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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



+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



-_ 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

[1] F. I. Niordson,

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[2] I. S. Sokolnikoff, [3] M. Dikmen,

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finite

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software

ol mechanical 1989.

and

in solving applied

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_