BTERATWE‘SOUJTIONS 0F PLANE EMSYOSTATK PROBLEMS

Thesis fax the Degree of Ph. D.
MIG-1.3%!“ STATE UNWERSITY
Chesfier L“ Davis
1965

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THESIS

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This is to certify that the

thesis entitled
ITERATIVE SOLUTIONS OF

PLANE ELASTOSTATIC PROBLEMS

presented by

Chester L. Davis

has been accepted towards fulfillment
of the requirements for

__P_h_.I_)_,__ degree in Miss

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Ma'or rofessor
g E. Malvern

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Date November 18, 1965

       

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LIBRARY

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University

 

 

 

 

 

 

ABSTRACT

ITERATIVE SOLUTIONS OF PLANE ELASTOSTATIC PROBLEMS

by Chester L. Davis

Three objectives of this thesis are: to compare the efficiency of
three iterative methods of solving biharmonic-finite—difference equations,
to report on ISOPEP, a system of computer programs designed for the
numerical solution of biharmonic plane elastostatic problems, and to
demonstrate the utility of this system of programs and the advantages
and limitations of numerical solutions by examples.

Three matrix iterative methods considered are point successive
overrelaxation, the alternating direction implicit method, and the
cyclic Chebyshev semi-iterative method. These are compared in terms
of computer storage and time required for the solution of a model
problem. Numerical results indicate successive overrelaxation is best
unless the mesh is refined so the number of points exceeds 350. Then
the alternating direction implicit method is superior.

ISOPEP, a system of FORTRAN II subprograms for the iterative sol-
ution of plane elastostatic problems, is explained. Documentation of
ISOPEP, including listings of source decks, specifications for input
and the output from a sample problem, is provided.

Discrete values of the stress functions and stress components
for six ISOPEP problem solutions are provided. These problems include
three notched tensile specimens, an infinite plate with a square hole

and a semi-infinite plate with a uniformly distributed load along a

Chester L. Davis
portion of the edge. The numerical solutions of the six example
problems indicate that a high speed digital computer with a large
main memory provides an effective and economical means for the analysis
of plane elastostatic problems. Good agreement as shown in the com-
parison of the numerical and eXplicit stress solutions for some of these
problems. Use of numerical solutions for the investigation of stress

concentrations is shown in several of the examples.

ITERATIVE SOLUTIONS OF PLANE ELASTOSTATIC PROBLEMS

By

Chester L. Davis

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and Material Science

1965

ACKNOWLEDGMENTS

I wish to express my appreciation to Professor Lawrence E.

Malvern for his suggestion of the problem.and his significant con-
tributions to the study. Also, I wish to thank Professors William
A . Bradley, George E. Mass, Charles S. Duris and Robert H. Wesserman
for their service on my graduate committee.

Financial support was provided by the National Science Found-
ation, the Consumers Power Company, the Ford Foundation and a Michigan
State University Graduats_Research Assistantship. The facilities of
the Michigan State University Computer Laboratory and the University of
Toledo Computation Center were used. The substantial support of these
contributors is acknowledged with gratitude. I wish to thank Mr. John
W} Hoffman, Director of the Engineering Research Division, for his
assistance and support.

To my wife I wish to express my appreciation for her aid in typing

and counsel throughout this endeavor.

ii

TABLE OF CONTENTS

Page

I. INTRODUCTION . . . . . . . . . . . . . ...... . . . 1
II. THE DIFFERENTIAL EQUATIONS . . . . . .’. . . . . . . . .h A
III. DERIVATION OF THE DIFFERENCE EQUATIONS . . . . . . . . . 9
IV. ITERATIVE METHODS . . . . . . . . . . . . . . . . . . . 20

A. POINT ITERATIVE METHODS

B. METHODS OF BLOCK ITERATION
v. COMPARISON OF THREE ITERATIVE METHODS . . . . . . .h. . 52
VI. SOLUTIONS OF PLANE ELASTOSTATIC PROBLEMS . . . . . . . . 59

VII. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . 93
BIBLIOGRAPIIIY O O O O O O O O O O O O O O O O O 0 0 O O O 98

AWDICES O 0 O O O O O O O O O O O O O 0 O O O O O 0 O 101

iii

TABLE

1.

3.
h.

5.

7.

8.

9.
10.
11.

12.

13.

1A.

LIST OF TABLES

Comparison of iterative subroutine characteristics .

Comparison of machine time and number of iterations
required for convergence . . . . . . . . . . . . . .

Summary of problems solved with ISOPEP program . . .

Comparison of the ISOPEP program for the IBM 1620
and CDC 3600 computers . . . . . . . . . . . . . . .

Stress function at points adjacent to the semiu
Circula'r mesh 0 O O O O O O O O O O 0 O O O O O 0 O O

sttress component at points adjacent to the semi:
Circular nOtCh O 0 O O O O O O O O O O O O 0 O O O 0

Y—stress component at points adjacent to the semi-
CirCUlar nOtCh e e e e e e e e e e e e e e e e e e e

Xlwshear stress at points adjacent to the semi-
circular nOtCh e e e e e e e e e e e e e e e e e e e

Stress function at points adjacent to the V-notch . .
X-stress component at points adjacent to the V-notch
Y-stress component at points adjacent to the V-notch

XY-shear stress component at points adjacent to
th. V’DOtCh e e e e e e e e e e e e e e e e e e e e o

Stress concentrations factors for the V-notch
tCHBil. BPCCimQD e e e e e e e e e e e e e e e e e e

Comparison of boundary values of C75 obtained frmm the
iterative solution with those from Savin's solution . . .

iv

Page
52

5h
56

58

7k

7k

75

75
76
77
77

78

80

91

LIST OF FIGURES

FIGURE Page
2.1 Boundary forces . . . . . . . . . . . . . . . . . . . . . 6
3.1 Rectangular mesh . . . . . . . . . . . . . . . . . . . . . 9
3.2 A regular point . . . . . . . . . . . . . . . . . . . . . 10
3.3 (a) Unequal mesh spacings (b) Subregion l . . . . . . . . 13
3.11. Dualmeshes....................... 17
3.5 Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Twelve cells for P5 . . . . . . . . . . . . . . . . . . . 19
h.1 The model problem . . . . . . . . . . . . . . . . . . . . 20
h.2 Mesh for the model problem.. . . . . . . . . . . . . . . . 20
h.3 Successive overrelaxation - Namber of iterations vs.63 .-. 30

h.h alternating direction-implicit method - Number of
1t.r‘t1°n. '391' e e s e e e e e e e s s s e e e s s e e 3“

‘05 Two-m. block. 0 O O O O O O O O O O O O O O O O O O O O M

h.6 Cyclic Chebyshev semi-iterative method-Number of
itCr‘tian. 7's 9 e e e e e s e e e e e e e e s s e e e s ‘8

h.7 Comparison of iterative methods
(a) Iterations vs. number of interior points . . . . . . . 50
(b) lichins time vs. number of interior points . . . . . . 51
6.1 Boundaries of the model problem.. . . . . . . . . . . . . 60

6.2 Distributions of the stress function and 6} for
hobl“ 1 O O O O O O O O O O O O O O O O O O O O O O O O 62

6.3 Problem.2 - Semi-infinite plate . . . . . . . . . . . . . 63

6.4 Comparison of numerical and exact solutions for
hablm z 0 O O O O O O O O O O O O O O O O O O O O O O O 6A

LIST OF FIGURES

FIGURE Page

6.5 Distributions of the stress function and a; for
”Oblom 2 O O C O O O O O O O O O O O O O O C O O O O O O 66

6.6 Solution domain for Problems 3, l. and 5 . . . . . . . . . 67

6.7 Distributions of the stress function and d} for
PTObl'fllBeeseeeeeeeeeeseeeeeeesee 68

6.8 Distributions of the stress function and a} for
Pr°blm 1‘ O O O O O O 0 9 O O O O O O O O O O O O O O O O 69

6.9 Distributions of the stress function and a": for
HObICMSeseeeeeeseeeeeeesseeeees 70

6.10 Stress concentration in a semi-circular notch . . . . . . 72
6.11 Stress concentration in a Vinotch . . . . . . . . . . . . 73
6.12 Stress concentration in a rectangular notch . . . . . . . 79
6.13 Stress concentration factor It for a notched flat bar . . 81
6.1h Boundary points for the semi-circular notch . . . . . . . 83
6.15 Treatment of boundary points . . . . . . . . . . . . . . 83
6.16 Interior points near a boundary . . . . . . . . . . . . . 8h
6.1? Problem 6 - Infinite plate with a square hole . . . . . . 85
6.18 Approximation of the square hole . . . . . . . . . . . . 87
6.19 Domain of solution for Problem.6 . . . . . . . . . . . . 88

6.20 Stress function and 0; distributions for Problem 6 . . 90

vi

LIST OF APPENDICES

APPENDIX Page

A. CONVERGENCE OF ITERATIVE METHODS . o . . . . o . . . 101

B. ISOPEP _ A FORTRAN PROGRAM FOR THE ITERATIVE
SOLUTION OF PLANE ELASTOSTATIC PROBLEMS . o . . . . . 115

vii

I. INTRODUCTION

Determination of the stresses in a plate under conditions of
either plane strain or plane stress is a fundamental problem for the
structural engineer. The literature of the classical theory of
elasticity includes exact solutions of numerous problems. For an
account of the mathematical theory see Timoshenko and Goodier (1951)
and Mbskhelishvili (1953). However, exact solutions are available
only for those problems which have rather simple geometric shapes
and boundary constraints. Finite-difference equations have been used
as an alternative for the analysis of more complex practical problems.
This dissertation compares several iterative finiteodifference
solution methods with respect to their efficiency in treating plane
elastostatic problems.

The finiteudifference solution of a boundary-value problem
is a two-step procedure. First the partial differential equation
and the associated boundary conditions are replaced by difference
equations which relate the discrete values of an approximating function
at a finite number of points. 1 regular mesh of lines is superimposed
on the domain of the given boundary-value problem. A finite system
of linear equations is formed by writing a difference equation for
each node of the mesh. The second step is the solution of this
system.of simultaneous equations. It is often necessary to solve
for values at a thousand or more points.

The relaxation technique was employed by Southwell (1946) for
the solution of the biharmonic difference equations. Though Southwell

and his colleagues solved a number of complex engineering problems,

1

2
the relaxation method has not. been readily adapted for digital computer
solutions.

Iterative methods have been used extensively for computer solutions
of systems of equations. These methods make repeated use of simple
algorithms which at each application provide an imoved approximate
solution at one or more of the mesh points. The exact solution is the
limit of the sequence of the adjusted point values. Though there are
many iterative methods, the three which seemed to offer most promise
for the solution of the biharmonic finite-difference equations were:
the technique of point successive overrelaxation introduced by Frankel
(1950) and Young (1951.), the alternating-direction implicit method of
Cents and Dames (1958), and the cyclic Chebyshev semi-iterative method
proposed by (h-iffin and Yarga (1963). The comparison of these iterative
schemes is the first objective of this dissertation.

Using the computer time required to solve a given problem as the
measure of the efficiency of an iterative method, the numerical results
obtained indicate that for the biharmonic equation there is a critical
mesh spacing 11* such that the point successive overrelaxation iterative
method 1. the best of the three methods tested for h 2. 11* mm. for h< h“
the alternating-direction implicit method is best.

lach of these methods uses a parameter for accelerating the con-
vergence. Formulas for the bounds of optimum parameters are included for
rectangular regions. For the more general case of irregular boundaries
there are no convenient relationships for estimating the optimum para-
meters at the start of the iterative solution. The procedure for deter-
mining the accelerating parameter is different for each of the three meth-
ods. There existed the possibility that the choice of the most efficient
method might be more dependent on the method used for determining the

3
acceleration parameter than on the performance of the iterative method.
To investigate this possibility the optimum parameter for the solution
of a model problem.was determined on an empirical basis for each method.
The results were consistent with the other comparisons of the three
methods. However, this study did reveal that the machine time required
could be reduced from 18% to 42% by starting the solution with an
optimum parameter.

Another objective is the preparation of a set of computer programs
for solution of plane elasticity problems. Six FORTRAN-II routines have
been written for the Control Data Corporation 3600,and slightly modified
versions have been run on an IBM 1620. The solution of a particular prob—
1em.requires the preparation of a pair of routines for the given boundary
conditions. The main program which provides linkage of these subroutines
is called ISOPEP. A full description is provided in Appendix B.

The third objective of this dissertation is the demonstration of
the utility of the ISOPEP program. Solutions in terms of stress funew
tions and stress components are provided for six problems: (1) A square
plate with uniformly distributed loads on portions of two edges, (2) A
semi-infinite plate with a uniformly distributed load applied on a saga
ment of one edge, (3) A flat-plate tensile specimen with two semim
circular notches, (A) A flat-plate tensile specimen with two V-notches.
(5) A flat-plate tensile specimen with two rectangular notches, (6) An
infinite plate with a square hole. Several of the problems were select-
ed as examples of the numerical calculation of stress concentrations.

Two of the problems are included for comparison of the exact and numerw
ical solutions. The numerical solutions obtained with sufficiently small
mesh intervals provide stress components which are in good agreement

with values from.the exact solutions.

II. THE DMD]. EQUATIONS

Plane stress is the state of stress approximated in a thin plate
which has loads applied only on the boundary and parallel to the plane
of the plate. In a three-dimensional Cartesian coordinate system a
state of planes stress exists if the stress components 03,7;2 ) 7}:
are sero at every point.

Consideration of static force equilibrium under conditions of plane
stress leads to the equilibrim equations.

3o? }
‘53—+—7-’;1+X-

(1)
L? + 11v +Y: .
where I and I m the components of body force per unit volume.
For plane stress the Hooke's law relationship between stress and
strain is,
E G): l’ d} '- VO' +- BK
5’ f (2)

56,8 0} *1/07 4- Eofi‘
E 33.9 I 2 ( H1073“; .
where 3 is foung's modulus, Vie Poisson's ration, §(x,y) is the differ-
ence bet-sen the local current temperature T, and the original tempera-
ture To, and (is the coefficient of thermal expansion.
it a point in the plate the strains are defined:

EX=§§r, 67:15,};y7-‘fi1-fi3f‘ (3)
where u and v, the components of the displacement, are continuous
functions. Since the three strain components are expressed in terms of
the two functions u and -'9 they cannot be taken arbitrarily. By

differentiating and combining Equations (3) it is possible to obtain the
compatibility equation,

z
.55.... 751+ my = a"! (h)

At each point in the plate Eq.'s (l) and ()4) must be satisfied.
0. B. 1117 derived a single differential equation in terms of a stress
function, ¢(x,y), which will satisfy both the compatibility conditions
and equilibrium equations. The stresses are determined by the follow-
ing: a u.
=‘g—3d-V, W=§X£+V9 Wiv=-§—xg4‘; (5)
where Y(x,y) is the potential of the bow forces.

The Hoohe's law relationships (2) are substituted into the compati-
bility equation (h):

231%;- Vo-y + Each 3—1[o--;:-1/c; +E«€]=2(1+;/)§;y (6)
It is advutageous to introduce the equilibrium equations by forming the
st- of the derivative of the first with respect to x and the derivative

of the second with respect to y.
2—? ”Ere +43%-
£95.23 +£75.51.” 251+ =0

°’ va=_[a ax +95%%+%¥1T

In.“ (S) and (6) can be combined so the term containingT :yis elimi-

(7)

mated.
51??” + E «£1 + 5715*; was.) - «leg—3 gag—+3
Th1. IWMiesto

(«$14. +§?)[dx+d’+fqrd+(1+1/)[ [3%(—+§ QY =0,

In terms of the two-dimensional Laplacian operatorv2 this can be written

V‘[r+o- o—+o(E€]+(I+‘/)[%§- +93%, =

Substituting the expressions given in Equations 5 for the stresses and

6

assuming body forces can beucxpressed in terms of a potential function
“1.1) the equation can be written

V’4>+Eacv1(‘+(z—V) V2V= o (a)
If the body force potential 1.: harmonic, the equation becomes
W + Ear V‘é‘ . o (9)
and if‘i'iti‘additifdn"iteniperature changes are negligible, it becomes
V‘tb = o (10)

where in Cartesian coordinates, 9 +47
’46 a
Vl’d’ = $79 +2999y=~+ "

a y ‘I-

The solution . of a plane stress problem, when the body forces can be
expressed in terms of apetential, thus consists of finding the stress
function fl which satisfies the appropriate one of lqfls (8), (9) or (10)
and the prescribed boundary conditions. _

As noted by Bakelnikeff (1956) , even in a thin plate the stresses
vary somewhat through the thickness . The two dimensional problem
formulated here, strictly speaking, applies to the averages through the
thickness and this is. often called a state of generalised plane stress.

For a state of plane strain instead of plane stress the governing
equatiens have a similar formulation . The appropriate equations are
obtained by replacing a! by V/(l-ll), s by l/(l-Vz) and 0( by (I (1+1!) 1.
has (2). (5). (7). (1) and (9).

Hi and? are the components

 

of external loads per unit area

acting on the boundary and (I and 5
are the direction angles which the
normal nabs withthe x and y-axes
respectively, then neglecting bow

forces

 

 

‘7

X!
Figure 2.1 Boundu-y forces

X: 1032 + M Ky
“1‘72 mm} + )7 73.3 (11)
where
= cos 0:. 3 731 = coo-{3
In terms of the stress function the stresses on the boundary are given by
Xflfl ,5. ”ii—3%,}, Y=W~°§$ri§$£ (12)

Introducing coordinate axes s and n, tangent and normal respectively to
the boundary, the boundary conditions can be written
X= %:$1‘3§"* xlafifé z 3'J(§§)
7:.- egg? 37%;}: a -33“; 5%)
These can be integrated along the boundary
é—f =- - I You i (if).
3435- - W + at.
... sat- hes-e
. integrating along the boundary yields
4> Jasmin” who I Yests + (i?) )(7- in (fines) + 49. (15)
where (our -g;3, 51M: -
substituting into the normal derivative of ¢,
#= §$+va

g9; :-— Cosxf Y4; +51» or] X45 +( @f’kosmfigg) 33nd (i6)
Iqfls (15) and (16) determine ¢ ande ~9- at every point of the boundary in

(13)

(1h)

terms of the boundary stresses 1, I and the constants of integration

(%%)a,(%3 )Oand ¢°. These constants of integration m be chosen arbi-

trarily since they do not appear in the expressions for the stresses.
If the constants of integration can be chosen so ¢ is symmetric

with respect to a line of physical synetry for the plate, then

Eggs) 2 o , a, (v‘dmgw : o (17)

for points (x,y) on the line of symmetry, where n is the normal to the
line of symmetry.

The problems herein considered will have boundary conditions as
given by Equation (17) on a line of symmetry of the body and of the

form

u” ’ = 5am), 1%” = flow) (18“)

which follows from Equation (114) or
WW) 313 (Km) 9 1 #3212: F (m) _ (1810)

which follows from lqfls (15) and (16) on an outer bomdary. The func»
tions f1, f2, f3, and fh are valid only on the boundary and could be
expressed in terms of the single parameters 3 n is the outward normal.

For problems with displacements specified on the boundaries the
Iavier Equations should be used rather than the biharmonic equation.
This pair of coupled second order partial differential equations for
plane stress conditions, obtained by replacing the stresses in Equation
(1) with expressions in terms of strains and then substituting deriva—
tives of the displacements for the strains as given in lquation (3). can

be written:

171E37I7[Z%+(’ V)‘a_1 ay +"*");§‘5‘9’]+X=0
W[<a-V)§:~g +21? +(;+V)9_1]+W

(l9)

III. DERIVATION OF THE DIFFERENCE MMTIONS

Though analytical solutions have been obtained for certain special
cases of the biharmonic boundary value problem, the use of approximate
numerical methods is often necessary. Finite—difference methods are
readily adapted for solving the problems with high speed computers and
attention will be directed to these methods.

The governing partial differential equation is replaced by a
finite-difference approximation. 1 rectangular mesh is superimposed on
the region and the intersections M
of the horisontal and vertical
lines inside the region are /F
called nodes or mesh points. / 12 fl
Boundary points occur at the
intersection of the mesh lines \

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

J

:x

 

with the boundary. It is
convenient to use a uniform mesh , Figure 3.1 Rectangular mesh

 

 

spacing, say h. it each interior point the function ¢(x,y) is replaced
by an approximating function U(P), where P is an interior mesh point
(prp) . The function U(P) is defined only at the mesh points. Discreti-
nation of the problem is accomplished by replacing the partial differen-
tial equation in term. of ¢(x,y) by a finite system of equations in terms
of U(P). The equation for_U(P) is given in terms. of the values of U at
neighboring points. Thus the problem is reduced to solving a set of
simultaneous finite-difference equations .

Three different derivations of the pertinent differencexequatione
will be considered. These are based on Tyler's series, integration

and a variational formulation. Each has distinct advantages and contri-
9

1.0

butes to a better understanding of the problem.

Tiler Series

 

Let f(x,y) be a function of two variables, which is continuous in
the neighborhood of the point (a,b) and has continuous partial derive-a
tives up to order n in the neighborhood of (a,b). The Taylor's expansion

of f(x,y) about the point (a,b) is given by:

ht
fix,y)=f(a,s)+(fi (aw+fyeb))h+(&.(a,w + 21361 b)+fyygb))27 + ' ‘ ‘ ‘ *7?»

 

 

 

 

 

 

'here K» = :7” ,X+,,)f(a+a,hb+e/1) 05902“
X=<3+h, J— 57‘" 4'7

and subscripts denote partial

derivatives . In the region of the 8 2 6

17 Plans where a solution of the c; 3 o (2):? I 3‘

biharnonic equation is sought, a

nesh point (raga) will be called ’0 4 ’2

a regular point if the neighboring +11

points shown in Fig. 3.2 are all Figure 3.2 A regular point

interior or boundary points of the region and each point is at a distance
h fron the adjacent points shown.

A Taylor's expansion can be written for each of the points in the
neighborhood of (x 0970) and these expansions can be combined to find
difference quotients corresponding to an partial derivative with respect
to n or y.

Consider

U: = U.*MLB+(W‘.%1+(U£)03;+ (”1"")07I‘f;+ +(onF) JJT( + (UX6 )OIZhT
U3: uo—(W0h+(UX1)Q-h!-‘i (W90 é; +(UX )0: :(Ul‘llfi + (UX‘)ee-£ 6!

Ufl‘U’: 'ZUo+(Ux‘)a%f +(UX:)o-2'7h; "A.

 

11
*
where (Ux6)0,1 a: Uxabc +9h,y ) O i 6 f 1, and M, f max (UX‘IIJ alurg
the line between points 1 and 3
a
5-:(UxéL‘5. "Z?“

' (9917” (WEN) IT;
92% “e 6W"

U5: Uo+(U,)o(2A)+ (ux) igi} (maxi/1L (U
U1=Uoo)+‘(ux)(2(m )+(UX)021 (UP) 322.“ “(U 12’3“ «)5;er

U+U9=2U Hedi—1' iuxI )3:*”+ #Me

where M2 5|Ux‘(5,9' Subtracting (+(U1 + U3) from (115+ U9) we find
Uf+ U7-LI(UI I U3) +6Uo + [3“ TMa :(aJfl).

Thus,
(Ux’),=i‘I[U5+ U7 +6U. - 9U, — ‘IU,] - légiflfi (20)
Similarly it can be shown
(uyI)=—I[u +u, +6U LfU-qu]'136—TILM3 (21)
where 113$ maxlUytb 11. For 9 X1; y, consider
U =u + (ugh (UIJHHKUX)+2(ny)p+(Ug)9%1] + I—+2-.>’.§u
+I.. 377 Uo%{+Iex 91).er + In «tafwgogj:
u,.= U. - (U006 (Uy)L+[qu:)+2(UI>+ (:uy )]2, (+39: 3—,)U 3:7

”37‘ +3?)+Uo‘éf; “)5U051+(39X + Dy)‘ U0 Io Q!
U“? U’O‘ZIUO = [(Ux‘)‘, +2 (0+ij) (Uyl’ a)i]%.hk+ [(U I‘ll”. 4(UX331)¢ +‘(UX (1)9
. he
+‘I(nys)o+(U 90] if + gig—'7 M,

where Mhsmaxhlx ”MO, 40 n + m = 6. Similarly,
Us* Um ZUD‘: [(Ux1)o+2(ny)e+( Uy‘lfi fiHHUx") flux?) 0‘01'6(sz7)

“WWW (oUII’IiJZ-fivf 557” M5

where 145. < maxlanym'812, n + m = 6. Adding we find

Up U8 + U,O+ U,2- ‘IU :2[( me) )022h! + (UxI)2—-h:+( +(2)Uy 57+ “HUME
6
+(Uxay‘)ofi+ + 2'2"? —"'"M6

 

 

where M6 < mex|ux.y,.|, n + m = 6 along all lines through the mesh pomt.

12

Substituting from above
(Uny‘L' ”I'I'IIUvUe +Um+Uu+ IUa~2ILI+IlgIIJj+L§g ~_- léfi‘M‘ I (22)
Hence '1‘ I 9.
-, ‘7’?) = eafi‘tzfgefifiro
ia approxinated by
(0x90 +2 ”1%),“ U190 9" M UI+UI+U9+ Lin "' 21"”? Us" qe“ Um)
, . ~8(U, IUZW5 +U.,) +on,J+R,,—-—o (23)
Ihere R7, LS 'fijé-th‘ .
Kantorovioh and mm (1961:) give a, _<_ $9- 112)! where l in obtained by
replacing all the derivatives of higherarder in lode (20) 9 (21) and (22)
by their maximum absolute values on the meeh lines Joining point 0 and the
surrounding pointee , '

The remainder 95 providee ‘a bound for the di, ecretication error, that
ie, the difference between fixer) and the truncated eeriee approximation”
U(n,y) at the nodal pointee Thin indication of the order of the approxi-
nation in a major advantage of the Tyler" e eeriee nethod for deriving
the finite difference equation.

