman,
Proc.
Rayna,
[8] A. C. Hearn, using
[9] A.
Hearn,
simplification", (edited
[10]
[11]
Interactive
Jouko
and its implementation",
[12] Joel Moses, Stockholm,
[13]
"MACSYMA
• software
comm.
systems
for on-line
Properties
reference
computation",
interactive
system
Academic
Applied Press,
ACM
New
system",
• principal
74, pp 24-25,
13
thesis,
M.I.T.,
9, No.
manipulation
in symbolic
1965.
1966.
"ANALITIK
Manual",version
Vol.
assistance
Nov.,
9, pp 573-577,
oriented
"MACSYMA - The fifth year", pp 105-110, Aug 1974.
ACM,
of Elementary
for Experimental
EUROSAM
thesis,
Algebraic
machine
pp. 117-126,
of the SCHOONSCHIP
Seppanen,
M.S.
S. M.
"ALGY-An
Prentice-
1961.
of the ACM,
user
computer",
manipulation",
for algebraic
of Algebraic
and J. Reinfelds),
"presentation
Husberg,
program
"REDUCE-A
by M. Klerer
H. Strubbe, 59, 1974.
Nisse
" REDUCE
computer",
J. R. sanford,
vol. 27, pt.2,
"computation
a digital
C.
FJCC,
Fundamentals",
computer",
for polynomial
19, pp 389-392,
"MATHLAB-A
computations",
[7] Gerhand 1987.
voi.
and
by a digital 1953.
on a digital
"PM, A system
E. D. Callender,
program",Proc.
• History
Springer-Verlag,
Particle
for
Mathematics, York,
algebraic pp 79-90,
1968.
EURUSAM
features
Reactions
74, pp 55-
of the language
1974.
SIGSAM
ten, Mathlab
Group,
bulletin
MIT,
EUROSAM'74,
1983.
[14]
Richard
Pavelle,
engineenng
[15]
and the sciences",
W. A Martin on symbolic
[16]
Robert
Richard
Vol.
EUROSAM
Hornfeldt,
tensor
calculus,
EUROSAM
"A system
Andrzej general
[23]
Bruce
Char,
system",
[24]
[25]
Bruce
Char,
MAPLE
: a Compact
Algebra,
proceedings
[26] Stephen computer",
press,
symposium
1971.
language
and
system",
physics,
Geddes,
generation
and
(1965-1977)",
IBM
(1977-1986)",
IBM
algebraic
of tensor
Symbolic
SHEEP,
computation
algorithms
and indicial
and algebraic
computation,
vol.
University
of Stockholm,
program
Gonnet,
"The
17, pp 31-42,
1983.
W. Morven and
Eurocal'83,
What it can and cannot
vol. 17, pp 12-18,
Gaston
portable
Hodgkinson,
Project
symbolic
of sums",
system
SIGSAM,
Geddes,
Project
1979.
algebra
SIGSAM,
Keith
C. Woof f, D.
1979",
"ORTOCARTAN---A
Keith
ACM
Academic
substitution
ACM
2nd
1979.
of Theoretical
relativity",
March
algebra
of the SCRATCHPAD
"CAMAC
79, pp 279-290,
Kransinski,
II computer
for automatic
including
relativity",Inst,
ACM,
proc.
15, 1986.
Pless,
'"Fhe computer
pp 59-75,
in
MA,1985.
system",
of the SCRATCHPAD
79, pp 249-257,
Lars
[21] I. Frick,
"A history
Vera
Cambridge,
to problems
15, 1986.
1, no. 3, May
S. Leon,
Inc.,
applications
voi. 12, 1985.
"A history
D. Jenks,
and
'"Fhe MACSYMA
Scratchpad
vol. 1, no. 2, Jan.
newsletter,
[22]
"The
H. Griesmer,
[19] Jeffrey
Symbolics
manipulation,
85, pp 32-33,
newsletter,
: capacities
R. J. Fateman,
S. Sutor,
[17] James
[20]
and
and algebraic
EUROCAL
[18]
" MACSYMA
powerful
computer
pp 101-115, "muMATH
1977.
for algebraic
calculation
in
1983.
MAPLE
Gentleman,
do in general
symbolic
Gaston
Gonnet,
algebra
computation
'"/'he
design
system",
Computer
algebra
system",
of
1983.
: A microcomputer
1987.
Wolfram,
"MATHEMATICA---a
Addison-Wesley,
1988.
14
system
for
doing
mathematics
by
CHAPTER
SURVEY
OF
THE
LITERATURE
ON
II
SYMBOLIC
AND
ALGEBRAIC
MANIPULATION
2.1
Introduction
The
documents
plentiful
as those
existing
documents,
directions divided
and
published
in the area of numerical one
content
finds
Most
SAM of them
focused
published
on the following
(a)The
introduction
manipulation
after a careful
history
In general,
is closely
field are not as classification related
the documents
about
of the
to research
SAM
may
be
are :
itself --- More
were
However,
that the developmental
They
system
and algebraic
analysis.
of the publications.
into four categories.
1. About
in the symbolic
than half of the existing
in the period
of the first
papers
belong
generation.
to this class.
The
contents
are
functions,
etc.
topics"
of the new
SAM
system,
including
the capacities,
[1][2][3].
(b) The technical
reports
(c) The data structure,
.
Applications SAM
requirements relativity paper
impetuses especially
mechanics.
[10]. One of them
Henrard
[8][9].
--- This class of publications
scientists,
and celestial
[4] [5] [6] [7].
and implementation
One of the major from
by Hearn
by Deprit,
language
to science
papers.
of softwares
is the second
in developing in the fields
Some
and Rom in 1970 [11] The others
analysis
of the Feyman
problem
[14],
diagram
and Roberts'
[13], Rudiger
and Boris'
Loos'
on the solution
[15].
15
systems
of elementary
of the famous
is the recalculation
SAM
largest
examples
of the existing was due
particle,
moon
are such as Campbell work
about
of partial
general
were collected
of Delaunay's
in the
coordinates and Hearn's
Archimedes' differential
to the
cattle equations
.
Applications
to engineering
analytically.
Therefore,
numerical
engineering
problems.
This
SAM
and more
Due
of a solution
engineering
problems written
Anderson
by Madson,
[19],
discussed
has became
Korncoff
in more detail
Application
in the other
algebraic
fields
Fenven
of
applications
of
[16],
[20],
which
in engineering.
first is
approximation
strict and
recently.
and Hoff
in the solution
there are two factors
to be solved
of symbolic
--- In addition
has
[22], education
As time goes on, more
of numerical
becomes
more
involves
more
usually
algebraic
of solutions.
manipulation
to
of the publications,
[17],
Steinberg
The
requirements
Some
Levi
not solvable
Wilkins
and
such
[18], Noor
Roache
[21],
and
will
be
in the next paragraph.
manipulation
management
Smith and
manipulation
more popular
are
predominates
However,
due to the more
the applications
problems
why the engineering
is that the problem
manipulation
factors,
usually
and algebraic
requirement
algebraic
to these
as those
approximation
strict today. The second
sophisticated
engineering
as those on science.
the use of symbolic
that the accuracy
most
is one of the reasons
has not been as popular
necessitate
.
--- In fact,
been
to science
applicable
in other
[23] [24] and business
and more applications
and engineering, fields,
such
Symbolic as
and
information
[25].
will be reported
in various
fields.
This is
due to the fact that :
1. The
ongoing
personal
improvement
However,
of variously
in order
the people
in the
tool. As the discussion about
20 percent
Reviews
(1) Computer
A paper computerized
educational
space
SAM
algorithms
of hardware
systems,
especially
systems.
is replaced
assume
by the computer
the responsibility
by Richard
Pavelle
field are aware
of utilizing
in 1985, points
of the existence
screen
in all
this new
out [25], only
of SAM
system
and
use them.
applications for solving
and
paper
written
in the related
[ 16] published symbolic
SAM software
field should
in the paper,
of these actually of
sound
to see that scratch
of people
less than a quarter 2.2
memory
computers.
2. The availability
areas,
in the
in engineering
non-linear
problems
in 1965 is believed algebraic
manipulation
16
to be the earliest to solve
document
an engineering
which
employed
problem.
The
authors,W. A. Madson,L. B. Smith andN. J. Hoff, developedtheir own softwareto find the solution for the post-bucklingbehaviorof thin-walledcircular cylindrical shells underaxial compression. The majorcommandsdevelopedby themwere: *SERIESMULT : To expandtheexpressions, e.g.(a*sin(x)+b*cos(y)) *TRI.GSPAND
:
To
sin(x)*sin(2x)*cos(y)
*SEARCHSTORE then store them.
etc., into
double
: To solve
commands
The application assumption
non-double
: To search
*NEWTNRAPH last three
treat
of radial
As
the nonlinear
of the above
bending
strain
potential
energy
(therefore energy
was
were
mentioned
the story circular
relationship
and
of SAM
solution
was
further
lowering
conducted
from
function,
the calling
of the
method.
to the shell post-buckling
function)
of axial
problem
started
at the
when
publications only
application shell under
of normal
not found
until
as new terms
in seeking
This
J. Kempner
increased
17
Only
The
energy,
resultant
total
displacement
w.
by calling
the
be solved
REDUCE
and
post-buckling
inconsistency the series then,
two terms
in the into
Von
three
additional
But finally
FORMAC
he described load for a thin
1941, Von Karman
with only
solution.
in the engineering
[17] by I. M. Levi,
from
load in 1954. Since into the series
membrane
application
theoretical
Starting
displacement.
be obtained.
mechanics
of the paper
be found
could
of radial
then could
existed.
the minimum
load could
the
be derived.
manipulation
systems
axial compression.
were added
could
to structural
At the beginning
known,
to the coefficients
of symbolic
a few SAM
of the post-buckling
law, the stresses
the minimization
by computer-applications
Ay )
were
load
with respect
after
that a low post-buckling
expression
sin2(x),
like cos(y)*cos(x).
obtained
iteration
and Hook's
stress
the potential
in this paper.
cylindrical
indicated series
in 1971,
as
of like trigonometric
equations
._Aifcos( ixt/ Ax )cos(jny/
the Airy
One of the earliest field
system
commands
was then minimized
algebra
the coefficients
such
displacement,
The system equations obtained NEWTNRAPH command.
(2) Symbolic
terms,
terms
by using the Newton-Raphson
by the strain-displacement the stresses
trigonometric
and collect
w=t Then
trigonometric
n.
and Tsien
included
in the
Karman-Tsien's
terms
and found
investigations
everybody
were
was limited
a
by their
inability
to solve
minimum
load
wrote
special
the
minimum
the complex
did not end
until
program
load approached
algebraic
1965 when
equations
Madson,
in ALGOL
to increase
zero,
revealed
which
without
Smith
and Hoff
the series
the basic
error.
into
The drive of Stanford
14 terms
fallacies
to seek the University
and
found
in the application
the
of the
Karman-Tsien procedure. The second a compatible example using
topic in the paper
triangular
expressed
fortran
This
codes
REDUCE
were
strain
started
trigonometric
in the substitution printed
briefly
element
of creep
computation
as a double
and ended
about
the derivation
by symbolic
and algebraic
rate in plate and from
series,
the
of the above
quantities
out by REDUCE.
The
paper
ended
for
Then an
was demonstrated
of stress
by the calculation into strain
matrix
manipulation.
shell problems
x, y components
followed
of a stiffness
which
of equivaleat
rate equations.
were stress,
The resultant
with a brief discussion
on the
capacities.
(3) Applications This
of symbolic paper
[18]
Dynamics/Convair calculate
plate bending
of the calculation SAM.
talked
algebra was
Aerospace
the strain energy
manipulation
published Division,
in
language 1973
1::olt Worth,
for an anisotropic
by
for composite Dick
Texas.
shell. The strain
J.
Wilkins,
The author energy
structures Jr.
analysis of
General
used PL/I FORMAC
can be written
to
as
!r x
N Y
"E
x
Cy
N_
g_y
M_
K" x
My
Ky
M_
K_y
dO
Where • N • stress
resultants.
• M • moment
• e • mid-plane *
resultants.
strain.
K • curvalure.
• Q: shell sttrface
area.
18
(2.1)
• V" strainenergy. and the constitutive
equations
are expressed
as follows
'A, I
A12
A
At2
A 22
A13
A 23
• BI!
B12
B13
A 23
Bt2
B22
B23
A33
BI3
B23
B33
13
_X
Ny
N._ M_
BII
BI2
BI3
Dll
Di2
Di3
812
822
823
DI2
D22
D23
Bl3
B23
Dr3
D23
D33
My
Where equation
• Aij , Bij , Dij are the pertinent
(2. l) and (2.2) can be combined
V
Then
the displacements
v
constitutive
are approximated
K'y K._
of Hooke's
j
Law.
{r }r [D ]{ r } ]d Q
_
,-.,,_
_
The
(2.3)
as
_C:.XmY.
_,_
-
components
(2.2)
K" x
into the form
= _fa[{e}r[A]{e}+2{e}r[B]{x}+
w - _
theory,
B33
Ey
(2.4)
OX.,
OY.
_C,,,X,,.---_--
(2.6)
Where the Cij are constants to be determined the strain-displacement relations are
by Rayleigh-Ritz
method.
By Vlasov
shell
au
(2.7)
Ex
--
-_"-
dv
w
e,
""_"+
T
3u
(2.8)
Ov (2.9)
oXw _ x
"
I( y
==
w R' 3Zw
r_
=-2
(2.10)
t_X 2
oxoy
(2.11) 1 3u
ROy
19
1 o_ +
R Ox
(2.12)
WhereR hereis theradiusof shellcurvature. With the assumptionof a symmetricconstitutivematrix,thecalculationof the integrand of equation(2.3) wasdoneby FORMACby the substitutionof equations(2.4) to (2.12) into (2.3).The Rayleigh-Ritzmethodwasthenappliedto takethepartialderivativewith respectto all undetermined constantsin the displacementseries. This was also manipulated by FORMAC.
The resultant
and their derivatives. that time, variation
Since
the informal
by a symbol
integrals.
There
The
expressions
final
and suitably
and an appropriate
were
Symbolic
In the beginning symbolic
and
algebraic
capacities
and special
The second
is in terms
the symbolic
different
to allow
@X ,_
@Y 4
---_-,
T
integration
type was found,
for the non-dimensionalization
integrations
modified
of
in the energy
into FORTRAN
at
each term in the energy
Each time a certain
code
variation by adding
it was of the
equation. DO loops
by hand.
Manipulation
in Structural
of the paper
[19],
manipulator
MACSYMA.
commands,
Mechanics
A. K. Noor
These
as well as associated
--- Progress
and Potential
and C. M. Anderson included
the brief
introduced history,
the basic
packages.
part of the paper gives three applications
using MACSYMA.
which
was done by examining
constant
then slightly
the indices
variation
of performing
of derivatives.
were a total of twelve
(4) Computerized
in the structural
mechanics
field by
They are •
1. Generation
of characteristic --- There
of basic integrals (a) Linear
was incapable
integration"
combinations
changing
element
were the energy
FORMAC
"symbolic
for a specific
replaced
expressions
arrays
were three
of finite
elements
types of basic integrals
for geometrically
nonlinear
problems.
for a shear for linear These
flexible
problems
shallow
shell
and three
types
were
problems
A '1 .
B ,, O
,,N'NJdI2
..
f
-f
(b) Geometrically
,, N
(2.13)
tgaN Jd [2 i
,, o_,,N
nonlinear
(2.14)
3aNid12 problems
20
(2.15)
C
qk
-f
D u_t
,jk
.. f
qkn
evaluation
analytically (2.18) the
mass
equations
were
elliptic
Selecting
were then plotted
(c) Evaluating
of the
system
analysis
analysis
started
was
functions with
for the specific coefficients.
specific
undetermined
strain
strain
each
and kinetic
of
the
energies
energies
strain
finally
energy)
obtaining
the
in MACSYMA.
analysis
of laminated
fundamental and
with respect
were unknowns
developing
as quadratic
The
facility.
in this application
coefficients
and kinetic
the evaluation
as well as effective
graphic
to the free vibration
for
for repetitive
as soon as the governing
also done
out by MACSYMA's
analytic
functions
of the
to the undetermined
symbolically.
stiffness
and mass coefficients
has been done successfully,
see the details
21
by performing
in chapter
integrations
three.
to
and Dihedral
included
and
in
(numerical
models
coefficients
to
of integrations
the thermoelastic
--- The tasks done by MACSYMA
amplitude
(b) Differentiating
(with
and thermal
technique
number
of continuum
(2.15)
determinant
approach
by MACSYMA
energy
Lagrangian
This numerical
approximation
expressions undetermined
coefficients
plates
manipulations
performed
of equations
hybrid The
be
by the help of permutative
The numerical
of the Rayleigh-Ritz
displacement
2This
shapes
the
can
of a Jacobian
was proposed.
reduced
(2.14)
the integrand
2 Therefore
of stiffness
the
and
due to the existence
of strain
equations.
obtained.
of mode
composite (a)
calculation
forming
(2.13)
and mass coefficients
computation
differential
3. Application
stiffness
(2.18)
in general
manipulation)
--- The symbolic
governing
results
However
can be substantially
coefficients,
OpN"dl-2
in equations
integrand
energy,
(2.17)
k
cgrN
of the
components,
and kinetic
l
cgoN
plus symbolic
structures
of strain
i
exactly
of effective
(2.16)
J o_ N kd g'2 r
be integrated
denominator
be performed symmetries.
Nkdg2
0#N
by MACSYMA.
quadrature
lattice
'
,,O_,N
#
of integrals
cannot
2. Evaluation
3,NJO
f ,.aoN
Ec, arp "
The
,.N
a
over
volume.
(d) Simplifying the expressionsfor the nonzerostiffnessand masscoefficients and developingFORTRANcode. After stiffnessandmasscoefficientshadbeenevaluated,thevibration frequenciesand modeshapescould beobtainednumericallyby usingany schemefor generalizedeigenvalue problems. The last part of the paperdiscussedthe problemswhich limited the applicability of computerized symbolic manipulation.The major problemsmentionedin the paper were summarizedas follows. 1. Production
of large
expressions
during
the computation
(intermediate
expressions
swell),
2. Slow
speed
of symbolic
3. Low portability 4. Need
computation.
of large symbolic
for analyst
interaction
5. Inability
to estimate
6. Problems
associated
In addition,
during
the storage
systems.
the symbolic
requirements
with interface
the authors
manipulation
between
suggested
computation.
and CPU time for symbolic algebraic
the directions
and numerical
of future
computations.
calculations.
research
in this field.
They
were
1. Reduction
of a general
computational 2. Hybrid
symbolic
generation
and S. J. Fenves
to assist analysis.
integration
of rational
of structural
mechanics
problem
to its
level.
of finite element
As the title implies WA)
formulation
computations.
3. Approximate (5) Symbolic
(tensor)
These
included
[20], the authors
(Carnegie-Mellon
in the development
stiffness
of a software the construction
functions. matrices
A. R. Korncoff
University) to generate
(Boeing
used the symbolic the stiffness
of the strain-displacement
22
computer
service,
processor
matrices matrix,
Seattle
MACSYMA
for finite calculation
element of the
determinantof the Jacobian,andmultiplicationof relevantmatrices.The integrandwasthen integratedsymbolicallyif it wasintegrable(e.g.theconstantstraintriangleelement).Otherwise it would beoutputasthe functionof the problemparametersfor further numericalevaluation (e. g. four-node quadrilateralelement)3. In addition, the softwaregavetwo options in the materialpropertymatrix. Onewas "user-supplied"materialproperty.The other was "library supplied"which provides one-dimensional elasticity,plain stress,plain strain,axisymmetric, and3-D linear isotropicelasticity The exampleof constructingtheisoparametricformulation for constantstraintrianglewasalsoshownin theappendixof thepaper. (6) Symbolicmanipulationandcomputationalfluid dynamics The authorsS. SteinbergandP. J. Roache[21] employedthe symbolic andalgebraic manipulatorVAXIMA, a VAX versionof MACSYMA, to transformthephysicaldifferential equationand the boundaryconditionsinto the rectangularregionandthenconstructedthe so calledstencilcoefficientmatrix for a finite differencescheme.The majorideascamefrom the generalelliptic problems. In physical coordinates,the linear elliptic equation could be expressedas /1
Lf
Where
/1
-
O=f OX _dX
aij, bi, c, d were
was to find a numerical boundary conditions.
given
+
b,
+ cf
+ d
Ox
and were
approximation
(2.19)
the function solution
of coordinates
which
satisfies
in general.The
equation
(2.20)
and the given
L f=0
Since
coordinates
difference
scheme
calculations coordinates
(2.20)
the physical
physical
domain
into could
the
I1
3 The
rectangular,
matrix,
in general,
computational easily.
it is necessary coordinates
This coordinate
its determinant,
and
to transform in which
transformation
cofactors.
The
the finite
involved equation
the
the
in new
#I
,a,,&,&,
integration
obtained
is not regular
be constructed
of the Jacobian would become
-,
problem
exactly
(2.15)
.
(2.21)
ae,
for a four-node
and will be discussed
isoparametrical in detail
23
in chapter
quadrilateral
element
three of this report.
has been
Wherethetilde denotedthatthequantitieswerefunctionsof computationalcoordinates. After the transformationof equationand boundaryconditions had beendone, the centereddifferencemethodwasemployedto constructthefinite differenceschemeasfollows" X
C,.j._(et,
eve3)g(e
l+iAe
ee2
I,Iq, Iqkl,,3
Where
the coefficient
manipulation.
Taking
+j Ae z, e3 +k
ci,j, k were called
advantage
the stencil
of the symmetric
Ae3)=
property,
the number
(2.22)
ve 3)
and was constructed
ci,j, k could be dropped to 10 from 27. The resultant expressions in the FORTRAN language for the next numerical scheme. 2.3
R(et,e
by symbolic
to be computed
of ci,j, k then could
for
be coded
Conclusion
After following
.
making
a survey
conclusions
None
are drawn
of the papers
solutions_
They
approximation solving .
The finite
discussing
element
matrix,
kept
partial
analysis
etc. The
same
on symbolic
and algebraic
manipulation,
the
'
applying
function
generally
papers
of the publications
SAM
to engineering
traditional
methodology
to get more
accurate
differential
equations
the application
by
solutions. or integral
of symbolic
stop at the step of making situation
problems
also occurred
tried
to get closed-form
increasing
the
of
the
This is due
to the difficulty
in
equations
and algebraic a local
in finite
terms
analytically. manipulation
stiffness
matrix,
difference
on the
local
analysis.
mass
This
was
because (a) The
finite
element
methods.
The
error
integration
or
and
accuracy
error
Although
the
the
manipulation (b) In general in FDA of
FEA's
Therefore
difference
of results
manipulation. solution,
finite
methods
depends
which
can
on many be
consideration
problems,
are huge in dimension. FDA)
job
it is necessary
by
to be finished
24
by
to improve
the accuracy
mathematical
The limitation
impossible
and
of the tedious matrix
by numerical
and
space
makes
algebraic
analysis.
algebraic
and
algebraic
equations.
in FEA and stencil
of memory
symbolic
on round-off
symbolic
symbolic
the stiffness
approximation
not just
in applying
was to help in the formulation
(or
cured
themselves
factors,
it was also one of the purposes
major
engineering
are
coefficient
the execution manipulation.
of
(c) Althoughmostof theSAM systemsalsopossess the capacityof numericalanalysis, the executionspeedof numericalanalysisin symbolicandalgebraicmanipulatoris slower in comparison to that in pure numerical analysis. The difference of efficienciesbetweenthemis remarkablewhenthejob is big. Thereforeit is bestnot to haveit donecompletelyin symbolicandalgebraicmanipulation.As thedocuments showed,nobodydid thewholeFEA or FDAjob in symbolicmodealone.
25
2.4
References
[1]
A.
C.
Hearn,"REDUCE--A
simplification", Klerer
user
interactive
system
and J. Reinfelds),
[2] Joses Mose,"I'he 1974.
fifth
oriented
for experimental
pp 79 -90, academic
year---MACSYMA",
[3] J. A. Van Huizen,"FORMAC No. 1, pp 5-7, 1974.
interactive
today,
applied
press,
ACM,
[6] Edward W. Ng,"Symbolic No. 3, pp 99-102, 1974.
[7] John Fitch,"A 1974.
simple
[8] A. C. Hearn,"Structure algebraic 1985.
computation
integration
method
[10]
by./.
[11]
A. Van Hulzen),
A. Deprit, Science
compu.
,/. henrard,
(edited
REDUCE
A. C. Hearn,"Scientific and scientific York, 1976.
nth roots
• The key to improve
[9] ,/. P. Fitch,"Implementing (edited
of taking
by computers
(edited
1970.
26
ACM,
vol. 8,
and T. Soma),
ACM,
ACM,
vol.
vol.
vol. 8,
8, No. 4,
symbolic
and
pp 215-230,
computer
algebra
computer
science
voi 162, 1983.
of symbolic
by ,/. M. Ortega).,
A. Rom,"Lunar
168, pp 1569-1570,
ACM,
computation",
by N. Inada
Aug.
differentiation",
functions",
of integers",
algebraic
105-110,
in REDUCE",
on a MICRC-computer",
pp 128-136,
applications
of algebraic
of a class of algebraic
by M.
1968.
to an orphan",
program
algebraic
(edited
vol. 8, No. 3, pp
or what can happen
limit evaluation
for
mathematics
New York,
[4] B. F. Caviness, H. I. Epstein,"A note on the complexity ACM, vol. I 1, No. 3, pp 4-6, 1977.
[5] Steven ,/. Harrington,"A symbolic 13, No. 1, pp 27-31, 1979.
system
ephemeris.
computation", pp 83-108,
Delaunay's
academic
theory
press,
New
revisited",
[12] G. H. Harrison,"Acompactmethodfor symboliccomputationof RiemannTensor", J. of ComputationalPhysics,4, pp 594-600,1969. [13] J. A. Campbell, A. C. Hearn,"Symbolic analysis of Feynman Diagrams by computer,J. of ComputationalPhysics,5, pp 280-327,1970. [14] RudigerLoos,"Amthor's solution of Archimedes'cattle problem", ACM, vol. 11, No.1, pp 4-7, 1977. [15] K. V. Roberts,J. P. Boris,"The solution of partial differential equationsusing a symbolic style of Algol", J. of ComputationalPhysics,8, pp 83-105, 1971. [16] W. A. Madson,L. B. Smith, N. J. Hoff, "Computeralgorithms for solving nonlinear problems",J. of SolidsStructures,Voi. 1, pp. 113-138,1965. [17] I. M. Levi,"Symbolic algebraby computer-applicationsto structural mechanics", AIAA paperNo. 71-363,pp 19-21,April 1971. [18] Dick J. Wilkins, Jr.,"Applicationsof a symbolic algebramanipulationlanguagefor compositestructuresanalysis,Computer& Structures,Vol. 3, pp 801-807,1973. [19] A. K. Noor, C. M. Anderson,"Computerized symbolic manipulation in structural mechanics---progressand potential",Computer& Structures,Vol. 10, pp 95-118, 1979. [20] A. R. Korncoff, S. J. Fenves,"Symbolicgeneration of finite element stiffness matrices",Computer& Structure,Vol. 10,pp 95-118,1979. [21] S. Steinberg, P. J. Roache,"Symbolic manipulation and Computational fluid dynamics",J. of ComputationalPhysics,57, pp 251-284,1985. [22] S. Bandyopadhyay, J. S. Devitt,"The role of symbolic computation in the management of scientificinformation",EUROPCAL,pp460-461,1985. [23] P. S. Wang,"Implicationsof symboliccomputationfor the teachingof mathematics", ACM, vol. 10, No. 3, pp 15-18,1976. [24] R. H. Lance,"Symbolic computation and the instruction of engineering undergraduate",ASME, HTD-vol. 105,AMD-vol. 97, pp 21-24, 1988.