Inflation Techigue _

The use of integrals for the derivation of finite difference equa-
tions has not been need an extensively as the other methods. Verge (1962)
providee a general introduction to the method and indicate: it has been
need in nuclear reactor design computer. codes for several yeereo Use
of this method for the biharmonic equation is made by Griffin and Verge
(1963).

The aatiefaction of the partial differential equation (8) at every
point of a region R is equivalent to the satisfaction of the integral

equation
IVV1(V2¢+BV*C€)J‘4> :- 0 (21:)
'R

13

where B = (1mV), C -CKE, for every arbitrarily chosen subregion Ai'
Hence, for any arbitrarily chosen subregion Ai bounded by 31, Green's
theorem gives
2 )de 3953- ‘ I Cfl'fi-o
f/v(v2~<p +BV+C§ JV “MWIBW Id - <25)
Hi I

Discretization is accomplished by substituting a set of discrete
quantities U1, (1!- 1,2,3,--—m) for the values of the continuous function
¢(x,y) at the mesh points corresponding to i - 1,2,3,--m. A major

advantage of this method is the eimpli city of the treatment of irregular

 

 

 

 

 

 

 

 

mesh spacings. In Fig. 3.3 3 1a
the mesh Spacing to a point P1 (ha r.§%&_nm1
I
adjacent to a general interior 3 hoa hm , :3 :33 lg»
o . 0 i v
point P0 is hoi. The subregion E '
hot/- “‘39:?"F1H
about point P0 has sides sci, °
if.
1 = 1:213:40 4 (
(a) (B)
501 = 503 z %( ho2 + hot) Figure 3. 3

(a) Unequal mesh spacings

_ _ 1 (b) Subregion £0

Central difference quotients are used to approximate the line integral

in Equation (25) about the subregion A0 of Fig. 3. 3 (h)

I30:___. ‘59!
:(VZUI V Us )7;— :[BW - V0 ) + C(é‘,- $0 )] he! (26)
where V2 U1 is the value of V2 ¢ at point P1' Approximate values osz Ui

and V2 00 are obtained by using Green's theorem again.

ffsz, dx dy== ag’ds (27)
an
F
The discrete value of‘v'7-Uo is taken such that
2 =_L. aLbd _ J. . ’§“' 28
VU°H£BII‘130‘H%:(U U°)TT2' ( )

A0 = (801)(502_) = (he; + h09)(ho[+ h63)/1#

14
At point Pi as given in Fig. 3. 2
, .S‘. I;
V2U;=‘[(Q5-" U)? , +(U4‘-' U')-§M-+(U, U)??- 5 o +(U”! W6II2]H,
WhOrO H, = (155")(5'55)

For the general case

,Z—fV‘U V‘U >iii= if; iigu, Uy-M ML] ifm- Mg] (29)

If the mesh spacing is constant, Equation (29) reduces to the form
of Equation (23).

‘The evaluation of the right side of Equation (23) requires 12 addi-
tions and 3 multiplications. (Ihen the mesh spacing is uniform.the
multiplication by h"2 is not-necessary.) The evaluation of the right side
of Equation (29) requires 29 additions and 25 multiplications if the
ratios giéa. are computed once and stored. For the IBM 1620 the time
required for 10 floating point additions is equivalent to that required
(for one multiplication. Thus on the 1620 Equation (23) could be evaluated
for approximately 7 mesh'points in the same time required to evaluate
the right W use of Equation '(29) for a single mesh point. Any
iterative method or solution of the finite-difference equations requires
the evaluation of one of these expressions at every interior node at
least once during each iteration. Several hundred iterations may be
required. The rate of convergence of an iterative scheme which uses
Equation (29) would have to be 7 times the rate of convergence of a
scheme which uses Equation (23) for the same number of mesh points before
Equation (29) would be preferred. In some problems it is possible to
reduce the number of mesh points significantly through the use of
arbitrary mesh spacing. Regardless of whether it is possible to

establish the superiority of Equation (29) over Equation (23) for use at

15

every point, Equation (29) provides an excellent method of handling
irregular boundaries and the changing of mesh size from one subregion
to another within R.

Variational Formulation

Application of the variational method for deriving finite-differ-
ence equations can be found in Courant and Hilbert (1953) and Forsythe and
‘lasow (1960). Ehgeli, Ginsburg, Butishauser and Steifel (1959) show its
use in deriving biharmonic difference equations.

The variational formulation of difference equations for plane
elastostatic prdblems is especially convenient for prdblens given in
terms of two displacement functions. Griffin (1965) has shown the
advantages of this approach. The basis for this derivation is the
Principle of Stationary Potential Energy which states: Fran the set
of continuously differentiable displacement distributions which satisfy
the given‘boundsry conditions of an elastic body, the displacement
distribution which actually occurs is the one which makes the potential
energy stationary. n

For a plate subjected to plane stress, take 1,! as the body forces
per unit volume and if the surface forces per unit area. Then the
strain energy V, per unit volume is

V: fiféf+ £37 + 21’Exe'y + 11:12! ‘6};
The strain energy per unit volume can be expressed in terms of displace~
ments (u,v) if these are continuously differentiable functions of (x,y)
by substituting
5x2??? €y=§a§ra Ey=(%1+35%
v= m%a[ei->*+er+~2~—%i + "flee—Ir] ‘3”
‘1 change or variation of total strain energy will occur'with any arbi-

 

"16

trary variatiu of the displacements 8n, 8v,
“(7 = Sffdexdydz = f/ SVdXd)’
where for convenient?” the s-dineneio: is taken as one and it is assmd
that v does not vary with s.
The virtual work 6"“;" done by the external forces under the virtual
displaoauts Sn, {v is given by
5 We“ affix“ +Y5v)Jny+ {(5582: +YJrHJ

the potential energy 1. defined ‘
Gaffvdxdy - [fix-u +Yr) )clxdy- -f(Xu+YI})JS (31)

Applying theR principle of stationary potential energy and considering bow
forees ad surface forces constant SQ = 0

A m =1)” svelxdy - [{sz +Y§r)Jny ‘JKXéatY-Eflda’ (m
This states that the-Zion‘s in strain energywill be «mil to the mark
dementhebetbyentenalforoee for an arbitraryvirtnal dieplnoe-
meets (n.8v,.fh1swillbetrneonlyif w arethe actualelastie
displse-snte predued by the external forces. .
the usual procedure in the calculus of variations is to consider
the eonditions imposed on the integrand of Donation (3!) and thee
derive a lineu- partial differential equation of the form
W} = 0 __ (33)
there “(a,v). Ilse additional bondary conditions arise which are
called natural boundary conditions. Instead of finding the differug
and mum of the fern of lquation (33) and then act-emu. emu
differences the variational form of location (32) will be solved
snorioany. There are two major advantages of this approach. first
the nestles v which satisfies Iquation when: automatically «um
thenatu'alboudaryoeaditione sethat fin-thereonsueratieneffie

17

natural boundary conditions is not necessary. Second, the discretization
of the problem leads to a linear system of‘equations that has a coeffi-
cient matrix which is symetric and positive definite. These relation»
ships are important because they are part of the criteria for the
convergence of iterative methods for solving linear systems of equations.
(See Appendix A.)

For a plate in the region R of the xy plane and unit thickness the
total potential energy is

Q - 2,, ,4, —5—~[/M33‘ ”fa-351+“ V){(§%)+(%19, }+ " 11W; #5?an

 

 

 

 

 

 

 

 

 

 

- [/(querMny - 2’3 (23, affine (3“)

The stationary value of potential energy is a Smininum for stable
equilibrium under specified boundary conditions.

a rectangular mesh and its dual
are imposed on the region B. The y
dual mesh lines (indicated nith ‘ i If
dash lines) are' parallel to and b—[w — T __ t .
spacedhalfwaybetween the lines —.%--.._§-- .:L ,__4:__
of the primary mesh. .L J. : :

_-l__._1_..‘.... '_...J_._+.....

The variational. problem : i : :
Equation (32) is given in tons L—E—.._-.E}.-i .é. “1r“
of a continuous displacement ' l ‘ 1 4-”
distribution u(x,y) and v(x,y). Figure 3.14 Dual meshes

This is replaced by a problem in which the displacements have discrete

values “1 and vi at the mesh points. The potential energy of each mesh
point is approximated ,and the sum taken over all mesh points represents
the total potential energy 0(u1, v1) . The problem is reduced to finding

the unknown displacements “i and 7i at each node which will make Q(ui,vi)

18

stationary. This requires

‘1’: =0, 3’-”233"-“~—”

“‘ . (35)
g 80’ Ja‘,2,3)..~——77)

'3

where n is the number of nodes at which “i is unknown and m the number
where vi is when. Equations (35) represents a system of n+m linear
difference equations. ,

hgeli, Stiefel et al (1959) show that a variety of quadratic
functions may be used to approximate the potential energy. The
polygons formed by mesh lines and dual lines will be called cells.
A primary mesh point will occur at one vertex of a cell. A regular
interior point will be the? canon point of four adjacent cells as
shown in Fig. 3.5. The potential energy integral (31.) is approximated

for each cell under the assumption that

 

 

the functions a and v and their 20
h.
derivatives are uniform over the f ----- . -2- "I
l
3. “all 5 1 hol L
cell. The functions take the ‘ i i
l
w l
discrete values at thenode r0. L__-______j
. . . h...
The derivatives are approximated a;
J»
for cellB by Figure 3.5 Cell 6
. 9X 30’ 5 y - hoz ‘

Th0 approximation for the potential energy is
Q-OZI gflEW[V(‘kJ’*%)+<I-w{( lfi); K-Z/sfi+__ “‘4 .4164?
‘fiTflb-qr-“h” -(Xu,+ YW'M“ :(Xusdehath (36)

The last term is included only if P0 is a boundary point. At a boundary

point specifed derivatives of u and v are introduced in Eq._ (36).
The potential energy at the node P0 is the sum of the potential

energy in the four cells adjacent to Po

19

go - °o21 + Qo23 + 0.31. + Qohl
and the total potential energy is the sum of the potential energy in all

 

 

cells
Q s' ZQigk
The difference equations are obtained from Equation (35).
Consider all the contributions 5 2 5
to the potential energy q which Q» \\ \V
xx- --\ - —
3 M\\ K V!
involve the displacements at a \\ x
\ \
single mesh point no, v0. There Sh § _:
are twolve cells as shown in Fig. J \1
1-...-- .9

 

 

 

3.6 which would use “os'o at a
regular interior mesh point. Figure 3.6 Twelve Cells for P0
%%=nr%=r[“"”btfl”(-“fi‘+1f§?)i(b..+h.)(-‘%,%“45%) m,
we‘i’mflé-u-ufli-ACH,‘ 0
'hm 1. - BLBquz‘ffioz/ig +5.35... +I‘0'f1’01]
9%? 7%["'“’h’%m(%+%l+wwl%+rf§ o.)
_ f' “#WH “7"“. ‘".)]+Kfl.=0
set. the relationship to the Navier Equation (19). when the solution. of
the finite-difference equations (37) and (38) is obtained, stresses can
be approximated using an area weighted average of the stresses over all
cells which have one node as a common vertex. The stress components in
terms of displacements are
“)7 = 77-57%}; ”if
GP 175173 [-3-3; +1! as] (39)

-_E_. 9 2r
Ki‘zmvfl’alf -r 2x}

None of the problem solutions included in this dissertation use

Equations (37) and (38).

IV. ITERATIVE METHODS

_- 1 simple example _ 1V

 

 

 

 

 

 

 

 

 

 

 

 

 

considered by Timoshenko i - Q s:

aInd Goodier (1951) will 117—6: LIP P
illustrate several _ 1
details of the solution. 0 “5:”
Given a unit square E
plate which is sub- ._L ‘1‘!" ‘ . E1}
Jected to boundary . .5} E; f

loads as shown in Figure 11.1 The model problem

Fig. 11.1, find a numerical solution which approximates the stress func-
tion “1.7) on the region of the x,y plane occupied by the plate.

5“
l
.1
.9!
I
‘F‘
1

Th. boundary r--- 41:1 - —-'le— .151... ..
l 1
‘ ' i I 1

Min 'fnL iii—#15:;—

. I
conditions are of :
I
I
1&0- __,_fl.* U11 U12. Uni UH U15 flL _ '53-,

6"
o

 

-..- -.... ....J

the form given in
Equation (12), but
are readily trans-
forud into the
form of Equation
(18) which leads

 

 

_ 31 _._ £30 U21 U11. U23 1.23.]- U25 {16""1627

 

 

 

 

 

5:29-- {to U31 U32. UL L131- Us!: A

 

 

!

. ' . 1

1;“ 159/ 1%. i345. 61.44. _151;5_ .11‘. ‘ E
I 3 I

 

to the problem "’i — _ .
statanant .. ...... 1&1_ €52. _'8'53_.1553._1§5.£ _._. .. ..
VWWQ): 0 (1:0) Figure 11.2 lesh for the model
problem

for o<x.<a.,. ~15a<41<J§Q
subject to boundary conditions

¢(s)=f<s) f 9
I, I! orxyon } (1‘1)

$251.1): n(n,y) the boundary
n
20

21
A simple way to introduce the boundary derivative is to solve for a point
one mesh space outside of the boundary in terms of the derivative and the

point inside the boundary on the same mesh line, eg,
31=U11'+Vo1(2h)’

where Uij is the approximation to the stress function fl. If the constants

of integration are selected so Equations 17 are satisfied then, by symmetry,

= U i=l,2,3,—--5

341 21 and 351 = ”11

The partial differential equation (40) is replaced by Equation (23) at

each interior node.
”11:1/20[8(U12+U21+f1o+fo1)'2(U22+f02+foo+f20)'(U31+U13+81+810)1’

etc.

The system of equations can be written in matrix form

A E = E (42)
where W , 1
.4(f01 + £10)“ .l(f00+f02+f20) - .05(g1 + 310) U11
.4 f02-.1(f03+f01)-.05 g2 U12
.4 f03—.1 (f02+f04)-.05 33 U13
.4 fog-.1 (f03+f05)-.05 g4 Ulu
.4 (f05+f16) ~ “1(f04+foe+f26)'°05 (gS + 317) U15
.4 £20111 @0140; 1651;23505' ' ' - 31;;
-.05 £02 U22
-.05 £03 y_= U23
E_= -.05 f0“ U24
.4 f26-.1(f15+f35) -.05 (f05+ g27) U25
1'4 (f30+341)‘°1(f20+f40+842) "05 (330 + $51) U31
-4 842 U32
.4 gg3 U33
-4 8H4 - 03k
'4 (845 +f3e)"1(f26+f46+844) ‘°05 (355 + 337) . _Uas .

 

 

 

 

and

(1.

F1

22
1 a.h .05 o o =.b .1 o o 0 .CS 0 o o o.I

01 ”oh cl 0 0

I
I
I
I
0 01 “oh 9]. O I
I
O 0 01 .013 91 I

I

I

O O 0 01 ”all

0 oOS‘oh 1 “oh
0

~.h 1 «oh .05 o
.05 -.h 1 «.1. .05
o .05 -.h 1 -.h
JD 0 _ o o .05! o o o .1 -.h o o .05 -.h 1_J
The original boundary value problem, Sq.“ (110) and (141) 9 has been

I
I
I
I
I
I
I
I
I
I
I
i I 0 01 ”oh 01 o : 005 “ch 1 ”oh 005
I
I
I
I
I
I
I
I
I
I
I

O
O 0 O 005 0 0 0 c1 ”all e].

 

 

replaced by the problem of solving a system of. linear algebraic equations
as given in matrix form by Equation (112). Direct methods have been

used for solving systems of linear equations. Recentureviews of these
methods are found.in Fox (1963) and Forsythe andfflasow (1960).

Faddeeva (1959) provides a detailed account of a variety of methods.
birect methods for problems which require a large number of mesh points,
several hundred or even several thousand for example, are rarely used
for two reasons. First,if Gaussian elimination is used it is necessary
to store all the elements of matrix i. For N mesh points this requires
the storage of H2 elements. Iterative schemes require the storage of

only a small integer multiple of N such as 3N or SN. The second

23

complication arises from the roundpoff error in the direct solution of
some systems of linear equations. Iterative solutions have the advantw
age of being self-correcting and round—off error is minimized.

If the non-zero elements in 5 are sparse and arranged in bands
parallel to the main diagonal” the storage requirements for a direct
solution may be substantially reduced. Comock (1951;) gives an
improved method for the direct solution of the biharmonic equation.
Equation (112) can be putibged into subnatrices as indicated by the
dotted lines. The subatrices have a more convenient form if a scalar

factor of 20 is introduced.

 

 

20;g.20§
I- 1 - - - —
!§3.[. 91 El
2052 = 2.!2. 92 = 1.12 (1‘3)
LIE! .313. _§3_

 

 

 

 

where E is the m unit sub-atrir, n isthe number of interior mesh
points in each row. The 93 are the column subnatrices composed of ale»

ssnts U13 along one row of grid points. The E]: are the corresponding
eolmn subnatrices of 20 1.

30-8100- 182000—
-820-810 2-8200
gal-82041 a. 02-820
01-820-8 002-82
1.001.820 _0002—8

 

 

 

 

For the general case of a rectangular region of n columns and p rows in

the mesh, the submatrices g, _B_, l are all nxn if the difference equations

are written for successive mesh points starting at U11 and sweeping across

the first row then from left to right across successive rows.

t

Cornock

211

shows how to determine the elements of a matrix Q such that
EAEKEE-Q
reduces to the form

311912

 

 

 

 

 

 

0 U
921 9.22 0 9.2 9-2
931 23. r 9., - 9., (mo
.. _ \ ._ ._
.. _ \ .. -
.- _ \ .. ..
.991 992 in -99. -92.
The system
911. 9-12 2:L __ 9-1
921 922 E2 92
is solved by direct elimination. Iith g1 and 112 known, 113..- ”lip can

be found by back - substitution.

One alternative to direct elimination is the ”relaxation“ technique
of Southwell (191:6). An initial solution is assumedst the points of the
grid superimposed on the plate. Using difference equations a new value
is computed at each point and residuals are determined at each point.
The largest residual is identified and the initial guess is modified
systematically so all residuals are reduced to zero. This procedure
has not been used much for digital computer solutions, because it is
most effective when the succeeding modifications are Judged by a
skilled practitioner. To date the logic of these decisions has not been
efficiently programmed. However, some aspects of this type of decision
making have been incorporated in direct search methods as discussed by
Hooks and Joevis (1961).

As a second alternative, better adapted to computer solution, consis-

der an iterative method. Equation ([12) can be written

25

(5-D 11+ 1 i=2.
(145)
01'. g=(;-_l_)y_+g
In initial vector approximation _U_(°) to the solution 31 is selected and
a sequence of successive vector iterates Uh“) are calculated using
Eh”) : ($- -Amfin) + .F. m > o . (1.16)
This scheme is known as the Richardson iterative method (or method of
simultaneous displacements, point Jacobi, or point total—step method.)
See Varga (1960). It requires all elements of the vector iterate U“)
for the computation of 9-01-11) . Other iterative methods introduce the
new values of the eluents of the vector as they are determined, and these
are used in the computation of successive elements. Methods of the
latter type use only half the storage required for Richardson’s method.
Iterative methods are characterised by the repeated application of
a computational scheme which yields an approximation to the exact answer
as a limit of the sequence of successive vector iterates. 1 basic
question which must be answered affirmatively for any useful method is 3
does the sequence of vector iterates converge? To answer this an error
for each vector iterate is defined
Eh) = Eh) _ P. n_>_ o
Subtracting Equation (I45) from (I46) we find
EIm-tl) . (1:. .. 5) 2(3)
Repeated application of this relationship, starting with the initial
vector 1.1“) gives
. ems-(14)" 155°) n> . (m
considering a single element 310') of 30-), if the H; 010!) . U,- it is
necessary that $111“) and 135031“) exist and 31.313031“) as 0.
This condition will hold for all elements only if

26

$134; - i)‘ of” = 9 74 a _. (us)
for any arbitrary vector E (c). Q is the null vector of 1: elements.
Iquation (ha) is valid than only if

gig-we - i)‘ - .9. (1:9)
where 9 is a square null matrix. . '

hilne (1953) shows that Equation (19) is the necessary and sufficient
condition for convergence of an iterative method and this is assured if
all the eigenvalues of the matrix 2:, e _I_ - a are less than one in absolute
value. _ ..

Iindsor (1957) has shown by derivation of the eigenvalues of 21 for
a rectangular plate that Riehardson' s method is not convergent for the
biharmonic equation. In the remainder of this section iterative methods
will be reviewed and those’which are especially useful for solving the
biharmonic equation will be identified.

a. Log-:31 - Iterative lethods .

The general form of Equation (1:3) for the biharmonic difference
equation on a rectangula- region with n colums and p rows of interior
mesh points is of the form

ii-:_ , , so
where l is a m: sparse, non-singular matrix, Q and g are column vectors
of 1: components and k '. up.

The derivation of the matrix form of some iterative methods is
simplified if we tales

A a Q 'I' 2 + E . (51)
where g is a strictly lower triangular matrix with the elements
‘13 (i) 3) below the diagonal of _a_ and all other elements zero. ]_J_ is

formed of the diagonal elemmts ‘ii of _1_ and g is a strictly upper

.27

triangular matrix with the elements ‘13 (i < 3), Q, 2 and g are lock
matrices.

Richardson' 3 Method

U(m+l) - - - (52)

where 2‘1 is the inverse of Q, a diagonal matrix with elements l/an.

For the model problem, Dal, n25, ps3 the point value is
Inn)” on) m) UM) ( '27)
U33- =53 MU}? 3+3? 3+U +£33.31 .UI .+-U ”"~ +U-,333

m3} m H
+Ui+:}n]- .05[U”3 +Ubj;z+U,(;2j«/-IJ Mafia], (53)
1525.7), Isl—P.

Though the introduction of exterior mesh points may not be the best. way
to account for the normal derivative boundary condition one advantage of
this approach is that the term F13 will be automatically included when
the calm subscript of an term is not within the range 1 to n and when
the raw subscript of any term is not within the range 1 to p. Hence the
1'13 can be dropped from Equation (53).
Gauss-Seidel Ile___t___hod ‘

Varga (1962) identifies this also as the Liebmann method, point
single min method and the method of successive displacements. The main
difference between this and the Richardson method is the i-ediate intro-
duction of the adjusted point values into a single vector iterate. The
matrix form illustrates how this is accomplished.

(2+9) 2-21-11 9.

2“”) = (earl i—(aurl s i“) (51‘)
For the example problem, the point values are computed with
Us”: 3 ru-+u33:;’+u3‘::,’3 11:33-33:23: (1.31:7:
33-3333; 33 can-5.33:3" 333" L332;- U313; ‘5"

28”

Here too, the 1'13 will be automatically included if point. values have
been provided on the boundary and on the first exterior mesh line. ‘
Starting atUil and sweeping along columns successivelyche required
(n+1) iterate values will be available as specified by Equation (55).
Boundary values computed initially remain fixed. Exterior point
values can be recomputed at the end of each iteration.

This method is convergent for the biharmonic difference equation
but the rate of convergence is so slow that it has limited usefulness.
for a problem with 66 interior mesh points 1098 Gauss-Seidel iterations
satisfied the same convergence criteria as 22h iterations of the
successive overrelaxation method.

Successive Overrelaxation

This is also known as the extrapolated Liebmann method, Parter
(1959) . systematic overrelaxation and the extrapolated Gauss-Seidcl.
Young (1951:) proposed an acceleration of the convergence of the Gauss—
Seidel method based on an examination of the changes introduced in a
given vector iterate and then introducing a multiple of the change at
each point. The point value obtained by the Gauss-Seidel method will
be designated 31““ ),1 5 i 5 1:.

Then
Uimu) ___ Uilez'. ”{Uilan Ufm} : ‘ I _ 00) ugh») + w Uitm-a) (36)
The quantity 6) is the relaxation factor. For overrelaxation, l< “<2.
‘hen e a l the method is Gauss-Siedel. The matrix representation must
account for the prior application of Equation (56) at all preceding
points. Hence,
D Ulm0+ §LZ""”')= f _Hylm) (57)
U’”*” = UW-t ML] ""21! W) (58)

29

Eliminating Em” between qu's ,(57) and (58) we find the matrix tom
of the successive overrelation aethod

(Q+w§)g‘”’*" =[u-w)p- tutu Um-t cuff (59)
The component U13 of the vector iterate at a point where 15 13 n,l g j g p

is given by
“a, b”) mu “fl its“) 3+1)

(m H), (m) Um

(7n) (7») My) he)
FUH’jH: -bUl'flfltl) .Q 5(U2-2 J +U51-2+Ui+z,,+Uy+2UiJ (60)
The rate of convergence of Equation (59) depends on the value of a).
The optima value of w is given by the formula

2
w = —-—————
b [+1/ I .. 92. (61)
where f3 is the spectral radius of the point Jacobi. method.

a," =. IE‘"’II/||§“”"’ll

62
m «° = w A»: N. _ _‘ ’
Any nor. of the error vector 2 could be used. One readily cmpnted 1.
“Em H= 2:: hi ml a , i (see Appendix A)

a canon proc;dure for the deter-nation of the relaxation factor is
to set it initially to one; and, after a number of iterations, say 100,
use Dee's (62) and (61) for calculating a not w. Subsequently m can be
recoaputed every ten or weety iterations. Forsythe and Issue (1961)
pp. 368-372 describe two alternative approaches and indicate that the
determination of a good estimate of Us early in the computation is an
investigation in which there is continuing interest e

The relationship between the estimated spectral radius and the
nmer of iterations required for convergence of the model problen of

Fig. h.1, p.20 is given in Fig. ho3.

30

 

U1
.8

g :

   

Iodel Problem

5 :‘s eOOOOI

6 x 13 flesh

The "' e05 SEC/IT.

§

NO. OF ITERATIONS
l-' w
.8 ?

 

 

 

:9? 598 .. 597 :96 [95 I91; ,_
' Estimate of Spectral Radius 9
Figure h.3 Successive overrelaxation
Number of iterations vs. P
B. lethods of w Iteration
The iterative methods considered thus far used an explicit formula
for the calculation of each component of the vector iterate. Is it
possible to use direct methods to find s block of components of the
vector iterate? Consider the model problem in terms of submatrices,
Equation (13). Taking arbitrary values for the column submatrices Eh

and g3, the components of 32 can be determined by solving

E 92 (“1): .122 - .3. Hi ('0 - a 23‘" (63>
share 92 a. {U21, U22, 1123, Ugh, U25}. Since a is an (nxn) matrix,
substantially smaller than _1_ which is (Ink) the direct solution of
Equation (63) will not require unreasonable blocks of computer memory
and is an acceptable procedure. Using this method the point values of
the vector iterate are not determined explicitly one at a time; instead
11 components are determined simultaneously. Hence, this procedure is

called the simultaneous displacement method and is classified as implicit.

. . v .IV. '1 4 ." 11?».

tlll‘lvr'

vi. Ill-Ills!!!

I!

I‘ll

31

The matrix form of certain block methods assures faster average rates of
convergence. See Appendix A, Section 5.

lrms, Gates and.Zondek (1956) and Keller (1958) have investigated
block methods using the components of the solution vector on a line as
the basis for partitioning the matriximo For the biharmonic difference
equation, "two line“ schemes and the alternating—direction implicit
method have been studied and appear to have advantages over other block

methods. See Parter (1961A)

 

The AlternatingéDirection Implicit Method

Peaceman and.Rachford (1955) found that the rate of convergence of
a "line" method could.be substantially improved if after sweeping all
rows using the simultaneous displacement method and a relaxation factor,
the next sweep of all the mesh points‘was made by columns.

Coats and Dance (1958) derived a convergent, alternatingndirection
iterative method for solving the biharmonic equation. This method is
similar to the alternating-direction method for solving Laplace's
equation proposed‘by Douglas and Rachford (1956). The derivatives in
the biharmonic equations are replaced using central difference approximam

tions

      

   

9X? ~ ij = 8:Ul'j =Uz'J'-2 RUUZJ-I +6U _LfUz J-H +UIJJ+l
93V~zLflU1U= 8;sz—Ui-)2J‘ PUU-IJ+6U:J-4U?+l) J+UZ+11J

.2142. u w ”1
2’3 L~IEU 2851 U "'2 L+U" 2[Uz’-I,J +UX+IJ+ 1,.J'l 1.1
X 9‘1 1'“: X J 2J g (J + UHIJ.“ + UH J'H +Ul+')JI”1-UI'U"J‘]

'4
WW) 2-. 5x U23 + 2 5x151; Uz'J' + 3‘} Uz'j
Is shown for the model prdblem, Equation (h2), introduction of the

difference equations reduces the problem.to the solution of a linear

32

system in k unknowns—

53>

Id
ll

l’zj

which can be written
(5 + 9. + 2) u E
where l}: flij’ [g gjij and L gjij respectively represent the components
of g g, 911 and £11 at the mesh point (i,j).
For the alternating direction implicit method, Equation (1:3) is
replaced by a pair of matrix equations
'<r§+.I.) 9.: (yrs-r2) in:
<r9+r2+2>11=<£~ rs) E+r§
where r is any positive scalar. The iterative scheme in the form pro-
posed by Peaceuan and Rachford would appear

3 +1) 1.1““) = (_I_ - rll g) 31“)+ rlll 2 (6h)

(rm-H H ‘H
(rmi'l 9- + rn+1 .P- + D Elba) '-'-' (I " rn+1 fl) Ehfié) + r.+1 E. 565)

9- - rmH
The first sweep of the mesh is by rows and only Equation (614) is solved
for the vector iterate 9542'). This is an implicit method since all the
components of Eh'w) in one row are determined by_ (614). It is necessary
to retain all the components of pf.) while solving for 6‘99. During
the second sweep Equation (65) is solved one column at a time for the
components of 2“”). The solution along one row or one column is
obtained by direct elimination. Conte and Dames use a factorization
techniquemhich is well adapted for the case of Fig.4.2, where there are,
at most, five unknowns along a mesh line.

The Douglas-Rachford method is a variant obtained by changing (65)
to a form which does not contain E. Substituting for g _t_I_(‘+'/2) from (61;)
we obtain

(rm-l Q + I'mi-l 2- + -I-) E-(mfl) 1’ 2 ._ Haw/2) "° (1 " I.1114-1 9 " rm+1 2) 2(a)

33

Coats and Dames use a simplified form not containing 3

use) = “(as ._ rm [9 Hum) .. Q Hm] . (6'6)
The use of Equation (66) for the column solution provides a. Simpler
computational procedure than the one associated with Equation (65).

The determination of optimum acceleration parameters requires
consideration of another form of the system of linear equations. This
is found by combining Eq. (6h) and (66) into a single equation.

2W1) = are; 11‘“) + an r; (67)
where gr c (_I_+ rg)'1[(r§+;_)’1'(;-rg- r2) + r51]
r = (l + marl '(ra + 9‘1 . A

The difference between the nth vector iterate 2(3) and the solution
vector 2 is the error vector 1!“). It can be shown (see Appendix A) that

gun) =_ 5,.“ 50-) (55)
If g“) is the initial, arbitrarily selected vector iterate then
3(0) 3 2(0) .. 1.] and

§(l) . firm 0 firm-l eoeeeefirz oar]. 2(0)

For a convergent iterative scheme Elm) 92 as m 960 , and convergenceof
this method can be accelerated by the choice of rm for each iteration.
However, the relationship of this iterative method to others considered
is less complicated for the Specialease when a single value is assigned
to the scalar r. Then i

g _ 15‘“) 4353‘?) f _ (69)
and the eigenvalues of the cents-Dames matrix gr provide the basis for
detemining the optimum acceleration parameters. See Fig. tub vhich
shows the relationship between the acceleration parameter r and the
amber of iterations required for convergence of the model problem.

Fig. 4.1, p. 20.

 

 

500. Model Problem

U“ 5 a .00001
2’ 4p 6 x 13 Mesh
o boom Tine . .098 SEC/IT.
22
0’ 300-
DJ
F.
-200-
LL
(3

.100n
(D
Z

 

 

 

I I t I I l l
0 b0 60 _. 80 10° 120

”A

.Ualues of Acceleration Parameter r
Figure heh Alternating direction implicit method
Number of iterations vs. r
Choice of optinnn values of rm for a square plate has been considered

by Conte and Dames (1958) and Faimatherand Mitchell (1961;). For this

particular geometry the eigenfunctions of Equation (68) can be expanded
in the form

E3) = a sin (pfiih) sin (qfljh), (p. q .. 19 2. °°°°. 104)

Consider an amplification facecr
) = if”
Ft ELI”
If this is substituted into Equation (68) we find

(1 a 16r +3 Se 33)?

 

(16 rml))\p,qe1+16rm1r(t§;h,g;) + 256 ,mlg :3? (70)
where
Sp : sin P3212 , 5Q 2: sin i192, h is the mesh interval.

The error associated with the initial vector iterate can be

35

expanded in the for.

u—

(O) h-’ (0) ' n . '
E" 2%-97’1?‘ SMOOTH/05m (3/106),
, _

th

and the error vector at the n iteration can be written

(a) d (1») , , . ‘
E5] zg’C”, 5/”(f7NIIISIM (fly ’1’,

where
no = (a)
It follows fro- !quation (70) that Of (16r)! A3351 for all p and
q if r! is positive. Hence, after 11 iterations each compontlt of the error

decreases by a factor
m .
1U: (167')! A?) 3/

if r is positive. The minimum value of (16r) A found from
I X P"!

Equation (70), is zero and occurs when
I
lér = S’s-5;.

This indicates that an appropriate choice of the 1:? can be nade so all
components of the error vector will vanish and the exact solution can be
obtained. Rather than attempt to find this optimum I} , a less conplicated
procedure consists of choosing the r]? which optimise the rate of conver-
gence of the method. _

Consider a slightly different expression for an amplification
factor

 

- -__-_- (I—lérglf’sle 2-
’67'Afz "’ (Hursfiiflz

Since

36

'6'; +33? 2. 25,2592.
we find from Equation (70)
fl I16r~7| 2 pl IérM
for all p and q. The analysis of the problem of finding an. optimum set
of acceleration parameters is treated by Doug Lea and Rachford (1956) .

The factor by which the error is decreased after I iterations is
2. a 2

2,0936?) ’67:?) = I:— jyy/MIQ) = l: [2%f:+ 5:?)1]
It is necessary to find the set oer( (=1, 2 ,-—-—-t) giving a
maximum value of 2t which is as 8.311 as possible for (p,q :3 l, 2, 3
---k-l) . The Douglas-Rachford solution treats this as aChebyshev
minimal: problnpwhilo- Conte. &1DCI08 reconend a set of acceleration
parameters of the fora

1611, “((14), X: l, 2, -f-, t

where 0< or < 1. This permits the determination of an upper bound on

2t

l-( -TL
Z. (5' 3,, I671) < Titan): [W e 1 (71)

The formal procedure, outlined by Conte and Danes, starts with a choice
of Pt ((1) which permits the determination of the number of cycles of t
double sweeps of the mesh required for the selected reduction factor.
However, the choice of Pt ( or) is subject to the empirical observation
that best results are obtained for or< 0.2. The number of iterations, t,

per cycle is conputed from

t _>_ I + Wee/5:341) (72)

2; = T s X= 52,3,-~,t (73)

37

gives the value of rk for the gthiteration in the, cycle.

Is an example of the method consider a 20 x 20 grid on which it is
desirable to reduce the initial error by a fector 10-6. ”Selecting or a 0.2
makes Pt” 0.01. The ntmber of cycles required is determined from

(Pt)n == 10"6 _ .
Thus, 11 g 3 is the number of cycles. 'The number of iterations per cycle

is

> [05521919 ~
{I‘— l+ 087.2) ~ 7'35

ortg8and

Factcrisation Technilue .

Il’he difference equation (6h) can be written in the form

Mb“ (11+Vr)(*'+Vz) (we) I’M-Va): ,
Pg.) ‘6.)- 4U]- J6] +(6 +— 7M) U5). L/Lj).§b/. + Ui+2)J Fé') (71‘)

where
Uh”) m) ('m)_‘_ (7»
’33' '(?.,l,,—, "WI 3-W( I+IJ+U5-I,J')+ )8(UI', 1,13: +U1IJ:‘))
m (on) (M (7" 3") ’”
- 2 ( Ll“, J+I +U -I,J'H+ UH l-l “(limp/'1) U1) Iii-2- ~ U61";

similarly, difference equation ( 66) can be written in the fore

(n+I)_ Um“) (”11‘") Uhs-II) I‘m-H) ‘
U,,_z +l6 6+%—IU,,, LILU,,,,.+U,,,,,= F2, (75)
where
l’” 4%) Urm) (1n)+ (ML (7»;
re = —,— ,, . ":‘wa.’ +6 Us 4-4113"; Use.

Usins (7h) at all mesh points in a row leads to a 'quidiagonal"
system of linear equations of the form

38

" cl -h 1 "U11 l ”F111 7
~15 02 4| 1 U12 P32
1 --h 03 -h 1 U3 r33
1 -h 0,, at 1 111,, rm,

1 =21 6n -l ‘1‘ Ui,nol FBn-l
l -h on Uipn rfln

 

 

 

 

 

 

D

.J l-

where the C's all have the value (6 + l ). 1 system of linear equa-

rn+1
tions of the same general form is obtained when (75) is applied at all

 

mesh points in a single column. The factorization technique is an effi-

cient direct method of solving a quidiagonal system of equations. Take

'1 = 01, 'q 8 Cq " gq_2 _. quQ"1
BO :09 Blg-h/Cls qu-(h-tdq gq_1)hq, Bn=o~
‘0 I 0’ 81 a: l/Cl, . {q : lhq , (n a ‘11-]. I O
t - “b - Bq-z

h0 =3. 0, h1 *Li'm/QI’ hq = [FR - no:2 - dq hq_1 1/ .WQ’ 2g q 5n
Starting with the specified initial values the n values of h are computed.

Then the values of U are computed by back substitution

Un=hn

q = ha " Bq ”w ~' " 8.9 ”q+2 ’ q = “‘1’ “‘2’
The alternating-direction method was initially applied to the

‘1

U ,1

 

solution of parabolic and elliptic partial differential equations of the
second order. For Laplace“ 8 equation with sufficiently small mesh spac-
ing it provides a significant increase in the rate of convergence per

iteration over the point successive overrelaxation method. This advant-

39

age must be weighed against the requirement for the double computation
in a two sweep scheme and the need for storage of the (n+2?) vector
iterate. Compared with recently developed block successive overrelaxau
tion methods, superiority of the rate of convergence is not rigorously
established. Varga (1962) notes that the convergence of the Peaceman-
Rachford and similar alternating direction methods has been established
only for rectangular regions. Though there has been some success in
applying alternating direction implicit methods to more general regions,
the convemence in the general case is yet to be Justified. There is
no general theory for the determination of optimum ‘ convergence para-
meters except for rectangular regions. These same limitations apply to
the solution of the biharmonic equation and in addition Keller (1961)
demonstrated for several block methods that corresponding biharmonic
schemes converge more slowly than Laplace schemes.
Semi-iterative Methods

A system of linear equations

5 y. " E:

can be solved by an iterative method of the form

31“”) g g E“) + E m a o (76)
it a .-.-.- _I_ - g is a positive-definite nan matrix. See Appendix A. as
m as, 11(3) converges to the unique solution of the system of equations.
A semi-iterative method uses an algebraic combination of solution vector
iterates EU!) as a means of increasing the rate of convergence.

Starting with an initial estimate 9(0), the error vector associated

with the vector iterate 2(11) is given by
30“) . U0“) - U . m 1“ E(°)

)40

A linear algebraic combination of the vector iterates Eh“) is introduced
1““) :3 f: pjtml 2(3) 1:120 (77)
J=0
where the coefficients pj(m) are selected so that each [(13) is a weighted
average of the 2(3) and a better approximation to the solution vector
y than 2011). For the special case 2(0) 2:, g it is necessary that
X | »" f filmbl (78)
i=0

Then 10“) g g for all m _>_ 0. The requirement of Equation (78) will be .

imposed on the constants p.) (m) for any arbitrary choice of 11(0).
The error vector associated with 3:0“) is denoted iii/(m)
2"!“ g :6“) - y, = fpjon) 11(3) - g (79)
_ .. . i=0.
Since the Constants pj(m) must satisfy Equation (78)
’ I... '1
5"" — z: srwfl” ~ ( i F} WW
I . -0
EW= z: firmIflw—g]
E/(m’= Z 8’ (M)_E_:(J) ‘
0m): j (a)
( gm M )E (80>

0.
H

0

If we introduce a polynomial in a component of 115, defined by
Pm(u) s: 22: p3(m) u:3 m a O (81)
J=o

then we can write (80) in the form

1.35m) ___ Pm (E) 2(0)
where Pmogg) is a polynonial in the matrix 2°. The condition imposed by
Equation (78) on :ijm) requires Pm(l) g 1.

Using the definitions of matrix and vector norms given in Appendix

in

A, we can write

IIE"’"’H =IIBfM)E°’ll i N am: HU’H ' m

For “I309” < l the average rate of ccnvergence of the semi~iterative

IV
:3

method for m iterations is defined as

._ .— E MN
11 [ 13(3)]: _XnJJFLi
If the polynomial is selected so Pn(u) e um, the vector Em) is

identically 11f.) Ind the average rate of convergence can be readily

simplified.

ghnqfl. ~121LEmlfllfl. W ._-_._ RPM”?

7n 3' --'m v" 4

and, as shown in Appendix I, it is equivalent to the convergence rate of
the basic iterative formulation of the problem. The average rate of
convergence of I“) will be optimised by finding the minimum of II Plum/l
under the restriction 241) = 1’. See Varga (1962) Chap. 5.
Chebyshev Semi-iterative lethods

Golub and Varga (1961) identified a polynomial which satisfies the

requirements for optimising the average rate of convergence. They used

 

 

 

[ 2ub- (b+a)
Pm(u)-:m[2-(b+a)'| 5 11112.0
m H J
Cos (m Basal 5), =1 5 3 5 1: m .. 0
where on“) : Cosh (m Cosh-’1 s), ‘ > 1,

are the Chebyshev polynomials
The Chebyshev semi-iterative method for the problem specified in
Equation (76) has the form
(n+1) (n) (maul) (zeal) tn
1 =wm1[!.1 +z-z 1+1 (6:)

1:2

where (See Varga (1962) p.138)
2 Om l ‘

w,.1, “n+1- 1 p , m_>o
and p is the spectral radius of 1. lots should be madeof the absence
of the vector iterate E“). It is not usedim the computation. The
vector iterate [(#1) is formed directly from the preceding, vector
iterates 35') and 10"“. It is necessary to store both I“) and
10"“. Spectral radius is defined in Appendix A, Sec.2.
E2129. £22: 2 man 11:22

Consider a slightly different approach to the solution of lquation
(70). Let

Id
II
[N
[’1

I +
, t
or equivalently

IN
IS
Id
'1".

!-!!+9 an
where

‘-" [fl 1'" [1! 2J' , 9' E]

If _I_ is an an convergent matrix, Equation (83) has a unique solution

IO
II

E and the subvectors E and E are equal to the solution of Equation (76).
Since the solution of. (83) would require“ the manipulation of.(2n12n)
matrices, it is not reco-mnded as a practical method. However ,' it does
provide the basis for an improvement of the suit-iterative method.

Issue that E in Equation (76) is m nxn, Hermitian, convergent
matrix with the special fen

4:. it]

where the sub-strides are square and gl- is the conjugate transpose of g.

1:3

A matrix [which can be expressed in this form is said to be weaklycyclic
of index 2. l weakly cyclic matrix 5 of index k :is such that 51' has real“
non-negative, eigenvalues. For the assumed form of g the successive over-
relaxation method for Equation (76) can be written (Varga, (1962) p, 11,9)
21““) = w [a 22“) + :1- 91 m] + 21““).
gab-pl): “@(fll)+§_fi2(m)]+ge(m), mgo _
where g and g; are partioned- into 111, He and :1, 32 respectively.
1'_h_g m Chehz‘ “ shev Sent-iterative M
The Chebyshev semi-iterative method corresponding to this special

matrix ! can be written 1 (Verge (1962) p.150)

9.1"”) = am [3 22“) + £1 - 21““1’] + QM)»

22‘”1’=w.+ [s2“’+za-U“’9J +95“)? 1:1

Since the iteration parameter a) “1 is a function of the number of
iterations n, the cyclic characteristic of _l_ per-its the following fore

5‘2”” :- n+2 “1L- am” ”1;- 42‘2”] +952” --:°

IPhis is hows as-the cyclic Chebyshev semigiterative method. bus to skip-
ping half the vector iterates the rate of convergence otthis method is
twice tint of tin Chevyshev semi-iterative method. Basically though, this
method is Just a variation of the successive overrelaxation method. Varga
(1562) shows that the average rate of convergence of the cyclic Chebyshev
semi-iterative aethod is better than the average rate of convergence of
the successive overrelaxation method. The cyclic Chebyst semi-itera-
tive nethod will be used to solve the model problen and the convergence
rates of successive overrelaxation, the alternating direction implicit

method and the cyclic Chebyst seniaiterative methods will be compared.

hh

M Chebyshev §eni-iterative Hethod for the Biharmonic Elation
Consider the solution of the biharmonic equation for a rectangular
region with a grid of n interior mesh points in the x direction and p
interior mesh points in the y» direction.
The difference equation
can be written in the form f J

 
 
 

§H:£ . ‘ (8h) '

where N a up, _A_ isan Hid!

+~=w
l

coefficient matrix, .3111.
the solution column vector

and {is a colunn vector

N\°
A
l I
IP""T—-

which accounts for the
specified boundary
conditions.

The for-ulation'of the Figure in; m line blocks _ .. _ __
model problem in terms of sub-utricesfiquation (113), can be extended for

the general case

.. .. ‘- - r' -
I l
emtgggg, 9. 21 Eli
I l
21.22.29. 9. 92 :2
---r-1--I-fi-e~d p- --
. I ,.
OI'BsusI! 0 U = r
::'..‘.."..:::;..-:. :5 :9
L. : : 4
99. ----- _I. 2.! 2,, lip
L. _ _ _J _ J

 

 

 

 

 

 

The l, g, 1; are nxn matrices and El: and El: are n conponent coluln sub-

natrices corresponding to the solution vector components along one row.

hfi

Griffin and Verge (1963) select a two res block of the couponcnts
of g and show that the assoc1utod matrig Q is of the form required for
the cyclic Chebyshev scheme. The number of rows of interior mosh points;
p, must be cvono This permits the selwciifin ot t e E two row column suh~

matrices for the p rtioning of y,

 

 

 

 

 

 

{“ " , 7
E 5 Q 0 2; #i
T '1
- .1: 2S, .9 2‘2 Lé
T
0 K L o T e F‘
- - - ., —3 - -3
(85)
9. 2.2? I: T F'
L _. intJ _ht .
‘where
1!. 2 SF. 9. E. B. ,. /
E = D E = 9 ET 2 9 Q :3 (1‘2"
B n B I o I q , /
_,._ .. _ —. - q 2

 

The difference equations (8h) can be expressed in the form
T
L - ' - T - 6
~29’Eq E-q-l Elm-1 (8)
or since §.is a 2nx2n symmetric positivexdefinite matrix, which assures

the existence of the inverse Lfl, it can be written:

T g gfl F' - L“l KT T - Lfl g

T
"q "q "" "‘ 'q-l ”gel

‘which establishes the fern required for use of the cyclic Chebyshev
semi-iterative method.

Assuming the solution is known at all mesh points except those in
the two-row'hlock q, Enuation (86) provides the basis for the determine:
tion of the remaining unknowns. The forn.of the matrix §_is conveniently
simplified if the components of Tq are selected alternately fron.the two

TOWBe

1:6

 

  

Take I? = {Uh};Ua,5+|2Ua,j1U2,J-Hs “““ ”‘1 UmjiUmJ‘“; ’
than 20 -8 ~3 2 1 O
.3 20 2 m8 0 1 0
-3 2 20 ~8 =8 2 1 0
1 O l
L,: 0\\ l\
__o o
-8 l 2
O O 1 O 0
2 o -8- 1 2
5 = o O 0 O l 0
o 0 2 0 ”'5‘ 1; 2
\“ \\~-~ \
\\‘\ L v ‘\\.\ " ‘
e \~-. - \ \\ ‘ C.
2 f*o

 

The matrix E is symmetric with nonmzero elements appearing only in

 

 

 

_ (2nx’2n)

“\as ‘~1 (2nx2n)

the main diagonal and the eight adjacent bands. Hence 9 the linear system

of equations (86) can be solved by direct methods. The square—root method

can be used towadvantage. See p. 1.8.

After obtaining a method for the solution of (86) for_one block,

the full cyclic Chebyshev semiaiterative method is introduced

h?

' I\(1»III)_ (1233)] (am) 0 5n; g; “($1 -
TW ri’... T 51:21 I, a z <8“
TQM“): (2)11-1)+ *l2M+')_I1-I-(2m I) >
‘wlw I2R+l+ w"”+l[-I 2m: -- 2H! J’ 777 ‘ O
and
“(2314-2) - mm“) I (1)144!)
— >
LI“ =2.f:’k —KT _._—2", gig/3+, 3 ”-0, (88)
am I i) um‘f' H2?" 42) (23;) ¢ -t
.J’s / ,
“1:22 :1; wzmn[- Tzk Tzn ]’ ’ \ A

The Chebyshev method consists of solving the difference equations (87)
over the first two-row block, and all subsequent odd-numbered blocks,
then difference equations (88) are solved over all even-numbered blocks.

The iteration parameters (02 m+1’ (0sz are computed recursively

using

 

(0‘1“:[I— Eli-us] ‘For 5?. 2

share f’ is an approximation of the spectral radius of A. If _1_ is the
coefficient matrix for a rectangular region, _ an approximation to the
maximum eigenvalue, given by Griffin and Varga is

inwéfyi

where a?“ .

76: (it)? 5.1%“ + '5") + 3 will

(89)

 

h is the uniform mesh spacing in both the x and y directions and a and
b are the dimensions of the plate. The relationship between the number

of iterations and the value selected for f is shown in Fig. h.6, from the

model problem of Fig. n.1, p. 20.

he

 

 

 

 

lodel Problem
2501, 8 : 000001
6 x 13 Mesh
m. Time = .16 SEC/IT.
Z 200,
9
E
o: 150.
Lu
._ #4,—
— lmw
La.
0 5
O' 0..
Z

 

 

 

.99. .96 . .9“; - .95
Estimate of Spectral Radius {9
Figure 13.6 Cyclic Chebyshev semi-iterative method
3 _ Number of iterations vs. ’0

zaageaeama

l direct-method algorithm for solving a system of linear equations,
the square-root method, which is suggested by Paddeeva (1959) as one of
the most efficient, is applicable only to symmetric systems. In Equa-
tion (36) which can be written

ag=a I
the matrix 9 is nine-diagonal symetric and positive definite. Hence,
1;. can be expressed .as the product of two triangular matrices, one of
which is the transpose of the other

a=¥s

—9

' "I
811 312 313 81h 815 0 - - - _
o 823 ”an 325 826 - - - -

    
  

  

822
\\ \ O \
\

\\\ \\
S : 0 \ \ I. ’2n-h,2n
O\\:\~.. 92n-3 ,2n
“ \‘x‘ t’2n-2, 2n
0\ \\: 2n-l,2n
s2n,?.n

 

 

$— _

“cording to the rules for matrix multiplication the following relation-

ships hold
213-51; 8132* ‘21 ’23 + " ' ' ‘ ; " +811 313' 1‘3 (9o)
111"]2.i+§21""'f"“11’ 1‘3

The I“ are ooppnted by recursive use of equations (90).