27
[25] Richard Pavelle," MACSYMA • capacities and applications to problems in engineeringandthe sciences",SymbolicsInc., Cambridge,MA, 1985.
28
CHAPTER
CAPABILITIES
3.1
OF
THE
SYMBOLIC
III
AND
ALGEBRAIC
Introduction
The capabilities Roughly
speaking,
the functions match,
of the symbolic
a system
which
of differentiation,
variable
substitution
MATHEMATICA,
have
and algebraic
is designed
for general
integration,
matrix
and equation
solver.
a lot of built-in
manipulator
operation, Some
mathematical
This chapter manipulators
will demonstrate
which
REDUCE
are available
is the oldest
demonstrated demonstration
system
MACSYMA
although
What
matrix
can
systems,
the
in The
manipulators
some
and
polynomial
substitution
and built-in
at the place
where
and
functions.
also
Therefore,
algebraic
by examples.
solving, rational
operations
treatment operation,
The strategies
they are necessary.
29
They
and particular
allow
and algebraic
be employed
and
doesn't
include
do
the
Since will be
to help
the
version
of
the graphics of the results
packages. ?
in symbolic
and
are differentiation,
code
to get
may need
mechanics.
function,
and
the users
most examples
graphics
pattern
as MACSYMA
of applied
of trigonometric fortran
possesses
the postprocessing
manipulators
useful
usually
of the symbolic
will
the graphic
dependent.
like REDUCE,
of Michigan,
will be done by other
in detail
equation
examples
of Michigan
of the most
are demonstrated
solving,
possess
University
Others,
MACSYMA
on the market.
symbolic
algebraic
Of course,
which
system
manipulation,
such
functions
at The University
usage
polynomial
capabilities
by solving
doesn't
it is available
section,
operation,
equation
used
and algebraic
In this manipulation
available
REDUCE
being
symbolic
3.2
at hand
by using REDUCE. if it is necessary.
Unfortunately,
package,
the fundamental
are quite
purpose
the answers by just calling the appropriate command once. users to write a short program to get the same answers.
from
MANIPULATORS
integration,
function, output,
techniques
algebraic
differential
number
system,
are also mentioned
3.2.1
Differentiation
All differentiations,
without
necessary,
REDUCE
knows
capabilities
of this
analytical
differentiation However,
in many the wrong
how
finding the first derivative follows"
to apply
can be done
the chain
differentiation
cases, results
exception,
to solve
will probably
such as the evaluations may
rule
analytically
be obtained
of x xx with respect
by REDUCE.
problems.
replace
by careless
or naive
to x. Two different
The powerful
traditional
of the Jacobian
and users.
solutions
If it is
numerical
Hessian For
matrices
instance,
in
may be obtained
as
1: on time; Time: 134 ms 2: df(x**(x**x),x); X X 2 X
+X*(LOG(X)
*X + LOG(X)*X
+ 1)
..................................................
X Time:
383 ms
3: df(x**x**x,x); 2 X X *X*(2*LOG(X) Time: 233 ms
Here the important
the first solution tasks in symbolic
problem-related
is correct.
cases,
throughout
How
manipulation.
fields is very helpful
In some to be retained
+ I)
the unevaluated
A sound
in checking
the computation.
the correctness background
of the results
knowledge
form of functions,
6: p:=a*x*y; P := A*X*Y Time: 150 ms
+ DF(Y,T)*X)
30
in the SAM
and
dx
such as "k" is desired
This also can be done as follows
5: depend y,t; Time: 83 ms
is one of
them.
differentiation
4: depend x,t; Time: 84 ms
7: df(p,t); A*(DF(X,T)*Y Time: 133 ms
to judge
•
3.2.2
Integration
In REDUCE, integrations REDUCE
with
all the
the integration
(the closed-form
on the screen. lower limits
integrations constants
solution
The definite
by the
discarded.
may be obtained
after the indefinite
command
INT
If the function
may exist theoretically),
integration
into the results
performed
indefinite
is not integrable
the original by further
are
by
form will be displayed
substituting
the upper
and
integration.
%A non-integrable case. 8: int(sqrt(a^2-x^2),x); 2 INT(SQRT(A Time: 950 ms
2 - X ),X)
%An integrable case. 9: int(1/(aa2+xa2),x); X ATAN(---) A A Time:
466 ms
In most manipulation. fundamental
and
substituted will become After
the non-integrable
This pre-treatment calculus.
of the symbolic tireless
cases,
errorless by a'cos(t),
manipulator of
computer.
of -a2*sin2(t)dt
expression
in terms
expression
to get the original
the original
of x. The check
variable
The
10: int(-aa2*(sin(t))^2,t); 2 - T)
...............................
Time:
For
with respect
procedures.
A *(COS(T)*SIN(T)
of changing
2 2134 ms
31
appropriate
the integrating extend
variables
in
the capabilities
human
example,
if the
x in command
8 is
_
x2 with respect
to x
of
to t, which
by skeptics three
intelligence
is integrable
x may be substituted
following
after
combining
and the integration
may be done
integrand.
integrable
users can substantially
by suitably
then dx=-a*sin(t)*dt
is done,
will become
the technique
the intelligent
advantages
the integration
the integration
involves
Sometimes
and algebraic
integrand
back
by REDUCE.
to get the desired
by differentiating commands
with the
the resultant
demonstrate
these
11: sub(cos(t)=x/a,sin(t)=sqrt(1-x**2/a**2),t=acos(x/a),ws); X ACOS(---)*A A
2
2 - SQRT(A
2 - X )*X
..........................................
2 Time:
716 ms
12: df(ws,x); 2 A-X
2
2
2
SQRT(A - X ) Time: 450 ms
3.2.3
Matrix
operation
Matrix manipulators. matrices
operation These
and scalars,
the trace, computing and so forth.
3.2.3.1
Matrix
The matrix from
d is obtained
include
of the
powerful
the addition
inverting
matrices,
the eigenvalues
capabilities
and multiplication
calculating and associated
of
symbolic
of matrices,
the determinant eigenvectors
and
multiplication
of a square exactly
algebraic
matrix,
of
finding
if they are available
multiplication
three
body
multiplication. which
is one
rigid
rotation
In robotics,
the rotation
angles
from the product
it is necessary
is a good
to find the analytical
of each arm can be computed. of three consecutive
%Declaring 13: matrix
four matrices. a(3,3),b(3,3),c(3,3),d(3,3);
%Inputting
matrices.
14: a:=mat(( 1,0,0),(0,cos(q A(1,1) := 1 A(1,2) := 0 A(1,3) := 0 A(2,1) := 0 A(2,2) := COS(Q1) A(2,3) :=-SIN(Q1) A(3,1) := 0 A(3,2) := SIN(Q1) A(3,3) -= COS(Q1)
1-2-3 in dynamics
1),-sin(q
1)),(0,sin(q
32
direction
1),cos(q
1)));
example
of the utility
form of final orientation
The final direction cosine
of
matrices
cosine
a, b, c.
matrix
Time: 700ms 15: b:=mat((cos(q2),0,sin(q2)),(0,1,0),(-sin(q2),0,cos(q2)))$ Time:367ms 16: c:=mat((cos(q3),sin(q3),0),(-sin(q3),cos(q3),0),(0,0,1))$ Time: 383ms %multiplicationof threematrices. 17:d:=a*b*c; D(1,1) :=COS(Q2)*COS(Q3) D(1,2) :=COS(Q2)*SIN(Q3) D(1,3) :=SIN(Q2) D(2,l) :=-(COS(Q1)*SIN(Q3)-COS(Q3)*SIN(Q 1)*SIN(Q2)) D(2,2) :=COS(Q1)*COS(Q3)+SIN(Q1)*SIN(Q2)*SIN(Q3) D(2,3) :=-COS(Q2)*SIN(Q1) D(3,1) "=-(COS(Q1)*COS(Q3)*SIN(Q2)+SIN(Q 1)*SIN(Q3)) D(3,2) :=-(COS(Q1)*SIN(Q2)*SIN(Q3)-COS(Q3)*SIN(Q 1)) D(3,3) :=COS(Q1)*COS(Q2) Time:617ms The terminators$ in commandlines 15and16prohibittheprintingof resultsandsave almosthalf of the time comparedtocommandline 14whichusestheotherterminator. 3.2.3.2 Matrix inversion Unlike numericalanalysisin whichthetime-consuming operationof matrix inversionis to beavoided,to find theinverseof a matrix symbolicallyis oneof thesignificantandsimple tasksin symbolic andalgebraicmanipulation.Oneimportantapplicationof it is in solvinga systemof linear equations.For example,the problemof finding a curveto fit the given setof databy the leastsquaremethodresultsin solvinga systemof linearequations.The coefficient matrix hereis theHilbert matrixwhich is usuallyusedto investigatethephenomenonof roundoff error accumulation.REDUCEcansolvethis problemexactly.The numericalsolutionand symbolic solution aretabulatedin Table3.1,3.2, 3.3 for threedifferent Hilbert matrix sizes. As the tablesshow,the numericalsolutionis notcapableof producingaccurateresultsevenfor the caseof the 7*7 Hilbert matrix. The deviationbetweenboth solutionsis also plotted in Figure3.1.The significanceof symbolicandalgebraicmanipulationis evident. 8: matrix h(40,40),x(40,1)$ 9: for i:=1:40do forj:=l:40 do h(i,j):=l/(i+j- 1)$ Time:30917ms 10:for i:=1:40do x(i,1):=i$ Time: 583ms 11:h:=(1/h); 33
H(1,1) :=1600 H(1,2) :=-1279200 H(1,3) :=340267200 H(1,4) :=-45113759600 H(1,5) :=3573009760320 H(1,6) :=-187.583012416800 H(40,39) :=-1141149866470104951399125616277120066810096080000 H(40,40) := 58520505972825894943544903398826670092825441X)_ Time: 1355067ms 12: x:=h*x; X(1,1) := -64000 X(2,1) := 102272040 X(3,1) X(4,1) X(5,1) X(6,1) X(7,1) X(8,1) X(9,1) X( 10,1) X(11,1) X(12,1) X(13,1) X( 14,1) X( 15,1)
:=-40781023920 := 7204667408120 :=-712815447183840 := 44879235720719400 := - 1948531047013671120 := 61638279321584754360 := - 1478390066741437724160 := 27706961874232024704320 := -415343205971360018352000 := 5073605405648309638180800 := -51267503668234803803526400 := 433831946060133824442926400 := -3105695134246771063279900800
X(33,1) X(34,1) X(35,1) X(36,1) X(37,1) X(38,1) X(39,1)
:= -275148980879869194392659362117120 := 129027970745724768036578024940720 := -49525830202172298308155472545440 := 15151287095628987186696245706000 := -3551734684918636715496119886400 := 598923394712817307441549441200 := -64662238360272798660726511200 X(40,1) := 3356375056654755429131776400 Time: 96000 ms
34
x
Exact REDUCE
x(1)
solution
Gauss
125
elimination
solution
124.7899166408321
x(2)
-2880
-2875.997324887610
x(3)
14490
14472.49323423709
x(4)
-24640
-24613.29019994230
x(5)
13230
13216.83350607823
Table
x
3.1 • Comparison
Exact
x(1)
REDUCE
of solution
solution
-216
x(2)
for 5*5 Hilbert
Gauss
matrix
elimination
solution
-204.4675087167038
7350
7027.565254454758
x(3) - 57120 x(4)
- 54968.63675793860
166320
x(5)
-201600
x(6)
85932 Table
x
3.2 • Comparison
Exact REDUCE
160780.3224850843 - 195532.2818417542 83555.641 of solution
solution
for 6*6 Hilbert
Gauss
56991533
matrix
elimination
solution
x(1) 343
x(2)
131.3201346851223
- 16128
x(3)
-7941.23464123
5595
177660
100320.8278414759
x(4)
-772800
-4761 54.4030660979
x (5)
1559250
1020735.514691367
x(6)
- 1463616
- 1001760.776649569
x(7)
516516 Table
3.3 • Comparison
365761.52976973 of solution
35
for 7*7 Hilbert
matrix
52
10 27 .
lo2s 107--3: 1021"
lo19: L
lO17 1015 m
t#
1013. 1011
10 9
L
@
10 7 10 5 1(13 101
'
0
3'o
10 rank ¢_I-Ii]_rt
Figure 3. I : Error between
3.3
Eigenvalues
analytically examples 21: matrix
5O
the solutions of REDUCE
and Gauss
elimination
and Eigenvectors
To find the eigenvalues application
'
40
matrix {r)
of symbolic
and algebraic
for an arbitrary only show
and eigenvectors
the exact
manipulation.
dimensional solutions
matrix
for a matrix Since
is another to solve
is theoretically
important
the eigenvalue impossible,
task in the problems
the following
for a 3 by 3 matrix.
s(3,3)$
22: s:=mat((sxx,sxy,sxz),(sxy,syy,syz),(sxz,sxy,szz))$ 23: mateigen(s,eta); 3 2 {{ETA-ETA *SXX-ETA
2
2 *SYY-ETA *SZZ+ETA*SXX*SYY+ETA*SXX*SZZ-ETA*SXY 2 -ETA*SXY*SYZ-ETA*SXZ +ETA*SYY* SZZ+SXX* SXY* SYZ-SXX*SY y*SZZ 2 2 2 -SXY *SXZ+SXY *SZZ-SXY*SXZ*SYZ+SXZ *SYY,
36
2
ARBCOMPLEX(1)*(ETA*SXZ+SXY*SYZ-SXZ*Syy) MAT(I,1):= ........................................................................... 2 2 ETA -ETA*SXX-ETA*SYY+SXX*SYY-SXY ARBCOMPLEX(1)*(ETA*SYZ-SXX*SYZ+SXY*SXZ) MAT(2,1):=........................................................................... 2 2 ETA -ETA*SX.X-ETA*SYY+SXX*SYY-SXY MAT(3,1):=ARBCOMPLEX(1)}} Time: 1283ms 24: s:-mat((5,1,0),( 1,2,4),(0,4,3))$ Time:284ms 25: mateigen(s,eta); 3 2 {{ETA -10*ETA +14*ETA+53, ,
MAT(l,1)
MAT(2,1)
4*ARBCOMPLEX(2) := .............................. 2 ETA - 7*ETA + 9 4*ARBCOMPLEX(2)*(ETA := ........................................ 2 ETA - 7*ETA + 9
- 5)
MAT(3,1) := ARBCOMPLEX(2)}} Time: 666 ms 26: trace(s); SXX + SYY + SZZ Time: 100 ms
As MATEIGEN
the command contain
(a) Characteristic
(b) The
number
lines
23 and 25 show,
three parts. They
the
solutions
from
are
equation.
of repetition
roots. It's one in the above
(c) Eigenvectors.
37
examples.
REDUCE
by calling
If the eigenvaluesare required, addition.
The equation
requirement.
The
"arbitrary
TRACE 3.2.4
constant".
numbers
as shown
which
which
appeared
in example
S. The
24 and
equation
needs
will be demonstrated
to be solved later can meet
this
is referred
to as
in the eigenvectors
25 also
trace of the matrix
show
results
in
from
is also evaluated
REDUCE
by
by the command
26.
solver
REDUCE
can solve
the imaginary
following
SOLVE
Examples
into matrix
Equation
includes
command
ARBCOMPLEX(1)
complex
substituting
solver
the characteristic
example
27:solve(ETA_'*3
the polynomial
part,
solves
equation
the "I" will
show
the characteristic
- 10*ETA**2+
up to order
three
up to represent
equation
obtained
exactly.
If the solution
the imaginary
symbol.
The
above.
14*ETA+53=0,eta); 2/3
{ETA=-((63*SQRT(229)*
1-691*SQRT(3))
*SQRT(3)*
2/3
I+(63" SQRT(229)*I-691, 1/3
SQRT(3))
-20*(63*SQRT(229)*I-691_SQRT(3))
1/3
2/3
•3
*I+58"2
1/3
1/6
*2
-58*2
1/3 "3
2/3
'3
1/3 )/(6*(63*SQRT(229)*I
- 691*SQRT(3))
*SQRT(3)*
1/3 *2
1/6 "3
2/3 ETA=((63*SQRT(229)*I
- 691*SQRT(3))
*SQRT(3)*I
2/3 SQRT(3)) 1/3 3
1/3 + 20*(63*SQRT(229)*I-691*SQRT(3))
2/3 *I-58'2
- (63*SQRT(229)*I 1/3 *2
1/3 *3
1/6
2/3
*3
-58*2
1/3 )/(6*(63*SQRT(229)*I-691*SQRT(3))
- 691"
1/3
*SQRT(3)* 1/6
*2
*3
),
2/3 ETA=((63*SQRT(229)*I
Time:
1/3 + 10*(63*SQRT(229)*I
I/3 1/6 2/3 1/3 *2 *3 + 58*2 *3 )/(3*(63*SQRT(229)*I 5717 ms
If the numerical turned
- 691*SQRT(3))
on, the
mode
NUMVAL,
numerical
solution
precision by default. The to the round-off errors.
imaginary
complex
of three
1/3 - 691*SQRT(3))
switch
eigenvalues
parts in the following
28:solve(ETA**3 - 10*ETA**2 + 14*ETA + 53=0,eta); {ETA=4.830750950611553d0 + 2.991124223331416d-7"I,
38
- 691*SQRT(3))
COMPLEX can
*2
and FLOAT
be obtained
example
1/3
1/6 *3
mode
in sixteen
are very small
are
digits
and are due
ETA=-( 1.6167630306368408d0 + 6.960060167347475d-8"I), ETA=6.786003556862447d0 + (-2.295118206596669d-7)* I} Time: 1567ms REDUCE can solvesystemsof linear algebraicequationsexactly. The limitation is determinedonly by thememorycapacityof thehardwaresystem.The following examplefinds theminimizationof a quadraticfunction Q=kl*yln(2}+k2*yl
*y2+k3*y2h{2]+k4*y2*y3+kS*y3_2]_k6,y3
29: Q: =kl *y 1* *2+k2" y 1*y2+k3* Time: 550 ms 30: a:=df(q,yl); A := 2*KI*Y1 Time: 167 ms
+ K2*Y2+
y2* "2+k4"
y2* y3+k5*
subject
y3** 2-k6'
to yl +y2=2
y3+y4* (y 1+y2-2) $
Y4
31: b:=df(q,y2); B := K2*Y 1 + 2*K3*Y2 Time: 167 ms
+ K4*Y3
32: c:=df(q,y3); C := K4*Y2 + 2*K5*Y3 Time: 166 ms
- K6
+ Y4
33: d:--df(q,y4); D:=YI+Y2-2 Time: 150 ms 34:
soive({a=0,b=0,c=0,d=0},{y
1,y2,y3,y4}); 2
{{YI=
4*K2*K5 - 8*K3*K5 + 2'K4 .....................................................
- K4*K6 , 2
4*Kl*K5
- 4*K2*K5
+ 4*K3*K5
8*Kl*K5 - 4*K2*K5 Y2= ....................................................
- K4
- K4*K6 , 2
4*Kl*K5
- 4*K2*K5
+ 4*K3*K5
- K4
2 16*K l*K3*K5-4*K
l'K4
+2*Kl*K4*
K6-4'
2 K2 *K5-K2*K4*
Y4= .........................................................................................
}} 2
4*Kl*K5
- 4*K2*K5
+ 4*K.3*K5
39
K6
- K4
Time: 1400ms After the substitution •
follows
of yl to y4 into the quadratic
function,
the minimum
is found
as
35: q:=q; 2 Q:-(
16*K 1*K.3*K5-4*K
l'K4
2 +4*K 1*K4*K6
-K 1*K6
2 2 K2*K6 -K3*K6 )/(4*Kl*K5-4*K2*_*K3*K5-K4 Time: 267 ms
3.2.$
Treatment
REDUCE REDUCE
does
trigonometric
of the
doesn't
even
possess
the
functions
know
without
cosine
d in command
)
function
potential
teaching
*K5-2*K2*K4*K6+
2
an equation
can be handled
For example, matrix
trigonometric
2 -4"K2
as simple
to learn easily
the operation
it. Due
by just
as sin2(q)+cos2(q)=l. to this
teaching
REDUCE
rule of trigonometry,
17 is as follows
powerful
However capability,
the operation
the determinant
the rules.
Of direction
:
36: det(d); 2
2 2 2 *COS(Q2) *SIN(Q3) +COS(Q1) * 2 2 2 2 9 2 2 COS(Q3) *SIN(Q2) +COS(Q1) *SIN(Q2) *SIN(Q3)'+COS(Q2) *COS(Q3) * 2 2 2 2 2 2 2 SIN(Q1) +COS(Q2) *SIN(Q1) *SIN(Q3) +COS(Q3) *SIN(Q1) *SIN(Q2) + 2 2 2 SIN(Q1) *SIN(Q2) *SIN(Q3) Time: 483 ms COS(Q1)
2
*COS(Q2)
After simple.
Note
2
*COS(Q3)
teaching
REDUCE
2
+COS(Q1)
the appropriate
that the time consumption
This is due to the extra internal swells.
work
needed
operation
in command for simplification.
rules,
38 is longer
4O
becomes
than that in command
It also reveals
37: let cos(ql)**2+sin(ql)**2=l,cos(q2)**2+sin(q2)**2=l,cos(q3)**2+sin(q3)**2=l. Time: 650 ms 38: det(d); 1 Time: 584 ms
the solution
the phenomenon
quite 36. of
3.2.6
Solving
differential
REDUCE
is unable
possess
this capability.
uniform
load q [Figure
The
equation
to solve
the differential
following
3.2]. The differential
dx 2 - s*w
Where well
s and r ,in general,
as boundary
end, the s becomes
example
conditions.
+ r*x
equation is a problem
equation
(x
directly,
while
of beam
MACSYMA
deflection
does
w(x)
under
is of the form
- L)
are the function
In the case of small
(3. 1)
of Young's
modules,
moment
deflection
with simple
of inertial
supported
as
on both
zero, and r=q/2EL
W
q
X
Figure
The command
(Cn)
and (Dn)
3.2 • Beam under uniform load. The boundaries specified to find general solution.
in the
following
is the solution
(C 1) depends(w,x); (DI) (C2)
(D2)
examples
is the MACSYMA
given by MACSYMA.
[W(X)]
diff(w,x,2)-s*w-r*x*(x-l)=0;
- RX(X-
(C3) ode2(d2,w,x); Is S positive, negative, P;
2 dW L) + .......... 2 dX
S W =0
or zero?
41
are not
prompt
for
inputting
the
SQRT(S)X (D0)W
-SQRT(S)
= %K1%E
2 RSX-LRS
X
X+2R
+ %K2 %E 2 S
Where
the %K1
%E here is the symbol
Although complex,
, %K2
are constants
of exponential
to be determined
by boundary
conditions.
The
function.
the next example
(no specific
it only takes 3.6 milliseconds
physical
problem
associated
with it) is more
in MACSYMA.
(C4) depends(y,x);
(134)
W(X)I
(C5) diff(y,x,2)+(2/x)*diff(y,x)-(2/x**2)*y-( dY 2--dX
2 d Y
1/x**2)*sin(log(x))=0;
2Y
SIN(LOG(X))
(135) 2
X
2
dX (C6)
2
X
X
ode2(d2,y,x);
(D6)
Y=
3 SIN(LOG(X)) + COS(LOG(X)) .......................................... 10
%K2 ....... 2 X
+%K1X+
(C7) time(d3); Time: (I37)
[3.6d0]
3.2.7
Polynomial
and
Polynomial symbolic First,
calling
and rational
and algebraic
in most cases
Therefore
operations
the problems
the appropriate to specific
operations
manipulation.
it is necessary
are restricted
rational
forms
is no way to avoid
employing
of the expansion,
factorization,
two polynomials,
obtaining
There
are two occasions
to be solved
to manipulate REDUCE
is one of the most important
commands
for particular this package.
in employing
are not as simple
the formulae
usage.
them.
In order
The capabilities
and cancellation
the part of polynomial
42
Second,
sometimes
to get the appropriate
of common and rational
factors,
include
determining
functions,
in
functions.
demonstrations.
manageable
of this package
functions
these
as the above
into a machine
to solve
and useful
forms
before
the solutions forms,
there
the controls the GCD
and so forth.
of
%Turningoff expansionswitch. 39: off exp; %Inputtingp andq polynomials.Thenumericalcommonfactors %will beautomaticallyfactoredout. 40: p:=(x-1)*(5*x-3)**2*(4*x+8)^3*(9*x-6); 2 3 P := 192"(5"X - 3) *(3*X - 2)*(X + 2) *(X - 1) 41: q:=(x-1)*(5*x-3)*(4*x+8)*(x+4); Q := 4"(5'X - 3)*(X + 4)*(X + 2)*(X - 1) %Gettinggreatestcommondividerof p andq. 42: gcd(p,q); 4'(5"X - 3)*(X + 2)*(X - 1) %Turningon expansionswitchandcheckingp,q. 43: on exp; 44: p; 7 6 5 4 3 2 192"(75"X +235'X -163"X -723'X +392'X +664'X -624"X+144) 45: q;
4
3 2 4"(5'X + 22"X - 5*X - 46"X + 24) %Defininga fractionr. 46: r.=p/q; 7 R :=(48'(75"X
6 +235"X
4 (5*X
3 +22"X
5 -163"X
4 -723'X
3 +392"X
2 +664'X
-624"X+144))/
2 -5*X
-46"X+24)
%Turning on the greatest common divider switch and rechecking r. %The common factors have been cancelled as shown in command 47. 47: on gcd; 48: r; 4 48"(15"X
3 + 41"X
2 - IO*X
- 52'X
+ 24)
....................................................
X+4 %Getting the denominator 49: den(r); X+4
and numerator
of fraction
43
r.