Since L3,: . a4“ a s4”: . a",J . o,

'1131/111“, ”133%: ZSJSh

11 31/711 “:53 ' 1) 1'
'1J=MS"S“ , '1+h.>_dzi

 

'13 .-_- 0, DJ _

The solution of _I_.. 11: g g is obtained by solving two systems
fast . . éz-z

Il’he components of the vector 1! are computed with the recurrence

formulas

Number of Iterations

50 5-:
p1 p1 .— 2:_ 821 m2 .
“1...... ms 9%: ’
Similarly the components of‘l are detegmined with
m2!) mi - :— Sig-T2-

T2n 82n,2n , T1 811 , i 2n

 

 

5000-

30004 Successive Overrelaxation X

___ Cyclic Chebyshev Semi-iterative Method n

3000“ """""'
_______ Alternating Direction Implicit Metgod o

1000]
800‘

600‘

300?

200‘

 

 

L4 L 4‘
V fi"

0 100 200 300 LOO 500
Number of Interior points

(a) Iterations ve. number of interior points

Figure A.7 Comparison of Iterative methods

Execution Time in Minutes

10.04
8.0~

6.0<
5.01

Loo‘

300‘

2.0+

51

 

 

r
f” Successive Overrelaxation X
/ _____ Cyclic Chebyshev Semi-iterative Method a
I Alternating Direction Implicit Method 0

 

* x . L L4 . A 1 4
0 1 00 200 300 A00 500

 

Number of Interior points
(b) Machine time vs. number of interior points

Figure h.7 Comparison of Iterative methods

V.

COIPIRISON OF THREE ITERATIVE METHODS

A prime objective of this study is the identification of the rela-

tive merits of three iterative methods for the solution of the biharmonic

difference equations.

Commenting on the problem.of selecting the best

iterative method for solving engineering probless Varga (1962) p.2h5

makes several observations.

(1) For the general case there are no

theoretical arguments which rigorously establish the superiority of any

one method.

on the numerical experiments of many investigators.

(2) The present evaluation of these methods has been based
(3) The numerical

results indicate that for each two-dimensional second-order partial

differential equation boundary-value problem there is a critical mesh

spacing he such that the two line cyclic Chebyshev semi-iterative method

is superior for all mesh spacings h 2 he, while for h < he a.multip1e-

acceleration-parameter PeacemanéRachford method is better.

Table 1. Comparison of iterative subroutine characteristics

I

_.Y_

 

Number of Storage of 1 Time ‘sec.[itgratéon}
subroutine FORTRAN arrays for
Name Statements hSO points Mesh_pts. Mesh pts.
80R 32 one .050 .223
am 89 1221. .098 .585
CHEB 162 1126 .167 .612

 

 

80R - Successive overrelaxation method

ADI - llterna'ting direction implicit method

 

6333- Two line cyclic Chebyshev semi-iterative method

 

The evaluation of these iterative methods for biharmonic difference

equations is more complicated.thsn for Laplace's difference equations.

For Laplace's equations, Golub and‘Varga (1961) indicate that the cyclic

Chebyshev semi-iterative method requires effectively no more additional

52

53

arithmetic operations or vector storage than other iterative methods.
This is not true for the biharmonic difference equations as shown by the
comparison in Table l of the three iterative subroutines included in the
ISOPEP program of Appendix B.

The overriding consideration for many is the actual machine time
required for a solution which satisfies a specified convergence criterion.
Convggence criteria V _ _

The norm H E“) ll II of the error vector E“) is often used for
terminating the iterations; the norm is defined by

H Em) “II I ._ ‘Eémfl
A value is assigned to a parameter 52,- and when

. . Ila“) Hus a

the ‘solution vector iterate gun) is accepted as the solution. Griffin
(1963) indicates that for the two-line cyclic Chebyshev semi—iterative
method, if the nuber of two-line blocks is large the average difference
between corresponding values of U1 in successive iterations will be
approximately 2 8/1, where k is the number of mesh points at which the
stress function is unknown.

The criterion used in the 1801!? program of Appendix B , which includes
the subroutines 8G, ADI and cm, is

I‘ (I) la: 5 5
with 8. 105. for the model problem '01 |m s 3.6.

The three iterative methods were used to solve the model problem
of Section VI - 1. “try conditions were used for the stress function
at the plate centerline. rm. is a sunny of the results in Table 2
and comparison of the methods in terms-of number of iterations and machine
time required for different mesh intervals in Fig. h.7.

Sh

Table 2. Comparison of machine time and number of iterations required
for convergence

P—r Number NEST-er of E 55

 

 

 

 

 

 

 

lesh " of ' Iterations _ .7 Execution time,

”It“ :3: 13663:? son I 0031 %m%‘ig’fii“
1/10 5:11 50 138 93 91 .070 .108 .116

1/12 6:13 72 19h 9s 99 .161 .156 .283 as
1/12 60:13 72 336 128 123 .280 .205 .31.2

1/20 10:20 190 636 317 - s- 1.08 1.35 - is
1/20 10:21 200 500 33h 357 .87 1.1.7 2.08

1/30 15:30 105 1981; 712 - s 7.25 6.73 - a
1/30 15:31 1.50 21.116 7M 1053 9.10 7.29 10.75

 

 

 

 

e There must be' an even number of rows for the SCSI subroutine.
el- Optimum relaxation parameter used

32;: of numerical: results .

Numerical experimentation can provide insight for_the appraisal of
iterative methods and may provide a basis for theoretical investigations.
The results obtained are based on solutions of the model problem only
and conclusions should be qualified accordingly. 1 number of observa-
tions are presented for consideration.

For the model problem the cyclic Chebyst semi-iterative method
is iteratively faster than point successive overrelaxation for all mesh
sises considered. As shown in Fig. h.7(a) it is also iteratively faster
than the alternating direction implicit method for mesh spacing h>l/16 9
or for less than 125 points. In terms of machine time required, suc-
cessive overrelaxation is best for less than 350 mesh points or h'31/26.
If h <1/26 the alternating direction implicit method is best.

The rélaxation factor or acceleration parameter was selected on the

basis of the" treatment given in the discussion of each of the three

methods in Section IVmB. For successive overrelaxation an initial value
“)0 g 1.5 was selected, and this was changed after every 10 iterations.
As shown by the second and third problems in Table 2 this will not
necessarily assure a good approximation of the optimum value ofw-b.

When the cyclic change was used for the 6x13 mesh size problem approxia
mately 70% more iterations were required then when the optimum re] exam
tion factor was used. Another indication of the variation which can be
expected when using this procedure is shown by the scatter of points in
Figures ho7(a) and (b) for the 50R subroutine. This contrasts with the
curves fitted to the points for the other two methods. It should be
noted that a random selection of values for a) will produce greater charge
in the number of iterations required for convergence for the successive
cverrelaxation method than for either of the other methods. Comparison
of Figures 11.3, 11.11 and 11.5 shows that the alternatingmdirection implicit
method and the cyclic Chebyshev semiaiterative method do not impose as
severe a penalty on overestimation of the appropriate parameter as does
point successive overrelaxation. For a considerable range of values
above the optimum, the number of iterations increases only slightly above
the minimum for these two methods , while the minimum in Fig. 11.3 for
point successive overrelaxation is much sharper.

Another important consideration is the computer storage required by
each of the three subroutines. The data in Table 1 clearly identifies
the successive overrelaxation method as the one which requires least
'storage for both the arrays of data and the sequence of instructions.
Because of the relative simplicity and minimal storage requirements the
successive overrelaxation method was incorporated as the main iterative

method in the ISOPEP program.

56

Since the number of iterations required for convergence appears to

increase significantly with the number of mesh points, attention was

directed toward the possibility of improving the rate of convergence by

first solving the problem with s relatively coarse mesh, then interpolating

between thee vslues to generate a better initial solution vector for s

finer mesh.

some of the results are given in Table 3.

 

mu 3.

 

' 01'.

  

.. elv t ISOPEP

‘0'.

l subroutine CHANG was written for this interpolation and

     
 

' Estimate

 

Prob. Converg. Ilesh Number of Time Time

No. criterion sise Iterations ‘75-‘15?- LHours) of f
101 10'"5 5x10 118 3.31 .9571
102 10'5 10x20 521 51.9 .9801;
103 10"5 15x30 15h8 5 38.3 .99112
201 10‘5 3x6 39 .85 .719h
200 10-5 6x12 811 2.62 .9556
203 10-5 12x21. 86 6.31 .9906
205 10"5 7x18 276 16.9 ‘1 .9737
206 10-6 11x28 1551 5»- 29.7 ‘7 .9900
303 10"5 6x12 225 2.33 .9506
306 10-6 21:18 1038 h 1.9 .9793
801 10-5 6x12 179 2 .30 .9670
I102 10-5 12x21. 126 5.67 .96118
has 10"6 2141118 166 1 3h.7 : .8862
501 10-5 6x12 135 I 1.80 .9h76
502 10"5 121211 295 12.59 .9781
506 10"6 2111118 1157 5 117 .6 .9858
601 1:15 15x15 176 1 20 .9802
601 10'7 name 7072 57 _ 11.2 .9998

 

 

 

 

 

 

 

57

Problems 101, 102 and 103 in Table 3 were solved using the subroutine
CHANG. The final solution of Problem 103 required a total time oi” 6
minutes, 311 seconds and s. total of 2187 iterations for the three problems.
The same problem solved with initial values of the vector iterate set to
.1, required 7 minutes 15 seconds and 19118 iterations as listed in Table 2.

The other sets of problems listed in Table 3, such as 1101, 1102, hoe,
were solved using the iterpolation procedure as the mesh was refined.
For Problem 1108. only 166 iterations were required after the last refinem-
ment of the ash while for Problem 506, which has the same number of
interior points, 1157 iterstions were needed. Both of these problems
have more than twice the number of mesh points used in Problem 103 yet
the solution time is approximately the same for 506 and substantially
less for 1108.
mobilities g; the Computers used

 

- Two computers have been used in‘this investigation. One, the
Control nets Corporation, 3000 at llichigsn _State University Computing
Center, is very fast and has a large main memory of 32,000 words of 12
decimal digits eech. The floating. point multiplication of two numbers
with lO-digit mentissas requires less than six microseconds. The other
computer is the 18! 1620. at the University of, Toledo Computation Center.
The 1620 performs e. floating-point multiplication of two numbers with
eight-digit nentisses in approximately 12 milliseconds. The main wry
provides for storsge of 20,000 decimal digits which is about 1/19 of the
3600 memory. However. the min memory is supplemented with en 13! 1311
Disk Pile which-provides 2,000,000 decimal digits of secondary storage.
Several problems were solved on» both computers end e. comparison is
provided in Table 11. The slight difference in the number of iterations

58

required for convergence is possibly a consequence of the difference in

the mantissas of the floating point numbers used.

The difference in the speed of the two computers contrasts sharply.

In fact for problems with a hundred or more mesh points the cm 3600 is

approximately 1200 times faster than the 13]! 1620.

Lt rates of 3375.00

and 830.00 per hour for the 3600 and 1620 respectively, the cost of

solving problem 1.102 would be $8.82 on the 3600 and $572.10 on the 1620.

m qualifications of the comparison should be noted. First, the time

required for compiling a representative combination of the ISOPEP W

subroutines on the 3600 uses between 20 and 30 seconds.

Compilation time

on the 1620 ranges between 10 and 15 minutes. Forthis purpose the 3600

is only about 30 times faster than the 1620.

A second consideration is

the time saving possible on the 1620 through use of a machine language

successive overrelaxation subroutine.

The introduction of such a sub-

routine for the iterative solution of Laplace' s equstion resulted in a

50$ reduction of machine time required.

mu. 1;. Comparison of the 130m program for the IE! 1620 and one 3600

 

 

 

 

 

Computers

Number of CDC 5635 Tfie IBM 1355 '—
Prob. flesh , Iterations Total beaution Time
No. Size 3600 1620 Min. Sec. Kin. Sec. Hours lethod
1.101 51:10 118 116 25 3.35 I .97 SOR
2 .101 5x10 90 9O 29 6.0 l . 78 ADI
1.102 10x20 636 622 l 25 1 5.0 19.07 son
2.102 10x20 317 311 2 2 1 21. 27.20 ADI
1.103 15:30 1981; 19714 7 36 7 15. 158.00 son

 

 

 

 

 

 

 

 

 

VI. SOLUTIONS OF PLANE ELASTOSTATIC PROBLEMS

The second of three objectives of this dissertation is a general
computer program for the iterative solution of plane elastostatic problems.
The program ISOPEP includes six subroutines which contribute to this
objective. The program is not completely general, however, since sub~=
routines _for the boundary conditions must be added for any specific problem.
1 flow diagram, listings of FORTRAN source decks, description of input
preparation, and the output of a sample problem are provided in Appendix
B. The analysis of physical phenomena with mathematical models, the
analytical or numerical solutions of the models and the subsequent test»
ing by comparison of the solutions with experimental observations, consti-
tute major pursuits of scientists and engineers. The creation of the
ISOPEP program contributes to the analysis of plane elastostatic problems
and this computer program is considered a major part of the dissertation.

Duonstration of the utility of iterative methods for the solution
of several problems is the third objective of this dissertation. Problems
number two and six are included because analytical solutions are available
for comparison with the numerical solutions. Problems three, four and
five are representative of the practical problems for which analytical
solutions are not available. The treatment of boundary conditions for
irregular regions is included in the discussion of problem three.

1. T_h__e 29331 Eoblem .

The square plate and edge loads as shown in Fig. 6.1 are synetrical
with respect to the x—axis. By proper choice of constants of integration
the stress function will also be symmetrical with respect to the x-axis.
Starting at the origin and proceding counter-clockwise the boundary

conditions are obtained from Equations (11;) and the requirements for

37.91370
59

60

 

 

 

 

 

 

 

 

Ali
F e
HP - ‘T
______________ a V ’a
3 *—
‘——«
fl-d
”—1
k—4 .44
H1
5 c 1
a: a 7 (x

 

 

 

 

Figure 6.1 Boundaries of the model problem

The constants of integration 1, B, C, D. E, and F in the following
equations are determined so as to produce the desired symmetry of 0 and
consistent values of ¢ at each of the points b, c, d, e, f, g in the
functional expressions for 95 along neighboring line segments. Two of
the constants are arbitrarily chosen to make ¢ 3 0 and if: 0 at b.
Along ‘53

a“? ' a _

nay-'0' 3945-0;

along-Ed. ‘
§§=—ry+(3$)c=-m, gamma
4) = 'P92/2 ”7 : ‘Pyz/a ‘IzIPcz1 ,‘

alongd—e
%§=(}$)J= “'44P," ‘3‘?"0 .

_._ 45: "9QP9+B 2 ‘— 1+pr + o/anP 2'
along ef

M= (M)e=—,‘+a¢>5 =0

59

as

)

<1) I: C 2 (436') : -o.oaa‘p;

61

was?!
§$=*‘+PV+D $2.0,
4: = ~2ry‘+Dy+E = ~27°y1+126dry‘-32“a?;
maxi?
.. ad a .. ...
‘37: (5)3“0’ 53?" ’
¢=F.= 00

Values of the difference-fequation approximation of the stress
function 013 and nodal point values of thenormal stress component 0:11
are given in Fig. 6.2. The point values of the stress function are
approximated

2 3...... —
where C -'-’ 32 . The constants p and a have been assigned values of unity.
2. gemi-infinite 23333 33333 uniform 393.3 93 33:3 156mm" ' ‘

The exact solution of the problem of a semi-infinite plate with a
uniformly distributed load of intensity 'Po acting on an interval of the
bemdary -e 5 x 5 c, is given by Timoshenlao and 00on (1951). The
stress function has the form

9 m l (r20 - r191)

lumerical solutions of this probl- have been reported by Veyo and
Hormbeck (1961:), who used point successive overrelaiation of the biharmonic
finite-difference equations, and Pisacane end lalvern (1963), who used _a_
numerical mapping technique for application of the luskhelishvili complex
variable method. For comparison with the other numerical solutions a
. square region, 2 units by 2 units, of the semi-infinte plate will be
considered. The intensity of the distributed load P0 is chosen as {one
and the value of c isfi. For the assigned values of c and Po the stress

{motion can be written

62

FIGURE 6.2 Dlstributlone of the Stress Function and 0. for Problem 1

COLUMN
0 0 ' 1 0 7 C 1 10 11 13 11
:‘00000 101110 1.0300 3.1910 20.000 3.0710 109.00 1.1990 07200 000000 07300 11.
«210'u"1106'"'1100'"°1165"332566°.3210."Hudunflod""1210"00000.“:«100 6:

31.1101 1.1111 1.1910 1.1120 2.0111 2.0129 1.9100 1.1111 .1120 .0021 :1200
-5,“ 000:,“ 000:}.‘,00: £5461...:!;6 ...:5“‘..-:,z‘,...:}’i’..::‘16‘..: 28“’OOOE°1.’
1.1119 1.0111 1:1102 1:1110 2:0001 2.0012 1.9101 1.1011 .0911 -.0001 .1200
‘.32tt133:3221233:32221.33 32112333: 12113313221133z3211133: 31.2533z3111233: 3.1113333..:.
31.0202 1.1020 1:2121 1:0101 2:1290 2.1112 1.0199 1:1021 .0101 -.0002 .1200
'-32:1233:31:1133:31:2133:: 212233: 31:1133zz212133: 3221;33:3112133: 331211333321113333....
1.1100 1: 2120 1:1000 2:9111 2:0001 2.2010 1:0000 1:2010 :0121 - :0120 .1200
3-3111 3333211133:321:133zz211133zzzss 3333121 33:3111133: 3112 33::112133::1121333:.7:s
131010 1:1010 1:0211 220210 251010 2:1190 1:1110 1:2100 30100 -:0190 .1200

’ -:122233=3122133::211133::112133::222133: 3J12a233 321113333121133::112133z3112133331...
1:0101 1:0000 250019 250992 250119 2.0912 1:0110 1:1100 :0000 0.0219 .1200
'-3221233=3121133:3112133z3111133::212233::212233::22213333222133zzxtst333311113333:...
’2.0100 2.0010 2.1190 2.1000 2.1102 1.9919 1.0020 1.1119 .1002 -.0122 .1200
3J1115" 3111233:3111132 3221533331.: 33:3121 33:3222133z3221233:3122 333311113333:1ez
2:1012 2:0120 2.1111 2:0109 2:1911 1:0919 1:1290 1.0010 .1010 -.0101 :1200
3° -3122133: 3112233:3111133: 3111;33:3121233zz2o2s32 3212133::212 33:32:513232221333:21..
2.1121 2:0009 2:0011 2:2001 2:0111 1.1090 1.0100 130110 .1102 —.0011 S1100
“.:121133::112133::115132H131233 3112533z3111232 3212133=:112133=:121133::1151333:25.:
231000 252000 2:2090 2:0001 1:9090 1:0100 1:1019 :9000 :1109 -:0110 :1200
3.3111133::111133:311.133z314212311113 :31121 3:3112133z3112133:32.1:33:32as23333...1
2.0110 2.0101 1:990? 1:0902 1§1011 1:1192 1:2010 :9109 :0002 -.0021 .1200
" 3121 "131116"131111"131516"1:ii. 3313111133:3211232 “21523313212233 322133333....
1.0110 1.0102 1:1091 1:0010 1:1011 1:1912 1:1191 :0091 20010 -.0121 £1200
3-:221233zzat1133::222233z3111132 3111133::125133::212z33::121233::111133331112333z....
1:1009 1:1100 151219 1:0011 1:1010 1:2110 130119 E1090 E0002 ~50000 :1200
" 3222233::222133::211133::.21133:3111133: 31112333:22.133°3122133::122133::21223"3....
1:2011 1:2001 1:2101 1:2111 1:1121 1:0120 :9010 E0190 .1101 -:0902 :1200
3‘ 3211233=3222233z3211133:32222'333222233 :3112133:3121233:3111132 3121233:::21 3333....
:9990 :9902 :9001 :9010 :9221 :0110 :1091 :1111 .1010 -:1111 .1200
"-:222233=:211133::212233:322123 :3221233:3112 33:3111133:3211 33:3112 33:312213333....
.1100 .1110 51101 .1010 :0001 .0091 :1011 .0190 .2110 -.1110 :1200

*' -32112323222133::211133:311:233z32.1133:.3211233::111233::212233::211233::11123333....

.000. ...90 3.002 ..soo ....7 ..301 ..072 .131. .1..a -3191: 21200

*’ :66!!°":6611"::6315'"36611"136111"=36111":31119"531119'231111°'1§3‘11'":.122
.2221 .2210 :2219 :2100 :2100 .2119 g2122 .1991 .0910 -.1102 :1200
1°. :6“‘000066‘,000: .6“ 00006650000066. 000:6" 00006,; 000:,6‘ 000:“: 000:,‘§ 000:’,°,
.0011 :0011 :0010 $0011 :0110 €0101 :0010 :0109 .0101 -:1091 51200
3‘ 3.1213333221133332.113333221233332.2233:. 3251133zza12133zz112133:::11233::22213313....
22 :0000 0.0000 0.0000 0.0000 0.0000 0.0000 0:0000 0:0000 0.0000 -.1000 .1200

38666'°6:6666"836666°'616666"6:6666° 6.6566"5.6666"6.6556'°"1565°°1'6666" toooo

2 -== #2[{(x-2)‘+y3}ta2" iii/3*) 3‘ {02011232714 1%)“g *5"?

 

Values of the stress .. - _-

 

 

 

,00‘0

 

function on the bound»

(DP

arise of the square
region which are inside
of the semi-infinite
plate can be determined
from the exact solution .
The values of 91 .2 points
one nesh interval outside

the square region can be

 

 

 

similarly computed or

 

 

T__’_""'

‘1

Figure 603
derivatives of the exact 3_ Problem 2 - Semi-infinite plate

obtained from the normal

solution at the boundary. Due to symmetry it is possible to determine
the solution by considering one half the square region. Along the
y-axis which 1. the .111. of sy-etry, 211. nor-.1 derivative, g} g o,
is used to determine the values of 0 at points exterior to the solution
0min.

Alon; 3310 use Equations (11:) and find
2. .- at
%—$="R“" 373%:‘0:
_. ... 9 _.
'33?- X+B° 793412.39

(15 = -£X2+Bx+ C .
Sy-etry of ¢ with respect to the y-axis requires 3.0. Two of the
constants of integration may be arbitrarily selected. Let A u. c n 0.

6192

Thonnlongo—n-

§2Q=-X, c$13$=09 ¢=-PLX2'
é¥=(%¥)a=-’4o §%=(§$)Q=O2

‘ ¢=—j¢x+D=-hx+V32

Thovolno o£D=1/32 numwmo row-mt gag-U32 intho

1103135

moo-1m m- ¢ um. "53 moi-5'. Iota tho hat to tumor tho mu
Motion - é- : ond 1/3! on nooouory on n oonooquonoo of tho arbitrary
oohotion of I and c. _

Tho vnlnon of tho nor-:1 oonponont of mono 0")" nnd tho ntrooo
{motion nro 31m in 11:. 5.5 for 9 M8 nub. rho oonvorxonoo ,
oritorion nood m 10", nnd on mum «unto at tho solution notor
tor tho 111128 noon m obtunod fron intorpolotion bottom tho oolntion

moor M for o 7:11: noon. Tho total nubor of notation roquirod;
V’

 

 

 

PH
.1-»

02‘ '

y - 1/19, h - t/n.
Exact Solution

 

 

V Numerical Hopping

0 Succoooivo Ovorrolmtion of
Finite-differenco Equations

 

m. 6.1: cup-risen of mnoriool uni mot 50le for Problon t

65

for both problou too 1830. A flat distribution of 0.1 ot an interior
pointoyoo {nod for tho initiol ootinoto for the 7x11; nooh. Voyo and
Bamboo]: roportod opproxinotoly 3000 itorotiono woro roqnixod for tho
111128 looh with on initinl volno of -.0625 oooignod 'ot oll intorior
point- ond o oonvoo-gonoo oritorion of .5210'7. though point voluo of
tho oolntion of tho problon oolvod by Voyo ond Hombook voro not ovoilnblo
for oonporioon, it no not poooiblo to diocorn ow oignifioont littora-
onoo in tho rooulto moon in Fix. 6.14 Ind oinilor grnphiool rooulto in
thoir "poflo _ _ > .

rho «notion of tho itorotivo solution at tho mot. «liftoronoo
oquotim iron tho mot oolntion in greatest in thooo. rogiono whoro
tho otrooo groiionto oro motoot. Tho otrooo nodionto oro lone ot
tho tint row of intorior pointo_ bola! tho boundu-y onbjootod to tho
diatributod load. This dotribntion of 0'? for tho onct ind finite -—
ution-moo solution! at y» l/lh io ohoon in 11¢. 6.14. Tho nonoriool
uppu. oolntiono 0,! mooono no mm no o1oo ohooh. It is opporoht
thot tho um upping toohniqno .providoo only ought]; bottor
roonlto in tho rogion oirootly mdor tho oppuoo zooo thou-o rtrooo'
notionto *oro highoot ond noh pooror roonlto .noor tho bonnduy x g 1.
Both nuoriool nothodo provido bottor ”promotion to tho onot
oolntiono ot ran of nooh pointo oorrooponung to y a ah.
Problonob Egg-Iota“ tonsilom.

rho problolo of prootiool intoroot to tho onginoor froqnontly hovo
more!» bond-tion Ind ottontion in ofton oontorod on Aotrono oonoontron-
tion. on innotuotion o: otrooo oonoontrotiono hoo hooo oohomtoo by
oolving oooh notohod tonailo opooi-on problon for ditioront nooh mom
Throo diftoront tnoo of notohoo homo boon oonoidorodx

66
FIGURE 6.5 Distributions of the Stress Function and cry for Problem 2

COLUMN
Row 3 6 5 5 7 5 9 1o 11 12 13 16 15 16 17
.00000 -.ooz55 -.o1ozo - .02295 - .06017 -.05503 -.o7589 - .09575 -.11156 - .1296o .16732 -.1o517 -.15355 -.2ooo9 -.21e75
~116666°1i16666'1116666 11165666 11666661116666"116666'°116666 1116666°'116666°°'16666 '1" .0666 11:666611116666"11oooo
. C O
: z . . : C I z . : C C O z :

:600251 -600506 -601256 -602682 -606086 -605821 - 607593 -609376 -611161 -612966 61o132 -616518 -610306 -1zoo9o -62101?