50: num(r); 4
3 2 48"(15"X + 41*X - lO*X - 52"X + 24) %Gettingtheleadingdegreeof numeratorof fractionr. 51"deg(num(r),x); 4 %Thenumerator 52:on ifactor; of r in command49 alsocanbefactoredasthemultiplicationsof eachfactors. 53: factorize(hum(r)); {2,2,2,2,3,X + 2,X + 2,3"X - 2,5"X - 3} 3.2.8
Fortran
code
REDUCE
can automatically
(default), input
and REDUCE
to the fortran
written
output
form,
the REDUCE functions
code.
while
REDUCE
following
examples
is useful
can be made
style expression in making
natural
style
as a subroutine
and
allows
a REDUCE
subroutine
from REDUCE
are generally
make
the switch
from symbolic
and algebraic
This
not
only
saves
out all the possibilities
The output
understanding
effort
of error
be directly
introduced
these functions.
forms are in natural
style of human
as hand-
for input
into
lengthy,
the
very
manipulation
in symbolic
about
7 6 6 52 5 P := A +7*A *B-7*A *C+21*A *B -42'A *B*C+21*A
expression
it to be looked
the results
will give a clearer
%Inputting the polynomial. 54: p:=(a+b-c)**7;
code
expression,
Since
smoother.
but also rules
the fortran
The natural
code
main program.
analysis
manipulation,
The fortran
main program.
of code-conversion
numerical
produce
and
by hand
being
to
algebraic typing.
The
writing.
52 43 42 *C +35"A *B -105*A *B *C+105"
4 2 43 34 33 322 3 3 34 A *B*C -35"A *C +35"A *B -140*A *B *C +210*A *B *C -140*A *B*C +35"A *C 2 +21*A
5 24 23 2 2 2 3 2 4 2 5 *B -105*A *B *C +210*A *B *C -210*A *B *C +105*A *B*C -21*A *C +7*
6 A*B -42*A*B
5
4 *C+105*A*B
7 6 +B -7*B *C+21*B Time: 1434 ms
5
2 3 3 2 4 *C -140*A*B *C +105*A*B *C -42*A*B*C
2 4 3 3 4 25 *C -35"B *C +35"B *C -21*B *C +7*B*C
%Turning on the fortran-code 55: on fort;
conversion
switch.
44
67 -C
5 6 +7*A*C
%Checking the output of fortran code. 56: p; ANS=A**7+7.*A**6*B-7.*A**6*C+21.*A**5*B**2.42.*A t*5*B •*C+ 21.*A**5*C**2+35.*A**4*B**3-105.*A**4*B**2*C+ 105. *A**4*B*C**2-35.*A**4*C**3+35.*A**3*B**4-140.*A**3*B *'3"C+2 lO.*A**3*B**2*C**2-140.*A**3*B*C**3+35.*A**3 *C*'4+21.*A**2*B**5-105.*A**2*B**4*C+210.*A**2* B**3*C *'2-210.*A**2*B**2*C**3+ 105.*A**2*B*C**4-21.*A**2*C **5+7.*A*B**6-42.*A*B**5*C+ 105.*A*B**4*C**2-140.*A*B *'3"C*'3+ 105.*A*B**2*C**4-42.*A*B*C**5+7.*A*C**6+B** 7-7.*B*'6"C+21.*B**5*C**2-35.*B**4*C**3+35.*B**3*C** 4-21.*B**2*C**5+7.*B*C**6-C**7 Time: 850 ms %Changing the number 57: cardno!*:=10$
of continuation
line in fortran
code.
58: p; ANS 1=7.*A*B**6-42.*A*B**5*C+ 105.*A*B**4*C**2-140.*A*B • *'3"C*'3+ 105.*A*B**2*C**4-42.*A*B*C**5+7.*A*C**6+B** • 7-7.*B*'6"C+21.*B**5*C**2-35.*B**4*C**3+35.*B**3*C** • 4-21.*B**2*C**5+7.*B*C**6-C**7 ANS=A**7+7.*A**6*B-7.*A**6*C+21.*A**5*B**2-42.*A**5*B • *C+21.*A**5*C**2+35.*A**4* B**3-105.*A**4*B**2*C+ 105. • *A**4*B* C*'2-35.*A*'4" C**3+35.*A**3*B**4-140.*A**3*B • *'3"C+210.*A**3*B**2*C**2-140.*A**3*B* C*.3+35.*A*.3. • C*'4+21.*A**2*B**5-105.*A**2*B**4*C+210.*A**2*B**3*C • *'2-210.*A**2*B**2*C**3+ 105.*A**2*B*C**4-21.*A**2*C • **5+ANS1 Time: 1034 ms %Turning off fortran-code 59: off fort; %Turning off the 'natural 60: off nat;
conversion
style'
function
switch.
switch•
%Checking the output of REDUCE code. 61: p:=p; P := A**7+7*A**6*B-7*A** 6*C+21*A**5*B**2-42*A**5*B*C+2 l*A**5*C**2+35*A**4*B**3-105' A**4*B**2*C+105*A**4*B*C**2-35*A**4*C**3+35*A**3*B**4-140*A **3*B**3*C+210*A**3*B**2*C**2-140*A**3*B*C**3+35*A**3*C**4+ 2 l*A**2*B**5-105*A**2*B**4*C+210*A**2*B**3*C**2-2 IO*A**2*B *'2'C*'3+ 105*A**2*B*C**4-2 l*A**2*C**5+7*A*B**6-42*A*B**5*C + 105*A*B**4*C**2-140*A*B**3*C**3+ 105*A*B**2*C**4-42*A*B*C**5 +7*A*C**6+B**7-7*B**6*C+21 *B**5*C**2-35*B**4*C**3+35*B**3*C *'4-21" B**2*C**5+7*B*C**6-C**75 Time: 816 ms
45
3.2.9
Number
system
In addition do numerical
to symbolic
analysis.
(a) Integer
There
--- In general,
the value
manipulation, are three there
the symbolic
and algebraic
ways to treat numbers
is no practical
of 2 l°°° gives 255 digits.
manipulators
in REDUCE.
limit on the number
They
can also
are
of digits.
For example,
It only takes 383 milliseconds.
62: A:=2** 1000; A: = 107150860718626732094842504_ 181056140481170553360744375038837 03 5105112493612249319837881569512759467291755314682518714528569231 40435984577598574803934567774824230985421 0746050623711418771821530 464749835819412673 987675591655439460770629145711 Time: 383 ms
(b) Fraction numbers) telling
number
--- Numbers
are represented
that aren't
by default
integers
and
by the quotient
operated
with
of two integers
symbols
(or
with message(s)
users the conversion.
63: a:=0.999*b*c; *** 0.999 represented 999"B*C A :--............. I000 Time: 200 ms
by 999/1000
64: a:=0.999"0.5; *** 0.999 represented by 999/1000 *** 0.5 represented by 1/2 999 A := ........ 2000 Time: 217 ms
(c)Real
number
approximations
--- It is also to numbers
%Turning on the numerical 65: on numval; %Turning on the floating 66: on float;
possible
to ask REDUCE
with arbitrary
precision
mode.
system
switch.
67: pi; 3.141592653 589793d0 Time: 150 ms 68: on bigfloat; *** Domain mode FLOAT
changed
to BIGFLOAT
46
to work
with specified
the
floating
numbers
point
of digit.
% Specified50digits. 69: precision505 70: pi; 3.1415926535897932384626433832795028841971693993751 Time: 217ms 3.2.10
Substitution Therearetwo substitutionfunctionsin REDUCE.Oneis for local substitution,andthe
otheris for globalsubstitution.The differencecanberevealedin the following examples. %Defininga functionf. 71: f:=6.*A**2*R I**2*U2-3.*ATAN(U 1/A)*A**2* RI*R2*U 1-9.*ATAN(UI/A)*A**2*RI*R2*U2+3.*ATAN(UI/A)*A **2*R2**2*U I+3.*ATAN(U I/A)*A**2*R2**2*U2-2.*ATAN(U 1/ A)*RI**2*U I**3+6.*ATAN(U I/A)*RI**2*U 1"'2'U25 Time: 1234ms %makinga 72:
local substitution b:=sub(atan(u 1/a)=k,f);
and calling
2 B :=-(3*A
it as B.
2 *K*RI*R2*UI+9*A
2 R2 *U2Time: 433 ms %Checking
2 *K*RI*R2*U2-3*A
2 *K'R2
2 *U1-3*A
*K*
2 2 2 3 2 2 6*A *R1 *U2 + 2*KZR1 *U1 - 6*K'R1 *U1 *U2)
the original
function
f after the local substitution.
It's unchanged.
73: f;
-(3*ATAN(
-3*ATAN(
U1 2 .... )*A *RI*R2*UI+9*ATAN( A U1 2 2 .... )*A *R2 *U2+2*ATAN( A
U1 2 .... )*A *RI*R2*U2-3*ATAN( A U1 2 3 .... )*R1 *UI - 6*ATAN( A
2 2 - 6*A *R1 *U2) Time: 367 ms %Making global substitution. 74: let atan(ul/a)=g; Time: 133 ms %The original 75: f;
function
f has been changed.
47
U1 2 2 .... )*A *R2 *U1 A
U1 2 2 .... )*R1 *U1 *U2 A
2 2 2 2 2 2 -(3*A *G*RI*R2*UI+9*A *G*RI*R2*U2-3*A *G'R2 *U1-3*A *G'R2 *U2 2 2 2 3 2 2 -6*A *R1 *U2 + 2*G'R1 *U1 -6*G'R1 *UI *U2) Time: 283ms 3.2.11
Built-in
The
built-in
general
purpose
suitably
combined
functions
which
MACSYMA (a)
functions
functions
usage,
commands allow
of REDUCE.
users to simply are shown
evaluation
automatically
system dependent.
there are not many built-in
functions
Limit
are quite
apply
---
functions.
On the contrary,
call commands
in the following
REDUCE
However
is designed
for
they can be obtained
MACSYMA
has many
once to get the solution.
by
built-in
Some of these
paragraphs.
If it is necessary,
L'Hospital's
Since
the
function
rule to evaluate
LIMIT
in MACSYMA
will
the formulae.
(C8)limit(sin(x)/x,x,0,plus);
(D8)
1
(C9) time(d8); Time:
(I39)
[2. 116d0]
(C 10) limit((6*x*2+3*x-4)/(x(D10)
1),x, l,plus); INF
(C 11) limit((1-x)**(1/x),x,0); -1 (Dll)
%E
(C 12) time(dl Time: (DI2) (b)
Laplace
1); [2.75d0]
transformation
--- The LAPLACE
functions
in physical
domain,
and ERF,
into the s domain.
into algebraic available.
equation.
(C 13) laplace(
l/sqrt(t),t,s);
(D13)
The
command
such as EXP, In addition, command
SQRT(%PI) ............... SQRT(S)
48
LOG,
in MACSYMA SIN, COS,
it also can transform for
inverse
SINH,
can transform COSH,
a differential
of Laplace
transform
the
DELTA equation is also
(C14)time(dl3); Time: (D14)
[0.05d0]
(C 15)laplace(((c-b)*exp(a*
t)+(a-c)*exp(b*
B-C
A-C
........
+
S-C (DIS)
t)+(b-c)*exp(c*
t))/((a-b)*
C-B
........
+
S-B
.......
S-A
........................................
(A(C 16) time(d9); Time: (D16)
B) (B - C) (C-
A)
[0.384d01
(C 17) laplace(sin(a*t)-a*t*cos(a*t),t,s); 2 A
2S
..........
(D17)
2
1
A( .......................
2
2 22 (S +A)
S +A
)
2 2 S +A
(C 18) time(dlT); Time: (D18)
[0.233d0]
%Inverting the Laplace (C 19) ilt((s+6)/(s^2+4*s+ -2T (D19)
%E
(C20) Time:
time(d4);
transform. 12),s,t);
2 SIN(2 SQRT(2) ( ......................... SQRT(2)
T) + COS(2
SQRT(2)
%Transforming a differential equation into algebraic (C21) laplace(diff(y(x),x,2)-3*diff(y(x),x)+2*y(x)=0,x,s);
equation.
(D20)
[0.933d01
d (D21)
T))
.... (Y(X)) dX
v [ - 3 (S LAPLACE(Y(X), I !X=0
X, S) - Y(0))
2 +S (C22) Time:
LAPLACE(Y(X),X,S)+2
LAPLACE(Y(X),X,S)-Y(0)
time(d21);
49
S=0
(b-c)*(c-a)),t,s);
(D22)
[(3.15dOl
(C23)
laplace(diff(y(x),x,2)+w^2*
y(x)-b*sin(w*x)=0,x,s);
!
d
!
(D23) .... (Y(X)) dX
2
! +W t _X=0
2 LAPLACE(Y(X),X,S)+S
LAPLACE(Y(X),X,S)
BW .......... 2 2 W+S (C24) Time: (D24)
Y(0) S - 0
time(d23); [0.217d0]
(c) Series
expansion
Taylor
series
--- The MACSYMA
and power
series
expansion
(C25)
taylor(%e^x,[x,0,7]);
version
series expansion
is available
on the market,
in University
capabilities.
Although
it is not available
2
3 4 5 6 7 X X X X X +--- +--- + .... + ...... + ....... 2 6 24 120 720 5040
X (D25)/T/
(C26) Time:
I+X+--
time(d25);
(D26) (C27)
(D27)
[0.567d0] powerseries(%e^x,x,0); INF =--= \ > /
I1 X ..... Il!
I1-0 (C28) Time: (D29)
time(d27); [0.45d0]
5O
of Michigan
+...
the function
here.
provides
the
of Fourier
3.3
References
[1] A. C. Hearn,"REDUCE CA 90406-2138,
reference
version
[3] Stephen
[5]
Richard
Gilbert 1986.
[7] Thomas
Publishing
Burden,
publishers,
J. Douglas
Peter
book
Pavelle,"
laboratory
Co., Santa
for computer
Monica,
science,
R. Spiegel,"mathematical
outline
series
in mathematics,
Wadsworth
Analysis",
Publishing
Prindle,
Weber
and
mathematics",
David
wellesley-Cambridge
A. Levinson,"Spacecraft
press,
dynamics",
1983.
• capacities Inc.,
and Analytic
Murry
by computer",
1985.
W. Likins
Symbolics
mathematics
mathematics",
to applied
Company,
for doing
Faires,"Numerical
edition,
MACSYMA
and the sciences",
[10]
RAND
Inc., 1988.
Engineering
Third
R. Kane,
[9] AI Shenk,"Calculus
3.3, The
group
A system Compan,
Strang,"Introduction
MacGraw-Hill
[8] Richard
The mathlab
Wolfram,"Mathematica-
L.
version
1983.
Peter V. O'Neil,"Advanced Company, 1983.
Schmidt
[6]
manual",
10, January
Addison-Wesley [4]
manual",
July 1987.
[2] "MACSYMA MIT,
users'
and applications
Cambridge,
Geometry", Handbook McGraw-Hill
51
MA,
Scott,
to problems
in engineering
1985.
Foresman
of formulas Book Company,
and Company, and 1968.
tables",
1984. Schaum's
&
CHAPTER
APPLICATION
OF
SYMBOLIC
AUTOMATIC
4.1
IV
AND
ALGEBRAIC
PROBLEM
MANIPULATION
TO
FORMULATION
Introduction
One of the most impcx_t is to automatically Modern
formulate
scientists
complicated
examples
easily
and correctly.
1. The derivation
3. The
mathematical
are continually
This
approximation
being
to a function
template
are dependent similar.
making with
manipulators,
any errors.
more
and
more
most problems
six automatic
can
formulation
They are :
by Fourier series.
for the iteration
method
matrix and mass problem.
the use of symbolic
mathematical
challenged
manipulation
in dynamics.
equations
on the problem
to be solved,
52
numerical
construction
manipulators to any fields.
the basic
commands
analysis.
for 6-node
for 4--node isoparametrically
and algebraic
can be extended
in nonlinear
matrix
6. Finite element stiffness matrix construction element in a plane elasticity problem.
formulate
without
will demonstrate
manipulation.
of motion
and algebraic
for the shell problem.
5. Finite element stiffness element in a heat transfer
Of course,
equations
and algebraic
chapter
and algebraic
of equations
formulation
4. The formulation
gained from symbolic
With the aid of symbolic
done by symbolic
2. Tensor
lengthy
and engineers
formulas.
be treated
advantages
as tools Although used
triangular
quadrilateral
to automatically the methodologies
in programming
are
4.2
Derivation
4.2.1
of equation
of motion
by SAM
Introduction
The derivation Figure L.
4.1 involves
of equations finding
of motion
kinetic
energy
by the Lagrange
T, potential
energy
method
for the system
V, and therefore
shown
the Lagrangian
M,
i Figure
The
4.1 • Dynamic system for demonstration and algebraic manipulation
Lagrangian
then
is partially
coordinates
and the rate of generalized
after
the time
taking
Mathematically,
potential
derivatives
the Lagrange
I
T V 4.2.2
REDUCE
.
"iM2(x
coordinates.
to appropriate
equations
where L is the Lagrangian energy V.
differentiated
I
--Mtgr(l_cosO)_M2g(r_rcosO program
and
terms
and
1: on time; Time: 83 ms
53
to both
of motion
assembling
generalized
will be obtained
the necessary
terms.
between
kinetic
T and
energy
.2
2 + _IO
(4.2) +xsin
solution
respect
in the form of
as the difference
1
+ tO) 2 + _M_(rO)
with
The equations
are expressed
and is defined
of symbolic
0)+
tk(rO)2
(4.3)
in
%declaring theta and x as a function 2: depend theta, time; Time: 67 ms
of time.
3: depend x,time; Time: 50 ms %Calculating the total kinetic energy of system. %The velocity v of body 2 need be evaluated later. 4: re:=( 1/2)*i0*(df(theta,time))**2+(
I/2)*m
l*r**2*(df(theta,
2 DF'(THETA,TIME) *I0 + DF(THETA,TIME) TE := ............................................................................... 2 Time: 500 ms
2
time))**2+(I/2)*m2,v**2;
2 2 *R *M1 + M2*V
%Calculating the position vector of body 2. 5: P:=(r*cos(theta)-x*sin(theta))*j+(r*sin(theta)+x*cos(theta)),i; P := COS(THETA)_'I*X+COS(THETA),J,R+SIN(THETA),I,R.SIN(THETA),.I,X Time: 333 ms %The velocity vectors is then obtained by taking the denvative %the position vector with respect to time. 6: dp:=df(p,time);
of
DP := COSfTHErA)*DF'(THETA,TIME)*I*R-COS(THETA), DF(THETA, TIME)*J*X+COS(THE A)*DF(X,TIME-')*I-DF(THETA ,TIME)* SIN(THETA) *I * X-DF'(THET A,TIME)*StN(THET A )* 2*R.DF(X,TIME) *SIN(THETA)*J Time: 234 ms %Getting the magnitude square of velocity 7: v** 2:=lcof(dp, i)**2+tcof(di:Q)**2; 2 V :=COS(THETA)
2 *DF(THETA,TIME)
2 *X +2*COS(THETA)
DF(X,TIME) Time: 650 ms
2
+DF(THETA,TIME) 2
S1N(THETA)
2 *R +COS(THETA)
2
2 *DF(THE_A,TIME)
2 *DF(THETA,TIME)*DF(X,TIME)*R+CC_(T'I-tETA)
2 DF(X,TIME)
2
of body 2.
2 *SIN(THETA)
2 •
2
2
_*R +DF(THETA,TIME)
2
* 2
*X +2*DF(THEI'A,TIME)*DF(X,T1ME)*SIN(THETA) 2 2 *SIN(THE--TA)
%Teaching REDUCE the tri'gonometric rule for simplification 8: let (cos(them))* *2+(sin(theta))**2= 1; Time: 150 ms
54
of %formulae.
*R+
%Inputtingthe
potential energy of system. 9: ve:=-m 1* g'r*( 1-cos(theta))-m2* g*(r* ( 1-cos(theta))+x*sin(theta))+( VE := (2*COS(THETA)*G*R*MI+2*COS(THETA)*G*R*M2-2*SIN(THETA) 2 2 • G*M2*X2*G*R*M1 - 2*G*R*M2 + K*R *THETA )/2 Time: 483 ms %calculating 10: la:=te-ve;
1/2)*k* (r* theta)**2;
the Lagrangian.
LA := -(2*COS(THETA)*G*R*MI+2*COS(THETA)*G*R*M2-DF(THETA, 2 2 2 TIME) *I0 - DF(THETA,TIME) *R *M1 - DF(THETA,TIME) 2 2 *M2 - DF(THEI'A,TIME)
2
*M2*X
2
2 *R
-2*DF(THETA,TIME)*DF(X,TIME)
*R'M2
- DF(X,TIME) *M2 -2*SIN(THETA)*G*M2*X-2*G*R*M12 2 2*G*R*M2+K*R *THETA )/2
Time:
617 ms
%Deriving the equation %any simplification. 11: e l:=df(df(la,
of motion
df(theta,
for theta coordinate
without
time)),time)-df(la,theta);
E1 := -(COS(THETA)*G*M2*X+DF(THETA,THETA,TIME)*DF(THETA, TIME)
2 *M 1
*I0+DF(THETA,THETA,TIME)*DF(THETA,TIME)*R
2 *M2 +DF(THETA,
+DF(THE'FA,THETA,TIME)*DF(THETA,TIME)*R 2 THETA,TI
ME)* DF(THETA,TIME)*
M2* X +DF(THETA,THETA,TI
ME) 2
*DF(X,TIME)*
R* M2-DF(THETA,TIME,2)*I0-DF(THETA,TIME,2)*R 2 *M 1-DF(THETA,TIME,2)*R *M2-DF(THETA,TIME,2)*M2*X DF(THETA,TIME)
Time:
SIN(THETA) 783 ms
* DF(X,TI
ME) * M2* X- DF(X,TI
*G*R*MI+SIN(THETA)*G*R*M2-K*R
%Deriving the equation of motion for x coordinate 12: e2:=df(df(la, df(x,time)),time)-df(la,x)+f;
without
2 -2*
ME,2) * R* M2 + 2 *THETA)
simplification.
2 E2 := DF(THETA,TIME,2)*R*M2-DF(THETA,TIME) *M2*X-DF(THETA, TIME) *DF(X,TIME,X)*R*M2-DF(X,TIME,X)*DF(X,TIME)*M2 + DF(X,TIME,2)*M2 - SIN(THEq'A)*G*M2+F Time: 483 ms %Teaching
REDUCE
the 2nd derivative
of theta w/t theta and time.is
55
null, and so does x.
13:let df(theta_time_theta)--__df(x_x_time)--__df(theta_theta_time)---__df(x_time_x)--_; Time:350 ms %turningoff theautomaticexpansionswitch. 14:off exp; Time:50 ms % %Startingpolynomialmanipulationto simplifytheequationsof motion. % %Gettingthecoefficientof angularacceleration term. 15:a1:=lcof(e1,df(theta,ti me,2)); A 1 := (M1 + M2)*R Time: 617 ms
2 2 + I0 + M2*X
%Getting the coefficient of sin(theta). 16: a2:=lcof(e 1,sin(theta)); A2 := - (M1 + M2)*G*R Time: 550 ms %Getting the leftover after taking off the above two terms. 17: a3:=e 1-a l*df(theta, time,2)-a2*sin(theta); A3 :=-(COS(THETA)*G* M2*X-2*DF(THETA,TIME)*DF(X,TIME)*M2*X 2 -DF(X,TIME,2)*R* M2-K*R *THETA) Time: 533 ms %Rearranging the equation of motion for theta coordinate. %The results are simpler than those of in command 11. 18: e 1:=al*df(theta,time,2)+a2*sin(theta)+a3; 2 2 E1 := ((MI+M2)*R
+I0+M2*X
)*DF(THETA,TIME,2)-(COS(THETA)*G*M2* 2
X-2* DF(THETA,TIME)*DF(X,TIME)*M2*X-DF(X,TIME,2)* *THETA)-(M I+M2)*SIN(THETA)* G* R Time: 400 ms %Getting the coefficient of x acceleration 19: c 1:=lcof(e2,df(x,time,2)); C1 := M2 Time: 350 ms %Getting the coefficient 20: c2:=lcof(e2,df(theta, C2 := R'M2 Time: 333 ms
of angular time,2));
term.
acceleration
term.
%Getting the leftover terms by removing the above 21: c3:=e2-c l*df(x,time,2)-c2*df(theta, time,2); 2 C3 := - (DF(THE/'A,TIME)
*M2*X
R* M2.K*R
two terms.
+ SIN(THETA)*G*M2
56
- F)
Time:350ms %equationof motionfor x coordinate. 22: e2:=cl*df(x,time,2)+c2*df(theta,time,2)+c3; 2 E2 := -((DF(THETA,TIME) *M2*X+SIN(THETA)*G*M2-F)DF(THETA,TIME,2)*R*M2 - DF(X,TIME,2)*M2) Time:317ms 23: bye; The examplesshown above are the demonstrationof obtaining the left hand side formulaeof equation(4.1).The equationsof motionof systemcanbesimply doneby setting the resultsof command18 and22 equal to zero. As the systemis complex, the analytical derivationsof equationsof motion by handwill becometediousandprone to error. For the casesof complex systems,The applicationof symbolicandalgebraicmanipulationwill be moresignificant. 4.3
Automatic
4.3.1
tensor
Preliminary
Tensors situations,
are
tensor often
and algebraic problem
covariant
notation
of coordinate
are quite
This advantage
form
and using SAM
the Latin indices
formulation
equations
in the related
this difficulty
to expand
shell
metric
tensor
special
problem,
the
surface.
the benefits
fields,
the expressions
gained
by
of tensor
by the use of symbolic
by formulating tensor
physical
the thin shell
equations.
For the sake
1, 2, 3 and the Greek
indices
are
stress.
only
Based
necessary on parametnc
inputs
are
equations,
three the
are calculated
3___ 3u _
a = det (a_)
The contravariant
Although
the resulting
without
aab and its determinant
fl._.. aqn "/ix, Ill J ,,. 3u a
describing
can be avoided
for the range
fi(u 1,u 2) of the shell middle
tensor
concisely
will be demonstrated
paragraph
of the thin
for
transformations.
will be referred
of 1, 2 in the following
metric
problem
entities
significant
manipulation.
On the
shell
mathematical
Fortunately,
of convenience,
parametric
convenient
rise to errors.
in tensor
in the range
for
formulation
that are independent
employing formula
formulation
(4.4) (4.5)
is therefore
given by
57
fi 2 ,1 u =const.
3
X
=Z
2 X X
Figure
4.2"
a '_ - cofactor
The
first
and
second
kind
Pictorically
vector
notations
of shell
(a,_)/a
(4.6)
of Christoffel
symbols
for
the surface
are defined
as follows
respectively"
[afl,y
] - _a,_,
fly
The tensor
covariant
-
differentiations
are therefore
.p + aa, .a - a_.,
-ia
(a
.r +a
of a covariant
)
(4.7)
,_ -a
vector
pr ,6)
(the
(4.8)
1st order
tensor)
and
the 2nd order
calculated
(4.9)
A p .,, " D ,_A Ij " A ¢ .a - { fl Ya } A r
A or ' *' "
When
the above
computed
" A or a
formulations
by the formulae
are done,
-
f 't a { '}a fl
a
the curvature
of
58
_r
y
tensor
ct
and
t_
its determinant
(4. 10)
are then
d_
Where equal
- X
f_
(4.11)
d - det (d_)
(4.12)
the X i are the unit normal
surface
[see Figure
tensor
and undeformed
The generalized
the middle
+p,_
the deformed
+p_
two-dimensional =Dav
K¢
+prapar
state. The a*al 3 is defined
gradientp,
(4.15)
and rotation
q are represented
- d_
(4.17)
displacements
and w is the out of plane
the bending
and undeformed
tensor
curvature
is equal
-(qP
+ e_e
tensor.
+ O/a)(d_ r_ qr P_)(D
as
+ Daq,,
r + dtj par)
a Pap - d tjpqa)]
O =det(Pa_
The constitutive
equations
between
(4.18)
d ctl3 is expressed
a I12_.i ,,. (-a'7) [( +p;
displacement
to the difference
- d_
tensor
as (4. 16)
a + w.a
tensor
curvature
d'_
tensor
as
+ qaqa
Z
The deformed
metric
a - d_w
of the shell. Similarly
curvature
as deformed
displacement
the vct are the in-plane
surface
the deformed
(4. 14)
is denoted
= a_
qa - d,_v
Where
between
- a_)
superscript
p,_
as half of the difference
,
- 2(a_
a'_
delta.
tensor. l
the asterisk
(4.13)
Kronecker
is defined
metric
E_
Where
4.2] and are
j k ,jkf.,f.2
a-l12e
eij k is the generalized The strain
Where
of the shell middle
to X ' -
Where
vectors
for isotropically
elastic
59
and thin shell are
of
N _
, 1---_[(1_
M
-
- v)E
(4.19)
"# + wt'_E_]
Eh 3
Where
the Young's
final equilibrium
Where
12(1-
Y
v:) [(1-
modules
equations
v)K_
(4.20)
+ va'_Kr]
E and strain
E co
tensor
shouldn't
be confused
here.The
are then
DaN
_¢ + 2dr
D,,D
pM _
DoM
+ M
-d,_dY#M
the F # and P are the external
D,,dr
'¢ - deN
forces
+ '¢ -P
applied
-0
(4.21)
- 0
(4.22)
in the in-plane
and out-plane
directions,
respectively.
4.3.2
REDUCE
The The
REDUCE
explanations
necessary. reference
program
The
and
program
shown
of the program resultant
resultant
below
expressions
is based
are also included
expressions
are too huge
on the above to facilitate
formulation
methodology.
an understanding
where
to be included
here and
[17].
ARRAY X(3),C 1(2,2,2),C2(2,2,2); OPERATOR U,V,W,F; MATRIX A(2,2),CONTRA(2,2),D(2,2);
% INPUTTING %
SURFACE
PARAMETRIC
FUNCTIONS
X(1):=U(1); X(2):=U(2); X(3):=CONSTANT; % % CALCULATING % FOR
M:=l:2
COVARIANT
DO FOR N:=l:2
METRIC
TENSOR
& ITS DETERMINANT
DO
A(M,N): =FOR I:= 1:3 SUM DF(X(I),U(M))*DF(X(I),U(N)); DETA:=DET(A); DEPEND W,U(1),U(2); FOR I:=1:2 DO DEPEND V(I),U(1),U(2);
%CALCULATING
CONTRAVARIANT
METRIC
60
TENSOR
COMPONENT
are available
it is in
FORL:=l:2 DO FORM:=l:2 DO IF L=I andM=I THEN CONTRA(L,M):=A(2,2)/DETA ELSE IF L NEQM THEN CONTRA(L,M):=-A(M,L)/DEI"A ELSE CONTRA(L,M):=A(1,1)/DETA; % %CALCULATING THE
1ST CHRISTOLFFEL
SYMBOL
% FOR L:=l:2
DO FOR M:=l:2
DO FOR N:=l:2
DO
C I(L,M,N):=(1/2)*(DF(A(L,N),U(M))+DF(A(M,N),U(L))_DF(A(L,M),U(N)));
%CALCUI_,ATING
FOR L:=l:2
THE 2ND CHRISTOLFFEL
DO FOR M:=1:2
C2(L,M,N):=FOR
I:=1:2
SYMBOL
DO FOR N:=l:2
SUM
(SEE
EQ. 4.6)
DO
A(L,I)*CI(M,N,I);
% %SUBROUTINE FOR CALCULATING %FOR THE 1ST ORDER TENSOR % PROCEDURE
COVD(L,
DF(VAR(M),U(L))-(FOR
DERI VATI VE
VAR(M)); I:=1:2
SUM C2(I,L,M)*VAR(I));
%SUBROUTINE FOR CALCULATINE %FOR THE 2ND ORDER TENSOR
PROCEDURE COVD2(L,FUN(M,N)); DF(FUN(M,N),U(L))-(FOR I:=1:2 SUM -(FOR I:=1:2 SUM C2(I,L,N)*FUN(M,I)); MATRIX E(2,2),K(2,2),P(2,2),FF(2,3),XX( % %CALCUL,ATING %ALSO CALLED %
THE COVARIANT
THE COVARIANT
DERIVATIVE
C2(I,L,M)*FUN(I,N)) 1,3);
REL.ATIVE ALTERNATE TENSOR AS GENERALIZED KRONECKER DELTA
PROCEDURE EE(I,J,K); IF I=J OR J=K OR K=I THEN ELSE IF
0
(I=l AND J=2 AND K=3) OR (I=3 AND J=l AND K=2) OR (I=2 AND J=3 AND K=I) THEN 1 ELSE - 1;
%CALCULATING %
TIlE
CURVATURE
TENSOR
61
& ITS DETERMINANT
FOR I:=1:2DO FORJ:=1:3DO FF(I,J):=DF(X(J),U(I)); FORI:=1:3 DO XX(1,I):=( 1/SQRT(DETA))*(FORJ:=l:3 SUM (FOR K:= 1:3 SUM EE(I,J,K)*FF(1,J)*FF(2,K))); FORI:=1:2 DO FOR 3:=1:2 DO D(I,J):=FOR 11:=1:3 DETD:=DET(D);
SUM
XX(1,I
1)*DF(FF(I,I
1),U(J));
_=====_=_
%CALCULATING % FOR
L:=l:2
THE GENERIZED
DO FOR M:=I:2
DISPLACEMENT
TENSOR
DO
P(L,M):=DF(V(M),U(L))-(FOR
I:=1:2
SUM C2(I,L,M)*V(I))-D(L,M)*W;
% %CALCULATING %
THE ROTATION
TENSOR
(SEE
EQ. 4.17)
ARRAY Q(2); FOR M:=l:2 DO Q(M):=DF(W,U(M))+(FOR
I:=1:2 SUM
D(M,I)*(FOR
J:=l:2
SUM
CONTRA(I,J)*V(J)));
i_---=_'====."
%DERIVING O'_
====
=
THE
STRAIN
TENSOR
--.,,
FOR I:=1:2
DO FOR J:=l:2
DO
E(I ,J): =( 1/2)* (P(I,J)+P(J,I) +(FOR I1:=1:2 SUM (FOR Jl:=l:2 OFF PERIOD; ON FORT; OFF PERIOD; FOR I:=1:2 DO FOR J:=1:2 DO WRITE " E(",I,",",J,")=",E,(I,J);
SUM
CONTRA(I
1,J1)*P(I,J1))*
P(J,I 1))+Q(I)*Q(J));
% %CALCULATING %
THE ABSOLUTE
2-D ALTERNATE
TENSOR
PROCEDURE EPS(L,M); IF L=I AND M=2 THEN SQRT(DETA) ELSE IF I_,=2 AND M=I THEN -SQRT(DETA) ELSE 0; % %CALCULATE %
CONTRAVARIANT
ABSOLUTE
PROCEDURE CEPS(L,M); IF L=I AND M=2 THEN I/SQRT(DETA) ELSE IF L=2 AND M=I THEN -I/SQRT(DETA)
62
2-D ALTERNATE
TENSOR
ELSEO; % %CALCULATING THE BENDINGTENSOR O_=-'_===========_-
FOR I:=1:2
DO FOR J:=l:2
DO
; DEPEND AD,U( 1),U(2); RAD:=DET(A+2*E)$ WRITE" AD=",RAD; % %DERI %
VING
THE CONSTITUTIVE
MATRIX N(2,2),M(2,2); FOR I:=1:2 DO FOR J:=l:2
%EFFECTIVE I_
MEMBRANE
EQUATIONS
DO
STRESS
TENSOR
.........................................................
; % %WRITING-OUT %
EXPRESSIONS
FOR I:=1:2 DO FOR J:=l:2
1)))
OF STRESS
DO
WRITE" NC,I,",",J,")=",N(I,J); FOR I:=1:2 DO FOR J:=l:2 DO
63
AND MOMENT
TENSOR
WRITE
"
M(',I,',',J,')=',M(I,J);
_0==========:
%DERIVING
THE EQUILIBRIUM
EQUATIONS
(_===='---=="
.mm
ARRAY Ml(2),Nl(2); FOR I:=1:2 DO MI(I):=FOR J:=l:2 SUM (DF(M(I,J),U(J))+(FOR I 1:= 1:2 SUM C2(I,I +(FOR I 1 := 1:2 SUM C2(J,I 1,J)*M(I,I 1))); MM:=FOR I:=1:2 SUM
1,J)*M(I
l,J))
(DF(M I(I),U(I))+(FOR 11:=1:2 SUM C2(I,I 1,I)*MI(I 1))); BEQ:=MM-(FOR I 1:=1:2 SUM (FOR I2:=1:2 SUM D(I 1,I2)*(FOR I3:=1:2 SUM CONTRA(I2,I3)*(FOR I4:= 1:2 SUM D(I4,I3)*M(I 1,I4))))) -(FOR I1:=1:2 SUM (FOR I2:= 1:2 SUM D(I 1,I2)*N(I 1,I2)))-PP; FOR I:=1:2 DO NI(I):=FOR J:=l:2 SUM DF(N(I,.I),U(J))+(FOR 11:=1:2 SUM C2(I,I 1,.I)*N(I 1,J)) +(FOR 11:=1:2 SUM C2(J,I 1,J)*N(I,I 1)) +2*(FOR I 1:= 1:2 SUM CONTRA(I,I 1)* (FOR I2:= 1:2 SUM D(I2,I 1)*(DF(M(I2,J),U(J)) +(FOR I3:=1:2 SUM C2(I2,t3,J)*M(I3,J)) +(FOR I3:=1:2 SUM C2(J,I3,J)*M(I2,I3)))))(FOR I1:=1:2 SUM M(I 1,.I) *(FOR I2:=1:2 SUM D(I 1,I2)*(DF(CONTRA(I,I2),U(J)) +(FOR I3:=1:2 SUM C2(I,I3,J)*CONTRA(I3,I2)) +(FOR I3:=1:2 SUM C2(I2,I3,J)*CONTRA(I,I3)))) +(FOR I2:= 1:2 SUM CONTRA(I 1,I2)*(DF(D(I 1,I2),U(J))(FOR I3:= 1:2 SUM C2(I3,I 1,J)*D(I3,I2))(FOR I3:=1:2 SUM C2(I3,I2,J)*D(I 1,I3)))))+F(I); WRITE BEQ," =0"; FOR I:=1:2 DO WRITE NI(I)," ----0_; BYE; 4.3.3
Remarks
The
REDUCE
the geometry equations stress,
other than plates,
in the beginning
moment
derivatives. series,
Fourier
of the scope
program
tensors
Therefore
and solution the program
of program.
and equilibrium we may assume
series etc.), of this report,
neglect
shown is easily
above modified
As the results equations three
show,
are functions displacement
the unnecessary
terms
it will be left to the interested
64
are just
for the case of plates.
by changing the formulae
the input parametric of strain,
bending,
of three displacements fields
and solve researchers.
For
as polynomial the problem.
and their (or power
Since
it is out
4.4
Approximation
4.4.1
of a function
in properties.
fields.
purpose
mathematical
series
belong
to different
etc.
of terms
evaluation
Since
even
manipulation, be attained.
Here,
to demonstrate The
the function include
as more
a function
approximations
an example
the advantages hat function
of Fourier
of making
without
of symbolic
is a fundamentals
of shape
errors
be for avoiding
and
the Some
Fourier
is up to the
and
and the desired
and algebraic
more Based
on
a high accuracy
to the hat function
function
easily.
is increased.
By the symbolic errors
to
in various
in the formulation,
theoretically,
series approximation
of application
engineers
the accuracy
are involved
different
However,
approximation
methods,
to achieve.
can be formulated
It may
series
can be approximated is difficult
of the other.
so that it can be solved
Taylor
terms
are apparently
by many
on the problem.
will turn out and possibility
the approximation
adopted
are the approximation
Of course,
though
of such
depends
spaces
one in terms
and is actually
a function
they
included.
expression
reasons,
to replace
or for simplifying
to approximate
expansion
complicated
is feasible
of approximation
difficulties
methods
number
in
algebraic
accuracy
can
will be shown
manipulation.
in finite
element
method.
It is
as •
Ill(X)(x)
Mathematically, from
series
which
impossible
one by the other
The
popular
two functions
It is theoretically
approximate
defined
Fourier
Introduction
Mathematically,
these
by
the sinusoidal
The approximation
Z l+x 1-x
it's a piecewise function
which
to the hat function
f(x)
,for ,for
ao - .-_-.+
-1-:x 0,:x
C l continuous
is C _ continuous by Fourier
_ a_ cost_), mix,
>; FOR I:=1:2 DO ;
the symmetric
material
matrix.
MATRIX D(2,2); D: =MAT((K 11 ,K 12),(K 12,K22)); MATRIX LO(6,6),NU(6,6),CC(6,6),SKE,(6,6); O'_
...................................
%Obtaining
the stiffness
matrix.
SKE:=(AREA/3)*(TP(NS1)*D, %Making
the appropriate
NS I+TP(NS2)*D*NS2+TP(NS3),D,NS3)$ substitution
COB:=-(X3*Y2-X3*Y 1-X2*Y3+X2*Y FOR I:=1:6 DO FOR J:=l:6 DO
to simplify
the final expression.
I+X 1*Y3-XI*Y2);
; %Outputting
the resultant
fortran
;CC(I,j):=LO(I,J)/COB;
subroutine.
ON FORT; OFF ECHO; OFF PERIOD; OUT "sym"; WRITE " WRITE" WRITE" WRITE" FOR I:=1:6
subroutine stiff(xl,y 1,x2,y2,x3,y3,dl implicit real*8(a-h,o-z)"; dimension ske(6,6)"; DJ=-(X3*Y2-X3*Y 1-X2*Y3+X2*Y DO FOR J:=l:6 DO
IF J>=I THEN ELSE WRITE
1,d12,d22,ske)";
I+X I*Y3-X
WRITE "SKE(",I,",",J,")=",SKF_.(I,.I) "SKE( ",I,",",J,")=SKE(.,J,,,,.,I ,")";
74
l,Y2),;
WRITE " return"; WRITE" end"; SHUT "sym"; OFFFORT; BYE; 4.6.3
Resultant
stiffness
matrix
subroutine stiff(x 1,y 1,x2,y2,x3,y3,d implicit real*8(a-h,o-z) dimension ske(6,6)
made
by
REDUCE
11 ,d 12,d22,ske)
DJ=-(X3*Y2-X3*Y 1-X2*Y3+X2*Y I+X I*Y3-X l'Y2) SKE( 1,1)=(D 11 *Y2**2-2*D 11 *Y2*Y3+D11 *Y3**2-2*D 12"X2' Y2 • +2*D12*X2*Y3+2*D12*X3*Y2-2*D12*X3*Y3+D22*X2**2_2*D22 • *X2*X3+D22*X3**2)/(2*DJ) SKE( 1,2)=(D11*Y I*Y2-D 11*Y I*Y3-D11*Y2*Y3+D I l*Y3**2-D12 . *X I*Y2+D12*XI*Y3-D12*X2*Y I+D12*X2*Y3+D12*X3*Y l+D12* • X3*Y2-2*D 12*X3*Y3+D22*X 1*X2-D22*X 1*X3-D22*X2*X3+D22* • X3**2)/(6*DJ) SKE(1,3)=-(D11*Y I*Y2-D11*Y I*Y3-D1 l*Y2**2+D1 l*Y2*Y3. D12*XI*Y2+D12*XI*Y3-D12*X2*Y l+2*D12*X2*Y2-D12*X2*Y3+ • D 12*X3*Y 1-D 12*X3*Y2+D22*X l*X2-D22*X l*X3-D22*X2**2+ • D22*X2*X3)/(6*DJ) SKE(1,4)=-(2*(D11*Y I*Y2-D11*Y I*Y3-D11*Y2*Y3+D11.Y3..2 • -D12*XI*Y2+D 12*X I*Y3-D12*X2*Y I+D12*X2*Y3+D12*X3*Y I+ . D 12*X3*Y2-2*D 12*X3*Y3+D22*X l*X2-D22*Xl*X3-D22*X2*X3+ • D22*X3**2))/(3*DJ)
SKE(1,5)=0 SKE( 1,6)=(2"(D 11' Y 1*Y2-D 11 *Y I*Y3-D 11 *Y2**2+D 11' Y2*Y3• D12*XI*Y2+D12*XI*Y3-D12*X2*Y l+2*D12*X2*Y2-D12*X2*Y3+ • D12*X.3*Y 1-D12*X3*Y2+D22*Xl*X2-D22*Xl*X3-D22*X2**2+ • D22*X2*X3))/(3*DJ) SKE(2,1)=SKE(1,2) SKE(2,2)=(D 11*Y 1"'2-2"D 11*Y I*Y3+D1 l*Y3**2-2*D12*Xl*Y 1 • +2*D 12*X 1 *Y3+2*D 12*X3*Y 1-2' D 12'X3' Y3+D22*X 1*'2-2" D22 • *X 1*X3+D22*X3**2)/(2*DJ) SKE(2,3)=(D 11*Y l**2-D 11*Y I*Y2-D11*Y I*Y3+D1 l*Y2*Y3-2* • D 12*X 1*Y 1+D 12*X 1*Y2+D 12* X 1*Y3+D 12* X2* Y 1-D 12*X2*Y3+ • D 12*X3*Y 1-D 12"X3" Y2+D22*X 1**2-D22*X 1 *X2-D22* X 1*X3+ • D22*X2*X3)/(6*DJ) SKE(2,4)=-(2*(D 11*Y I*Y2-D11*Y I*Y3-D11*Y2*Y3+D 11"Y3"'2 • -D 12*X I*Y2+D 12*X I*Y3-D 12*X2*Y I+D 12*X2*Y3+D 12*X3*Y 1+ • D12*X3*Y2-2*D12*X3*Y3+D22.X l*X2-D22*Xl*X3-D22*X2.X3+ • D22*X3**2))/(3*DJ) SKE(2,5)=-(2*(D 11 *Y 1**2-D 1 I*Y 1*Y2-D 11 *Y I *Y3+D 11 *Y2*Y3 • -2*D12*XI*Y I+D12*XI*Y2+D12*XI*Y3+D12*X2*Y 1-D12*X2*Y3 • +D 12*X3*Y 1-D 12*X3*Y2+D22*X 1**2-D22*X 1*X2-D22*X 1* X3+ • D22*X2*X3))/(3*DJ) SKE(2,6)=0 SKE(3,1)=SKE(I,3) SKE(3,2)=SKE(2,3) SKE(3,3)=(D 11 *Y 1** 2-2'D 11 * Y l *Y2+D 11 *Y2**2-2*D 12* X 1* Y 1 • +2*D 12*X 1*Y2+2*D12*X2*Y 1-2' D12*X2*Y2+D22*X 1*'2-2" D22
75
• *X l*X2+D22*X2**2)/(2*DJ) SKE(3,4)=0 SKE(3,5)=-(2" (D 11*Y 1**2-D 11*Y I*Y2-D 11*Y 1*Y3+D 11 *Y2*Y3 • -2*D12*XI*Y I+D12*XI*Y2+D12*XI*Y3+D12*X2*Y 1-D12*X2*Y3 • +D12*X3*Y 1-D12*X3*Y2+D22*Xl**2-D22*X l*X2-D22*X1,X3+ • D22*X2*X.3))/(3*DJ) SKE(3,6)=(2*(D 11 *Y 1*Y2-D 11 *Y 1*Y3-D 11 *Y2**2+D 1 l*Y2*Y3• D12*XI*Y2+DI2*X I*Y3-D12*X2*Y l+2*D12*X2*Y2-DI2,X2,Y3+ • D 12*X3*Y 1- D 12*X3*Y 2+D22*X 1*X2-D22*X 1* X3-D22*X2**2+ • D22*X2*X3))/(3*DJ) SKE(4,1)=SKE( 1,4) SKE(4,2)=SKE(2,4) SKE(4,3)=SKE(3,4) SKE(4,4)=(4*(D 11*Y l**2-D 11*Y I*Y2-D 11*Y I*Y3+D11"Y2"'2. DI I*Y2*Y3+D11*Y3**2-2*D12*XI*YI+D12*XI*Y2+D12*XI,Y3+ . D 12*X2*Y 1-2*D 12*X2*Y2+D12*X2*Y3+D12*X3*Y l+DI2*X3*Y2• 2*D 12"X3" Y3+D22*X l**2-D22*X 1*X2-D22*X 1*X3+D22*X2**2• D22*X2*X3+D22*X3**2))/(3*DJ) SKE(4,5)=(4*(D1 I*Y I*Y2-D11*Y I*Y3-D 1l*Y2**2+D 1 l*Y2*Y3• D12*XI*Y2+D12*XI*Y3-D12*X2*Y I+2*D12*X2*Y2-DI2*X2*Y3+ • D12*X3*Y 1-D12*X3*Y2+D22*Xl*X2-D22*Xl*X3-D22,X2**2+ • D22*X2*X3))/(3*DJ) SKE(4,6)=-(4*(D 11*Y l**2-D11*Y I*Y2-D11*Y I*Y3+D1 l*Y2*Y3 • -2*D 12*X I*Y 1+D 12*X 1 *Y2+D 12*X 1*Y3+D 12*X2*Y 1-D 12*X2*Y3 • +D 12"X3" Y 1-D 12*X3*Y2+D22*X 1**2-D22*X 1*X2-D22*X l'X3+ • D22*X2*X3))/(3*DJ)
SKE(5,1)=SKE(1,5) SKE(5,2)=SKE(2,5) SKE(5,3)=SKE(3,5) SKE(5,4)=SKE(4,5) SKE(5,5)=(4*(D11*Y l**2-D11*Y I*Y2-D11*Y I*Y3+D11"Y2"'2• DI I*Y2*Y3+D1 l*Y3**2-2*D12*Xl*Y I+D12*XI*Y2+D12*XI*Y3+ • D 12*X2*Y I-2*D 12*X2*Y2+D 12*X2*Y3+D 12*X3*Y l+DI2*X3*Y2• 2* D 12*X3*Y3+D22*X 1" *2-D22*X 1*X2-D22*X 1*X3+D22*X2**2• D22*X2*X3+D22*X3**2))/(3*DJ) SKE(5,6)=-(4*(D 11*Y I*Y2-D11*Y I*Y3-DI I*Y2*Y3+D11"Y3"'2 • -D 12*X I*Y2+D 12*X I*Y3-D 12*X2*Y I+D 12*X2*Y3+D 12*X3*Y 1+ • D 12*X3*Y2-2*D 12*X3*Y3+D22*X l*X2-D22*Xl*X3.D22,X2,X3+ • D22*X3**2))/(3*DJ) SKE(6, I)=SKE(1,6) SKE(6,2)=SKE(2,6) SKE(6,3)=SKE(3,6) SKE(6,4)=SKE(4,6) SKE(6,5)=SKE(5,6) SKE(6,6)=(4*(D11*Y l**2-DI I*Y I*Y2-D11*Y I*Y3+DI 1"Y2"'2• D11*Y2*Y3+D11*Y3**2-2*D12*Xl*Y I+D12*XI*Y2+DI2*XI*Y3+ • D 12*X2*Y 1-2*D 12*X2*Y2+D 12*X2*Y3+D 12*X3*Y l+DI2*X3_y2• 2*D 12*X3*Y3+D22*X l**2-D22*X l*X2-D22*X l*X3+D22,X2**2. • D22*X2*X3 +D22" X3**2))/(3*DJ) return end
76
4.6.4
REDUCE
program
for
mass
matrix
MATRIX N( 1,6),SQN(6,6),NS(6,6),DMASS(6,6),SDIF(6,6); N:=MAT((N1 ,N2,N3,N4,N5,N6)); % Calculating
the integmnd.
SQN:=TP(N)*N; % Inputting
shape
functions
N4:=4"S 1"$2; N5: =4* $2" $3; N6: =4" $3" S 1; N I:=S 1-(1/2)*(N6+N4); N2:=S2-(1/2)*(N4+N5); N3:=$3-( 1/2)*(N5+N6); % Evaluating
the integration.
LET $3=1-S1-$2; FOR I:=1:6 DO FOR 3:=1:6 DO IF I ELSE NS(I,J):=NS(J,I); % Outputting
the results.
ON FORT; OFF ECHO; OFF PERI OD; OUT "mass.ftn"; WRITE " SUBROUTINE MASS(X 1,Y 1,X2,Y2,X3,Y3,RO,CP, WRITE" IMPLICIT REAL*8(A-H,O-Z)"; WRITE" DIMENSION DMASS(6,6)"; WRITE" AREA=-(X3*Y2-X3*Y 1-X2*Y3+X2*Y 1+X I*Y3-X DMASS: =RO*CP* 2* AREA* NS; WRITE " RETURN"; WRITE" END"; SHUT "mass.ftn";
% checking
the correctness
of results.
..........................................