.6:6666' :66666'::66666'::66666'::16 616'::66666'::6166 '::66666'::66661'::66666' ':66666'::66666'::66666":66666"66....
o o o o o o o

: 6 o 6 o o o o o o O O o 0

-100522 -1o1ooo -1o1755 ~15259o -1otato -105977 -1o7o9o -1o9ooo -111zoo -1129eo 116759 -:1?5:o ~115521 -1zo1o7 o121591
1166666°1166166'1166666°1166666'1166565'111696 '1165665'1166666'1166666'1166666"166666"-16661 1166666'11666661'160259

~1o1555 -1o1505 -1ozoos -1o3555 -1oo773 -1ooza. -1o7910 -1o9ooo -111327 -115075 116556 -11oooz -115575 -1zo151 -121925
1H66665' '.6666 '11 6666'1166666'116666i°1166fi '111666 '1165656'116666 '1166666"161666'1166666'1166666'1166666"156011
O 0

-102672 -102657 -1o:235 -1o~1oo -10533o -1oo725 -105255 -1o9oo7 -111535 -1152oo 116972 -115717 -115671 -126253 -121999
1H56666':W66666°':66666'1166656'1166566'1166566'11i56i6°1116665'1166666'1166665"166666'1166666'1166665'1166665"-1oo~5o

-1o3650 -1o:oo3 -1oo116 ~1o5951 -1ooooo -1o7253 ~1057oe -:10239 -111562 -115696 115152 -116593 -115521 -126352 -122112
116666 '°'6o666°"66666"'66566‘1166566°"‘6656'116056 '1116666°116655 '116656 "16656 '1166651'116666 ’1161566'1161155

3105655 -1o¢o9o -1osooo -1o5755 -1oo7oo -1o793o -1o9252 -11o7oo ~11223o -113525 115655 -117155 -115556 -1ao5oo -122271
0 oo o oo o o Ooo I o o o ooo O O
-166666'1W666'1E66666 116566611135676'-.66166'1166766'1:15566'1E16666'11o6566°°:66665 -.66616 11666661-166666'1oozozs
O 0
: : : : I : : : : : : 0

-165571 -1o5o11 ~105025 -1oooo9 -1o7551 -105562 -1o9599 -111259 -112713 -115260 115522 -117oo5 -1191o1 -1ao751 ~122651

¥6656 '116666 '1166566'3 6666°1165fi '116666 '1163 6'116666 '1166666°11i656 "16666 '1166656'116666 '1166566°1Eoz969
-1oo529 -105555 -1o7o3o -1o7537 -105653 -1o9655 ~11ooo5 -111556 -113256 ~11~725 115265 -117521 -119o55 -121o7o -12276:
111666662 H66666'166666'1161166'1166566'116616 '1166H66'116666'1H665i5'166666'1166566'1165666 1166666'1166666'1153906

“607606 -60771, '600060 “600616 -60936I -61029§ -611370 '612573 ~613002 '615276 6167~0 -618260 -619825 -6Il¢20 '623056

11166666216661121616662166666 1166661'1166666'1166656'1166665'1116666‘111666 °°166666'1166656'1165166'1166656'1166561
C 0 O z 0 O O O O C
: : : : : z z : z : : O O C

'00.6’0 -.00196 'o09609 -o09620 '610315 'o16176 -.1218~ 'ol3317 -olfi557 'o65..7 567296 '010797 -oZOZTQ ':360’2 '623525
1166666 111665 6 11166666111666661166666 1116666 '11 656611166656'1H6566111666661116661611166665111666661116666611605729
'o097.’ .009.33 -o10172 ‘:106‘5 -066290 'o62095 'o13030 'olhlo.’ 'o15233 '51‘556 66.7097 'o69309 '02077. ':':390 ':230‘3
11166666116 6566611 W66 '116355611666665 1165 6666516666611166“ 11666656116666661166266911666662" ~169166 16655661166555?

~110555 —110975 -11126o -111ooo -112257 -115039 -115927 -616936 -11oo53 -117252 115552 -119911 -121329 -122797 -126369
1116666621 665 ’1166666'1166566'1166656'11 6666'116666 '11 66662116666'1H1656 "116666’111166 '116666 '1166666’1107255

‘611995 0612079 '612329 -6127§O -613303 -61b008 -61§Ib§ -615800 -616.60 -610015 619251 -620550 -621921 -6ll)50 -626020

1,:ooo‘oooooooooooo ...5...’;;I...:5.6......... o-o-ziooooooo oooo ooo ooooooooo Ooooo ooooooo

- 325 6 -.32505 -.6165 3 -. 7931 -.2557 3261 -.6656o -:65506'1 166661 .13965 1116166 -.1o555 -.o9I66°11o793o
O U C C . C C
: : : : : : O : : : I O O : O

1:115105 -113155 -113626 -113505 -11o333 -11o997 -115757 -11oo92 -1177oz -115566 119959 -121265 —1225?? -123965 -12537o
oo o o o o o o o lo
-1665662 :6666 '11 6666'1166663'1166666°1166666 1166666°1.66695'1.ii666'-1i66 '6 '166655'1166655 -111666 1166666" -1oo~95
.
z : : : : O O z I O O 6 I :

1;61¢226 -61¢297 -616517 -61001I -615376 -616003 ~616750 -617610 -618571 -619625 620161 -621971 -623267 -636502 -625960
o o ooo o oo o ooo o 0

-’66666'- .6666! 1165836‘-.66566".26§£6-:25706'-.11666'- .16663'1215617".16167"116.76 1116662 -111666' 1116161’1165976
O 6 : : : 6 6 6 I z : : : :

1:61936! -615662 -615610 -615999 -616629 ~617022 '617732 -6105b9 -619661 -620h15 621565 -622730 ~623961 -63525) -626590
00 000000 000 00000000... ooooooooooooooo oooooo ooooooo oozi ooooooo oOooooooooooobooo o 000

1165165 -:26616 -:666 95 -:2555o -.65155 -.22755 -:61666 1116521 -:i7506 -:66666 io5oo -.631oo -.11756 -.1o6oI -.o93oo
o o o o o o
z 6 6 o o o o o o o o o o o

'6IOQOQ ‘66‘529 -61672Q '6170Q6 -617¢91 -610056 -66|729 -61950. '62038b -621350 622397 -6Z3§18 -6ZQ707 '635950 '621263

1116666 “116666 '1166666'116666 '116665 '116165 ’116661 '1116666°1165666'11 6666"116666'1116666'1111666'1116561°11o9557

: : : : : : z z : : O : z : :
23117557 -117559 -117o33 -115139 -115551 -119o97 -119759 -126655 -121521 -122257 125252 -12~556 ~125652 -1aoo95 -127959
0 o. oo oo o o ooo o o no. oo 0 I o o
1666662166666'1166666'1166963 116666611.2ios6'1.l6fi '1.655352 165676'1116666°'. 15906 -166666 1112666 116066611109963
0 C .
: : : : z z : : : z : : : O .

-11o711 -11577o -1159oo -119255 -119635 ~620168 ~120711 ~121o72 -122275 ~123155 125132 -12517o ~125256 ~127655 -125556
1116666 111666661116 26661116666511166666111 6666'116616 11166665°11666i6°1166665'°166616°11162551116666611166666'1110139

~619l37 -619093 -620060 -6203!7 -620720 ~621207 ~121793 -122675 -623255 -626090 625036 -626036 -621109 -63|2§4 -629536
‘3666666': :61661'::66 666'::66666'::66666'::66666'::16666'::66666'::16666': :66666":16666'::66666'::66666'::16616'::1....
~62096¢ -621010 -621111 -621§41 -621001 -622273 -622036 -623687 -625226 -625050 625941 -626910 -627955 -619055 -630212
'1666666': 66666'::16666"'1o616"'16.66"'116.6' 1666 "'16 66':: 66:11' 16666"'66666':: 6666'::66666'::66666'::..m
-622092 -62214! -622295 -622960 -622090 -623365 -623082 -62b509 -625221 -626012 626879 ~621|17 -625822 -629OIO -631012
22:66616':: 666'" 6666'::16166'::66666'::6666'::16661': 16666"'1.666'::6666"'1616""16.16'::66111' :11111'::.....

'623221 -62326’ '623QIS '623657 -623993 -62#¢21 -626930 -6225Q1 -626226 -626969 627026 -6ZI713 -629706 -63o160 -631036

z116611621666662166616' 11i6666"—116666 1115656 1166666" -1i6695 '1116665 '1116696 '166666'1116 M6 6'11 666511166666'1116571
O C C
: : z : : O O . C O 6 : : : :

z-62Q’SO -62§3’6 '62b536 -62Q70. -615091 -625502 ~626000 -626500 -627261 -627977 62.7.6 ~62966¢ -630607 -631660 -632671

0.0.0....0.0.0.0....OOOIOOOOOOOIOOOOOOOOI......OOOOOIOOOOOOOOOOOOOOOOOOOCOOOOO0.0000............OCOIOOOOCIOOCOC H.

U
51y

67

Problen 3 _- Seu1circular notches, Problem 11 - V-notches and Problem 5 =—
Rectsnguler notches. Sketches of the specimens are included in Figures

6.10, 6.11, 6.120
E

 

 

 

1-66---” - --- ----g-o—H

23--------—---—-—-—7n

H

Figure 6.6 Solution domain for Problem 3, h and 5

Boundary conditions for the three problems are identical. Intro-
ducing e Cartesian coordinste system as shown in Fig. 6.6 integration
constants will be chosen so the stress function is symetricel with
respect to the x and y axes end it is necessary to solve: each problem
for only one quarter of the whole plate. It theend '53 e uniformly dis.-
tributed tensile loed of Po units intensity is applied. Using Equations
(11;) we find

“>461

94>...
'::FI, E“Ey+(%$l,’

But (6'35 3 0, if d is sy-etric with respect to the x-nds; hence
43= B‘f‘é + B = Elf/a.
The constants L and a are arbitrarily selected so sero. Then along 671’
93—)? = 0 5 ’3'? : 24 7; I
¢ Z ¢C : 2 a2 P0 ‘
Along the ho axes of sy-etry the nonel derivative conditions are used
to compute values of the stress function in two mesh ran exterior to

27
.1250
.1253

25
.1091
.1061

23
.0833
.0031.

i£&§'°‘255ii"°2655i"'2

21
.0626
.0632

19
.0“26
.0830

17
.0012
.0239
.0258

COLUMN
I
20002
:0010
:0070
I009.

I
0.0000

0.0000

13
-.0182
‘I0179
—.0170
‘I0156
‘.007‘

.
'I0003

11
-.0321
315"!

3

I

I

. .
‘.0205

I

I
-I0159

-2019.

3&1§"i

I
I
-:0317

-Ioaos

I
‘.0250

:
-:oazo
-Io.ov

I
-Iozes

I

I
-.0305

-I0025

0.1.
0355
3357"i

.
'.0316

I
I
.
I
I
I
'I0056
.

I
’.0992

‘.0497

-10355

-10399

”1:57.56”125356"i

“.0539
:
-I°53“
I
‘.0520
I
I
I
’.0690

“.0553
‘I0560
'.053I
-.0512
'.0609

I

I
-0036.

I
I

ROH

FIGURE 6.7 Distributions of the Stress Function and o, for Problem 3

0 6 0 b. 0 1. 0 1. 0 5 0 fly 0 6 0 0 0 3 0 7 0 2. 0 6 0 7 0 by 0 9 0 6
6. 3 Ra 0 E4 2 K. 0 ‘4 1 Eu 0 5. 6 5' 6 G4 6 H. 6 S. 0 fl. 1 ‘5 6 K: 6 ‘3 2 I: 3
2 2 .2 I 2 7 .2 9 2 1. 2 1 2 9 2 6 2 2 2 7 2 1. 2 5 .2 7 2 9 .2 0 .2 0
1 0 .1 1 .1 2 1 I. 1 6. 1 6 1 6 .1 7 1 0 1 0 1 9 1 9 .1 9 1 9 .1 0 .1 0

I II III III III II III II II II III III III III II III I. III III II III III III II II III II III III III II III.II III II III III I
I I I I I I I I I I I I I I 61 I1

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I
2.9 3.0 ‘.6 SI“ .6I1. 7.6 7.6. 0.1; 0.9 900v 9Ifiv 9IR‘ OIO- OI? Olfiv 006
6.0 6.3 6..“ 6.0 6.6 6.0 6.7 6.5 6.6 6.7 6.2 6.0 5.3 5.9 5.1 5.7
0.9 0.nv 0.0. 0.9. 0.0 0.6 0.2 0.0 0.3 0.7 0.1. 0.6. 0I6 0I7 0.9. 0.9
1.1 1.3 1.6. 1.6 1.5 1.6 1.7 1.7 1.0 1.0 1.9 1.9 1.9 1.9 1.9 1.9
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

I I I I I I I I I I I I I I I I

I I I I I . . I I . I I . I I .

I I I I I I I I I I I I I I I I
0.3 1.7 5.7 9.2 2.6 5.1 0.9 0.9 2.5 3.9 5..» 6.6 6.3 7.6 7.... 0.3
3.0 6.6 6.5 6.6 5.5 5.0 5.0 6.1 6.9 6.1 6.9 6.2 6.1 6.6 6.0 6.6
0.7 0.6 0.6 0.1 0.7 0.3 0.7 0.2 0.5 0.9 0.1 0.6 0.6 0.7 0.0 0.9
0.3 0.6 0.5 0.6 0.6 0.7 0.7 0.0 0.0 0.0 0.9 0.9 0.9 0.9 0.9 0.9
IIIIIIIIIIIIIIIIIIIII‘IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII.IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
I I I I I I I I I I I I I I I I

I I I I I I I I I I . I I I I I

I I I I I I I I I I I I I I I I
0.6 0.9 6.3 6.7 02 6.3 2.3 6.3 0.6 3.7 62 0.... 0.6 1.0 2.2 3.7
6.0 6.6 5.5 6.7 7.5 7.9 8.0 8.3 9.3 9.5 9.5 9.2 0.6 0.9 0.9 0.6
60 6.6 6.9 6.3 67 6.0 6.5 6.6 6.9 6.1 6.3 6.5 7.6 7.7 7.0 7.9
05 0.6 0.6 0.7 07 0.0 0.0 0.0 0.0 0.9 0.9 0.9 0.9 0.9 09 0.9
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I
209 6.2 9.0 1.1 2.9 2.6 0.6 7.6 6.6 9.6 6.0 7.2 0.7 3.3 67 5.6
5.0 6.0 7.6 9.2 0.6 1.1 2.5 2.9 3.2 3.6 6.5 6.6 5.5 5.6 5.1 5.7
I :1 6 2’ 6 I“ I I6 5 0! $5.9 81.0 5 :1 5 :5 5 7“ 5 33 filob ‘4I7 K'Ia 5 XV ‘4I9
0.0 0.8 0.0 0.0 0.6 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII.IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I
0 35 0 :1 6 30 11.0 9 36 91.7 6 :9 E’I6 6..“ 5.01 0 :2 14.7 “6.6 1.06 3 l5 hvo6

7 .3 9.3 1.9 3.0 6.5 6.2 7 .1 0.2 9.6 0.7 0.0 1.3 1.7 2.1 2.5 2.0
.2.2 .2Io .3.0 15.0 3.7 3.7, 3.7 3.7 3.7 5.7 “.0 “.0 .0.6 .9.9 6 A, 6.9
0.0 0.0 0.9 0.9 0.9 0.9 0 .9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

I1 I1 I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I

3 I“ 0 23 1..5 1..3 14.9 flvoo 6 96 9 :1 I.I0 I.I9 9 23 6 :5 5.I7 6 :1 0 22 1..7
22 6.7. 7.2 9 .5 1.3 3.7 B .3 5.3 7.5 0.0 0..» 9.1 0.9 0.9 1.9 1.9

1 .9 .....u. 1.1 1 .0 2.6 2.8 2.4. 2.2 2.1 2.0 2.0 2.0 3.9 3.9 39 3.9
0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.9 0.9 09 0.9
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIOIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

.1. Il IL I1 .1 I7. I1 I11 I1 III III. I1 I I I I

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I
2.... 7.6 5.1. 1.0 5.3 6.0 5.5 1.9 5.5 0.9 0.7 7.0 6.0 9.0 3.0 6.0
1.2 1.6 “.0 7 .1 9.9 1.1 3.8 5.8 6.2 7.8 0.7 9.0 0.2 0.6 1 .3 1.0
001 0.5 0.1 0I7 003 1.1 1.8 1.6 1I5 1.3 1I2 .101 «(III 2.0 2 .0 2.0
0.3 0.2 0.2 0.1 0.1 0.1. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
. I1 I1 I1 .1 71 I1 .1 .1 I1 .1 I1 I1 I1 21 '1 I1

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I

6 .0 1.6 9.5 9 .9 2.5 1.7 3.... 2.6 0.6 2.7 5.0 5.0 3.6 0.7 I .1 7.7
2.0 9.2 5.2 2.7 0.8 2.3 6.3 6.6 7.3 9.3 0.7 1.3 2.1 3.2 3.6 3.1
IIIB o :5 o :0 o :5 nUIg nUI6 o :5 o :U o :0 AVIb 7.I~ 1.I3 IIIZ \IIl 7II0 1.I°
0.3 0.3 0.2 0.2 0.1 0.1 .0 .1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
.01 _.1 ..1 _.1 . ‘1 .1 .1 I1 .1 .1 .1 I1 .1 I1 ,1 I1

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I
3.... 6.9 1.2 9.6 0.5 3.2 9.7 0.0 0.0 6.6 9.6 1.6 0.6 0.6 3.1. 6.1
1.6 7.1 6.2 0.5 3.2 5.3 2.7 0.6 1.0 2.3 3.2 5.6 6.9 67 7.0 7.2
2.3 1.0 1.3 1.0 0.6 0.0 0.6 .025 0.0 0.0 0.6 0.4 0.2 0.1 0.0 0.0
0.8 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
..1 ..1 _.1 ..1 .01 ..1 _.1 _I1 I1 .1 .1 .1 I1 91 '1 I1

I I . . . . . I I I I I I I I I

I I I I I I I I I I I I I I I I
6.1 0.0 1.5 7.0 5.5 7.9 1.6 0.6 0.3 1.6 6.6 5.5 5.0 3.6 9.6 3.9
7.0 3.1 0.3 6.7 3.2 0.0 0.2 5.7 3.6 2.0 0.6 0.3 1.6 2.1 2.0 3.2
2.5 2.1 2I6 1 ..1 1.7 1.3 0I9 OIS OIZ °I9 0I7 OI.) 0.3 002 001 OIO
0.6 0.4 0.3 0.3 0.2 0.). 0.1 0.1 0.... 0.0 0.0 0.0 0.0 0.0 0.0 0.0
II....II.....I.....II..I......I...............II.._.III..........IIII.....I..IIIIII...IIIIIIIII. .IIIIIII
..1 _.1 ..1 _.1 ..1 ..1 ..1 ..1 ..1 ..1 ..1 .1 .1 I1 91 .1

I I I . I I I I I I 6 I I I I I

I I . I I I I I I I . . I I I I
5.9 5.6 7.0 2.1 9.1 9.2 2.2 9.5 0.3 0.0 5.0 2.2 1.0 2.2 3.1 7.0
.1.0 .IIS .3.0 .0.“ 6.9 .3.6 .1.6 0.0. .6.6 .9.6 .3.0 .2.0 .109 0.1. 0.7. 0.3
3.6 2.2 2.0 2.3 1.0 1..» 1.0 0.6 0.3 0.0 0.0 0.5 0.3 0.2 0.1 0.0
0.6 0.4 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
.01 ..1 _.1 _.1 .21 ..1 ..1 _I1 .01 ..1 ..1 ..1 _.1 .I1 21 .1

I I I I I I I I I I I I I I I I

I I I I I I I . I I I I I I I I

705 7.6 9.0 3.2 02 0I1 37 9I6 0I9 °I0 6.3 1.03 °I6 III 56 1o].
2.... 0.9 6.6 1.9 0.6 5.... 2.0 9.3 7.9 6.9 6.3 3.0 2.0 1.6 0.1 0.3
3.7 2.2 2.0 2.3 1.9 1.5 1.1 0.7 0.3 0.0 0.0 0.6 0.6 0.2 0.1 0.0
0.6 0.6 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
. 1 . 1 . 1 _ 1 . 1 _ 1 . 1 _ 1 - 1 . 1 . 1 o 1 . 1 - 1 . 1 — 1

9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9

1 2 2 2 2 2 3 3 3 3 3 6 6 I 6 I

I
.1250
1.0000

.1050

1.0000

.0868
IIIIIIIIIIIIIIIIIIIIIIIII

I
1.0000

00703
.0000

"i

.0555

.0625

.0312

.0217

.0130

.0070

I0036

.0000

.0000

69

FIGURE 6.8 Distributions of the Stress Function and 0‘; for Problem 4

COLUMN
3 3 7 9 11 13 13 11 19 21 23 23 27
Rg'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIII.
1.3023 1.3203 1.3166 1.6963 1.7391 2.3066 3.6332
2 2 2 2 2 2 2
I I I I I . I
-00‘5I ‘OOIII -0060. '603I’ -0026’ -001.6 60°10 0°20.
IIIIIIIIIIIIIIIIOIIIIIIIIIIIIIIIIIIIIII 0 6060000666... 6
1. 166 1.3317 1.3963 1.3316 1.7936 2.2361 2.6331 .6169
2 2 2 2 2 2 2 2
-20639 —20627 -20392 -20331 -20263 -20126 20029 20213 .0616
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII
1.3361 1.3776 1.6606 1.3676 1.7729 5.9929 1.6333 1.0333 .3076
2 2 2 2 2 2 2 2 2
I I I I I
-2o613 -.06o3 -.o366 -20306 -20213 -.oo96 20031 .0226 .0620 .0623
IIIIIIIIIIIIIIIII'IIIIIIIIIIIIIIIIII II IIIIIIIIIIIIIIIIIIIIII II:I ”I I
1. 006 1.6166 1.6699 1.3626 1.6135 5.7266 1.3619 1.0919 65552 .3371
2 2 2 2 2 2 2 2 2 2
I
-20366 -2o373 -20336 -20213 -20163 -20066 20076 20263 20626 .0621 .0633
“1:11.11"1:1111°°1:1111'°1:1111'°1:1111'13111"1:1111"1:1111°°°:1111°'°:1111"221..
I : I I I I I I I I I
I I I I I I I I I I I
-.0332 -.o339 -.0302 -.0239 -.0130 -.0036 .0101 .0262 .0660 :06)! .0636 .1061
’ IIII HIIIIIII HIIIIIIIIIII'IIIIIIIIIIIIIIII III II “I IIIII IIIII ”I HIIIII ”I I
1.6396 1.6635 1.6393 1.6176 1.6131 1.6136 522556 1.0635 :7555 25552 :3055 .1103
I I I I I I I I I I I I
I I I I I I I I I I I I
-.0316 -.0306 -.0266 -.0206 -.o116 -.0001 .0126 .0261 .0633 .0660 .0637 .1062 .1230
1 IIIII IIII HIIIIIIIIIIIIIOIIIIIIIIII “III IIIIIIIIIIIIIIIIIIII:IIIIIIIIII IIIIIIIIII'IIIIIIIII
1. 6292 1.6255 1.6307 1.6239 1.3919 1.3263 1.1937 1.0216 6117 :6552 .6062 .2060 .0101
2 . 2 2 2 2 2 2 2 2 2 2 2
I I I I I I I I I I I
n-20200 -20266 -.0231 -.0111 -.0066 .0021 .0130 .0300 .0661 .0666 .0661 .1063 .1230
I I I I I Q I I II I I I
355 2252655522525565 2523555 25.5552225:2 25 2522525552 222522 002522225555222265252222559322225155 22.1260
I
2 2 2 2 2 2 . 2 2 2 2 .
I I I I I I I I I I
-20263 -20233 -.0191 -.0136 -.0036 .0066 20173 .0316 .0660 .0636 2°222 .1066 :1250
, IIIIIIIIIII IIIIIIIIII IIIIII. .IIIIII I IIIII HIIIIIII IIIIIIIIIIIIIII HIIIIIII HIIIIIIIIII
1. 3692 .3655 1.3360 5.3296 5.2630 5.2139 52 5561 .9692 .6679 .7026 .3319 .6110 22111
I
2 . 2 2 2 2 . . 2 . . 2 2

1-20211-20199-20165 -20101 -20026 20012 20196 20336 20692 20666 20669 21063 21230
H. .556IIIi:555,II!:ii£IIIi: .iai...izizii..!:i; ‘IIIlasaaIIIzasaiIIIIa‘ésIIII,5‘IIIIIII‘iIIII; IIIz‘o39