OFF FORT;
OUT "CHECK";
R:=FOR I:=1:6 SUM ; SHUT "CHECK";
of mass
matrix
SUBROUTINE MASS(X 1,Y 1,X2,Y2,X3,Y3,RO,CP, IMPLICIT REAL*8(A-H,O-Z)
77
DMASS)
DMASS)";
1,Y2)/2,;
DIMENSION DMASS(6,6) AREA=-(X3*Y2-X3*Y 1-X2*Y3+X2*Y l+X 1*Y3-X 1*Y2)/2 DMA SS(1,1)=(AREA*CP*RO)/30 DMASS(1,2)=-(AREA*CP*RO)/180 DMASS(1,3)=-(AREA*CP*RO)/180 DMA SS(1,4)=0 DMASS(1,:5)=-(ARF_*CP*RO)/45 DMASS( 1,6)=0 DMASS(2,1)=-(AREA*CP*RO)/180 DMASS(2,2)=(AREA*CP* RO)/30 DMASS(2,3)=-(AREA*CP*RO)/180 DMASS(2,4)=0 DMASS(2,5)=0 DMASS(2,6)=-(AREA*CP* RO)/45 DMASS(3,1)=-(AREA* CP*RO)/180 DMASS(3,2)=-(AREA*CP*RO)/180 DMASS(3,3)=(AREA* CP*RO)/30 DMASS(3,4)=-(AREA*CP*RO)/45 DMASS(3,5)=0 DMASS(3,6)=0 DMASS(4,1)=0 DMASS(4,2)=0 DMASS(4,3)=-(AREA*CP*RO)/45 DMASS(4,4)=(8*AREA*CP*RO)/45 DMASS(4,5)=(4*AREA*CP* RO)/45 DMASS(4,6)=(4*AREA*CP* RO)/45 DMASS(5, I)=-(AREA*CP*RO)145 DMASS(5,2)=0 DMASS(5.3)=0 DMASS(5,4)=(4*AREA*CP*RO)145 DMASS(5,5)=(8*AREA*CP*RO)/45 DMASS(5,6)=(4*AREA*CP*RO)/45 DMASS(6,1)=0 DMASS(6,2)=-(AREA*CP* RO)/45 DMASS(6,3)=0 DMASS(6,4)=(4*AREA*CP*RO)/45 DMASS(6,5)=(4*AREA*CP*RO)/45 DMASS(6,6)=(8*AREA*CP*RO)/45 RETURN END
4.6.6
Checking
In some above.
correctness
cases,
solution
to actual
problems.
individual
field.
problem to unit
the results
It is very important
with a known
above
from
REDUCE
are lengthy
require
some specific a knowledge
by the accessory
checking
constants.
78
The
program
procedures
of the entries
matrix
In general,
of a symbolic
of the specific
the summation
as the stiffness
their correctness.
is used to test the correctness
as an example,
multiplied
results
to find a way to check
In addition, They
of
field.
a small before
problem
application
are also available
in each
Takin$
matrix
the
of the mass matrix
symmetry
shown
of mass
matrix
mass
should
be equal
is proven
by
subtractingthe massmatrix from its transposeto get zerosfor eachentries.The REDUCE program to check correctnessis appendedin the programshownin subsection4.6.4. The following resultsinclude two parts.The first R is the summationof eachentriesof mass matrix. The secondSDIF(i.j) are the difference of massmatrix and its transpose.These checkingresultsconfirmthecorrectness of REDUCEprogram. 0"_
...............................................................................
% Checking the correctness of mass matrix by summing each entries % of mass matrix to make an unit multiplied by accessory constants i_
...............................................................................
R := AREA*CP*RO I_
.............................................................................
% Checking the symmetry of mass matrix % between mass matrix and its transpose. O_
by finding
.............................................................................
SDIF(1,1) SDIF(1,2) SDIF(1,3) SDIF(I,4) SDIF(1,5) SDIF(1,6) SDIF(2,1) SDIF(2,2) SDIF(2,3) SDIF(2,4) SDIF(2,5) SDIF(2,6) SDIF(3,1) SDIF(3,2) SDIF(3,3) SDIF(3,4) SDIF(3,5) SDIF(3,6) SDIF(4,1) SDIF(4,2) SDIF(4,3) SDIF(4,4) SDIF(4,5) SDIF(4,6) SDIF(5,1) SDIF(5,2) SDIF(5,3) SDIF(5,4) SDIF(5,5) SDIF(5,6) SDIF(6,1) SDIF(6,2) SDIF(6,3) SDIF(6,4) SDIF(6,5) SDIF(6,6)
:= := := := := := := := := := := := := := := := := := := := := := := := := := := := := := := := := := := :=
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
J
79
the difference
4.7
Closed
4.7.1
form
solution
quadrilateral shown
the methodology
element
the techniques
the Jacobian
However
the same
general,
determinant
advantage
the determinant
Jacobian
determinant
produces
a tremendous quadrature
difficulty
and
literatures, Huebner
of four-node
rule
element
the
introduction
inability
results.
the exact
closed-form Preliminary
& Carey
other.
the integration
is straightforward.
quadrilateral coordinates.
analytically,
this problem. rule
can
& Oden
element. Having
the of
the numerical
discussions
be found
in most
[11], Cook
In
transformation
therefore The
element
In the triangular
due to the coordinate
integration
integration
paragraphs
solution
isoparametric
of this relevant
[12], Reddy
[13],
[15].
analytic
The following
each
of the natural
quadrature
[10], Becker
to perform
from
for the isoparametric
to solve
of Gauss
& Johnson
different
of the integrand
introduced
of four-node
is the same as that of the triangular
is a function
in performing
is usually
matrix
and therefore
be gained
in the denominator
such as Zienkiewicz
The
are quite
Jacobian
difficulty
the stiffness
problem
is constant
can't
of the
[14], Weaver
element
to construct
for the plane elasticity
in last section,
element,
4.7.2
matrix
Introduction
Although
Gauss
of stiffness
introduces
the integration
will show that this difficulty
has been obtained
by appropriate
error in the finite
has been overcome
application
of REDUCE
and [8].
formulation
The local stiffness 1
matrix
for a 2-D isoparametric
• E • B,t,
Isl
Vdhere
• K" local stiffness
• t" thickness
matrix.
property
• IJI • determinant • _, r I • natural
element
is formulated
by
1
g -
• E' material
quadrilateral
matrix.
of Jacobian
matrix.
coordinates. of element.
8O
dO
(4.45)
• B ' strain-displacement In general, be expressed
matrix.
each entry
of strain-displacement
matrix
B is a function
of _, Vl, IJIand can
as :
b,_(_,'O,
lJI)-
c° +c,_:
+c2_?
+ c3_'_
Igl
(4.46)
Where
° bij" are denoted
by the entry in ith row and'jth
column
of matrix
B
• c i : are constants. For simplicity, independent
the material
of natural
property
coordinates.
matrix
The integrand
of _, r I and IJI, too. The entry of integrand
in equation
can be expressed
g,j(_,n,IJI) = a°+a'_ +a,o +a3_'+ a,_
Where
d 1, d2,. .... d8 are constants, The Jacobian
specifically
therefore
to be
will be function
as :
+ash" +a6_20 +a,_o _ +a,_'o" Igl
(4.47)
too.
[; 1I 1 _
m
JIt Jzl
"
the global
coordinate
functions
as those
shape
(4.45)
t are assumed
J is
J,
And
E and the thickness
variables of field
Jr: Jz2
x, y can be transformed
variables.
This
(4.48)
to local
is an intrinsic
coordinate
property
by the same
of an isoparametric
element. 4
x-
4
, i -1
y-2N, y, _ -1
Where
• xi' Yi " are global
node coordinates.
81
(4.49)
• N i • shape
functions.
Specifically,
the individual
shape
functions
N, - 42-(1- _)(1
- 1,/)
(4.50)
r/)
(4.51)
N 3 - 4L(l + _)(1
+ r/)
(4.52)
N 4-
+ r/)
(4.53)
N 2-
and the entries
of Jacobian
_X Jll
"
d_
J,l"
"_-"
J,2"
O"_''
''J'(Yl
"4-'(Y,-
J,,J22
function longer
(4.48)
to (4.53).
x4)+
_(-xl+x2+x3-x4)-a,,r
X4)+';(-x,-x2+x3+x4),,a,
Ya + Y,-Y4)
+ t4-'(-Y,-Y,
of the Jacobian
will be
(4.54)
I +b_,
I +by
(4.55)
_ +c,,
(4.56)
+ Y3 + Y4)-a_,_
(4.57)
+cy
- J,2J2, +
(axCy
ayc,,)r
-
I + (bxcy
+ b yc,)
(4.58)
U_ + V rI + W
the
This
of natural coordinates.
determinant
linearity
is expected
allows
in the solution.
an integration
respect
-
Xn-
U, V, W are independent
coordinates.
x3
- axby)_
Obviously,
these
+
(xm-x2+
•- (b,,ay
where
from equations
Y2+Y3-Y4)+_-YI+Y2+y_-y4)..ayr
the determinant
-
_(1 - _)(1
are derived
i -x2
ox
IJ I-
_(1 + _)(1-
r/
__'(x03 _ -
J_2"
Therefore,
are
to h can also two integrations
variables
be performed axe too tedious manipulator
computer.
In addition,
the solutions
FORTRAN-code
subroutine
parts
are demonstrated of commands
the exact
is only
integration
REDUCE,
Therefore
by the
in the next paragraphs and the time consumed
82
natural
the logarithm there
integration
operations
with
to finish
can be done
way and converted
All of these
by an example in each individual
is no
with the help of the
operations
in a systematic
of
to x is done,
Fortunately,
program.
and
the second
these mathematical
main
function
the algebraic
by hand.
can be organized
to be called
with respect
However,
to be handled
a linear
to be performed
As the first integration
analytically.
and algebraic
The explanations
Jacobian
h in the denominator.
symbolic
of solutions
of the
into a
procedures
of linear command
by
and
elasticity. are also
commentedfor reference.The total time consumedin this executionby REDUCE is around two and half hours.The resultantfortran expressionsfor just an elementof stiffnessmatrix occupy almost sixteenpages.Thesehugeexpressionsare the reasonwhy the closed form solutionwasnot availablebefore. 4.7.3 O'_
REDUCE
and
explanation
.......................................
% Turning O'_
program
on the CPU elapse
time.
.......................................
1: on time; Time: 133 ms I_
.....................................
% Inputting 4 shape functions. % p and q are natural coordinates. %p:xi %q: eta i_
.....................................
2: s 1:=( l-p)*(
l-q)/4;
P'Q-P-Q+ S1 := .................... 4 Time: 600 ms
1
3: s2:=(l+p)*(1-q)/4; P'Qp + Q - 1 S2 := ........................ 4 Time: 167 ms 4: s3:=( l+p)*(
l+q)/4;
P*Q+P+Q+ $3 := ........................ 4 Time: 167 ms
1
5: s4:=(1-p)*(l+q)/4; P*Q+PS4 := .......................
Q-
1
4 Time:
166 ms
% Expressing % and global 1_
...................................
x & y by shape function node coordinates. ...
.....
83
6: x:=sl*x l+s2*x2+s3*x3+s4*x4; X := (P*Q*X 1-P*Q*X2+P*Q*X3-P*Q*X4-P*XI+P*X2+P*X3-P*X4Q*X 1-QzX2+Q*X3+Q*X4+X 1+X2+X3+X4)/4 Time: 400ms 7: y:=sl*y l+s2*y2+s3*y3+s4*y4; Y := (P*Q*Y 1-P*Q*Y2+P*Q*Y3-P*Q*Y4-P*YI+P*Y2+P*Y3-P*Y4Q*Y 1-Q*Y2+Q*Y3+Q*Y4+yl+Y2+Y3+Y4)/4 Time: 267ms % DeclaringandInputtingmatrix % c % jac % % % % I_
: coefficient matrix : combination of Jacobian
elements
to calculate
matrix
sd : matrix containing the derivative of shape function. b : strain-displacement matrix. detj : determinant of Jacobian. j 11 ,j 12,j21,j22 : element of Jacobian matrix. .........................................................................
.
8: matrix c(3,4),jac(4,4),sd(4,8),b(3,8)$ Time: 550 ms 9: c: =mat(( 1,0,0,0),(0,0,0,1),(0,1,1,0))$ Time: 183 ms 10: jac: =mat((j22,-j 12,0,0),(-j21,j 11,0,0), (0,0,j22,-j 12),(0,0,-j21,j I 1))$ Time: 284 ms 11: sd:=mat((df(s 1,p),0,df(s2,p),0,df(s3,p),0,df(s4,p),0), (df(sl,q),0,df(s2,q),0,df(s3,q),0,df(s4,q),0), (0,df(sl,p),0,df(s2,p),0,df(s3,p),0,df(s4,p)), (0,df(sl,q),0,df(s2,q),0,df(s3,q),0,df(s4,q)))$ Time: 833 ms 12: b:=c*jac*sd/detj$ Time: 417 ms
% % % % O'_
the strain-displacement
Inputting material matrix D and calculating D : material matrix (assumed symmetric). Ga : integrand. th : thickness of element. ............................
.
integrand.
..................................
13: matrix d(3,3),ga(8,8); Time: 316 ms 14: d: =mat((e 11 ,e 12,e 13),(e 12,e22,e23),(e Time: 200 ms 15: ga:=tp(b)*d* Time: 8067 ms
13,e23,e33))$
b* th*detj$
84
.............................
matrix
B.
% Evaluating I_
each element
of Jacobian.
............................................
16: on factor; Time: 150 ms 17: on div; Time: 33 ms 18: on rat; Time: 50 ms 19: j 1 l:=df(x,p); 1 Jl 1 := ---*((X1 4 Time: 750 ms
- X2 + X3 - X4)*Q
20: j 12:=df(y,p); 1 J12 :=---*((Y1 - Y2 + Y34 Time: 550 ms 21: j21:=df(x,q); 1 J21 := .... *((X1 4 Time: 667 ms
Y4)*Q-
- X1 + X2 + X3 - X4)
Y1 + Y2+Y3-Y4)
+ X2 - X3 - X4) - (X1 - X2 + X3 - X4)*P)
22: j22:=df(y,q); 1 J22 := .... *((Y 1 + Y2 - Y3 - Y4) - (Y 1 - Y2 + Y3 - Y4)*P) 4 Time: 650 ms O'_
................
'. ......................................
%Making substitution for Jacobian matrix. % ax=(x 1-x2+x3-x4)/4, bx=(-x 1+x2+x3-x4)/4 % ay=(y 1-y2+y3-y4)/4, by=(-y l+y2+y3-y4)/4 % cx=(-x 1,x2+x3+x4)/4, cy=(-y l-y2+y3+y4)/4 ........................................................
23: j 1 l:=ax*q+bx; Jll := AX*Q+ BX Time: 333 ms 24: j 12:=ay*q+by; J12:=AY*Q+ BY Time: 117 ms 25: j21:=ax*p+cx; J21 := AX*P+ CX Time: 117 ms
85
26:j22:=ay*p+cy; J22:= AY*P + CY Time:
116 ms
%Calculating
the determinant
of Jacobian.
27: matrix j(2,2),ske(8,8); Time: 200 ms 28: j :=mat(tj 11,j 12),(j21,j22)); J(1,1) := AX*Q+ BX J(1,2) := AY*Q + BY J(2,1) := AX*P+ CX J(2,2) := AY*P + CY Time: 367 ms 29: detj:=det(j); DETJ Time:
:= - ((AX*P 333 ms
%Making
a further
+ CX)*(AY*Q
substitution
+ BY) - (AX*Q
+ BX)*(AY*P
and giving the lineality
+ CY))
of determinanL
30: let-ax*cy+ay*cx=al,ax*by-ay*bx=a2,-bx*cy+by*cx-_.a3; Time: 450 ms 31: detj:--detj; DETJ := - (AI*Q Time: 450 ms
+ A2*P+
A3)
32: on exp; Time: 50 ms
%Performing
the double
integration.
%Due to the complication of the expression in the integrand, %treatment" to each element of integrand before integration. %limitation of memory space and finish the job.
it is necessary to make the "preThis is a vital step to avoid the
33: for i:=1:8 do forj:=l:8 do ifj>=i then else ske(i,j):=ske(j,i); Time: 234766 ms
%Showing the results for the element %H l=log(al**2-a l*a2+a l'a3), %H3=log(a 1+a2+a3) , %HS=log(al-a2+a3) ,
in 1st row H2=log(-a H4=log(-a H6=log(-a
and 1st column of the local stiffness l**2-a l*a2+al*a3) l+a2+a3) 1-a2+a3)
...........................................................................................................
34: ske:=ske; 1 SKE(1,1)
-1
:= HI*TH*(---*A1 8
1 *A2*E13
....
-1 *A1
-1 *A2
2 *A3
*El3
+
8
1 -1 -1 2 1 -1 -1 .... *AI *A2 *A3*BX *E33 .... *A1 *A2 *A3*BX*BY*E13 16 8
...........................
+ H6*TH*(
1 -2 2 + .... *A2 *A3 *El3+ 16
1 2 -2 .... *AI *A2 *El3 16
87
1 .... *El3) 16
1 2 -3 + .... *AI *A2 48
matrix.
1 .................
1
Time:
4.7.4
79317
%Converting
....
-2
.... *A2 3 ms
Fortran
+ .... *EI3)+TH*( 16 1
*A3*BX*BY*EI3+---*A2
-1
-2
1 *EI3+---*AI 4 2
*A3*BY
*Ell)
6
subroutine
the results
1 *AI*A2 4
from
REDUCE
into FORTRAN
code
subrou ti ne(x 1,y I ,x 2,y2,x3 ,y3 ,x4,y4,e I ,e2,e3 ,e4,e5,e6,th,ske) implicit real*8(a-h,o-z) dimension ske(8,8) ax=(x 1-x2+x3-x4)/4. bx=(-x 1+x2+x3-x4)/4. ay=(y 1-y2+y3-y4)/4, by=( - y 1+ y2+y3- y4)/4. cx=(-x 1-x2+x3+x4)/4. cy=(-yl-y2+y3+y4)14. al=-ax*cy+ay*cx a2=ax*by-ay*bx a3=-bx*cy+by*cx h 1--log(a I ** 2-a I *a2+a I * a3) h2=log(-al **2-a l*a2+a l'a3) h3=log(al+a2+a3) h4=log(-al+a2+a3) hS=log(a l-a2+a3) h6=log(-al-a2+a3) ANS 14= I/4*A 1**(-3)*A3**2*Ay*CX*E 13- ltS*A 1"*(-3)*A3"*2 • *AY*CY*EI 1-1/16*A 1**(-3)*A3**2*CX**2*E33+ I/8*A 1_¢*(-3 • )*A3**2*CX*CY*EI3-1/16*A I**(-3)*A3**2*CY**2tE1 l-lJ16 • *A2**(-2)*A3**2*E13+ 1/16*E 13
ANS I=H I*TH*ANS2 ANS28=- I/4*A 1**(-3)*A3**2*AY*CX*EI3+ I/8*A I z* (-3)*A3"* • 2*AY*CY*E11+ 1/16*A I**(-3)*A3**2*CX**2*E33l/8*A 1"*( • -3)*A3**2*CX*CY*E13+I/16*A 1**(-3)*A3**2*CY**2*E 11+ 1/ 16*A2**(-2)*A3**2*E13-1/16'E13
ANS 15=H2*TH*ANS 16 A NS48= 1/8*A2**(-3)*A3**2*BX*BY*E • 2*BY**2*E11
13- I116*A2**(-3)*A3**
ANS29=-H3*TH*ANS30 A NS68=- 1116*A2**(-3)*A3**2*BX**2*E33+
1/8*A2**(-3)*A3
88
-I *A2*E13+
**2*BX*BY*E13-1/16*A2**(-3)*A3**2*BY**2*E11 Al'_S49=-H4*TH*ANS50 ANS75=1/16*A2**(-3)*A3**2*AX**2*E33-1/8"A2"* (-3)*A3"* • 2*AX*AY*E13+ 1/8*A2**(-3)*A3**2*AX*BX*E33+ 1/16"A2"*( • -3)*A3**2*AY**2*E11-1/4*A2**(-3)*A3**2*AY*BX*E13+l/8 • *A2**(-3)*A3**2*AY*BY*E1 l+I/16*A2**(-3)*A3**2*BX**2* • E33-1/8*A2**(-3)*A3**2*BX*BY*E13+I/16*A2**(-3)*A3**2 • *BY**2*E11-1/16'E13 Al'_S69=H5*TH*ANS70 ANS82=-3/16*A2**(-2)*A3*BY*CY*E11+1/16*A2**(-3)*A3**2 • *AX**2*E33-1/8*A2**(-3)*A3**2*AX*AY*E 13+1/8"A2"*(-3) • *A3**2*AX*BX*E33+ 1/16*A2**(-3)*A3**2*AY**2*E 11-1/4" • A2**(-3)*A3**2*AY*BX*E13+ 1/8*A2**(-3)*A3**2*AY*BY* • E1l+l/16*A2**(-3)*A3**2*BX**2*E,33-1/8*A2**(-3)*A3**2 • *BX*BY*E13+I/16*A2**(-3)*A3**2*BY**2*E11+1/16"E13 AI'_S83=TH*ANS84 ske(1,1)=ANSI+ANS 15+ANS29+ANS49+ANS69+ANS76+A NS83 return end 4.8
Significance Some
discussed
and
conclusion
conclusions
as follows
1. Improving allows
are drawn and the significance
of automatic
efficiency
solution
on-line
us to get a numerical
is faster
integration numerical large
than
--- the closed-form value
subroutine. the
Gauss
analysis
by simply
substituting
This procedure
quadrature
point to get a reasonable
dimension
formulation
is
•
into a FORTRAN-code this
problem
rule
the global
stiffness
nodal
usually
symbolic
needs
the significance
in improving
on-line
than
one
in the nonlinear
also plays the same role. In the case of a large number
size in matrix),
Of course,
more
template
matrix
coordinates
is done in just one step.
which
solution 4 . The
of the local
of elements
efficiency
(or
will be
greater• 2. Increasing There 4 Reduce special
the accuracy
is no integration
of solution error
integration is an exception case is ruled out here.
--- the closed-form
introduced
into the evaluation
and sometimes
89
results
solution
is an exact
of the stiffness
in Hourglass
drawback.
solution.
matrix. This
The
accuracy
of the
requires.
Therefore,
more
precise
3. Free from
approximation from using
than those of pure numerical and typing of motion
are too lengthy
by hand,
it would
be made
and algebraic inputs
the results
of equations
expressions
would
series
hand-calculation
derivation
done
Fourier
when
as high
as the user
and algebraic
manipulation
will be
analysis.
and stiffness
that
to key them
matrix
in the tensor
constructions
formulation,
show,
the algebraic
by hand. Even
supposing
that they could
nobody
guarantee
that
could
into the computer.
both difficulties
commands,
be increased
error --- As the results
be so tedious
manipulation,
the correct
symbolic
to be formulated
trying
can
With
are completely
no mistakes
the use of symbolic
solved.
there will not be any question
be
As long as the user
about
the correctness
of the
results.
4. Simplifying
FORTRAN
matrix
can
be obtained
when
Gauss
quadrature
integration. of various program
the
weight
analysis
makes stiffness
produced
achieved stiffness numerical
matrix
results from
command
in the
integration
becomes
once.
finite element
points.
symbolic
available and
This
method
stiffness isn't
true
to evaluate
and multiplications
These
will complicate
to be more
dominant
relocating
nodes
or even
does not possess.
9O
--- Sometimes
algebraic
numerical
For example,
will be smaller
method
of the local
the
more error sources.
in the incoming
by appropriately matrix
for different
feasible.
need
values
to have a "DO" loop, "CALL _ command
of symbolic
this analysis
the _CALL"
is employed
will produce
improvement
--- the numerical
using
integration
coefficients
expressions
dramatic
by simply
It is necessary
and therefore
5. Further
programming
so that
The
will
closed
that the diagonal
to improve
vanished.
manipulation
analysis.
suppose
the pre-analysis
terms
the ill-condition, the off-diagonal
This is the unique
form
lead
of to a
solution of global
this can
be
terms
of local
advantage
that the
4.9
References
[1] T. R. Kane Hill,
and
D. A. Levinson,"Dynamics
and
Applications",
McGraw-
1985.
[2] L. Meirovitch,"Methods
of analytical
[3] F. I. Niordson,"shell
H. Yang,"AM519
Michigan,
dynamics",1970.
theory",North-Holland,
[4] I. S. Sokolnikoff,"Tensor [5] W.
: Theory
analysis", course
Ann Arbor,
2nd edition,
note",
John
Mechanical
Wiley
& Sons
engineering
lnc,
,The
1964.
University
of
1989.
[6] P. V. O'Neil,"Advanced [7] N. Kikuchi,"finite
1985.
Engineering element
Mathematics",Wadsworth,
methods
1983.
in mechanics",Cambridge
University
press,
1986.
[8] A. R. Korncoff matrices",
Computers
[9] R. W. Luft, matrices",
and
S. J. Fenves,"Symbolic & Structures,
Jose M. Roesset J. of the structural
[10] O. C. Zienkiewicz,"The
vol.
generation 10, pp. 119-124,
and J. J. Connor,"Automatic division,
finite
proc. of ASCE,
element
of finite
method",
element
stiffness
1979. generation
pp 349-362,
of finite Jan.
element
1971.
3rd edition,
McGraw-Hill,
pp. 191,
elements
an introduction",
1977.
[11] E. B. Becker, Prentice-Hall,
[12] Robert
G. F. Carey
vol.
1,
1981
D. Cook,"Concepts
pp. 119, John
and J. T. Oden,"Finite
Wiley
[13] J. N. Reddy,"An
& Sons
introduction
and applications Inc.,
of finite
element
analysis",
2nd edition,
1974.
to the finite
element
method",
pp. 154, McGraw-Hill,
1984.
[14]
K. H. Huebner,"The Sons,
finite
element
method
1975.
91
for engineers",
pp. 190, John
Wiley
&
[15] W. Weaver,
Jr and Paul R. Johnson,"Finite
Prentice-Hall
[16]
Inc.,
A. C. Hearn,"
[ 17] W. L. Tsai, applied
REDUCE
"Applications
mechanics
and applied
elements
for structural
analysis,
pp. 111,
1984.
user's
of symbolic
problems",
mechanics,
manual",
Ph.D.
The University
92
Version
and algebraic thesis,
3.3, July
manipulation
Department
of Michigan,
1987. software
of mechanical
Ann Arbor,
Dec.
in solving engineering
1989.