-20119 -2o166 -20133 -20019 -20002 20093 20213 20369 20302 20611 20632 21066 21230
a’x:i“i"i:iafla..1.”I...i: 5;;,IIi:’ai’II‘:‘::,IIiIIIi‘IIII550IIIIIIiigIIII,7iIIII:;;;;III:;;‘; II:,l96

-20130 -20160II-201o7 -20033 20020 20113 20230 20362 20312 20677 20333 21061 21230

5 666666 ”I 666666666666666666666666666666 6666666666666 66666666666 HIIIIIOIIOIIIO 600.0
102I71 162555 .162320 1.20., 1.170. 16115.. .10016, 6 “I7 .IIII 6.033 67230 0663‘ :6179
2 . 2 2 2 2 2 2 2 2 2 2 2

1-2012t -2o116 ~20062 -20030 20061 20136 20263 20316 20320 20662 20636 21061 21230
122555225225552252 55522252 .5555225.' 5555225 2555 22525525222255552'22555222225525222255552222525522227006
I I I I

6 6 6 6

I 6 : 6 2 : 6 6 6 o I 6 6

o 6 6 6 o 6 6 6 6 6 6 6
“60101 "60091 ”60061 ”60°50 00060 60550 6025. 603II 60527 00“. 00.60 010‘. 61250

29 6666666 6666666666666666666666666666666666666 666 6666666666666 6666666666.66666666666666666666
53175I 1.1720 165655 161~36 161152 ‘007‘l i6°221 69651 69091 6.592 6.57. 67562 67702

6 6 6 6 6 6 6 6 6 6 6 6

I 6 6 6 o o 6 6 I o 6 I 6

-20061-2oo71-20061 20001 20016 20163 20269 20393 20336 20690 20662 21066 21230
1 IIIIIIIiIII I.IIIIII.IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII.II.I

1: 1662 :1555" 5255552252555522525555" 525366 1.0160 .9660 .9206 .6629 .6367 .6330 226292
I I I

I I I I I I I I I I I I I
-20066 -20036 40023 20022 20069 20113 20279 20601 20339 20693 20663 21069 21230

” OOOOIOiOUO 0'0 000060 00. 6600000 00 000000’ 000666066066 600060666600. 66666600660600...
15.1161611 52 55 55 956 5.0766 520655 1.005 .9613 ”5 .9031 .6662 .6163 .6793

I I I I I I I z I I I I I

33 40066 -20039 -20011 20033 20101 20163 20261 20601 20363 20696 20663 21069 21230
125555225255572 5255252252555522525555225255552252552222 22555522225552222252552222555 2222555522229213
I I I I I I I I I I I
: O o O o 0 6 o 0 I I o I
I

-20036 -.0026 020000 20061 20111 20196 20296 20612 20361 20696 20666 21069 21230

’11 III IIIIIIII II II .IIII'IiIIII III IIIIIIIIIIIIIIIIIII. IIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIII
6561.0665 525 625 5:03731.0655 5.0232 .9979 .9705 .969 .9376 .9331 .9399 .9332

I I I I I I I I I I

I I I I I I I I I I I I I

I I I I I I I I I I I I
-.0023 -.oo16 .0010 .0036 .0119 .0201 .0300 .0617 0330 .0100 .0666 .1030 .1230

9 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII II IIIIIIIIIIII: IIIII IIIIIIIIIIII IIIIIIIIIIIII
1.0669 1.0632 1.0669 1.0639 1.0330 1.0113 25555 .9132 9311 .9319 :5556 .9623 .9606

I I I I I I I I I I I I

I I I I I I I I I I I I I

I I I I I I I I I I I I I
-6001? '60008 6001. 60063 60526 60206 0030‘ 60610 60552 60701 60067 61050 61250

IIIIIIIIIII .IIII .II. IIIIIIIIIIIIIIIIIII'IIIIIIIIIIIIIIIIIIIIIIIIIIIIII. III II I
1. 0326 1.0555 2.55552 252 5526 1252552 25.0112 .9929 .9766 .9663 .96 3 .9102 .9793 .9976
I
. . . 2 2 2 2 2 2 2 2 2 2
-20010 -20001 20026 20069 20131 20210 20306 20622 20336 20702 20661 21030 21230
, IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII.II IIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII. II
1.0191 1.0212 1.0227 1.0223 1.01752 2525555 .9921 .9616 .9767 .9137 .9623 .99552 25.0061
I I I I I I I I I . I I I
I I I I I I I I I I I I I
I I I I I I I I I . I I I
-.0003 .0002 .0029 .0013 .0136 .0213 .0310 .0626 .0333 .0103 .0666 .1030 .1230
IIIIIIII .0 IIIII II.III.IIII.II ”II III II II... IIIIII IIIIIIIIIIIIIIIIIII III I.
1. 0102 1. 0555 2555522 1.0160 1.0105' 525555 29960 .9655 .9529 .9636 .9913 .95552 25.0060
2 2 2 2 2 2 2 2 2 2 2 . 2

~20002 20006 20032 i20076 20131 20213 20311 20626 20333 20703 20666 21030. 21230

7 III II. III. ..II.IIII.IIIIIIIIIIIII HIIIIIII II. III. IIIIIIIIIIII ”III II I.
120065 125552 1255552 20011 1.0036 1.0010 25556 .9925 25501.9931 .5570 .55552 25.0037
2 . 2 . 2 2 2 2 2 2 2 . 2

.9 .0000 20006 20036 20071 20136 20216 20312 20623 20333 20703 20666 21030 21230
1255552 221255522 25255552252: 552522 2555522 2555 222255552222555522225555 '2’2555522225555225255552 2520020
I
. . 2 . . 2 2 2 . 2 . . 2
I I I I I I I I I I I I I
.0000 .0006 .0036 .0076 .0136 .0211 .0312 .0623 .0333 .0703 .0666 .1030 .1230

51 III. ...:...6...:...6..i.656 IIiIII.IIIiIIIIIIIIIIIOIII

1.000 000 000 .0000 .0000 1.0000 5255552252555522525555225255552252555522520000

u

0‘:

70

FIGURE 6.9 Distributions of the Stress Function and er, for Problem 5

COLUMN
Row 3 5 7 9 11 13 15 17 19 21 23 25 27
-.0020 -.o0o3 -.0552 -.0007 -.0307 -.0191 0.0000
1. '5555'°i35555"535555"535555°' 35555"535555"539503
. . 3 3 3 . 3
s-30010 - 30002 -30551 -30000 -3o300 -.0191 030000
00 00000.0... 0.... 00.000.00.000... 0 .0
1.9257 1. 9027 1.9795 2.0227 2.05733 5. 0335 539052
3 3 3 3 3 . 3
-30013 .0590 -30507 ~3o003 -3o305 -30190 030000
1000000 .00....C0000000I0.00.0000000000..0..00.0
135510 1. 9099 1.9520 2.0115 2.0730 2.0902 2.0100
. 3 3 3 3 3 3
0 0 . 0 0 0 0
9-:°0°3 ”00501 '00,}. -00057 -9030} '00109 000000
0000.. .00000000000.000.00.00000000 00.00.00. 0
1:0307 1. 0500 1.9007 1.9029 2.0905' 2.1709 5.1219
. 3 3 3 3 3 3
-30507 —30572 -30525 -30007 -30335 -30107 030000
l100000000....000D0.0000.00.00.....000000000000000
157600 5.707; 10.307 109225 290709 202973 203626
3 3 3 3 3 3 3
-30505 -30551 -30500 ~30031 -303 0 -3o193 030000
oooooooo:ooo 900000900 000000000... 00 900059000
1. .0993 7155 1.7530 5.0272 1.9005 5.3020 3.0090
. . 3 3 3 3 3
-30537 -30523 .-30000 ~30007 -3o305 -30170 030000 .0200 .0010. .0025 .0033 .1001 .1250
5 0000000000000 ..0000000000.000.000.00000000000000000 00 .00 000 00. 00000000000000...
1.0020 1.0553 1: 0001 1.7302 1.0300 2.0027 0.2013 03550000 0.0000 5.5555 .0000 0.0000 0.0000
3 3 3 3 3 3 3 3 3 3 3 3 3
-30502 -30000 -30000 -30370 -30270 -30105 30010 30212 30010 30020 30032 31001 31250
H.000... H00 ”9 0000 9009909009909000000900000.900990 concoooooooooooooonzo 0.0.0000 H0000. ”OI.
130019 1.0155 1. 0020 1.0900 1.7009 1.9310 1.0000 .5500 .1530 .m5 .0250 -.0037 -30503
. . 3 3 3 3 3 3 3 . 3 3 .
. . . 0 0 0 0 0 . 0 0 0
-.0002 -30000 -.o007 -.0330 -.0200 -.0112 .0003 .0225 .0021 .0025 .0032 .1001 .1250
19 o ..i.... 09.0.. ...... o. ... 00. 0.0.0.000...000000000000:0. 0000: 0000000000 H00... ”0 I
1:5555 005 '1.0125 1.0525 5.7050 5.7000 1.0509 .0320 055 000 .0900 .0135 -.1130
0 . 0 0 0 0 0
0 0 : 0 0 0 0 0 0 : z : 0
0 I I 0 0 0 0 0 0 0 0 0
-.0010 -.0005 -.o300 -.0290 -.0201 -.0o70 .0071 .0203 .0031 30029 .0033 .1001 .1250
00.00.00.000 0000009 000000000000... 00 09.99099009905090009090900999090.90900050 000...... .0000
1 5005 1.5505 1.5755 1.5950 1.5975 3.5200 1.2907 .9210 .5000 .3510 .5000 .0053 -.1117
o o o o o O 0
0 0 0 0 0 0 0 0 0 : z : 0
0 0 0 0 0 0 0 0 0 0 0
-.0372 -.0359 -.o320 -.0250 -.0101 -.0o03 .0099 .0203 .0003 30035 .0035 .1001 .1250
135555 'i35iii"i35555"i35555 '533535"! I33555°°i': 51 35"‘355 55"‘3355 "°33555"'3 535‘ "35333" 330001
0 0
. . 3 3 3 . . 3 3 . 3 3
. 0 0 0 0 0 0 0 0 0
-0032. -0031, ‘00275 ‘00211 ’00122 '90009 00127 00203 :0... 0000’ 00.39 :5002 :1250
099000.000.0.900.000.00090900.00.00.05090000000955.0000...0000000509 00.0000 0.0000 00.00 I .00.
1.0039 1.0713 1.0720 1.0577 1.0117 1.3100 1.1027 .9000 .7505 055 .0030 .5055 .0001
0 . 0 0 0 . 0 0 0 0 0 0
0 O O 0 0 . O 0 0 : 0 0 .
-30201-30209 -30232 -30171 -30005 30023 30153 30303 30070 30051 30003 31003 31250
00000000. to. 00.o.0000000000000.090000000-00.0000000000000000000000000.000.900000000000000000000
1. 0120 133300 1.5100 1.3910 1.3015 1.2539 1.1200 .9090 .0009 .0370 .0092 .3023 .1952
3 . 3 3 3 3 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 0 0 0
29'90239 '00227 '90192 “00133 ”00°51 00°53 9017. 90322 000.3 90059 90.07 030.0 03250
135553"i35353°'535535°'I35553"535555'°1355533'535555"'35555"'35555"'33535"335553"'33355°"33200
0 0 0 0 0
0 0 0 0 z : 0 . 0 0 0 0 z :
,1-30199 -30100 -3o150 -30097 -30019 30000 30201 30300 30090 30007 30050 31005 31250
135555"i35555°'i35555' 5355553'535555"535335"i35555"'35555"'35553"'3'355'°'33355"°35555"°3051o
0 0 I 0
0 . : z : 0 : : 0 0 0 . 0
. 0 0 0 0 0 0 0 0 0 0 0 0
-.o103 -.0152 -.o120 -.oo05 .0009 .0100 .0222 .0350 .0507 .0070 .0050 .1000 .1250
33 0000.0 .0 .000 .00. 000000.000 0.0 .00.... 0.000000000000000... 0000000 000.00.000.00..0..00
135555 132503 5.2095 1.2270 531552 5.1330 1.0591 .9700 .0055 .7035 .7053 .0255 .5002
3 3 3 3 . . 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 . 0
-30131 —.0120 -.0009 -.0030 .0035 .0120 .0200 .0370 .0517 .0000 .0057 .1007 .1250
351.000.0600... 00 Hooooooooooooooooooooooo H00... 000.000....OOO0000.......OOQOI0000.00.00.00000000
:2022 1.2077 1.2021 1.1039 1.1513 131009 1.0501 .970 .9000 .0209 .7037 .7002 .0000
. 3 3 3 3 3 3 3 3 3 3 3 3
O c o o o o o O o
-.0102 -.oo92 -.0001 -.oo11 30050 .0100 .0250 .0303 30520 .0005 30059 31000 .1250
H0... 0.. Moo-0900000909990090.9000. H.000... ono.0000.o0000000000000000.000000.900000000000000
131555 1.1030 1.1590 1.1000 1.1102 1:00001.0317 .9777 .9101 .0591 .0150 .7737 .7525
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
-.oo70 -30000 -30037 30011 30079 30105 30271 30390 30530 30090 30002 31000 31250
000000000 000 005.900a90.000cocoo.u00090-00099000099099.00099.000000.000000.00900000000000.0000...
1.1190 131205 1.1215 1.1101 1.0093 1.0001 1.0215 .9001 .9302 .0095 .0000 .0330 .0279
0 0 0 0 0 0 0
0 0 0 0 z . 0 0 0 0 0 0 0
—30050 -30005 -3oo10 30030 30090 30100 30203 30003 30500 30090 30000 31009 31250
‘1 0.00 00000.0.0000....0000.0.00.00.00.000000000000.0.000000....00000.0000000000000...00000000000000
1. 0052 1. 0099 1.0001 1.0790 1.0002 1.0027 1.0130 .9032 .9009 .9100 .0992 .0007 .0910
. 3 3 3 3 3 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 0 0
h) ':0030 '90020 .°:°°°° 00000 00110 90393 00293 90011 00500 :0097 00.05 .1009 .1250
00.000 00.000 0.000000000000000000000 000.000.000.000. 00-00 .0 00.0 .00 0000000000 0
1. :5575 135555 13 0593 31.0539 1.0033 1.0200 31.0070 .9007 '.9029' 35003 ' .5510 .9205 '339013
3 . 3 3 3 3 3 3 3 3 3 3 3
0 o o 9 o 0 o o O O o 0 o
-.0021-:oo12 .0010 .0059 .0122 .0203 .0301 .0017 .0550 .0700 .0000 .1050 .1250
, 0000 0. 0.0.00 0.0000 0.0.000000.000000.00.000.00000000000000000.00000000. 0.0000. 000 0
135330 135555 1.0355 1. 5320 1.0259 1.0107 1.0037 .9900 .9750 .9011 .9557 :9555 .9770
3 3 3 3 ‘ 3 3 3 3 3 3 3 3
‘1-30009 -30001 30025 30009 30131 30210 30307 30021 30553 30701 30007 31050 31250
13555 "i35i55"535555'°535i55"i': Hi53'°535553'° 35555°"35553"°35555"'3555 55"‘3555 "'35 53" 39997
0 z 0 0 0 : 0 0 : 0 0 0 0
O o o o o O o o I 0 0 O O
-.0002 .0000 .0032 .0075 .0130 .0215 .0311 .0020 .0555 .0702 .0007 .1050 .1250
00.000.000.000... 00. 00000000000.00.0.00.000000000000000.000000000000000 000000... 0000000 00
1. 000 3 1.0003 135537 1.0007 1.0030 1.0020 9 .9909 .9901 .9550 .9939 .9932 1.0052
0 0 0
. . 3 3 3 . 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 0 0
.0000 .0000 .0030 .0070 .0130 .0217 30312 .0025 .0555 .0703 .0000 .1050 .1250
1 000.000.000.00... .00 ”O 00 0.0000000. 09.900.00.00... 00 o. 0000. 0000000000
1.0000 1.0000 130055 1.0555 1.0000 5.0000 1.0000 13 5555 535555 535555 1.5555 1.0000 1.0000

"9|:

71

the solution domain. Along the xsaxis, .3? :3 0, while along the yaaxis
g? = 0. Along the notch edge if or To: and 3?
53%va %§= (%:,4’)4=2a73

(b = ZQBy +C = 2aE(y—a)
The value of c is determined by the value of ¢d in the squetion used
along 3'5. The lost expression for g! is valid for :11 3 problems but
for Problem 5_ it simp1ifies to g s 0 along 3?. . '

The oolutions given in Figures 6.7, 6.8 end 6.9 were obtained with
boundsry oonoitions for which Po 9.! 1 end It “3;. Though a 12:21; mesh is
used for listing the solution in all three figures, the results are
token fro: the solution obtained for e. 2MB Iesh.

Stress concentretions occur at the base of the notch st y a e.

For the V snd seni-oirculsr notches the concentration occurs elong the
y—sxis. For the rectangular notch the concentretion occurs st 1 u s,

y g s. The oonsrisons of stress ooncentrstions for different choices
of nesh sise ere given in Figures 6.10, 6.11, 6.12. B. I. Peterson (1953)
gives sn snot stress oonoentretion of 3.08 st the bees of thee-i-
oirsulsr netshes in e tensile specinsn infinitely long. The stress eon-
osntrstion vould be slightly higher in e specimen of finite length.
Bouthsell (1956) determined s stress oonoentretion footer of 3.0 using

s 'relsxsticn' solution for e neoh of 116 points. .Solutions' obtsined
vith the ISOPEP progre- produce stress concentretions of 2.92 and 3.211
for neshes of- 65 end 1038 points respectively.

It the bees of the V-notoh or the corner of the reotsngulsr notch
even slight strsins induce stresses of high-ugnitude end the- stresses
detersined from s solution of the bihsrsonic difference squeticns would
herdly represent sn sctusl ptvsiosl stste of stress. However, stress

 

 

 

 

 

 

 

 

 

 

 

 

B
i? L
in
A
..l
3, « I:
* +—1
.__l
+4
3651’ ‘ Emu-eta
:4
O
U3
U3
:2 2. ‘
E—u‘
LC
4
A ififflrw
1 5” --«--———- ISOPEP - 1038 Mesh PtSe
J m~-~-- SOUTW‘IBLL - 116 Mesh PtS.
--------- ISOPEP - 65 Mesh P‘tS.
1. "
w _ g s ; A :1 - ; . . . 1*; y
0 1/6a 1/3a 1/2a 2/3a 5/6a a

ORDINATE AT 2: = 0
Figure 6.10 Stress Concentrationin a semi-circular notch

distributions obtained under the assumption of an ideally elastic body
are useful in the analysis of the plasticity problen. Exact solutions

of these problems would provide V infinite stress concentrations for these
two problems. The numerical solution provides a set of discrete values
of the stress function. The value at each nesh point is an average value
associated with an area surrounding the point. The area in general is
proportional to hz. As the uniform nesh space h is decreased, the stress
function solution at a point of high stress concentration would provide
a better approximation of the high stress. Figures 6.11 and 6.12 illus-

trate this phenomenon for the V and rectangular notches.

STRESS ox

 

 

 

 

 

 

 

 

 

H
g ‘3
9." L \ f
a_ . Fr
4—4 H
(—-u ....)
8.1 3,. 1' L4;
4—4 r-->
‘_1 -—>
*“ ,///l :: § ”’
7.’ * pl 3,. ‘ oil—.4
ISOPEP
6 {_ — 107A Mesh Pts.

0

... _ __ -i 267 Mesh Pts.

________ 66 Mesh Pts.

 

 

r'8.65

 

 

1/6a 1/3a 1/2. 2/3a 5/6a
’ ORDINATE AT x.= 0

Figure 6.11 Stress concentration in the V-notch

a

78

TABLE 5 STRESS FUNCTHON A16801N15 ADJACENT TO
THE SEME-CflRCULAR NOTCH

 

 

 

 

 

 

 

 

 

ne-e—e—e-sjg
NO‘WJ‘WN.

 

———u———— eQQLUMN
WP]; 13 111 15 16 17 18
3 -302569 -301823 ~300971 03 00000
4 ~302560 ~301816 ~300966 83 88080
5 -302533 -301795 ~300953 . 300002
6 -302h88 ~301757 -380928 300009
a -3OZ#23 -301703 ~388890 380025 '
-302380 ~301631 ~300833 308056 301842
9 -302239 ~3015h1 -308759 300187 381857 '
10 -302123 -301h35 -388678 300173 301093 302085 ’
11 -301992 ~3©1316 -3@0566 300256 3011h9 302107 303125
12 ~301851 -301186 -308&50 308353 301223 302155 3031h0
13 -301700 -3010#6 -3®032h 300863 301313 302221 303180
18 -301Sh1 -300898 -300189 300582 301115 302302 303239
15 -301378 -3007&5 ~300889 300788 301523 302392 303309
16 -301212 -300590 388098 300837 301637 302888 303386
17 -.01o&§J-.0013§ 00239 .ooagafl .01153 .02588 .03168
19 go 21 V_22 23 21 25""
30h166

30h180 305209 ' ' .
30h216 30522h 386253 307291 ' ‘
308266 305255 306269 387297 308338 309375 310%
308328 305296 306298, 307310 308339 309376 310
.0h389 .OSth .06326 .07338 .083h9 .09379 .101 6

 

 

 

 

 

 

TABLE 6 X-STRESS COMPONENT AT POHNTS ADJACENT TO
THE SEMB-CURCULAR NOTCH

M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

h _ 6 CQLJJMNB 2
1 1 1 ‘ 1 17 1 19 20 21 2

3 23th 23785 33238

8 23831 23701 23952

5 23369 23629 23576

6 23275 23506 23111

3 23158 23345 23061

23031 23130 23193 '

9 13917 13955 13911 13386 '

18 13805 13801 137h5 13668 13817

11 13708 13678 13611 13586 13367 13186 '

12 13628 13586 13516 13188 13213 3955 3695

13 13563 13518 13118 13338 13178 39th 3653 3839 '

18 13508 13h55 13380 13278 13131 3980 3711 3860 3238 '

15 13860 13h©5 13331 13229 13999 3936 3786 3543 3351 3612

16 13h17 13362 13288 13193 13078 3931 3769 3598 3131 3286

17 1.379 1.323 1.251 1.162 1.853 .927 .786 .638 .891 .358

TABLE 7 Y-STRESS COMPONENT AT POBNTS ADJACENT TO
THE SEMI-CERCULAR NOTCH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CQLHMN
13 10 15 16 17 18 19 20 21 22

ROW ' ‘~ . . .H H .3

3 3329 3209 0. 000

0 3328 3199 3000

5 3361.263 3110

6 3395 .326 3220

7 3025 .012 .305

8 .000 3002 .000

9 3350 .303 .357 .087

10 .312 3326 .372 .051 ‘

11 32713295 .308 .038 .580 ' '

12 3222 3200 .286 3350 3003 3590 3323

13 3169 3185 3210 .257 .318 3005 3500

10 3116 3126 3105 .175 .216 3268 3321 .352 ‘

15 3068 3070 3088 .108 .136 .170 3205 3226 3219 .180
16 .025 .030 .000 .055 .076 .102 .127 .107 .152 .135

TABLE 8 XY-SHEAR STRESS AT POINTS ADJACENT TO
THE SEMI-CHRCULAR NOTCH

0 13 10 1S 1% 17 1§_‘ 19 29 21 22

3 03000 03000

0 -3100 -3153

6 -3266 -3391

g -;300 -0‘95

-3307 -306 '

9 -3312 - .051 -3605

10 -3312 - .030 -3578

11 -3302 -.399 -3S11 - .633

'12 -3287 -3365 -3008 -3 35 - .628 - .578

‘3 -;275 -0337 -01‘01‘ ..o 70 -;530 ’:562 -

10 -3266 -3319 -3370 - .026 -3070 -3091 -3069

15 -3261 -3307 -3353 -3396 -3031 -3007 -3035 -.385 -.303 '
16 -.258 .298 .338 -.375 -.003 -.017 - .009 -.376 -.317 -.202

76

TABLE 9 STRESS FUNCTHON AT POENTS ADJACENT TO THE

THE V-NOTCH

 

QQLUMN

 

_ 12

HB

111

LJS

lg

H7

IQ

 

73
C)
i

‘d‘dddda

-;02fl87
-.02fl62
-;02095
-208997
-;on876
-;0fl7h@
-00“ 595
“50310152
-;03286
-00‘ ‘ 28
-;00969
-;00811
-;00656
-000353

‘.011570)
-;0fl5h0
-;oflh66
-;©U362
-301] 2h©
-;Oflfl06
-300965
-;008fl8
-300669
-;005fl8
-000367
-;00298
-000070

;00072

.002fl3

-900853
-;008U6
‘200737
-0 ©0631}
-0005fl 8
-;00392
--000260
-;00023
500035
.00355
.00295
.00h3h
;0057O
9007©h
.00834

0.00000
g00035
.003®h
.00fl9fl
.0029fl
'ZOOAOfl
.005fl7
.00638
.00763
.00890
OOUOUW
.0flflh3
.0fl268
.0fl390

 

\ #9

’0fl@60
'.0fl266

50357fl

003 O .022
-__?IT_—

.OUOhfl

.Oflflflfl
.0flfl8fl

.0fl36fl
.OflQGh

.Ofl682
0fl79h
:0fl906
.02®fl7
.0292?

201005

;02®83
502097
.02fi36
202193
.0226h

.023h5
.02h33

.02526
:02621
.02738
;028fi5
502911

.03125
;03336
;03167
.OBZUA
.03273
.033hl
303hn5
'.03h9h

.03575

'03557

.03739

:03832

 

agddfi

.0h966
.Ohfl75

.0h201
:0h239
.0h288
;0h3h5
30hh07
.Ohh72
.0h539
.0h607

 

ease
Nmmrwnaammugl‘Nmmkwnaemmw¢mrw

.0h67h

 

.05208
;05205
.05236
.05268
.05398
.OSSSh
;05%®#
.05#57
.QSSflU
.05565

 

.06250
.06255
.06272
.06298
.06330
.©6366
.06h®6
.06hh7
.06h88

 

.0729fl
g07296
.07309
.07329
.07353
0073811
.07h00
.07hh0

303333
308336
.083h6
008360
.08378
008397
.08hfl7

 

 

 

;fl0hfl6
;30h37
;30421
gfl0h26

.IOhBI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE 89 XMSFRESS COMPONENT A8 POTNTS ADJACENT TO
THE VuNOTCH

_ _ _« COLUMN ‘__ ._

83 8h 85 86 87 88 89 20 28 22

Row
3 2.306 3.858 8.653
A 2.350 2.952 3.557 8.628
5 2.288 2.608 2.633 8.539 .874
6 2.809 2.269 2.839 8.55C .958 .642
7 8.992 2.088 8.853 8.573 8.05 .708 .567
8 8.898 8.888 8.670 8.900 8.086 .798 .559 A82
9 8.725 8.672 8.582 8.335 8.098 852 638 .956 .337

80 8.622 8.562 8.886 8.280 8.085 .880 .699 .525 .375 .273
88 8.539 8. 975 8.372 8.235 8.072 .902 .736 .588 .938 .308
12 8.070 8. #07 8.385 8.896 8.059 .983 .767 .625 .492 .367

13 8.483 8. 352 8.268 8.868 8.085 .989 .789 .668 .538 .02
Th 8.366 8.306 8.230 8.837 8.032 .928 .805 .690 .577 .h67
85 8.326 8. 269 8.898 8.885 8.028 .928 .887 .783 .680 .509
16 8.292 8.238 8.872 8.096 8.088 .928 .826 .732 .637 595
87 8.263 8.288 8.850 8.079 8.003 .920 .839 748! .668 .576

TABLE 88 Y-STRESS COMPONENT AT POTNTs ADJACENT TO
THE V-NOTCH
COLUMN
83. 8h 85 86 87 88 89 20 28 22
Row
3 8.376 8.705 8.622
A 8.033 .983 .787 .874
5.665 .529 .882 .725 .682
6 .h28 .388 .289 .868 .573 507
7 .273 .282 .287 .325 .h26 .969 .482
8 .879 .887 .863 .233 .383 .365 .380 .337

9 .885 .098 .887 .867 .229 .278 .305 .388 .273 _
80 .066 .059 .076 .885 .86h .206 .236 .250 .258 .286
88 028 .026 .088 .072 .880 .886 .875 .892 .899 .896
82 -.002 - .003 080 .035 .066 .097 .823 .848 .809 .850
83 -.029 - 027 -.086 .003 .028 .955 .078 .099 .808 .807
In -.058 -.049 -.O39 ~.023 -.002 .089 .080 .056 .067 .078
85 -.O69 -.066 -.058 -.Ohh -.027 - .008 .008 .028 .035 .088
I6 -.O8h -.080 -.073 -.O68 -.047 - .038 -.085 -.008 .0098 .08;

 

 

 

 

 

 

 

 

 

 

78

TABLE 82 XY- SHEAR STRESS COMPONENT AT POINTS ADJACENT
TO THE V-NOTCH

 

 

 

 

 

The finest mesh used for the notched tensile specimen problems

"a We
gram.includes a subroutine STRESS for the calculation of any or all of

the stress components a}, a; and

 

The corresponding mesh interval h is 1/L8.

 

 

 

T;y.

 

 

 

 

 

C08.UMN

ROW '3 8“ 852 _86 87 flak 89 2O 2 22
3 0.000 0.000 0.000
h 0‘“, 0.000 -0672
5 - 0°96 -ofl2h _06S© ‘09®2
6 O. 000 -;228 -0563 ‘6773 “.698

- 7 "' .080 -0268 ‘0h98 -0650 -0653 -055h
8 ‘:‘]Jh -e286 "ekhe -056“ -0586 ’.5'00 -3160
9 -o“69 -0289 -0'O09 “.1895 -9525 -950“ ‘5‘th -0369
‘0 “.891 -0286 -0378 'ekhs -0u75 ‘0“67 -9h30 “037“ -0301

' In -020“ -028. -0352 -0“06 "oh3h -e"33 -e“07 -e36h “.380 -02“,
'2 “.28” ”027‘. -9330 -0375 -0399 -el§@2 -6385 -6352 -0307 -025“
‘3 -0236 -0267 -e3‘flk -0350 -9378 -0376 -036“ -03,8 -030“ ".255
I“ -.289 “.260 “e299 -9329 -03“? -0353 “63NS -0325 -029h -025“
‘5 '.22® -.255 '.287 -.301 -.328 -.333 “6328 -.381 -.286 -.251
‘6 -0120 -0250 -e277 -e298 -0301} “630$ ‘03“: "'e299 -0277 -02~7

The ISPOEP pro-

Values of all three stress coupon-

ents and the stress function in a region surrounding the point of stress

concentration are provided in Tables 5 to 12 for the semi-circular and V-

notches. See Appendix B for the finite-difference equations used. These

equations are valid at interior points only. Calculation of stresses
at points on an irregular boundary are not included in the subroutine

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘J
._ a 7: 6.28
6. , gr
.— +—->
«J .9
4—4 —->
w t4 -t X
5° ;_ f —+
‘— f H 4059
' (— 4 *—->
‘L _i ,
4° ‘ Li.

 

 

STRESS ox

 

996 Mesh Ptso
- ‘_ 2A6 Mesh Pts.

1; _'3"’_-

...._.__m 60 Mesh Pts.

 

 

O 1/6a 1/3a 1/2a 2/3a 5/6a a
ORDINATE AT x.= 8

Figure 6.12 Stress concentration in a rectangular notch

Though a solution obtained with a coarse mesh does not provide a
good.meesure of the stress at a point of stress concentration, the results
shown in Figures 6°10, 6011 and 6012 indicate a decrease in mesh interval
11 produces only slight changes in the stresses computed at all other pointso

Use of an average'value of the stress function over an area surround?
in; a mesh point depends more on the magnitude of the area than on the
geometry'of the region. For example at the'base of the Yanotoh the
area surrounding the mesh point is taken as (bout half the area for an
interior mesh pointo “Does this not imply that the solution obtained at

this point is a better approximation for s.tensile specimen'with a

r30

rounded rather than a sharp corner at the base of the notch? Some evim
dence for this conjecture is provided by comparison of the stress con»
centration factors fovmd from the numerical solations of Problems hOl‘9
1102 and [£8 with the stress concentration factors provided by Peterson
(1953) for notched flat bars in tensiono Peterson provides values of
the stress concentration factor Kt for tensile specimens with deep

notches with parallel sides and a semicircular base as sheen in Figure

6°13o The stress concentration is defined

653(3):

norm

where 0:0“ is the average stress across the minimum cross section of

Kt:

the specimen" The curve plotted in Figure 6013 shows the relationship
of xi to r/h where r isthe notch radius and D is the'width of the bars
This curve is based on the Neuberatheory solutions tabulated by Peterson
(pp 2647) for d/D s 05 where d is the minimum distance across the bar
at the base of the notch. The stress concentration factors for three

different numerical solutions of Problem )4 are listed in Table 133

Table 13 Stress concentrations factors for the V-notch tensile specimen
# w

 

 

 

 

 

 

 

 

 

Problem :32 h 0;: Ofnbrn Kt r/D
hol 6x12 1/12 h0362 2 2.181 1/12
1102 121211 I/Zh 60083 2 3 901:2 l/Zh
hoe 21am 1A3 80653 2 _ b.326 1/h8

Assuming the numerical solution approximates the

 

solution for a bar

with circular notch such that r g 119 the stress concentration factors

Kt for problems 401 934029 108 are plotted as three points labeled in

Figure 60139

and the Neuber theory solutionso

There is good agreement between the numerical solutions

Peterson reports that for notches

with inclined sides having an included angle (X and a circular are at

31.

the base, the notch angle has very little effect if 0° < (If 90°.

 

 

 

 

 

 

 

 

0 ”mm """“““'<' -.; ,r
6" “it a” I
“'T
(1 D
P 4. i :F
5o .. / ’X~_-i
X“ // U L‘\ r
S V '\ r Mino radius
is)
c: L” .-
.3
R; \
H
o.)
C'.
G.)
is.)
8 3°
m
w
3
u
U)
Y
29 ‘
Neuber theory solutions
7 Numerical Solutions
2;?
0 .01 .02 .03 .04 .05 .06 .07 .08 .09 .{0

r/ D

Figure 6.13 Stress Concentration factor Kt for a notched flat bar
Treatment g§_1rrg‘g;ar boundaries

The uni-circular notch ie typical of the irregular boundaries
often encountered in practical problem. Griffin and Verge (1962) chow
how different mesh intervals can be used so that each horizontal. grid
line crossing the irregulnr boundary intersects a vertical grid line
on the boundary. There are several adventegee to this approach, but
it doee increase the number of arithmetic operatione needed for the

calculation of each vector iterate. For this reason e uniform neeh

82

interval is usedin ISOPEP, andthe irregularhanndaryieamroxinated
with straight line segments, introduced arbitrarily, as shown in -

Fig. 6.1L, connecting nodes of the primarpmeah. , “(gmhe dual-mesh is
formed by lines halfway between lines [of theapttimary mesh.) The points
labeled B are on the boundary and valueeof "31.1. are conputeduein;
Equation (28) of Chapter III. The only complication is the evaluation
of VZUi for points on the boundary. A few nameless till illustrate

the technique.
- ..L 3 a
VZUO “ Ho 9g #45

‘oeH
‘4

d

 

1e"

17" ' / L:

16» C x ‘
158—3.! /l/
Th

fie A
J;

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13
f l ‘ J l I I. l -- _.
1+ 5 6 7 8 9 10 1.1 12

 

J-—y

figure 6.11: Boundary points for the semi-circular notch
It an interior point where no . h? the line integral is approxilated
(see Fig. .393 and Equation (27))
so a
Wfiiw we: +‘Uz-Uo)'fff + wa- uni—3 +(u4-Uo)%—z]
For the notch bounduy

9.15- a _
ay"/a, 33—0.

83

at a boundary point such as P153 (I a 15, J . 3), Figures 6.1!; and
6015<‘)a

[1st —-=—-<U.,~ua), Has a

M;
C
O
v

vzu = pm %(u,+u3)-2Uo+-H.

Thus the appreciaation to the line‘intecrale along 53' and ii: is m.
in term of central difference quotients at the. ends of the intervals
instead of at ' an interior‘ point. This illustrates an additional couplin-
tion of the irregular boundary, which appears at host or the boundary

 

 

 

 

 

 

 

 

 

 

P0111“. _ y
n ”1'“ P13 ’11. Hearse 6.1L and 6.150,), (a) p15“
3 d
[3» Jam" “' (Ug- U0)” [c3 gals—gm U) f a
L a» MJWU§$>¢cose *(§-§L$M]3%, = g ,
=I35° 2- ‘ (b)'P18,11 .
VZUO=F[UI+Uq-2U,+%Jo ,, A/C
At the point ’15 6’ Films 6.11; and 6.15“); o I ‘ ,
c 'b
L}; :15 z (‘f-U Us )9 f5 Sid!) z'(U'.-U°) . a flack/f

may)? W]

a y 4 cl c{
L-gcl‘szpii'é‘yUa-Uo) ., ; 3%®~M!

[_UuU- 2U 52(in m]
02' + MW

 

 

 

 

 

vzuo=

 

 

”#054917"
1+

 

Figure 6.15 Treatment of boundary points

. ll . IillI
i“ (in!

“111.31! ‘1! .

8h

The evaluation of V2U(15,6) is used in equation (28) for the finite-
difference equations at the point Pm’ée It is not possible to write
finite-difference equations for point ’15,,6 using Equation (28). .Hence
the evaluation of “(15,6) and the discrete values of the stress function
at other points labeled with a c in Fig. 6.11; are obtained by. interpola-
tion. Fox (1950) treats this problem with the Gregory-{Newton forward
interpolation forsula. A point 0 exterior to the boundary is introduced
as shosn in Pu. 6.16. Iith
the value of 03 and III; . (fig—)3

 

specified on the boundary, it is M g |~—h .4
possible to eliminate U o and O 1 2 3

+1;
V
x

determine 01 at the first interior

point in terns of the boundary
conditions and one or more interior

Figure. 6.16 Interior points
points as follows near a boundary

- (x-xa QQJX-XJX-x A3U.(x-x. x- - A’LL - - -
U‘Uo'i-AUO h + 2! +4 +T—W4-WWW4' """‘ .

Let s - (XE - xo)/h then

U8 2 Ua + (U,-U,)S+(U2-2U,+U.)‘i%;!fl + (U3- 3U1+3U,"U.)"" {SEEM-2) +

(Ut- ‘+U,+eu2-Lw,+u.)fl‘il}+§im +-— —-- ,

U2:(U'_U°)+‘UZ-2UI+U0)(S‘*)+(U3”3U3+3U,-Uo)'1§'2;3§!£:—2 +
(Ut-‘+U3+suz-I+u,+u.)iL-fir’—-L€"' ‘:’-=1 - +----

9

or
US = mu, +D2U, +1302 + D,U,+D5u., +H,
AU; = EnUO + EZUI + LUZ + Etua + E: U? + H2.
where the D1 and 81 are fourth degree polynomials in I obtained by
retaining differences up to fourth order while H1 and 32 involve fifth

8:3

and higher differences of U as well as higher degree terms in s. Neglect:-
ing H1 and Hz the external point value can be eliminated and the expres«

sion of U1 is

u.=[%z— as + (gs — %)Ua + (%-§%)U,+(-Ef-Bf>ut]/(-§é- 15-2)

. 0
Though it is preferable to retain differences of at least the fourth '
order, similar formulas can be written for higher or lower order differ»
ences.
Hoblem ._6_ - grectfiular hole _in an infinite plate

The solution for an infinite plate subjected to uniform tensile
stress Po at x 2.00 and x a ~00 is provided by Bevin (1951) for several
different approximations of the boundaries of a rectangular hole. The
stresses on the boundary of a square hole obtained from one of Savin's
solutions will be used for comparison with the stresses obtained from
a numerical solution. The problem solved numerically is for a 2 unit
by 2 unit region surrounding the hole. The hole dimensions are 2/3 by

2/3. Values of the stress 11
' JL

 

function and normal derivatives

 

on the outer boundary of the Pa 35 E
*- .

selected region are determined

 

from Savin's solution. The

 

nunerical solution of the

 

 

 

biharmonic equation provides a

basis for the determination of £9.

lo"

 

 

 

 

the stresses on the boundary of

 

the hole,and these are compared
with stresses from Savin's

solution for a curvilinear Figure 6.3.? Problem 6
Infinite plate with a square hole

86

approximation to the square hole.

Savin's solution is obtained using luskhelivshili's method. It is
known that the stress function 95 (x,y) can be written

¢ (m) - u. [‘z‘mz) + 5(a)]

where Re means the real part of the complex expression, (X (z) and 5 (a)
are analytic functions of the complex variable a = x + 1y. '2— a: x-iy.

Savin's solution is obtained by conformal mapping of the region
exterior to the hole onto the interior of the unit circle in the
complex ¢-plane by a mapping function 2 a w (g‘ ). The mapping function,
approximated by three terms of an infinite series, and the complex stress
functions a (fi‘) 5 (I: (1“? fl and the derivative of g (3‘) 5 @[OKS‘fl
are given by Savin (1961) pp 51-53 as

com) = RH? -— ZL¢3 +32% 3'71}

(fig) = BREW + o.426§'+0.046(’3+o.008$15+ 0.00‘I' V719

..~_.§_c/ CS")..- _:_ 0.55%?-0.657i'3—o.026<’5-o.0299'7l
LP(S')_, d? "' BRLQS, + /+ 0.55“, —-O,/25878 ’

where 735:4- i)? .-_- (9e19, (f: ,7? ), ( E, 6) are rectangular and polar
coordinates respectively in the complex q plane. Points on the circum-
ference of the circle correspond to points on the boundary of the square
hole, but when the series for Ca) ( $’) is truncated, the correspondence is
not exact. The Cartesian coordinates of any point in the plate in terms
of the polar coordinates in the complex plme are given by

x = R (+C059-g-3C0539 +—§.—>g7-C0,S' 76) ,

, = .3 (fysine + game -— sumo.
The boundary of the hole defined by these equations when p g 1 is not
an exact square. The edges are slightly bowed and the corners are
approximated with circular arcs of radius r e 0.02hsa, where a is the

distance along the x-axis from one edge of the curvilinear square to

87

the other. H is the length of one side of the square hole. The value of
_R a .39321508 is used in the problem solved numerically. This assures the

transformation of the hole “Y

—————--—~—~—

      

boundary at a distance of 1/3
from the origin onto the bounds-
arycfthe circle (Delinthe

 

 

 

 

 

l
1
g; I
complex plane. Thus a g 2/3, a l
’ l
and the corner radius of curva- E
I
ture for Savin's solution is :
' 1 l
r .— o.0163. 0 v"
. ' Figure 6.18 Approxima-
The equations given by tion of. the square hole

Savin do not permit the direct evaluation of the stress function along

the boundaries 1 g l and y ..-. 1. Derivatives can be found from
‘7'—

%¥2+‘2—§§ == «(3’) + tagger) + W?) ‘
See, for example, Savin (1961) p. 6 or Huskhelishvili (1953) Po 183.
Values of (3 which correspond to points on x a l and y a l, are deter:-
mined for 0° 5 9 5 90° at intervals A9 a 3°. For these values of
and 91% and .3; are calculated.“ The availability of a basic set of
complex variable subroutines and statements in 3600 ' roam simplify
these calculations. . The values of, the derivatives of ¢ thus determined
relate to points unequally spaced along x e l and y a 1. A five point
Lagrange interpolation formula is used to find values of the derivatives
at equally spaced points. Along the x—axis, quadrature of g—g provides
the point values of at up to an additive constant. The g; values specify
a normal derivative. 1. similar procedure is used to obtain values of ¢

and normal derivatives along the boundary y g l.

88

The general problem of a multiplyeconnected region is discussed by
Griffin and Varga (1962). The constants of integration can be arbitrarily
chosen on an exterior closed boundary. However, for each interior closed

bomdaly it is necesealy to determine “wig-21“” one point P]: on

ax’
the boundary in such a w that the components:6 of displacement u.v and
the rotation a), . J. (.._ 3' - 351;.) will be single-valued. The three
additional mlmowns for each interior boundary are related to the point
values of the stress function in the region surrounding the hole in a

manner which permits use of iterative methods.

 

 

 

 

 

Since the purpose of the “Y .
”sent nmerical solution was | d b B
only to test to. efficacy of the -'
finite-difference method in re- 33'
producing details of the rapidly- "
varying stress near a stress- &.L—F 3
concentration point by comparison ’

with Savin's solution, the proce- a, L, X
dure of Griffin md Verge for a '6 26 l

' ” Figure 6.19 nomain of solution
multiply-connected region was not 4 ., for Problem 6

followed. Instead the results of Gavin's solution were used to choose

the integration constants on the boundary x a lin such a m that {1 -,.°
at the point ( l/3,o ), and the assumption that d is sy-etric with respect
to the x and y axes will satisfy the required conditions of single-valued-
ness. The symmetry conditions then took care of the other constants on
the hole boundary, and it was possible to solve the problem Iith one
quarter of the. plate as. a simply-connected region, see Fig. 6.19. The
x-axis is a line ‘of. symmetry. Hence gg— - 0 along the x-axis. The

89

 

ywaxis is also a line of symmetry. Hence éég— = 0 along the y-axis.
Along Z? 2
_.JLQ ._ _ 312...
07""3y3-_o’ 7701“" QXAy—Os
22_ .21 - a _A _
ay ' (99):.‘0: x“(ax q— ,q
¢rB=o
The value of B is zero since it has been assumed that ¢a = 0.
Along 3?
y- 3X‘—0 5):“ jangyo -' 09
§$= (Hie-‘0’ 3.2-“ 53%):
¢=C=¢F=O
Hence i-O.

Along R and Ed boundary values of the stress function and its normal
derivatives are obtained from Savin's solution.

Iterative solutions were obtained for mesh intervals of 1/15,
1/30, and 1/h8. Values of the stress function and a; at the inter-
section of every fourth grid row and grid column of the M8 mesh are
given in Fig. 6.20.

A comparison of the stress concentrations found in the numerical
solution and those given on p. 53 of Savin (1961 ) is provided in Table
11.. Values of 6'; given by Savin are the values of 6} in a rectang-
ular coordinate system which has the origin at the point under investi-
gation and the y-axis tangent at the given point to the curve 6 - 1.
See Savin p. 8. The table includes comparable stresses, of, for the

iterative solution with a convergence criterion of ‘IO'6

and a second
set 65 for the solution obtained with a convergence criterion of 10-7.
Both are included because this appears to be an exception to the

general case in which stress concentrations increase as the convergence

90

FIGURE 6.20 Stress Function and 6‘: Distributions for Problem 6

    
  

COLUMN
no. 3 1 11 13 19 23 21 31 33 39 63 61 31
.00000 -.00266 -.00886 —.01661 -.02600 —.03201 0.06057 -.o6119 c.05652 g
’ ...-IIIIIIIIIIIIIIIIIIOIIIIIIIIIIII IIIIIIIIII ..IIIIII ......IIII...
.0000 .0110 .0009 .2013 .3216 .6300 .3230 .3911 is
I I I I I I I I
I I I I I : I I I
I I I I I I I I I
.00000 -.00260 -.00033 -.01309 -.02366 -.03136 -.o3016 -.o6311 -.03223
7 eeaeeeeeeeeeeeeeeeee eeee 000000;. eeeOee eeeeeleeeeeete 0.. as
.0000 .0266 .1165 :2366 .366 69692 .3666 .613 36
I I I I I I
I I I I I I I I I
200000 -200233 -200131 -201369 -202002 ~202663 -203311 -203933 -206320
11 eeeeeeeeeeeeeeeeeeeeeseieeaeeeeea ee 00 eeeeee eeeeeeeeoe eeeeeOeaneI
.0000 .0660 .2110 .3332 .6300 :3393 .6036 .6396
Q I I I I I I I
o e e e e I e e e
200000 -200139 -200662 -200031 -201316 -201613 -202326 -202060 -203363
1, eee-eeee aesaeeeeeaeeeeOeeeaseeeae eeaeeeeeeeeoee eoeeeee 00006
20000 . 123 .6011.3106 mi .6616 :6933 .9293
I I I I I I
I I I I z I : z I
.00000 0.00000 0. 00000 0. 00000 0200000 200163 200130 200033 -200191 -200303 -200031 -201239 -2o1633
19 e eeeeeee 00600.0 .6eeeeeeeeeeeeegoeeeeoeeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeeeeeeeeee
1.66662 266992 2121136 0363 6 6330 1.1013 .9119 .0660 .0119 .1999 .1992 .0032
. 2 2 2 2 2 2 2 2 2 2 2
0 a a
.00313 200302 200609 200610 200061 .01161 .01606 201301 201603 201361 201066 0200900 0200660
II... III... IIIIII IIIIIIIIIIIIIIIIOIIIIIIIIIII I IIIIIIIIIIIII IIIIIIIIIIOI
126116 1262962 21. 6609 1.6390 21. 3063 1.3063 1.1916 1.0336 .9936 .9263 :6969 .0022
I I I I I I I I I I I
z I I I I \ I I\ I I I I I I
202211 202296 202310J 202323 202190 203139 203610 203100 203031 203063 203011 203692 203323
21 e e eeeaeeeeeeeeeeeeee eeeeeeeeeeeeeeee eeaeeeeee eee eeeeeee so es 000 eeeee 000006000000...
1.6120 1.3006 1.6010 1.6210 1.3391 1.3220 1.966 1: 1326 126936 1.0196 :9130 .9600
: : : : : \ : : . : : . : :
I I

203010 203033 203166 203369 203661 .06030 .06393 206106 206936 201013 201132 201121 201036
2212661222123929221236662212322922222661221226722212232622121996221212222212676I22120296222299952222

200139 200101 200912 209136 209.39 209001 210116 210322 210000 211026 211161 211262 2112.0
22 1291692212 365622122666221229952212 26662 222952212266622222996222226962212699622126639221269562222

 

e I e e
e e e e e e e e 6 e e e e
e e e e e e e e e e a
.13370 :13615 213551 213715 .16076 .16626 .16797 .15155 15673 .1573? .15939 .16001 .1616?
9 eeeee ee eee eepeeeeeeeeeeeeeeeeeeeee..eeeee6006000600eeeeeeeeeeeeeeeeeeeee eee aeeeee'e ea
1. 2623 12 2321 1. :2273 1.2139 1.2053 1‘C953 1.1660 1.1653 1.1395 1.11 .0826 1.056
0 e e 0 e e e e e e e
e a e e e e e e e \\ 0 e e e
e e e '0 e e e e e e e e
.10.66 .10910 .1906! .19259 .19565 .19880 .20260 .20597 .20929 .21221 .2166! .21653 .21792
eeeeeeee eee eeeeeeeee eeeeeeee eeeee.eeeeeeceeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee see-eeeeeeeeeeeee
1.1936 1:1959 1.1051 1.1707 1.1731 1.1683 1.1612 1.1696 1.1326 1.1121 1.0900 1.0690
0 e e e e 0 e e e e e e
e e e 9|. 0 e \ . e e \6 e e I
e e e e e e e e e e e
225190 .25233 .2535 .25562 .25031 .2616? .26690 .2683. .27172 .27677 .27766 .2796? .20166
.1 ae eeeee eeeeeee 0‘ eeeeeeeee eeee eeeeeeeeee eeeeeeeeeeeeeeeeeeeee eaeeeaaeeealeee9000600600000
1.1599 w151.153‘1.1506 1.11616 1.1655 1.1616 1.1360 1.1229 1.1002 1.0916 1.0763
0 e e e e e e e g . e
e e e e e e e e e \\ e e
e e s C e O 6 a
.32320 .32359 232671 .32665 232916 .33209 .33532 .33866 .36196 236503 .36702 235026. .235232

‘ IIIIIIIIIIIIOIIOQIIIOIIII.IOIIOOQQIOOIICIOQII ......IIIIIIIIOIIII.I.IIIIIIIIIIIIIIIIIOIIII...