CHAPTER
APPLICATION
OF
SYMBOLIC
MATERIALLY
AND
NONLINEAR
V
ALGEBRAIC
PROBLEM
MANIPULATION
---
TO
RIGID-PLASTIC
A
RING
COMPRESSION
5.1
Introduction
The application
of the finite
devised
by Lee and Kobayashi
allowed
the deformation
during
the process
manufacturing
process
is recognized
of plasticity,
from
unconstrained
the
is very tedious.
with
respect
upper
analysis
This method
screen
techniques
to the industrial
equality
solution
stage by stage of die, and the
manufacturing
field
of the finite element
Especially
when
the interface
to velocity
fields
derivation,
REDUCE
to do the job
into subroutines otherwise
surface is always
this chapter
of formulation.
symbolically intractable
constraint,
application
boundary
difficult
to obtain
will utilize The quantities
at the the element tasks become
93
possible
constitutive conditions.
the symbolic involved
and
Therefore, algebraic
in the formulae
level to facilitate
the global
and free of errors.
of it
the performance
of the first and second by hand.
is
element
the formulation
is considered,
an
requirement
equations,
to the problem,
and the evaluation
into
by the finite
and boundary
condition
function
changed
As the stationary
incrementally
condition
objective
is then
the equilibrium
incompressibility
the friction
to an inequality
multipliers.
will satisfy
with the normality
problem
can be solved
the success
along
leads
constrained
Lagrange
problem
equations,
work and associating
of this method
This
bound
hand
As a result,
was originally
in the last decade.
the design
This contribution
of virtual
unconstrained
traditional
made
problem
on the computer
As a consequence,
by introducing
compatibility
Despite
of integration
the principle
problem
and
equations,
popular
of metal to be revealed
forming.
constraint.
the total
method
to the rigid plastic
[2]), and became
were improved.
the theoretical
an equality
reached,
method
to be very significant.
Starting
with
([1],
behavior
of metal
element
derivatives unlike
the
manipulator then can be assemblage.
5.2
Preliminary
formulation
Consider prescribed frictional velocity
a body
on surface stress fields
.f',
S u and
prescribed
will satisfy
1. Equilibrium
of volume
V with the essential
two separated
natural
boundary
on S F and S c, respectively
the following
relationships
boundary
condition,
conditions,
[Figure
5.1].
velocity traction
The actual
/7,
F
and
stress
and
:
conditions"
crj,j
,, 0
(neglecting
s (_
body
force
)
(5.1)
s F (F)
V
S Figure
2. Compatibility
3. Stress-strain
5.1: Configuration
and incompressibility
of domain
and boundary
conditions:
i j - _(u, .i + uj., )
(5.2)
e,, .- 0
(5.3)
rate relationship
Where
(]1 "t
•
'J • deviatoric
• '_ and 4. Boundary
stress.
E • effective conditions
conditions
stress and strain-rate, :
94
respectively.
cr o n, u,
- U
o jn,
Where
the relative
velocity
t-
=fj
between
fr
The
.. F 1
=_
die and deforming
cr,ji,aV
along
velocity
=
F.
frictional
stress
(5.5)
on
S,
(5.6)
on
Sc
(5.7)
body is defined
field u_', the virtual
_'dS
+
f-.
(fr;
piece
•
interface.
work principle
+ J)dS
gives
+
(_r n,)U
e
f" is defined
as follows
(5.8)
die and working
¥
The
Sr
_u..a _. Vrt~
here is the unit base vector
With an admissible
(frr)
on
dS
(5.9)
M
as follows
and is plotted
in Figure
5.2.
f (V) r
mg
"-
V r
-mg
Figure
5.2 • Plot of frictional
stress
vs. relative
velocity
Wr
f0z, - -,,, • g, where
the friction
considering
factor
the normality
m is in the range condition
of 0 < m < 1 and g is the yield
shear
stress.
By
of yield surface
(cr j - o j)i_j
:- 0
95
(5.1 I)
and inequality
equationS
=-0 The equation
(5.9)
(5.12)
becomes
(5.13)
All conditions minimum (5.3).
values
Since
admissible note.
with respect
there field
will be met when
By introducing
function.
-_---_[ gt
at the point
solution
not
where
the arbitrary
exaggerate (5.10).
stress
value
¢dS _ inside
for the sake
constraint
5.2],
for equation
(5.14). in terms
_{]_:ket. _
does In order
Since
0
the frictional and
stress
constant
a is several
of arctangent
(5.15)
is absorbed
5 See appendix
A for proof.
6 See appendix
B for proof.
orders
to reduce
96
(5.14) is not
this
numerical
[see Figure
5.3]
(5. J5)
less than
the die velocity.
the error between
into functional
is
the convergent
to overcome function
special
formulation
Xi, dV]
not exist
of arctangent
any
the
can be included
element
•
the
of equation
without
f (9' ). 9, dS +
reaches
of simplicity
equation
for finite
r:
[see Figure
is approximated
condition
discussions
problem
the square
(5.13)
= - m • g • [( 2---)inn -,(a._)]f-'
the argument
The equation
I7 r , 0
be available
the frictional
r(i7,.)
g.
of equation
the asterisk,
!, the equality
stationary
¥
side
as u for the following
multiplier
the functional
differentiable
difficulty,
The
agdV -
represents
will
be denoted
hand
the incompressibility
in omitting
the Lagrange
into the objective therefore
where
to U_" and satisfies
is no ambiguity _'° will
the left
equation
by the following
Its function (5.15)
inequality6
is to
and equation
-I
-Tan (x)
X !
-100.
!
-EO.
50.
!
100.
-0.5 --1.
-1.5
Figure
5.3 • Arctangent
approximation
of frictional
stress
(5.16) The
final form
of equation
(5.13)
is
11,0 (5.17)
velocity specified
*:*0
Figure
5.4"
Physical
configuration
97
for nng compression
problem
5.3
Matrix
method
If a specific traction
is due
for
problem
to friction
can be dropped.
the
ring
compression
problem
of ring compression
force only.
By the substitution
is considered
Therefore
the term for SF integration
of the following
equations
firt E'
,J_c',3
£
the stationary
requirement
(
)K
0
+
to equation
5.4],
the surface
in equation
(5.17)
(5.17),
m
- B * U
(5.18)
£ "I,t
of the functional
*_'dV
[see Figure
(5.19)
results
U r *K
in the following
*U
*-_dV
nonlinear
equation.
+Z*Q (5.20)
together
with an incompressibility
constraint
Ur
*Q
equation
-0
(5.21)
where • K
-B
° O
-fB
r *D
° C" proper
r
*B
• C dV
matrix
such that
implies
the C r d incompressibility
• D" flow matrix.
•U: vector The derivative
of velocity.
It is represented
part in the second
as u" previously.
term of equation
98
(5.20)
is equal to
condition.
a(_-)
1 ao
3U
The first term of (5.22)
is dropped
the infinitesimal strain change (denoted as Y) of the material.
AU
The
_ at
" "=--e 3U
equations
(5.20)
into the velocity
vector.
_
Y =
3U
for simplicity
[see References
and
(5.21)
at
e-'-2 OU
due to the assumption 2]. Physically,
are then
perturbed
This gives the following
+ AU)[
(
by introducing
_
is constant yield
a small
for
strength
velocity
'
+X • Q
)+ 3U
" - [TO-- +
that
_3 is the current
equations
[ 2_a + .b._O_(____)au ] • K • (U + AU )dr • K ,(U
(5.22)
('-_-)AUIdV e
au]
(5.23)
and
(U
+AU)
r ,Q
=0
(5.24)
where
- - (foo"J, e dVr)dS is contributed equations
by the friction
(5.23)
r
and (5.24)
+
traction.
After
neglecting
can be combined
and higher
into the following
forms
=-
f
OU 2 9*9
the second
9* 1
Qr U 9* I
order
terms,
the
•
(5.25) 9* I
where
Ps.s=2_ Hs. s ='
{1K
+[M
"_'-K *U
-(1)ZN]}
(5.26)
- "-_-E
(5.27)
g
99
24 E" - _"
Es.l
Qs.l
.2 + e_ .2 + ee .2 E,
" Br
=
fvn
T
C dV
(5.30)
*D • B
Ms. 8-(
) [-_--E
dr
(5.31)
-d -do} -.fg,,3 e,. d_
(5.32) -K
*UIE
r
(5.33)
T
_. - "_C c"
(5.34)
Ns. s - E
(5.35)
• Ur * K
k -U
r
_=-
[.is r
Finite
The domain
=j2 , -
Ks. s - B r
*K
*U
The
, (_)2
for a specific
element
domain
tan - i (a V, .---7_)dV,
(5.37)
ldS
material.
analysis
discretization
used by finite
the symmetric
(5.36)
floo "lm *g
m, a g are constants 5.4
(5.28) (5.29)
[2d, ""_'c" 4., " "¢_'!2d0
I
.2 d, do + 3_y,,
*A
. rag 1 A *.l
E',d, - E'_c"8 -
element
for ring cross
analysis
section
is shown
in Figure
is only the upper half of that given
5.5. The actual
in Figure
5.5 due to
geometry. finite
element
in total. For an isoparametric
ql" q3-
model contains element,
4_-(1 - s)(1
96 four-node
the shape functions
- t )
14-'(l+s)(l+t)
and
100
quadrilateral
elements
are
,
qz - 1 (l+s)(l-
,
q4-1(l-s)(l+t)
t)
with 117 nodes
4.
u (s ,t ) -
4
q,
,
i -I
z -I
4
r(s,t
)-
4
.q,
• r+
,
z(s,t)
= Eq
-1
, *z
t -I
Where
• s, t are natural
coordinates.
• r, z are physical
coordinates.
• u, v are the velocity
The
subscripts
components
in r, z, u, v represent
The strain
rate in axisymmetric
' 0. O
Figure
in the r, z direction,
the nodal
indices.
case can be expressed
in a matrix
form as follows
I
I 3.0
I.tl
5.5"
respectively.
4.0
Discretization of ring cross section. Due to the symmetry of geometry, only the upper half of this cross section is used for finite element analysis.
101
:
o_
it
0
_ II
d-
de
& 0
vU)-B
*U
(5.38)
Where
u T -{u,
n
.,
u, u. u_ u, u. u,}
Iilifo 0
mt
ql
0 qa
ql
0
0
qa
q_ 0
0
q4
q3
0
0 q4
(5.39)
1 ÷
-!
Ii
0
0
0
0 °
1
1
_Os
-% c_s
o
o
5?
"gr
o
o
0
0
0
0
dr
1.t 0
&
77"
77"
_
&
qt., 0 qt., ql,
q:_
0
q:., 0 0 q2., 0 q 2,
qs,
0
q4,
0
I
qs., 0 q4., 0 ] 0 qs., 0 q,., 0 q3_ 0 q 4t j
(5.40) The Jacobian
is"
J
/
(5.41)
The flow matrix
is ; "2
0 D-
0 2
_-
0
O"
0
0
2
O
0
_-
0
0
0
0
T
l
1
0
C r - (1
102
(5.42)
1
(5.43)
Sincebilinear velocity distributionsareassumedfor four-nodequadrilateralelement, therelativevelocitiesalongtheinterfaceareinterpolatedasfollows : V,
If die velocity
r - rz _. _ul
is specified
rI - • + r_- rzU2
(5.44)
as uJdl= 1, then the frictional
if" = - {m *g
stress
will be
• (_,-)tan-_(-_-_'-)}_
(5.45)
and
,Vr
m * g *(_)tan-
5.5
Application
The
of symbolic
difficulty
the employment goal is to make limitation form
a template
of memory
for individual
REDUCE
.
entries
Integrable
form
belongs
manipulation.
in element
of equation
in the last paragraph Based
level for global
in hardware
systems,
(5.25)
--- This is the form
analytically
on equation
assemblage.
the goal
only. There
Non-integrable
form
--- The
Moreover,
pages,
complete
the
Therefore,
equations
can be eased (5.25),
However,
is modified
by
the original due to the
to make
are three kinds of forms
a template produced
form
and the summation
Miscellaneous
forms
(5.29),
(5.32),
(5.27)
and (5.33),
and even
of equations for quantities,
of equations
cubic
by
(5.26)
and
such as K,
(5.35)
have nothing
and (5.36),
operations.
103
can be
and
for example,
are not
in the denominator
(5.27)
more are
M, N, k and are carried
than
not
of the sixteen
available.
E, are obtained out
numerically
to do with integration,
are obtained
(5.37)
expressions.
of _" occupies
as well as their integration which
(5.30)
in the resultant
expression
--- Other equations
(5.31),
coordinates (5.26),
the resultant
from the fact that integration
evaluation
of _', its square
integrands
the individual
symbolically,
as (5.28),
since
results
The
are no natural
due to the existence
integrands.
which
by REDUCE.
to this class. There
integrable
,
form
shown
:
performed
,
the equations
and algebraic
capacity
(5.46)
manipulation
of formulating
of symbolic
(--.ff-)dVr]dS
symbolically
such
by matrix
a4_
The evaluation is necessary symbolic
in equation
to discuss
this subject
algebraic
manipulation
and
evaluate
of _
_
(5.25)
requires
independently
here
will be revealed.
of the integral
with respect
to relative
2. Evaluation
of the integral
with respect
to interface
3. The results
then are differentiated
Theoretically,
there
with respect
is nothing
wrong
can only do the first evaluation.
treatments deviate
are necessary
from
to make
the formal
so that
the
Intuitively,
work
given above
pre-treatments.
fundamental there
are
nature three
steps
to
,
g *
above.
In fact,
as follows.
]dS
1"£7dS (5.47)
I
equation (5.44). Equation
from
given
[ tan -'(.a-)-_"-]dS . v, ,v, e
--Ou_ - -m a+
fields.
two are not feasible. Therefore, some prea,a to evaluate the 70". This will force us to
tan - (-a-)dV,
- - m * g * ('r2)fs
2 /*2s t. r,tan (+)Jo J,,
- ,+, (-a-)_rarau r-,,
(5.47)
becomes
,,^
t
=
-4.m ;7"_,_ 'fr
2[r2
tan- I(+) v
- r rz tan -t(v)
]dr
(5.48)
I
equation
(5.44)
into UI--I¢
V,
and substitute 0_
of
S in (5.37).
the methodology
v
- - m * g * (T)
Rewrite
It
Vr in (5.37).
domain
-_(_)dV,
- m *g *
7"0" can be evaluated
velocity
to velocity
with
The other
REDUCE
methodology
-
The term
of delicate
They are •
1. Evaluation
REDUCE
a number
it in equation
-
Since REDUCE
-+._.s ,'-_
|
r I_
- _--:_hr , +
l--_"
|t¢
I
fl--ti
-
Yr + Z
(5.49)
(5.48).
/"'[r j,
is still unable
2
t _ tart- (x-r
+ z)
to handle
equation
_ • rzta n
t04
(5.50),
-i r (rr
further
(5.50)
+ Z)]dr
treatments
are
required.
Let
W
-Tr
Y
Z
F
+7--Yr
I
+Z
(5.51)
then
dr
and equation
(5.50)
-
dW
r'
(5.52)
becomes 2
09
-4.
f_'[_
_ -z,
1
Note point,
that
REDUCE
the upper can proceed
Y'
and
lower
and back substitution
be calculated
in the
differentiation
of the results
fortran
solution
similar
are presented
limits
by itself. The
the integration
of integration tasks
W
]dW
(5.53)
The
are changed.
it performs
of the variables.
manner. of
r2 tan-l
Y'
second
the first derivatives.
in this specific
The other quantity, derivatives The
Starting
are
REDUCE
from
problem
such as _ simply program
computed
as follows.
.........................................................................
%REDUCE
program
to calculate
friction
part and its denvatives
.........................................................................
OFF EXP; A 1:=INT(ATAN(W)*(W-ZP)**2/YP-ATAN(W)*(W-ZP)* R2,W)*4*TM*TK/((R1R2)*YP**2); A2:=INT(ATAN(W)* R 1* (W-ZP)-ATA N(W)*(W-ZP)* *2/Y P,W)*4*TM*TK/((R1 R2)*YP**2); LET YP=(U 1-U2)/(A*(R 1-R2)),ZP=-(R I*U2-R2*U 1)/(A*(R l-R2)); LET LOG((A**2+U2**2)/A**2)-LOG((A** 2+U 1"'2)/A*'2) =LOG((A**2+U2**2)/(A**2+U 1"'2)); OFF EXP; ON FORT; OFF ECHO; CARDNO!*:=10; OUT "look.ftn"; WRITE " SUBROUTINE FRIC(U1,U2,A,TK,TM,R1,R2,B 1,B2,B 11,B 12,B22)"; WRITE" IMPLICIT REAL*8(A-H,O-Z)"; B 1: =SUB (W=U2/A ,A 1)-SUB(W=U 1/A ,A 1); B2:=SUB(W=U2/A,A2)-SUB(W=U 1/A,A2); B 11: =DF(B 1,U 1); B22: =DF(B2,U2); B 12:=DF(B 1,U2); WRITE" RETURN"; WRITE" END"; OFF FORT; SHUT "look.ftn"; BYE;
105
include etc.,
and
-
this
a part
can by of
C The fortransubroutinemade
by REDUCE
for frictional
part.
SUBROUTINE FRIC(U I,U2,A,TK,TM,R 1,R2,B 1,B2,B 11,B 12,B22) IMPLICIT REAL*8(A-H,O-Z) ANS4=LOG((A**2÷U2**2)/(A**2+U I**2))*A**3*R2**2.3.*LOG • ((A**2+U2**2)/(A**2+UI**2))*A*RI**2*U2**2+3.*LOG((A • **2+U2**2)/(A**2+U I**2))*A*R I*R2*U I*U2+3.*LOG((A**2+ • U2**2)/(A*z2+UI**2))*A*RI*R2*U2**2.3.*LOG((A**2+U2** , 2)/(A**2+U l**2))*A*R2**2*U 1' U2+A*RI**2*U l**2-6.*A*R1 • **2*U I*U2+5.*A*R!**2*U2**2+A*R I*R2*U I**2+6.*A*RI*R2* • U I*U2-7.*A*R l*R2*U2**2-2.*A*R2**2*U I**2+2.*A*R2**2* , U2"'2 ANS3=-6.*ATAN(U2/A)*A**2*R 1"2'U2+3. *ATAN(U2/A)*A**2* R I * R2* U I +9.* AT AN(U2/ A )* A **2*R I *R2*U2.3,* A T AN(U2/A )* A **2*R2**2*U 1-3.*ATAN(U2/A)*A**2*R2**2* U2+2.*ATAN(U2/ A )*R I **2*U2**3-3.* A TAN(U2/A )SRI *R2*U I *U2**2-A T AN(U2/ A)*RI*R2*U2**3+3.*ATAN(U2/A)*R2**2*U I*U2**2-ATAN(U2/ A)*R2**2*U2**3+LOG((A**2+U2**2)/(A**2+U 1"'2))*A*'3' R I**2-2.*LOG((A**2+U2**2)/(A**2+U l**2))*A**3*RI*R2+ ANS4 ANS2=6.* ATAN(U1/A)* A **2*RI **2*U2.3.* ATAN(U1/A)* A **2* R I * R2*U1-9.* A T AN(U1/A )* A **2*RI *R2* U2+3,* A TAN(U1/A )* A **2*R2**2*U I+3.*ATAN(U 1/A)*A**2*R2**2*U2-2.*ATAN(U1/ A )*RI **2*U I **3+6.W ATAN(U1/A )iRI **2*U I *t2*U2-6.* A TAN( U 1/A)*RI**2*UI*U2**2+ATAN(U 1/A)*R I*R2*UI**3-3.*ATAN( U 1/A)*R l'R2* U I**2*U2+6.*ATAN(U 1/A)*RI*R2*UI*U2**2+ A T AN(U I/A )*R2**2*U I **3-3.* A T AN(U1/A )*R2**2*U I **2*U2+ ANS3 A NS 1=2.* A NS2*TK*T M 13I=ANS 1/(3.*(U I-U2)*'3)
ANS I=2.*ANS2*TKiTM B 12=A NS 1/(U 1-U2)* *4 RETURN END
106
5.6
Numerical
evaluation
The numerical
scheme
increment
of 0.01
the elastic
ring compression
interface
nodes
different
small
and
employed
in each stage.
The
how
small
than
10 "12 will achieve
initial guesses Since
be. According
FRIC, of every
from
of zero velocities
relevant
with this scheme.
with a displacement
modified
the problem
to the experiments,
convergence
iteration
are slightly
the existence
the subroutine
to the r-component
they should
treatment
is the Newton-Raphson
problem.
will overflow number
result
the solution in r-direction
is modified
node.
The convergence
at
by assigning
a
that arises
is
The question
only the numbers
of
which
are smaller
criterion
used
here is
-ll[I-: 0.00005. The boundary 1. Symmetric specified
conditions
boundary
boundary
and the working Numerical
The
elimination
5.6 and
different
5.7, respectively.
in Figures
from
Figures
scheme needs
As the shapes becomes
5. 10 and
matrix
[see Figure
factor
for m=0.5
In order
show
are distorted, to continue
is performed
library.
show,
The velocity
also
in the negative
subroutine
As the figures factors.
surface
5.5]
are
by the 4-point
distributions
The neutral the effective
the error
and
the deformed
The equation
m=0.0 shapes
lines in both cases
the technique
in
of adaptive
+ 3z_z + 3rx2_) 2
states
are visible
and the convergence
I
107
solver
in the deformed
7 The effective stress is defined as + 3_
Guass
are completely
to be introduced.
- (r, cr_ - o_cr_
the die
are shown
stress 7 distributions
increases
the execution,
between
z direction.
is also done numerically.
for friction
5.11
at the interface
terms
5.8 and 5.9, respectively.
of elements
harder.
in z direction
as unit per second
from the IMSL
configurations
are also plotted
cases.
of a global
for low and high frictional
pictures.
--- The velocities
of non-integrable
method,
deformed
velocities
•
the z--O boundary.
piece are specified
integration
at two parts of the boundary
--- The
condition
rule. The assembly
is the Gauss
Figures
condition
to be zeros along
2. Die velocity
quadrature
are specified
for
two
of the mesh
< Z
5
8
!
0.0
LU
2.0
3.8
R COORDINATE
Figure
5.6
: Deformed
configuration
4. U
ON)
after
the
8th
stage
for mffi0.5
£
4.U
3. t,
R OOORI_CATE Flgure
5.7
: Deformed
eonflguraUon
108
ON) after
the
7th
8tt_e
for m-0.0
[]
109
110
III
Q
.g -q °..i
.g r_
°°
w
112
._
r,.,,.
5.7
References
[1]
C.
C.
Chen
and
compression",
[2] C. H. Lee matrix
S.
Kobayashi,"Rigid
ASME
and
AMD-vol.
Trans.
ASME,
solutions J. of Engrg.
[3] C. H. Lee and S. Kobayashi,"Analysis pressing
of solid cylinders
pp 445-454,
May
[4] S. N. Shah
and
method",
mathematical
3.2, April
the
15th
theory user's
ring
deformations
using
upsetting
method,
1973.
and plane-strain
J. of engineering
a
side
for industry,
analysis
of cold
machine
tool
heading
design
by the matrix
& research
conf.,
1974.
of plasticity",
Oxford,
The
Rand
pp 40,
corp.,
1983.
Santa
Monica,
CA 90406,
1985.
methods
D. R. J. Owen,
in elasticity
and plasticity",
Press
Edited
by
Swansea,
of the theory
E. Hinton,"Finite Limited,
J.
F.
Swansea,
T.
Zienkiewicz,"Numerical
3rd edition,
Pergamon
of plasticity",
Mir publisher,
Moscow,
in plasticity
• theory
U. K.,
& Noordhoff,
P. polukhin, Deformation",
elements
methods
and
practice",
U. K., 1980.
Pittman,
R.
D.
Wood,
in industrial
J.
forming
M.
Alexander,
progresses",
O.
Pineridge
C.
Press,
1982.
[11] W. Szczepinski,"Introduction
[12]
of
1982.
Pineridge
[10]
int.
manual",
[8] L. M. Kachanov,"Fundamentals 1974.
[9]
analysis
vol. 95, pp 865-873,
of axisymmetric
pp 603-610,
[7] K. Washizu,"Variational press,
for Ind.,
by the finite element
of
[6] A. C. Hearn,"REDUCE version
to rigid-plastic
S. Kobayashi,"Rigid-plastic
England,
[5] R. Hill,"I'he
element
197 1.
proceeding,
Birmingham,
finite
28, pp 163-- 174,1978.
S. Kobayashi,"New
method",
plastic
to the mechanics
of plastic
forming
of metals",
Sijthoff
1979.
S. Gorelik Mir Publishers,
and
V.
Vorontsov,"Physical
Moscow,
113
1982.
Principles,
of
plastic
[13]
L E. Malvern,"lntroduction Hall,
Inc.,
in engineering,
and problem
MacGraw-Hill
D. G. Luenberger,"Linear Wesley,
of a continuous
medium",
Prentice-
1969.
[14] G. E. Mase,"rheory
[15]
to the mechanics
of continuum
Book company,
and
nonlinear
t984.
114
mechanics",
Schaum's
outline
series
I970.
programming",
2nd
edition,
Addison-
CHAPTER
APPLICATION
OF
SYMBOLIC
AND
ALGEBRAIC
PLATE
6.1
from which and tries
many
by various
Koiter,
E. Reissner
and
class. The other dimensional
theories
are derived.
means
manifold.
manifold.
shell
these theories
dimensional
TO
THE
PROBLEM
to reduce
and
shell
worked
algebraic the capacity the simple
a large
none
manipulation. limitations and which
numerical
method
apply
deals
The
theory
derived
by one
direct
of publications
existing
methodology
the problems
region
Since
symbolic
and algebraic
at the point where
consequence of this work, pushed a step further.
the analytic
8 See next chapter for a detail algebraic manipulation.
study
discussion
element
by employing
the plate
to this
deformable
of vectors
two-
over
P. M. Naghdi,
this
and
W.
and
to solve
plate
the tool of symbolic
and
and
method
of software,
software
but also due to
8 . This chapter
shell problems,
to them,
manipulation
on the capacity
115
belongs
statements.
the finite
of plate
by W. T.
the real shell is a three-dimensional
manipulation
the symbolic
fields
point
in a two-
The works
of some
A. E. Green,
and algebraic
to solve
concepts
body as a starting
by this study
or more
theories.
using
in the symbolic
two distinct
theories.
adopted
a shell as a bounded
are called
with
are only
This is not only due to the late availability
universal
examples
consider
are called
has to rely upon some a priori
amount
there
to the form that can be expressed
on this class of shell theories.
approach
problems,
the problem
are supplemented
This class of theories
Despite
in literature,
One takes the three-dimensional
are in this class.
class of theories
the direct
exist
This class of theories
manifold,
L. Wainwright
and
MANIPULATION
Introduction
Although
body,
VI
and finally
is stuck shell
presents
switches
by its limitations.
problems
limitation
then
outlines
by computer
of symbolic
and
to a As a are
6.2
Preliminary
formulation
Stating
the equations
from D
1
of equilibrium
in three-dimensional
state,
GV(u=,z)+p'(ua,z),O
(6.])
WheFe
•Dj" operator • oiJ: •pi.
• u a"
of covariant
3-D state of stress external
volume
Gaussian
• z" normal
and applying
derivative ( i,j=l,
2, 3).
force.
coordinates
coordinate
of surface
( a=l,
2).
of surface.
the following
6va(u_',z _3(ua,
in 3-D space.
Kirchhoff-Love
) - (a_
hypothesis
- zd_)
• (6v,
to virtual displacements
- z6q,)
(6.2)
z ) = 6w
(6.3)
where Y
• aa " component
of mixed
metric
° da
of mixed
curvature
" component
Here the rotation
&/r
is defined
tensor.
as
6q,, - de/iv
Then, following the lengthy of virtual work gives
tensor.
# + (Sw.,,
derivations
by F. I. Niordson
(6.4)
in his Shell
Theory
[1], the principle
f fot_v_ ,Se_ +M '_,sK . jaa -
ffo[F°_vo
+p_w IdA
Where
116
+ _¢ [T"6vo
+ M_'&a
+ Q6w lds
(6.5)
• N ctl3" effective
membrane
• M ctl_ • effective
moment
• dF_,o.[3: virtual
strain
• dKctl3 : virtual
bending
• F °t, p : effective
tensor.
tensor. tensor.
force vector vector
• Q : supplemented
• dva,
tensor.
load components
• Tct : membrane • M °t : moment
stress
acting
shear
dw : virtual
in surface
acting
and normal
directions,
respectively.
on the boundary.
on the boundary.
force on the boundary.
displacements
in plane and
transverse
directions,
respectively.
• dret=_ctl]dql3 • e c_13: alternate
tensor.
Mathematically,
the strain
tensor 1
where
the superscript a
E¢
-_ 2(a_
K
=d"
asterisks
_ a e +Po_
indicate +Pt_
Ect[3 and bending
tensor
Kct[5 are defined
as
,,
- a_
)
(6.6)
-d
(6.7)
the deformed
+Pr,,Pt3y
state and can be derived
as follows
•
+ q"qtJ
(6.8)
1 Y
d_
- (_)'[(l+p_ -(qP
equation
+ eme
+ P /a)(do_ r_
qy p_
The generalized two-dimensional (6.9) are expressed as •
+ Dtjq,,
)(D ¢ p_
+ dcp_)
- d opq_,)]
displacement
gradient
(6.9)
p and
its determinant
(6.10) (6.11)
117
in
After linearization, the strain and bending tensorscan be expressedin terms of displacementsasfollows • E#
=,_(Dav # + D#v,_) - d_w
K
,_DaD#w
Considering be
+d_Dav
the isotropic
(6.12)
r +darDavr
thin elastic
+vrD
# d_
shell for simplicity.
-d
#rdawr
(6.13)
The constitutive
equations
will
•
The form
which
method 6.3
N "¢
Eh I " __'77r[( v )E "# + va _ E_ ]
M "#
Eh r'l -r = j_"_"_bt( v )K '_ + va _ Kr ]
substitution
of the last four
is expressed
in terms
to solve
equation
Methodology
After (6.5)
calculations of metric,
solving
shell
curvature
tensors,
derivatives
is complex,
manipulation, the surface.
the parametric
2 Calculating parametric metric
the
equations.
should not violate condition.
equations
Christoffel
and curvature
3. Substituting
equations
symbols,
the metric,
curvature
and strain-displacement
tensors, equations.
118
equations,
involve
in general, symbols.
giving
surface
equations
element
the calculations
If a given
of
as follows
tensors
based
correctly,
condition.
and Christoffel
equations :
shell.
and curvature
Gaussian
geometric
and algebraic
is presented
are chosen
the
the evaluations
the parametric
for a given
equation
to (6. 15) show,
the help of symbolic
should obey the integrability
the Coddazzi-Mainardi
(6.12)
shell problems
the metric
If the parametric tensor
With
by simply
of the middle
the finite
to discretize
relations
of Christoffel
to solve
in the weak
in the next paragraph.
Moreover,
will be tedious.
of methodology
applying
it is necessary
as equations
derivatives.
will result
FEM
and strain-displacement
tasks can be performed
Here the outline
Before
formulation,
the computations
(6.5)
is outlined
by
However,
and covariant
require
tough
problem
by FEM.
these calculations
these
1 Finding
shell
equations
vectors.
methodology
of mathematical
problems
of constitutive
of covariant domain
for
(6.15)
into equation
of displacement
(6.5), a universal
the preparations
to solve
equations
(6. 14)
symbols
the resulting
In other
equations
on the
words,
they
and the regular
into constitutive
4 Discretizing method.
the
The above can
be solved
symbolic
and
algebraic
form
algebraic
(6.5)
and
solving
for any shell problems.
way by simply
feeding
REDUCE.
in the following
and
of equation
is universal
manipulator
is shown
Symbolic
6.4.1
methodology
in the same
plate problem
6.4
variational
appropriate Based
it by finite
Different
parametric
element
shell geometries equations
on this methodology,
into the
an example
of
paragraph.
manipulation
application
to plate
problems
Methodology
1 Three
parametric
fl=ul,
f2=u2 '
2 Calculate
equations
as follows"
f3=constant
surface
calculations
are chosen
metric,
curvature
tensors,
are based on their fundamental
and Christoffei definitions,
symbols
given
symbolically.
These
by:
• metric tensor" I
a _ -- f'+,f,o • curvature
tensor
(6.16)
• i
d_
- X
!
f.,_
(6.17)
where
f.'l , f,+z and X '/are surface
Figure
5.2 pictorially,
Gaussian
and X/ is defined
and normal
coordinates.
They
are shown
in
by •
I
X +
•- a
-re
fJfk ,jk_,l_,2
• The 2nd kind of Christoffel
a
fl
The
REDUCE
presented
Y
}
symbols
is defined
.a_[fly,p].Ta
program
as follows.
(6.18)
The
I
_
for calculating results
follow
as:
[a-_ Oatj p + Oarp a,'
equations the program.
119
a, ,,r]
8a
(6.16),
(6.19)
(6.17),
(6.18)
and
(6.19)
is
O_OZZZ_Z_Z_ZZZZZZZZZZ_ZZZZZZZZZZZZZ_ZZZZZZZZZZZZZmZZZSS_S_8
% REDUCE % TENSORS
PROGRAM FOR CALCULATING & CHRISTOFFEL SYMBOLS
METRIC,
CURVATURE
_OZZZZZZZZZZZZ_ZZ_ZZZZ_ZZZZ_ZZZ_ZZ_Z_ZZZZZZ_ZZ_ZZZZ_ZZZZZ
C_
............................................................
%INPUTTING O'_
THE
PARAMETRIC
EQUATIONS
............................................................
MATRIX A(2,2),CA(2,2),F(2,3),D(2,2); ARRAY X(3),C 1(2,2,2),C2(2,2,2),N(3),E(3,3,3),U(2); U(1):=S; U(2):=P; OFF PERIOD; X(1):=U(1); X(2):=U(2); X(3):=CONSTANT; FOR I:=1:2 DO FOR J:=l:3 DO F(I ,J): =DF(X(J),U(I)); FOR ALL T1 LET COS(T1)**2+SIN(T1)**2= 1;
%CALCULATING
COVARIANT
METRIC
TENSOR
...............................................................
FOR M:=l:2
DO FOR N:=l:2
A(M,N):=FOR A:=A; DETA:=DET(A);
%CALCULATING {_qD
I:=1:3
SUM
DO DF(X(1),U(M))*DF(X(I),U(N));
CONTRAVARIANT
METRIC
TENSOR
.......................................................................
FOR L:=l:2 DO FOR M:=I:2 DO IF L=I AND M=I THEN CA(L,M):=A(2,2)/DETA ELSE IF L NEQ M THEN CA(L,M):=-A(M,L)/DETA ELSE CA(L,M):=A( 1, I)/DETA; O'_
.........................
7 ..................................................
%CALCULATING
THE
1ST & 2ND CHRISTOFFEL
SYMBOL
WRITE "THE 2ND CHRISTOFFEL SYMBOL'; FOR L:=l:2 DO FOR M:=l:2 DO FOR N:=I:2 DO C 1(L,M,N):=(I/2)*(DF(A(L,N),U(M))+DF(A(M,N),U(L)) -DF(A(L,M),U(N))); FOR I_.:=1:2 DO FOR M:=l:2 DO FOR N:=l:2 DO >; O_
...............................................
%CALCUI_.ATING
NORMAL
VECTOR
...............................................
FOR I:=1:3 DO N(I):=(FOR J:=l:3 SUM FOR K:=l:3 E(I,J,K)*F( 1,J)* F(2,K))/SQRT(DETA); 7o
....................................................
%CALCULATING _'_
SUM
CURVATURE
TENSOR
....................................................
FOR J:=l:2
DO FOR K:=l:2
DO
D(J,K):=FOR I:=1:3 SUM N(I)*DF(F(J,I),U(K)); FOR ALL T1 CLEAR COS(T 1)**2+SIN(T I)*'2; FOR ALL T1 LET COS(T 1)*'2- I=-SIN(T1)**2;
%OUTPUTTI _7_
NG RESULTS
.................................
OFF PERI OD; OFF ECHO; OUT "SURFACE"; WRITE " ........................................................ ". WRITE" THE COMPONENTS OF METRIC TENSOR "" WRITE " A:=A; WRITE " WRITE " THE COMPONENTS OF CURVATURE TENSOR"; WRITE " D:=D; WRITE " WRITE " THE CHRISTOFFEL SYMBOLS"; WRITE " FOR I:=1:2 DO FOR J:=l:2 DO FOR K:=I:2 DO .........................................
WRITE "CHRIS(" SHUT "SURFACE"; BYE;
The outputs
".
,I,"," ,J," ," ,K,")="
of REDUCE
,C2(I ,J,K);
are presented
as follows
.......................................................
THE
COMPONENTS
OF METRIC
.......................................................
A(1,1) A(1,2) A(2,1) A(2,2)
:= := := :=
1 0 0 1
121
TENSOR
•
D(1,2) D(2,1) D(2,2)
:= 0 := 0 := 0
THE
CHRISTOFFEL
...................
..
SYMBOLS
................
.__.
CHRIS(I, 1,1)-0 CHRIS( 1,1,2)-0 CHRIS( 1,2, I)=0 CHRIS( 1,2,2)=0 CHRIS(2,1,1)-0 CHRI S(2,1,2)-0 CHRIS(2,2,1)=0 CHRIS(2,2,2)=0
As the output case,
from
According
REDUCE
shows,
to the above
all of the Christoffel
solutions,
the metric
symbols
and curvature
vanish
tensors
in the plate
in matrix
form
are
[aqo ] = (10
3. After substituting are as follows
_,
[dqs ]." (0
the calculated
metric
and curvature
0_
tensors,
(6.20)
the constitutive
equations
• For membrane"
I!t f: 1
12
0
Eh
"77
• For bending
0
(6.21)
•
io 0], f
" lXl-,,"---'-_
/M_! The relationships are from
,'lf "l 0/E2_/
1-v
22
equations
1- V
o
between (6.12)
(6.22)
lie_
linearized
strain-displacement
and (6.13)
•
and
bending
curvature
tensors
I
E ,,_ -
"_( D ,,v p + D _v a ) - _vp.,, 1
K ¢ - D,,D c w . w.a _
+ v,,p)
(6.23) (6.24)
122
where
the disappearance
between
covariant
differentiation
unity of the metric tensors.
4. Finite 6.4.2
element
Finite
The (CST)
method
element
chosen
Cheung,
Where
= 2A[cr(_a_t
is
differentiation.
between
In addition,
contravariant
and
is the
(6.25).
Zienkiewicz
of the constant (CKZ)
will be discussed
1
later.
triangle
covariant
The shape
strain
triangle
element. functions
The for this
= _Y l-Y3 2a ,
Cl
_
the deflections
--
+ 7_,_2_:3)
- br(_,,_
of (1,2,3)
b
._.
z
o + 5"_,_2_,)]
and
no summation
2---_,
Xl-X ,
C2
3 -
x3Y
-y
X2-x
2A
2 +
x3y I
is applied
•
2
b3 =
3
="
convention
as follows
v
Y J-Yl
(6.25)
1
bi, ci, and A are defined
2
x2y
1
+ 7_,_2_3)]
2
permutation
b,
2A
2
- ca(_,._r
1
The constants
2A
'
C3
_xjy
u, v, w in local coordinates
==
3 +
I
2A
x,y
(6.26) z
-
Xzy
x, y, z direction
I
can be interpolated
by
3
u = Xu t
L
-I
3
v
=_v
L
(6.27)
I-I
o_ I _ +-.7. N _ + .2..l_Nay
w =waN
The substitution
fL
the
paragraph).
the combination
King,
_ + 7_,_2_3)
= 2A[b a (_a_r
X 3-X
Then
the distinction
_
(ct,13,¥)
equation
the distinction
this element
2
Nay
remove
partial
starts (see the following
2
N,,
in this case eliminates
and ordinary
for this topic
the
of selecting
_
symbols
discretization
with
considerations element are
Lot
tensors
element
element
of Christoffel
of equations
[B: OtBl+
B_D
(6.21)
2B2]dA
to (6.27)
"d
into (6.5) gives
= f fD rr dA
123
+
the discretization
fs r ds
+ f M r ds
form
(6.28)
in
Where
l--v
DI = F2"
(6.29)
0
D 2 " 12(_-__l
(6.30)
2_
0
_---
0
'g2
Bz"
, c2 c3 bl Ii R b 2 b 3 0 0 0 c,
b2 0 c2
b3 _ 0 1 0a c3 0
0 a
,
--_ a_ 2
0
oooo oooo oooo1
LI
0
0
0
0
L2
0
0
2_
0
0
L3
0
0
(6.31)
0
a
o I
B2
c2
c3
bl
b2
0
0
cI
c:
b3
_ 0
c3
0
bl
_
_
o _
M
-[M'
M2][_b '|
c2b z
C3]b, _,
124
b2
b3
C 2
C3
Jilt
(6.32)
t_,J
1_'
(6.33)
0
N 1
V 1
W t
N,_
Nly
0
W 1.x
W 1,y
1l 2
0
N 2 V 2
W
N2y
Nzx
2
W
2,x
W
0 2 .y
N3 1"13
N3x
3
W3
N3y
W3,
x
]
W3,y
(6.34) (6.35) (6.36)
m
Lt t
0 0
0
0
0
0
L 2 0
0 N,
0 Nix
0
L_
Nly
0 0
0
0 N 2
0
0
0
L 3 0
0
0 N2,
0 N_
Ls 0
0
0
0
N 3
0
0
0 l N3y
N3x
(6.37) 6.4.3
Numerical
Three
1. In-plane
boundary
uniaxial
contribute traction
results
and
conditions
tension
to the stiffness S only.
are presented symbolically.
post-process
are applied
[see Figure matrix.
The REDUCE as follows.
to the test problem.
They
6.1] --- In this case, only the membrane The
consistent
programs
In addition,
load
vector
to the stiffness the stress
is due
distribution
can
v
FEM domain
v
v
6.1: Plate with hole under uniform 100 N/cm**2
125
uniaxiai
tension
load
component
to the boundary
and the consistent
_---10cm
Figure
are •
also
load
vector
be calculated
]
%*********************************************************** % Symbolicalprogramfor makingstiffnessmatrixfor platetensionproblem matrix bc(3,6),In(6,15),d(3,3),b(3,15) ,s(15,15); arrayn(3); n( 1):=s1;n(2):=s2;n(3):=s3; In(1,1):=I ;ln(2,6):=1;In(3,11):=1;1n(4,2):=1; 1n(5,7):=1;In(6,12):=1; bc:=mat((bl,b2,b3,0,O,O),(c 1,c2,c3,bl,b2,b3),(O,O,O,c 1,c2,c3)); d:=mat((1,O,v),(O,(1-v)/2,0),(v,O, 1)); b:=bc*ln; s:=tp(b)*d*b*pj*ye*h/(2*(I-v**2)); bl:=(y2-y3)/pj;b2:=(y3-y1)/pj;b3:=(y1-y2)/pj; cl:=(x3-x2)/pj;c2:=(x1-x3)/pj;c3:=(x2-x1)/pj; for i:=l: 15do ; for i:=l: 15do ; on fort; off period; off echo; out "plane.ftn"; write " subroutine ske(xl,yl,x2,y2,x3,y3,s)"; write " dimension s(15,15)'; write" implicit real*8(a-h,o-z)"; write " pj=(x 1-x3)*(y2-y3)-(y l-y3)* (x2-x3)"; write ' ye=2.1" 1.Oe07'; write " v=0.29"; write " h=0.2"; for i:=l: 15 do for j:=l: 15 do ifj>=i then write " s(",i,",",j,")=',s(i,j) else write " s(",i,",",j,")=s(",j,",",i,")"; write " return"; write " end"; shut "plane.ftn"; bye; C_Z_ZZZ_Z*ZZ_*_ZZZ*Z*Z*_:_ZZZZZZZZZ_ZSZZZ_ZZSZ**ZZZS_IISZZZZSZ
% Symbolic
program
to calculate
the load vector
for plate tension
_ZZZZ*ZZ*ZZZZZgZZ*_cZZZZZZZZZZZZZIZZZZZ_,ZZZZIZZ_,Z_ZZZZZZ
matrix f( 1,2),sn(2,15),fn( 15,1); army n(3),ff(15); n( l):=s 1;n(2):=s2;n(3):=s3; f:=mat((fl,f2)); for i:=l step 5 until 11 do ; fn:=tp(f*sn); s2:=O; s3:=l-sl; for i:=l: 15do ; for i:=1;15 do ff(i):=fn(i,1); on fort; off echo; off period;
126
problem.
out "151oad.ftn"; write " subroutineIoad(x1,y1,x2,y2,x3,y3,f1,f2,f3,fe)"; write " implicit real*8(a-h,o-z)"; write " dimensionfe(15)"; write " rl=sqrt((x1-x3)**2+(y1-y3)*'2)"; write " h=0.2"; for i:=l: 15do write " fe(",i,")=",ff(i); write " return"; write " end"; shut "151oad.ftn"; bye; % REDUCEprogramto constructsubroutineto compute % stressdistributionfor platetensionproblem. matrix d(3,3
), bc(3,6),fn(6,1 ),stre(3,1 ); array n(3); n( 1):=s 1;n(2):=s2;n(3):=s3; bc: =mat((b 1, b2,b3,0,0,0),(c 1,c2,c3 ,bl, b2,b3),(0,0,0,c d:=mat(( 1,0,v),(0,( 1-v)/2,0),(v,0,1)); operator u; for i:=1:6 do fn(i,l):=u(i); stre:=d* bc* fn* ye/( 1-v* *2); b 1:=(y2-y3)/pj ;b2: =(y3-y 1)/pj ;b3: =(y 1-y2)/pj; c 1:=(x3-x2)/pj ;c2:=(x 1-x3)/pj ;c3 :=(x2- x 1)/pj; on fort; off echo; off period; out "st.ftn"; write " subroutine stress(x 1,y 1,x2,y2,x3,y3,u,st)"; write " implicit real*8(a-h,o-z)"; write " dimension u(6),st(3)"; write " v=0.29"; write " pj=(x 1-x3)* (y2-y3)-(y 1-y3)* (x2-x3)"; write " ye=2.1 * 1.0e07"; for i:=1:3 do write " st(",i,")=",stre(i, 1); write " return"; write " end"; shut "st.ftn"; bye;
The resultant matrix,
fortran
load vector
subroutines
obtained
and stress distribution
of stiffness
matrix
researchers
may refer
is quite
lengthy,
to the PH.D.
from the above are presented only
a part
thesis of Wen-Lang
SUBROUTINE SKE(X 1,Y 1,X2,Y2,X3,Y3,S) IMPLICIT REAL*8(a-h,o-z) DIMENSION S( 15,15) pj=(x 1-:,c3)*(y2-y3)-(y l-y3)* (x2-_) YE=2.1' 1.0E07
127
1,c2,c3));
programs as follows.
to compute Since
of it is showed. Tsai
stiffness
the expression The
[51] for details.
interested
V=0.29 H=0.2 s(1,1)=(H*YE*(V* X2"'2-2" V* X2*X3+V*X3**2-X2**2+2*X2* • X3-X.3**2-2*Y2**2+4*Y2*Y3-2*Y3**2))/(4*PJ*(V**2-1)) s(1,2)=(H*YE*(V*X2*Y2-V*X2*Y3-V*X3*Y2+V*X3*Y3+X2,Y2_ • X2*Y3-X3*Y2+X3*Y3))/(4*PJ*(V**2-1)) s(1,3)=0 s(1,4)=0 s(1,5)=0 s(1,6)=-(H*YE*(V*X 1*X2-V* X 1*X3-V *X2* X3+V*X3**2-X 1*X2 • +X 1* X3+X2*X3-X3**2+2*Y2*Y3-2*Y2*y 1-2*Y3**2+2*Y3*Y 1) • )/(4*PJ*(V**2-1)) s(1,7)=-(H*YE*(2* V*X 1* Y2-2' V*X 1*Y3+V*X2* Y3-V*X2*Y 1-2 • *V*X3*Y2+V*X3*Y3+V*X3*Y 1-X2*Y3+X2*YI+X3*Y3-X3*Y 1)) •/(4*PJ*(V**2-1)) s(1,8)=0 s(1,9)=0 s(1,1O)=0 s(1,11)=(H*YE*(V*X I*X2-V*X 1*X3-V*X2**2+V*X2*X3-X l'X2 • +X 1*X3+X2**2-X2*X3+2*Y2**2-2*Y2*Y3-2*Y2*y l+2*Y3*Y 1) • )/(4*PJ*(V**2-1)) s(1,12)=(H*YE*(2' V*X 1*Y2-2" V*X 1*Y3-V*X2*Y 2+2' V* X2*Y3 • -V*X2*Y 1-V*X3*Y2+V*X.3*Y 1-X2*Y2+X2*YI+X3*Y2-X3*Y I)) •/(4*PJ*(V**2-1)) s(1,13)=0 s(1,14)=0 s(1,15)=0 s(2,1)=s(1,2) s(2,2)=(H*YE*(V*Y2**2-2*V*Y2*Y3+V*Y3**2-2*X2**2+4*X2 • *X3-2*X3**2-Y2**2+2*Y2*Y3-Y3**2))/(4*PJ*(V**2-1)) s(2,3)=0 s(2,4)=0 s(2,5)=0 s(2,6)=(H*YE*(V*X 1* Y2-V*X 1*Y3+2"V'X2* Y3-2"V*X2*Y 1-V* • X3*Y2-V*X3*Y3+2*V*X3*Y 1-XI*Y2+XI*Y3+X3*Y2-X3*Y3)) •/(4*PJ*(V**2-1)) s(2,7)=(H*YE*(V*Y2*Y3-V*Y2*Y 1-V*Y3**2+V*Y3*Y l+2*X 1" • X2-2*X l*X3-2*X2*X3+2*X3**2-Y2*Y3+Y2*Y l+Y3**2-Y3*Y 1)) •/(4*PJ*(V**2-1)) s(2,8)=0 s(2,9)=0 s(2,10)=0 s(2,11)=-(H*YE*(V*X 1*Y2-V*X 1*Y3+V*X2* Y2+V*X2*Y3-2* V* • X2*Y 1-2*V*X3*Y2+2*V*X3*Y 1-XI*Y2+XI*Y3+X2*Y2-X2*Y3)) •/(4*PJ*(V**2-1)) s(2,12)=-(H*YE*(V*Y2**2-V*Y2*Y3-V*Y2*Y 1+V*Y3*Y l+2*X 1 • *X2-2*X l*X3-2*X2**2+2*X2*X3-Y2**2+Y2*Y3+Y2*y I-Y3*Y 1 • ))/(4*PJ*(V**2-1)) s(15,15)=0 return end
128
subroutine load(x 1,y I ,x2,y2,x3,y3,f IMPLICIT REAL*8(a-h,o-z) dimension fe(15) rl=sqrt((x3-x2)**2+(y3-y2)**2) h=0.2 fe( 1)=0 fe(2)=O fe(3)=0 fe(4)=0 fe(5)=0 fe(6)=(H* F 1* RL)/2 fe(7)=(H* F2*RL)/2 fe(8)=O fe(9)=0 fe(10)=0 fe( 11 )=(H* F1 * RL)/2 fe( 12)=(H* F2* RL)/2 fe( 13)=0 fe(14)=O fe(15)=0 return end
1, f2,fe)
subroutine stress(x 1,y 1,x2,y2,x3,y3,u,st) implicit real*8(a-h,o-z) dimension u(6),st(3) v=0.29 pj=(x l-x3)* (y2-y3)-(y YE=2.1 * 1.0E07
l-y3)* (x2-x3)
st( 1)=(Y E*(U(6)* V*X l-U(6)* V*X2-U(5)* V*X l+U(5)* V*X3+U • (4)*V*X2-U(4)*V*X3+U(3)*Y2-U(3)*Y 1-U(2)*Y3+U(2)*Y I• U( 1)*Y2+U( 1)*Y3))/(PJ*(V**2-1)) st(2)=-(Y E*(U(6)*V*Y2-U(6)*V*Y 1-U(6)*Y2+U(6)*Y l-U(5) • *V*Y3+U(5)*V*Y I+U(5)*Y3-U(5)*Y 1-U(4)*V*Y2+U(4)*V*Y3 • +U(4)*Y2-U(4)*Y3+U(3)*V*X l-U(3)* V*X2-U(3)*X l+U(3)* • X2-U(2)* V_'X l+U(2)* V*X3+U(2)*X 1-U(2)*X3+U( 1)* V*X2-U( 1)* V*X3-U( 1)*X2+U( 1)*X3))/(2" PJ*(V**2-1)) st(3)=(Y E*(U(6)*X I-U(6)* X2-U(5)*X 1+U(5)*X3+U(4)* X2-U • (4)*X3+U(3)*V*Y2-U(3)*V*Y 1-U(2)*V*Y3+U(2)* V_'y I-U(1) • *V*Y2+U(1)*V*Y3))/(PJ*(V**2-1)) return end
The
stress
stress should applied
distribution
concentration be mentioned
of plate under is visible
tension
in the top of the hole.
here is that the stress
load. This is contributed
is plotted
129
One
at the middle
by the bending
in the top fiber of the plate.
by PATRAN
effect
interesting
in Figure
phenomenon
top of the plate which
produces
6.2. The that
is less than the the compression
2. Four-edgesimply supportedplatebendingwith uniform transverseload [seeFigure 6.3] --- In this case,the membranecomponentof stiffnessmatrix is neglected.The REDUCE program to makethe bendingcomponentof stiffnessmatrix is shown as follows • % REDUCEprogramfor constructingplatebendingstiffnessmatrix arrayn(9); matrix sn(3,9),bc(2,3),bn(2,9),ssn(6,9),bcbc(3,6),bssn(3,9),d(3,3) ,ske(9,9),s(9,9); n( 1):=s1+s1* s2*(sl-s2)+s1* s3*(s1-s3); n(2):=pj*(c3*(sI * * 2"s2+s1*s2*s3/2)-c2"(s1* *2"s3+s1*s2*s3/2)); n(3):=pj*(b2*(s1.**2' s3+s1* s2*s3/2)-b3*(s1** 2' s2+s1* s2*s3/2)); n(4):=s2+s2*s3* (s2-s3)+s2*s1* (s2-s1); n(5):=pj*(c 1*(s2** 2' s3+s1* s2*s3/2)-c3"(s2**2*s 1+sI *s2*s3/2)); n(6):=pj*(b3* (s2**2* s1+sI *s2*s3/2)-b1*(s2**2*s3+sI * s2*s3/2)); n(7):=s3+s3*s1*(s3-s1)+s3*s2*(s3-s2); n(8):=pj*(c2" (s3* *2*s1+s1* s2*s3/2)-cI *(s3* * 2"s2+s1*s2*s3/2)); n(9):=PJ*(bl*(s3**2*s2+s l*s2*s3/2)-b2*(s3**2*s1+sl.s2,s3/2)); for i:= 1:9do ; bc:=mat((b1,b2,b3),(cI ,c2,c3)); bn:=bc*sn; for i:=1:9do ; s3:=l-sl-s2; bcbc:=mat((bl,b2,b3,0,0,0),(c1,c2,c3,b1,b2,b3),(0,0,0,c1,c2,c3)); bssn:=bcbc*ssn; D:=MAT((1,0,V),(0,(I_V)/2,0),(V,0,1)); ske:=tp(bssn)*d*bssn*pj$ for i:=1:9 do forj:=l:9 do ifj>=i then elses(i,j)=s(j,i); b1:=(y2-y3)/pj;b2:=(y3-y1)/pj;b3:--(y1-y2)/pj; c1:=(x3-x2)/pj;c2:=(x1-x3)/pj;c3:=(x2-x 1)/pj; on fort; off echo; off period; CARDNO!*:=IO; out "z.ftn"; WRITE " SUBROUTINE SKE1(X1,Y 1,X2,Y2,X3,Y3,S)"; WRITE " IMPLICIT REAL*8(A-H,O-Z)"; WRITE " DIMENSIONS(2,9)"; WRITE " pj=(x1-x3)*(y2-y3)-(y1-y3)*(x2-x3)"; WRITE " YE=2.1' 1.0E07"; WRITE " V=0.29"; WRITE " H=0.2"; FORI:=1:2 DO FORJ:=l:9 DO 131
IF j>=i THEN WRITE " ELSE WRITE " S(",I
S(",I,",",J,")=",S(I,J) ,"," ,J,")=S(",J,"
,",[ ,")"
,
WRITE " RETURN"; WRITE " END"; WRITE " SUBROUTINE SKE2(X1,Y 1,X2,Y2,X3,Y3,S)"; WRITE" IMPLICIT REAL*8(A-H,O-Z)"; WRITE " DIMENSION S(4,9),s 1(2,9)"; WRITE " pj=(x 1-x3)*(y2-y3)-(y 1-y3)*(x2-x_3)"; WRITE " YE=2.1' 1.0E07"; WRITE" V=0.29"; WRITE" H=0.2"; WRITE " CALL SKEI(XI,Y 1,X2,Y2,X3,Y3,S1)"; WRITE " DO 20 I=1,2"; WRITE" DO 20 J= 1,9"; WRITE" 20 S(I,J)=SI(I,J)"; FOR I:=3:4 DO FOR J:=l:9 DO IFj>=i THEN WRITE" S(",I,",",J,")=",S(I,J) ELSE WRITE " S(",I,",",J,")=S(",J,",",I,")"; WRITE " RETURN"; WRITE" END"; WRITE " SUBROUTINE SKE3(X1,Y I,X2,Y2,X3,Y3,S)"; WRITE" IMPLICIT REAL*8(A-H,O-Z)"; WRITE " DIMENSION S(6,9),s2(4,9)"; WRITE " pj=(x 1-x3)*(y2-y3)-(y 1-y3)*(x2-x3)"; WRITE " YE=2.1" 1.0E07"; WRITE" V=0.29"; WRITE " H=0.2"; WRITE " CALL SKE2(X1,Y1,X2,Y2,X3,Y3,S2)"; WRITE " DO 20 I=1,4"; WRITE " DO 20 J=l,9"; WRITE " 20 S(I,J)=S2(I,J)"; FOR I:=5:6 DO FOR J:=l:9 DO IFj>=i THEN WRITE " S(",I,",",J,")=",S(I,J) ELSE WRITE " S(",I,",",J,")=S(",J,",",I,")"; WRITE " RETURN"; WRITE " END"; WRITE " SUBROUTINE SKE(XI,YI,X2,Y2,X3,Y3,S)"; WRITE" IMPLICIT REAL*8(A-H,O-Z)"; WRITE " DIMENSION S(9,9),s3(6,9)"; WRITE" pj=(x 1-x3)*(y2-y3)-(y 1-y3)*(x2-x3)"; WRITE" YE=2.1" 1.0E07"; WRITE" V=0.29"; WRITE" H=0.2"; WRITE" CALL SKE3(X 1,Y I,X2,Y2,X3,Y3,S3)"; WRITE" DO 20 I=1,6"; WRITE" DO 20 J= 1,9"; WRITE" 20 S(I,J)=S3(I,J)"; FOR I:=7:9 DO FOR J:=l:9 DO IF j>=i THEN WRITE" S(",I,",",J,")=",S(I,J) ELSE WRITE " S(",I,",",J,")=S(",J,",",I,")"; WRITE" RETURN"; WRITE" END"; SHUT "z.ftn"; bye;
132
The resultant fortran subroutineis too large (around 135 pages) to be presented completelyhere.The following is onlya smallportionof it. SUBROUTINE SKE1(X 1,Y 1,X2,Y2,X3,Y3,S) IMPLICIT REAL*8(A-H,O-Z) DIMENSION S(1,9) pj=(x1-x3)*(y2-y3)-(yl-y3)* (x2-x3) YE=2.1' 1.0E07 V=0.29 H=0.2 ANS5=-4*Y3*Y 1"'3+2"Y 1"'4 ANS4=-4*X2*"2' Y2*Y 1+8'X2'* 2"Y3* '2+2' X2* *2*Y 1* *2-16* X2* X.3"3-16*X2*X3*Y2**2+32*X2*X3*Y2* y3.16*X2*X3* Y3 • *2+5*X3**4+8*X3**2*Y 2"'2-16*X3**2*Y2*Y3+ 10"X3"'2" Y3**2-4*X3**2*Y3*y i +2*X3**2*Y 1"2+5'Y2"4-16'Y2"'3" Y3-4*Y2**3*Y l+24*Y2**2*Y3**2+6*Y2**2*y 1"'2-16*Y2*Y3 • *3-4*Y2*Y l**3+5*Y3**4-4*Y3**3*y l+6*Y3**2*Y 1"'2+ ANS5 ANS3=2*X 1"'4-4"X l**3*X2-4*X l**3*X3+6*X 1"'2"X2"'2+6" X 1** 2"X3** 2+2"X 1*'2' Y2"* 2-4"X 1* "2' Y2*Y 1+2*X 1*'2'Y3 • * 2-4' X 1* *2*Y3*Y 1+4*X 1* *2*Y 1* * 2-4' X 1*X2"'3-4" X 1*X2* Y2* * 2+8' X 1*X2*Y 2' Y 1-4*X 1*X2*Y 1* * 2-4"X 1*X3* "3-4" X 1* X3*Y3* * 2+8"X 1*X3*Y3* Y 1-4*X1*X3*Y 1** 2+5"X2"'4-16"X2 • *3*X3+24*X2** 2"X3"2+ 10*X2**2*Y2**2-16*X2**2*Y2*Y3 +ANS4 ANS2=H**3*YE*ANS3 ANS I=A NS2/(18*PJ**3*(V**2-1)) S(1,1)=-ANS 1
133
Y
FEM domain
X
10cm
/'Figure
The
load
program
vector
L2x
6.3 • Physical
Z2x
configuration
of plate bending
in this case is for a uniform
and its output
are shown
transverse
problem
pressure
as follows"
************************************************ % REDUCE program to construct % for plate bending problem. army n(9),fe(9); n(1):=s l+sl*s2*(s
the load vector
1-s2)+s l*s3*(sl-s3);
n(2):=pj*(c3*(sl**2*s2+s l*s2*s3/2)-c2*(sl**2*s3+sl*s2,s3/2)); n(3):=pj*(b2*(s l**2*s3+s l*s2*s3/2)-b3*(s l**2*s2+s l*s2*s3/2)); n(4):=s2+s2* s3*(s2-s3)+s2*s l*(s2-s 1); n(5):=Pj*(c l*(s2**2*s3+s l*s2*s3/2)-c3*(s2**2*s l+s I *s2*s3/2)); n(6):=Pj*(b3*(s2**2*s l+s l*s2*s3/2)-b l*(s2**2*s3+s I *s2*s3/2)); n(7):=s3+s3*sl *(s3-s 1)+s3*s2* (s3-s2); n(8):=Pj*(c2*(s3**2*s n(9):=pj*(bl*(s3**2*s2+s s3:=l-sl-s2;
l+s l*s2*s3/2)-c l*(s3**2*s2+s l*s2*s3/2)); l*s2*s3/2)-b2*(s3**2*s l+s l*s2*s3/2));
for i:= 1:9 do ; b 1: =(y2-y3)/pj c 1 :=(x3-x2)/pj off period;
1);
l=0,tem3))*pj*
;b2: =(y3-y 1)/pj ;b3: =(y 1-y2)/pj; ;c2: =(x 1-x3)/pj ;c3:=(x2-x 1)/pj;
134
f3;
only.
The
REDUCE
off echo; ON FORT; out "zload.ftn"; write " SUBROUTINE LOAD(X1,Y write " IMPLICIT REAL*8(A-H,O-Z),,; WRITE " DIMENSION FE(9)";
1,X2,Y2,X3,Y3,F3,FE),,.
write " PJ=(x 1-x3)*(y2-y3)-(y 1-y3)*(x2_x3),,; FOR I:=1:9 DO WRITE FE( ,I, )= ,FE(I); WRITE " RETURN"; write " end"; shut "zload.ftn"; bye; N
Pt
tl
ft
SUBROUTINE LOAD(X 1,Y 1,X2,Y2,X3,Y3,F3,FE) IMPLICIT REAL*8(A-H,O-Z) DIMENSION FE(9) pj=(x 1-x3)*(y2-y3)-(y FE(1)=(PJ*F3)/6
1-y3)* (x2- x3)
FE(2)=-(PJ* F3*(2*X 1-X2-X3))/48 FE(3)=(PJ*F3*(Y2+Y3_2,y 1))/48 FE(4)=(PJ* F3)/6 FE(5)=(PJ* F3* (X 1-2*X2+X3))/48 FE(6)=-(PJ*F3*(2*Y2_Y3_y 1))/48 FE(7)=(PJ*F3)/6 FE(8)=(PJ* F3*(X 1+X2-2*X3))/48 FE(9)=(PJ*F3*(Y2-2*Y3+y I))/48 RETURN end
The
deformed
shape
6.5.
In addition, of solution.
Figure
convergence presented
3. Four-edge case
and
enforced
is shown
in Figure
6.4 and
three
different
sizes
They
are shown
in Figure
6.7 which
clamped
plate bending
the last
case
in the edges
is the
stiffer
phenomenon
is visible
of mesh
in Figure
under uniform
boundary
6.6.
135
tested The mesh
pattern
is in
to investigate
the
convergence
constraints.
The
it is expected
Figures
extra
slope
that the solution
shape
is shown
6.4 and 6.8.
trend
is
size.
load --- the only difference
case. The deformed by comparing
distribution
are
is the plot of error vs. element
in this case. Therefore
than that of the simply-supported
the stress
between
constraints
this are
will be stiffer
in Figure
6.8. The
°
I
|')
/
0"!
•
136
I
ffl
137
32 element
Figure
6.6"
64 element
98 element
Three different sizes of mesh for testing the convergence of solution of plate bending
$
0 3
n_m_ Figure
6.7 : Convergence
138
4
• of plate bending solution
£',,I
E_o L_
("q
I=
c7'
°,"_
,c::
. L--
I 0 N
o
°.
O0
!
0'!
139
6.5
References
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and
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CHAPTER
VII
CONCLUSIONS
7.1
Introduction
The algebraic
topics
manipulation,
SAM
in education,
Simple
examples
7.2
discussed
in this chapter
the difficulties contributions,
Advantages
of
symbolic
power---
manipulating computational intractable analytical
.
Reliability
--- A solution
obtained
is no round-off
--- The resultant
=
generation
if the input eliminates
symbolic
and algebraic
development
and
software,
and
the
application.
necessary.
and algebraic
intelligence,
has made
SAM
the
and
tireless
Due
algebraic
They
capability
in
tool in modern
to challenge
formulae.
by symbolic
manipulation.
an indispensable
the potential
sophisticated
forward.
will be correct code
new
work is pushed
There
of symbolic
It has created
and
of using
manipulation
human
numbers
community. problems
2. Accuracy exact.
and
of future
algebraic
with
symbolic
the points where
of application
Together
symbols
the prospect
and
the advantages
in running
to illustrate
There are many advantages can be classified as follows •
1. Tireless
existing
and
will be presented
include
both
to this
previously
advantage,
manipulation
the
is always
error accumulation.
expressions
obtained
by symbolic
and algebraic
manipulation
information
is right.
In addition,
the capability
of automatic
any typographic
debugging
the programs.
Efficiency
--- This is a new advantage
nonlinear
numerical
execution.
This advantage
analysis
can
is believed
found
errors
and substantially
in this research.
significantly
improve
to be a crucial
145
solution
reduces
The symbolic the
efficiency in the future
the time in
template of
in
program
for fields
in
which the developmenttime playsanimportantrole. Forexample,the realtime control will beoneof them. 7.3
Internal
swelling
As mentioned and algebraic
.
and
capacity
limitation
manipulation.
memory
needed
space
system
there are some
The followings
and algebraic
one
limitations
in the last chapters,
manipulation.
Memory
mathematical
--- A large amount This
for running
(hardware
are the detail
and
software)
sizes of memory
space,
amounts
of memory
for the same software.
same
hardware
whether
system
external
(about
815
Michigan
Terminal
involved
in the
included The
Apollo The
University
workstation
provisions
during
recommended
Unlike
execution
of hardware,
mega
MTS,
execution
that the symbolic
will
and algebraic
an example
the mechanism
is given
to calculate
of internal
Engineering
834560
be aborted
on RAM
REDUCE
in the
performance
is
should
be manually
(about
1 mega bytes). is up to seven
to run REDUCE Network
requirement
automatically.
manipulator
in the
bytes
can be automatically the space
different
depending
the computation
When
to guarantee
be implemented
in an
(CAEN)
of
extended
up
is beyond
Therefore
it is
on a machine
a safe execution.
swell in symbolic
the factorial
bytes
are provided
this space
with at least one mega bytes RAM capacity To demonstrate
during
Aided
if necessary.
package
of
need
running
spaces
If integration
to 1048560
bytes
at the Computer
of Michigan.
bytes
three
may provide
for running
integration
will be extended
hand,
For example,
it is invoked.
that MTS can provide
On the other
memory
(Fa11,1989)
the external
space
space
or not.
when
systems
Even the same software different
amount
varies a lot from
software
systems
for symbolic
The
manipulator
Different
hardware
need
provided
(MTS)
computation,
Domain
to six mega the
are currently
and the memory
bytes.
may
are connected
System
total memory
mega
sometimes
packages
K bytes)
and different
is required
limitations.
and algebraic
to another.
in the symbolic
:
space
is one of its fundamental a symbolic
existing
discussions
of memory
different
space
difficulties
of a number.
and algebraic Mathematically,
manipulation, a factorial
deft ned as :
n!-
The LISP
n1 (n -
function
1)!
ifotherwise. n _.0;
for this problem
is as follows
146
(7.1)
:
is
(defunfactorial(n) (if (= n O) 1 (* n (factorial(1- n))))) When the functionis called to calculatethe factorialof four, the building processwill occurfirst andthenthecollapsingprocessfollows9 . (factorial4) -> (* 4 (factorial3)) -> (* 4 (* 3 (factorial2))) -> (* 4 (* 3 (* 2 (factorial1)))) -> (* ,4 (*
3 (* 2 (* 1 (factorial
-> ('4('3('2(*
0)))))
11))))
-> ('4('3('21))) -> ('4('32)) -> (* 4 6) -> 24
The
internal
swelling
unquestionable given.
used
occurs
that this phenomenon
Moreover
continue
phenomenon
if input
infinitely.
number
Of course,
during
will become
is negative,
the execution
the more
process serious
the recursive will be aborted
of
if a larger
process when
building.
will
It is
number
is
theoretically
the provided
space
is
up.
2. Mathematical completely
limitation classified
the analytical
9 In some
cases,
solution
--- Strictly
as limitation
speaking, of symbolic
for the general
the collapsing
will last to the end of execution
a mathematical
process
5th
and algebraic polynomial
may be impossible
if it is not beyond
147
limitation manipulation.
is proven
not be
For example,
to be non-existent.
and the swelling
the capacity
should
phenomenon
of the hardware
system.
Thereforeit is alsoimpossiblefor symbolic andalgebraicmanipulationto solve this problem.However,in additionto theabovementionedexample,difficulties in solving mathematicalequationswhoseanalyticalsolutionsexistaresometimesencountered. The occurrenceof this phenomenonis quite systemdependent.In general,mostof such occurrencesare encounteredduring integrationand equationsolving (algebraic and differentialequation). 7.4
Symbolic
The
and
impact
publications
algebraic
of SAM
to such
So far, schools
content
of courses
University
include
of Michigan.
Richard
mathematics
Professor
system
Rand
element
MATHEMATICA, by Professor
Kikuchi
around
The introduction be encouraged.
University
of Michigan.
1. Since always
there
2. Most
of symbolic
not until
ran on
the
since
into the Macintosh
and algebraic
views
have
analyses.
The
of
system, laboratory
and
Professor
method.
into the education reported
At The
courses
other
and finite element
been
the
of Pennsylvania,
P. Papalambros
manipulation
that
engineering
II in the computational
design
The
introduced
to facilitate
1985.
such as Professor on optimal
and
mainframe,
of approximate
has used REDUCE
in the
the fall of 1984
and AT. At The University
Kikuchi
critic
of Pennsylvania,
which
is far
manipulation
into the sophomore
in the instruction
Others
So far, some They
system
mathematics
in the courses
field should
by students
at The
are :
in learning
course
in symbolic
the symbolic
and algebraic
and algebraic
manipulation,
manipulator
itself
students
rather
than its
to the subject.
of manuals
MACSYMA,
1988.
is no introductory
struggle
application
applied
was also implemented
R. Scott also used REDUCE
certainly
Noboru
mechanics.
was the first system
It was
many
community
and algebraic
system
are
and fluid
educational
University
of 1983.
There
theory,
from
MACSYMA
on IBM-XT
MACSYMA
and
The
MACSYMA
Professor
methods
relativity
the muMATH
the
is significant.
symbolic
University,
was implemented
of Michigan,
include
University,
introduced
Unlike
engineering
the response
in the winter
H. H. Bau employed
University finite
course,
course.
muMATH
Cornell
At Cornell
the graduate-level
Professor
officially
education
mechanics,
applications,
which
and
and
on celestial
successful
behind.
into
to science
of its application
Compared
mafiipulation
of symbolic
are unfriendly
new terminologies
and
algebraic
to the users.
in such a short time.
148
manipulators,
It is difficult
such
for new users
as REDUCE to understand
and the
,
There
is no
appropriate
manipulation
and its application
and algebraic carefully
pages
4. Qualified
.
The software
some
of outputs
cannot
use.
for public
to overcome
education
be revised
symbolic
manipulation to design
systems
Contributions
The study
swelling
'0') or several
problem.
the MACSYMA
some
algebraic
and numerical
they
It is not easy
are
in some
proposals
listed specific
should
that the existing symbolic software,
analysis
should
to use because manual.
offices
and is not yet
as follows
start
from
"Numerical
The
•
course
concept
and the complementarity
for such a new course.
The
the early
analysis"
manipulation".
be taught
at The
in the
are suggested
manipulation
and
system
of
between
in the course.
The existing
manuals
need
be
should
be rechecked
and made available
to the public.
study
in this report
and symbolic
is believed
manipulation.
to have
made
three original
contributions
of symbolic
and algebraic
They are •
mechanics"
this study,
manipulation solution.
when
the use of available
of this
presented
mechanics
1. Before
(say
is not
for easy accessibility.
3. The software
A). To applied
of symbolic
if assignment
as a symbol
For instance,
even
analysis
a textbook
algebraic
found.
It is recommended
"Numerical
manipulation,
2. It is urgent
and
and
the amounts
Therefore
at all due to the internal
these problems,
period. into
analysis,
as simple
is only implemented
of symbolic
symbolic
to applied
out just
be found
available
undergraduate
7.5
turn
symbolic
has just a half of its full capabilities.
system
revised
numerical
may not be fully operational.
functions
to teach
are not predictable.
or nothing
MATHEMATICA
The
l0 . Unlike
are not easily
of Michigan
In order
to facilitate
results
it could
instructors
University
,
manipulation
designed,
hundred
textbook
all of the advantages
were either In addition,
in handling
this
report
10 The one written by Gerhand Rayna application of REDUCE in applied
from
the applications
lengthy points
out
formulae, for the
or in increasing first
is rather an experimental mechanics.
149
the accuracy
of
advantage
in
time
a new
book
of REDUCE
than an
improvingtheefficiencyof theexecutionof a numericalprogram.It is believedthatthis advantagewill be crucial in suchapplicationsthat the developmenttime plays an importantrole. Forinstance,if the templateis preparedin symbolicform beforehandand implemented in a chip, the data receivedby a heat-seekingmissile can simply be substitutedinto thetemplate.Theresponsetimewill besubstantiallyshortened. 1
The
closed-form
element
solution
was not available
of it. The contributions First,
allows
global
matrix
known
proposes
Prospects
and
Some
o
strict,
The
trend
more
However,
change
implemented
swelling
simple
problem
has given
method
of
this
subroutine.
substitution
This
of nodal
and the assemblage
of the
of results
of symbolic
of SAM after
and mathematical the remedy
to avoid
these
limitation
for it. This
difficulties
and
report then
research
(in both industry
trends are as follows
convenient
gradually
by
and academia)
that more and more sophisticated
the mainframe
Applications
into a fortran
Secondly,
the problems.
of the quality
of future
smaller,
.
it to solve
expectations
are multifolds.
are more accurate.
is simplified
no one
pre-treatment
it is expected
design
field,
continuations
As the criterion
coded
solutions
•
of the internal
in the SAM
applies
and the solutions
programming
the first analytical
by this breakthrough
to be obtained
manipulation
a sysmatic
successfully
analysis
isoparametric
is expedited.
the difficulties
well
matrix
quadrilateral
presents
be automatically
the fortran
and algebraic
Although
and more
can
of a 4-node
This dissertation
is eliminated
stiffness
Of course,
B). To Symbolic
7.6
solution
stiffness
were
error
the element
coordinates.
before.
matrix
to the finite element
the integration
the closed-form
.
of a stiffness
and
for
SAM system in industry the teaching
systems
personal
will still co-exist
are scare
150
will
will be produced.
continuously
computers
or
to process
at this time.
of symbolic
in schools.
more
•
algebraic
packages
formulations
becomes
However,
and algebraic
even
go towards calculators.
large-expressions. this situation
manipulation
will
is actually
,
The
relationship
smoother
in the future.
vice versa) 4. The
between
is expected
gap between
numerical
The switch
analysis
from symbolic
to be automatic
theoretical
and
analysis
symbolic
manipulation
manipulation
to numerical
will
be
analysis
(or
eventually and computational
experiments
will be smaller
and
smaller.
,
The reconsideration Regardless
of whether
used to check become
of every
solvable
of higher
tasks
debugging
function
the sysmatic
research.
of symbolic
by SAM.
will become
or not. The solved The
necessary.
problem
unsolved
problem
problems
will become
can be
might
then
of SAM.
for applied
mechanics
popular
systems.
program
and to check and
of symbolic
checking
of results
the correctness
algebraic
manipulation.
manipulators from
of results
It is expected
will be developed
symbolic
are the important
and algebraic
soon.
that the
self
A technique
manipulation
for
should
be
be continued
in
in the future.
The following future
of SAM
the symbolic
associated
available
order terms
and formulation
been solved
of solutions
with the employment
due to the availability To debug
equation,
they have already
the correctness
6. The inclusion
,
problem,
They
1. Extension quadrilateral
three topics
are closely
relative
to this study,
and should
are :
of methodology element
in constructing
a stiffness
matrix
for 2-D
isoparametrical
to a 3-D problem. !
a
2. Inclusion
of the higher
This was neglected 3. Extension
derivative
in the original
of methodology
term of
formulation
presented
in section
151
o
--'7_ ---) 0t, r in Equation
(6.23)
by Lee and Kobayashi 7.3 to solve
the general
to Equation
(6.25).
and in this thesis. shell problem.
7.7
References
[1] P. H. Winston, [2] Robert
R Kessler,
Company,
[3] Editor transfer",
B. K. P. Horn, "LISP,
objects
"LISP",
2nd edition,
and symbolic
Addison
Wesley,
programming",
Scott,
1984. Foresman
and
1988.
H. H. Bau, ASME
[4] P. H. Winston,
T. Herbert, HTD-Vol.
"Artificial
"Symbolic
computation
105, AMD-Vol.
intelligence",
152
in fluid
mechanics
97, 1988.
2nd edition,
Addison
Wesley,
1984.
and heat
APPENDIX
Proof
of
A
Equation
(5.12)
V
Given:
Prove:
ff(17,)--m*g
ff(17
Proof
).
*_t-
17;-ff(17_).
17_ >0
: Let m*g =1 for simplicity,
/_
mF(17r)*V'_--ff(17;)
vr
°
_1¢;
v;t--
v;t ) (A. 1)
There
are three cases
Case
for discussions
•
1 • when V _ > 0 ,equation
(A. 1) will be
Z0'0,
Case 2" when
V _ - 0, equality
is hold.
Case 3 • when
V _ < 0, equation
(A. 1) will be
F - V'(-
Therefore,
when when
ff(fr
).
1-]--_),V,
17;-ff(17_).
{>0, 0,
when when
17_ a O is proven
153
V, V,
V, V,
_0 >0
aO oIf V,
There
of
B
0,V,
- V,)=fight
side of (B.1)
< 0 case.
Leftsideof (B.1)=IV;l_ Iv,I- v; +v, •¢ - ]_(
3. V_