\

 

| COLUMN //’Ex
“0% 13 16 11 10 19 20 21 22 I /’0y
\ .ggooo "861 010 -:03052 -. .colgo .03 34/ / .1,
I, \ :,z§50ezazgg’aeee:iazaeeeaéeaeeeeel 3/ X/ x y
.0000 .0620 .1612 .2231 .2 9—”
1
3

200000 -200013 -200061 -. 230916 -22
I

U
0’ (DO (I: ~M— N

l .00 0660 10
II eeeeeeeeeeeeeeeeeeee anOeeealese
..620 .3091 -.5029 .1693 -.1
:0000 0100 0 192 ’:3000 e
00 -. 000 -2 22 .0 030 - 66
0 22916 .3631W6 203 6
l7 eeseeeeeaeeeaee 000:00000006
-.0916 -.g162 -.1112 -.16 6
0 e 725 e 172 0397 0‘3 9
a e e O
00 006 .00 13 . 0021 00 26
I 0 23900 .0326 .8600 1206 2
1. eeeeee eeeeeeeeeeeeo 0000000
9 -.1321 -.o306 -.1292 -.1121
l .0000 .3113 .6639 .3031 .30 2
. 8881° 2- 88° 598881 528823 53%? .‘181?

9 aeeeeeae: 6e e :0 see 2 a: es s.

’ -8888 - 8’"81::"iiiéi"'-292§'°-‘sz1
I I I I I I
200066 2 00 200133 20021. 200260 200313

201.9901 2. 0 1.1333 1.3603 1.6031 1.2166
IIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIII
Ms - 7 11:2 use we:
I000 I 7 I a I I1 I‘96
I I I I I I
.00116 . 02 .00366 . 0613 0 619 o 3.0
1.0003 1. . 1.6611 1.2607 126291 123 90

21 eaaiaeeee 62222166922222612.212162522 e1988
23916 2 0 23663 29330 23231 23163
290309 2 31 280605 2 0619 203133 23002

2260 60‘ 1e I 10 565 60 132 le‘ .9 1e .72
eeee eeee eOee 0666600 00000 eeee
.3336 . 5 .;.12 .661; :3996 -.581

2 2 . 3 22 3 001 9
2o 610 2 61 2 933 1 11 201 91 2 1161
I I I

23.6690 12 3 1.2865 122631 1.6316 1. 063
229962222 52222199622 2262992222 .§599221.1266
eZIII e 7 e299. .2920 e .7. a 325

 

91

 

 

 

 

Table 16. Comparison of boundary values of cr' obtained from.the
iterative solution with those from Savin's solution.
Savin's Solution ISOPEP Solutions
9' x; y' Cfis C51 (TE 1; y
0° .3367 o «.936 -1.005 -.902 .3333 w o
-.991 -.895 .3333 .0833
-.973 -.883 .3333 .1250
-.966 -.861 .3333 .1667
—.897 -.828 .3333 '.2083
-.801 -.786 .3333 .2500
-.152 -.118 .3333 .2917
35° .3361‘ .2952 -.566
.170 .185 .3333 .3125
60° .3352 .3166 .605
65° .3296 .3296 6.368 6.78 6.52 .3333 .3333
50° .3166 .3352 6.660 .3333
3.602 3.281 .3125 .3333
55° .2952 .3361 2.888
2.710 2.665 .2915 .3333
2.176 2.057 .2500 .3333
1.979 1.869 .2083 .3333
1.880 1.775 .1667 .3333
1.819 1.722 .1250 .3333
1.780 1.690 .0833 .3333
90° 0 .3367 1.760 1.718 1.668 0 .3333

 

 

 

 

 

 

 

The stresses OE and 0" are compare 10 to 05 for iterative solutions
with the convergence cr teron of 10' and 10-7 reapectively.

92

criterion is decreased. The values of cf; and ng along x = .3333

for y‘<.3333 are values of (j; computed from the point values of the
stress function. Along y = .3333 for x<.3333 the values of UK and
073 are the values of 5; computed from the stress function solution.
At x.= .3333, y = .3333 Which corresponds to 6 = 45°, the values of
CyELgiven by Savin would be along a line which makes an angle of 45°
‘with the xpaxis. Values of’cy;} C5; and ’1gy computed at the corner
of the square were used to compute values of (TR and C75 in the same
direction as 0'3. The values, 0‘6 - 4.368, a; = 4.78 and 53 = b.52
are the largest stress concentrations occuring at corresponding points
in the solutions considered. The fact that slightly different values are
found for the numerical solutions might be expected from the different

representation of the boundary at the corner in the numerical solution

and Savin's solution.

VII SUMMARY AND CONCLUSIONS

The objective of this study has been three-fold; (1) The identi-
fication of efficient iterative methods for the solution of plane
elastostatic problems; (2) The preparation of a system of computer
programs for solving this class of problems; and (3) A demonstration of
the use of iterative methods.

An investigation of the numerical solution of elliptic differential
equations resulted in the selection of three matrix iterative methods
as the alternatives which should be considered for inclusion in a
digital computer program for the solution of plane elastostatic problems.
Computer programs have been written for the point successive overrelax—
ation method, the alternating-direction implicit method and the cyclic
Chebyshev semi-iterative method. Solutions of a model problem for
various mesh intervals and convergence parameters are used for compara
ing the methods. The model problem.is a square plate with uniformly
distributed loads on portions of two edges. See Section VI-1.

The superiority of one iterative method over another may be judged
by comparing the number of iterations required to satisfy a given con~
vergence criterion. The results given in Fig. 4.7 (a) show the cyclic
Chebyshev semi-iterative method is iteratively faster than the other
methods when the number of mesh points is lessthan approximately 125.
This number of mesh points corresponds to a mesh interval h = 1/16. For
h <i1/16 the alternating-direction implicit method is iteratively faster.
Whether one method is iteratively faster than another may not be an

adequate basis for selecting the best method for a computer program.

93

94

Cost,which is directly related to the computer time required,may be a
better measure of the superiority of a particular method. The results
given in Pig. h.7 (b) show the point successive overrelaxation method
is better than the other methods for a mesh interval h =r1/26. For

a finer mesh, say the number of interior mesh points is more then 350,
the alternating-direction implicit method is the best for the problems
examined.

The results given in Fig. L.7 were obtained using various methods
for approximating the optimum.relaxation factors. How do these three
schemes compare when the optimum.parameters for accelerating convergence
are used? 1 series of problems were run in an attempt to answer this
question. A plot of the number of iterations against the associated
parameter for accelerating convergence is given for each method. See
Figures n.3, L.h, h.6. In Table 2 the data for the second and third
problems provide a comparison of the number of iterations and machine
time for a problem with mesh interval h - 1/ 12. Optimum parameters were
used in the second problem.and the standard approximations given in
Section IV were used in problem.3. Use of the optimum.parameters pro-
duced substantial improvement in the machine time required; h2$ for
successive overrelaxation, 24% for the alternating-direction implicit
method and 18% for the cyclic Chebyshev semi-iterative method. These
results show the magnitude of the improvement which could be made in
iterative methods if better estimates of the optimum.parameters could
be found. This is a problem which warrants further investigation.

Another important consideration in the selection of an iterative
method for the solution of plans elastostatic problems is the storage

requirements of instruction and arrays. A summary of the storage

95

requirements for each method is provided in Table 1. The successive
overrelaxation method requires less than 50% of the storage needed for
either of the other methods. Using_successive overrelaxation, problems
with up to 435 mesh points have been solved on an IBM 1620 with a core
memory of 20,000 decimal digits. Comparisons~of the times required to
solve problems on the CDC 3600 and the IBM 1620 are given in Table A.

A system of FORTRAN computer programs for the solution of plane
elastostatic problems has been written and tested. Flow diagrams, list-
ing of FORTRAN source decks, specification of input and the output for
a sample problem.are provided in Appendix B. A subroutine for each of
the iterative methods is included. In addition there are subroutines
for: calculation of stress components; calculation of initial values
when the mesh is refined; input and output. The main program ISOPEP
provides for the linkage of these subroutines and an additional pair of
subroutines which account for the boundary conditions of a particular
preblmm. The boundary condition subroutines for the six problems dis-
cussed in Section VI are also included in Appendix B.

A set of six.problems has been solved using the ISOPEP program.
gThe first problem.is a square plate with edge load on two sides as
shown in Fig. h.1. This is a model problem.used to compare the selected
iterative methods. Distributions of the stress function and.cr; for
Problem.1 are given in Fig. 6.2.

The second and sixth problems were selected because they provide a
basis for comparing analytic and numerical solutions. Problem.2 is a
semi-infinite plate with a uniformly distributed load along a segment of
the edge. The stress function and cr§,distributions from the numerical

solution are given in Fig. 6.5. A comparison of the distributions of

96

c77}1at one mesh interval from.the plate edge is provided in Fig. 6.4;
values from.an exact solution, from the ISOPEP solution, and from.a
numerical mapping solution are compared. Problem.6 is an infinite plate
with a square hole. Comparison of the analytical solution of G.N. Savin
with the iterative solution is provided in Table 1A. Stresses on the
boundary of the square hole are used for this comparison. The numerical
solution shows a slightly higher stress concentration at the corner than
Savin's solution, but the agreement is fairly good, considering that the
effective rounding of the corner implicit in the finite-difference solu-
tion approximates the boundary in a way different from.Savin's truncated
series mapping.

The other three problems are notched tensile specimens.‘ Problem.3
has semi-circular notches, Problem.h has V-notches, and Problem.5 rect-
angular notches. These problems were selected for an investigation of
the numerical determination of stress concentrations. Stress function
and <77; distributions are given in Figures 6.7, 6.8 and 6.9 for Problems
3, h and 5 respectively. Additional details of solution values at the
base of the semi-circular and V-notches are given in Tables 5 through
13. Values of the shear stress "I3,y and a} are included.

Comparisons are made of the stress concentrations computed with
different choices of mesh intervals h. See Figures 6.10, 6.11, 6.12.

In addition, for the V-notch the stress concentrations for the ISOPEP
solutions are compared with the Neuber theory solutions provided by

R. E. Peterson (1953). Under the assumption that the numerical solution
at the base point of the V-notch represents a solution for a specimen
with a semi-circular notch of radius equal to the mesh interval, the

two solutions show good agreement. See Fig. 6.13. This agreement may

97

be somewhat fortuitous, but it does lend support to the usefulness of
numerical analysis in the vicinity of a singularity. In any actual
body the notch would be somewhat rounded, and the numerical analysis for
a sharp notch approximates the actual state of stress in a rounded notch
with radius equal to the mesh interval of the analysis.

The numerical results for the six problems solved indicate that
high speed digital computers provide an effective means for the
analysis of plane elastostatic problems. The stress distributions
obtained from.numerical solutions compare very well with explicit solutions.
Though it is necessary to use a fine mesh in the neighborhood of a singul-
arity, the ISOPEP program.for the CDC 3600 permits the use of over
3600 mesh points. This should permit the analysis of quite complicated
problems. ISOPEP solutions on the CDC 3600 are reasonable in cost. At
an hourly rate of $375.00 per hour for computer time, the cost of solving
a problem with 200 mesh points would be $5.41. Th; cost for the solution
of a problem with approximately 1100 mesh points would be $36.23.

The possibility of augmenting the ISOPEP program.has been con-
sidered. Since ISOPEP is a set of linked subprograms, additions could
be easily'made.. The additions which have been considered include: (1) A
better treatment of boundaries formed with circular arcs, perhaps,
through use of polar coordinates; (2) Use of a refined mesh in a sub-
region of the solution domain; (3) Use of a different formulation of the
differential equations. The variational method for deriving difference
equations as used in Section III provides a finite-difference form of the
Navier equations, see Eq. (37). A study of the iterative solution of

these would be of value.

B 13le

Arms, R. «1., Gates, 1.. De, and Zondek, B. (1956), “A lethod of Block
Iteration; J. Soc. Indust. Appl. lath. _h_, 220-229

Birkhoff, Garrett; Verga, Richard 8., and Young, David I., Jr. (1962),
'Alternating Direction Implicit lethods ,9 Advances in Computers,
Vol. 3, Academic Press, New York.

Unless, I. A. (1961), “Solution of ”steam of Ordinary and Pntisl
Differential Equations by Quasi-Diagonal Iatrices,‘ Computing
Join-nal 3, 54-61

dents, Samuel D. and Dues, Ralph T. (1958), 'An Alternating fiireetion
lethod for Solving the Biharnonic Initiation," Isth.T Tables Aids
A Osnot. 12, 198-205. ,

dents, sum 9. and Dames, Ralph T. (1960), 'On An Alternating Direction
lethed for Solving the Plate Problem with lined Boundary donditions,‘
J. Assoc. Conut.lech.1, ash-273.

dos-nook, A. F. (1:95“. "The numerical Solution of Poisson's ad thi ‘
gm“ mttionl by latrieee,‘ Pros. denbridge Philos. cos. 2.
3 .

count, 3. and Hilbert, n. (1953). letheds of lathematisai m4
Inter-cine. publishers, Inc... mew or 1m pp.

5

cuthill, msabeth a. and Varga, Richard s. (1959). '1 Iethod of
normalised Block Iteration,“ J. Assoc. donut. lash. _6_, 236-2111;.

mum, ’1', Jre m WON He 3., no (1956), .m th. ”Mica
Solution of Heat Conduction Problems in he or Three Space Vari-
_ ables,“ Trans. Amer. lath. Boo. §_2, M14139.

Ingeli, IL, Ginsburg, T.; Ruthishduser, 8.; and Stiefel, I. (1959),
Refined Iterative Hethods for C tatisn of the Solution and the

en use 0 o 1E'E'oun nemesis, [Itte'fi'ungen ‘
sti'tu swan enatIE E. 3, Eirkhauser Verleg

_u Edsel/Stuttgart, 107 pp. . .- _ .
Faddee'rvi, v. n. (1959) . computational lethsds of Linear gym- a, Trans-
lated by Curtis '0. Fenster. fi'fifiiafl'ons, Em, New York,
252 ”e

Fairieather, a. and utensil, a. a. (1961:), 'a Generalised Alternating
Direction lethod of Douglas-Beehfsrd Type for 80le the Bihu-monie

lquatiem,‘ The donoter Journal 3, 21.2-21.5. ‘ . f
Forsythe, G. B. and faces, I. R. (1960), Finite Difference lethods for

Partial Differential Equations . John‘III'ei Win—Tc— . , es
Wfl, W ”e

98

99

Fox, L. (1950) , “The Numerical Solution of Elliptic Differential Equations
‘Ihon the'Boxmdary Conditions Involve a “Derivative,“ Philos. Trans.
Roy. Soc. London, Ser. A £113, 31.5 - 378

Fox, 1.. (1963). Numerical Solution 3; Ordinary and Partial Differential
auctions, Iaflson-IesIey Elishing Company, Inc" Reading, lass.

 

Frankel, S. P. (1950), “Convergence Rates of Iterative Treatments of
Partial Differential Equations,“ Math. Tables Aids Comput. A, 65-67.

Griffin, D. s. (1963), A Numerical Solution for Plate Bending Problems,
film-230 AEC Research and DeveISpment Report-{‘59 pp.

 

Griffin, D. S. (1965), Private Communication

Griffin, ‘D. S. and Varga, R. S. £1962), _A_ Numerical Solution for Plane '
Blastici Problems, HPD-Z 6, ABC Research" and fiveEpmiFt’ fiport,

Pp.

Griffin,"D. 8. and Verge, R. S. (1963). “Numerical Solution of’Plane
Elasticity Problems,“ J. or Soc. Indust. Appl. llath. g, 10116-1062.

Golub, Gene]. and Verge, Richard S. ( 1961A) , “Chebyshev Semi-Iterative
lethods, Successive Overrelaxation' Iterative Iethsds, and Second
fiderlslzichardson Iterative Ilethods, Part I", Numerische lath. },

7" e

dolub, Benin. and Virga, Richard S. (19613), “chebyshev Semi-Iterative
Methods, Successive Overrelaxation' Iterative lethods, and Second
guiégicherdson Iterative lethsds , Part II" , Nunerische lath. 3,
7" o -

Heller, J. (1960) , “Simultaneous, Successive, Alternating Direction
Iteration Schnes,“ J. soc. mdust. appl. nth. _§. 150-173. A.

Hooks, R. and Jeeves, T. A. (1961), “Direct Search Solution of Nmerical
and Statistical Problems,“ J. Assoc. Comput. lach._8_, 212-229.

Householder, A. S. (1953), Principles of’Numerical Anal sis Icaru-
Hill Book do., Inc., New Yen's, 2717px). ,

ficusehslddr, A. S. (1958), “The Approximate Solution of latrix Problems,“
J0 ‘:.”.e amt“ ”who _59 201-$4,430

lanterovich, In. Yo and Krylov, V. I. (19611), 1 Kill“ [CM 9!;
Haber I221 sis, Translated from the 31$ch 5? Master,
o ey an Sons Inc.,- New York, 681 pp. .- _
Keller, Herbert B. (1960), “Special Block Iterations ‘lith Applications
. to Laplace and Biharmonic Difference Equations,“ SIAM Rev. g, 277-287

Iilne, ‘Iilliam ldnund (1953), Numerical Solution of Differential w
tions, John I'iley and Sons, _Ec., New Tori, I75 pp. _.

 

luskhelishvili, N. I. (1953) , Some Basic Problems g_f_’ the"lathematieal
Theory of Elasticity, P. Rosalie—ff: Groningen, "(UH-pp.

 

100

Nat. Phys. Lab. (1961), Modern Computing Methods, Philosophical Library,
New York. , .

 

Parter, Seymour V. (1959), “On Two-Line Iterative Methods for the Laplace
and Biharmonic Difference Equations,“ Numerische Mathé, 2110-252.

Parter, Seymour V. (1961A), “MultimLine Iterative Methods for miptic
Difference Equations and Fxmdamental FroQuencies,“ Numerische

Math. 29 305““3190

Porter, SeymOur V. (19613), “Some Computational Results on Two-Line
Iterative Methods'for the Biharnonic Difference Equation,“ J.
Assoc. Comput. Mach. Q, 359-365.

Peaceman, D. I. and Rachford, H. 14., Jr. (1955), “The Numerical Solution
of Parabolic and Elliptic Differential Equations," J. of Soc. ‘Induste
Appl. Math. _39 28"]410 .

Peterson, R. E. (1953), Stress Concentration Design Factors, John'iley
and Sons, Inc., New York, 155 pp.

Pisacane, V. L. and (Malvern, L. E. (1963), “Application of Numerical
Mapping to the Muskhelishvili Method in Plano Elasticity,“ Journal
of Applied Mechanics, 2L2, woe-1.14..

Savin, c. N. (1961), Stress Concentration‘gn the Bound 95 Holes,
Government Publication, Mo scowaIEEgTadTESl—Tflussian) ,
English Translation, Pergamon Press, NewTork, 1.30 pp.

Sokolnikoff, I. S. (1956), Mathematical Theo _. of Elasticity, McCraw-
Hill Book Company, Inc. , New TorE, E76 p

O

Southwell, R. V. (19116), Relaxation Methods _i_n_ Theoretical Pflsics,
Clarendon Press, Oxford, 2118 pp.

 

Todd, John, Editor (1962), Surv '23. Numerical Analysis, McGraw-Hill
Book Company, Inc. , New YorE.

 

Timoshenko, S. and Goodier, J. N. (1951), Theo of Elasticity, McGraw-
Hill Book Company, 1236., New York, 506 pp.

Varga. R. S. (1960), "Factorization and Normalized I+erative Methods ,"
Boundary Problems in Differential Equations, R. E. Longer
(Editor), The University of Iisconsin Press, Madison, ‘Iisconsin.

Varga, Richard S. (1962}, Matrix Iterative Anal sis, PrenticemHall, Inc. ,

Veyo, S. E. and Hornbeck, R. I. (19611), “Numerical Solution of the
Biharmonic Equation Using an Automatic Iterative Procedure,“
Journal of Applied Mechanics _l_1, 7011-405.

 

Iindsor, Edith S. (1957), Iterative Solution of Biharmonic Difference
Equations, M. S. Dissertation - Dept. of Mafienatics, NYU Libraries,

University Heights, 39 pp.

Young, David M. (19514), "Iterative Methods for Solving Partial Difference
Equations of Elliptic Type," Trans. Amer. Math. Soc. ]_6, 92:111.

APPENDIX A

CONVERGENCE OF ITERATIVE METHODS

The convergence characteristics of an iterative solution of a

system of linear equations of the form
AH=£ (so

can be judged by examination of the matrix.§, The system of linear
equations associated with the biharmonic difference equations can be
written so 5 is positive definite and of the form

a=2+§+a we)
where Q is a nonsingular block diagonal matrix and the form.of‘§‘+ g
associated with a particular iterative method establishes the convergence
of the method. The basic conditions for convergence are given in
Theorem.A.5 of Subsection 3 below3while estimates of the relative rates
of convergence of different methods are cited in Subsection 4. Theorems
A.8, A.9 and A.1O give some alternative convergence conditions for the
Richardson's, Gauss-Seidel and successive overrelaxation methods.
Subsection 5 is concerned with the selection of optimum relaxation
factors.

For fuller discussion, with proofs and bibliography,see Varga
~(1962) and Faddeeve (1959). Specific page references in these readily
available sources are cited. No attempt has been made to give a hist-
orical account here with credits‘bo the originator of each result. The
purpose of this appendix is a summary review of basic matrix properties

and the convergence of matrix iterative methods.

101

102

1. Matrix and Vector Norms, the Spectral harm of 8. Matrix

 

Definition: If A is an n x n matrix, then A is convergent if the sequence

3

of matrices A, £2, 5 ....... converges to the n x n null matrix, _0_.

 

Otherwise A'is divergent.

In the investigation of iterative schemes it is important to
judge not only the convergence of the solution vector y, error vector
E, and associated matrices D, but also the rgtggflgf convergence. Norms
of vectors and matrices are of importance in the discussion of rates of
convergence. The norms have been defined in many different ways. Some
of these norms are discussed below. Except in this section IIXJI will
always denote the Euclidean vector norm defined as llel III below and
the matrix norm "All will mean the spectal £95! "All III' We will also
use the term spectral radius, denoted by 63(5) for the magnitude of the
largest eigenvalue of the matrix, defined in the next subsection.
designated ||§J| which satisfies

(a) Hall > o for an 9,... Hell - 0.

M Hall = lClll all where C is any ...m,

(c) ”2+1”: ”g H + “I“, the triangular inequality.

There are three norms of interest. For the vector
§g= {X1, X2, X3 ---- In; , we define

Io Ill HI = max lxil,

IL HEHII = |x1| + lle 1' -... +lxnl:

III. ”gunI =1/lx1l 2 +--- + |1rn|2 , Euclidean norm

 

Since norms are often used as bounds it should be noted that
urnI .<. ”sun .<. n “anI

103
Fadeeva (1959) states that a necessary and sufficient condition

for a sequence ;(m)

of vectors to converge to a vector §_is that
“_X-(m) =§H-—> O as m——>OO
for any norm satisfying conditions (a), (b) and (c); he gives a proof
for the three norms I, II, and III.
The norm of an n x.n matrix §,is a nononegative number [lg H
which satisfies
(a) IIAII> 0 1r Meandnai: =0.
(b) Hcall = ICIHaH
(c) Hysll Still! +1133.“
(a) Heal! .<. Hall Hail
The necessary and sufficient condition for convergence of a sequence
of nxnmatrices 5"“) is that Ham) - £||~+O as m—eOO. Any
norm which satisfies (a), (b), (c) and (d) can be used to establish

(m) all—*0 then I|g(m)|l->l1e!lo

convergence. If ['5
As in the case of vectors it is possible to introduce matrix norms
in a variety of ways. But for the purpose at hand it is convenient to

introduce a matrix norm which satisfies some special requirements. One

of these is the requirement for the matrix norm to be compatible with a

 

given vector norm. This condition holds if
MAX.” 5 Hall H ..X.l! . Compatimutx condiiiaa
It is possible to determine a matrix norm which satisfies (a), (b),
(c), (d) and which is compatible with a specified vector norm. For

example, as a matrix norm compatible with the Euclidean vector norm

 

X we define the s ectral norm 3A” of a matrix A as the
H" “111’ p I’ "III ...
least upper bound of the norm of the vector 5,; as ; runs over all

vectors of norm unityg

 

11111111111
1.11:1

101»

llélllll = max ”5. 39,111 With H E’HIII =1

2. Eigenvectors and eigenvalues
An eigenvector (also called proper vector, characteristic vector
or latent vector) of a linear transformation A is any non-zero vector
x such that
as= A;
where A may be a complex number and is known as an eigenvalue (or proper
number, latent root or characteristic number).

If a vector x is an eigenvector it must satisify

aux: +a12X2+ ‘ " ’ ’ “" amX”=;\x'

QZle +QZZXL+ - - -- - - -- aznxn=XX1

amx. + anaxa+ - - ‘ ' - - ' ‘1an ’2’“?

a system of homogeneous equations, which can have a non-trivial solution

only if
an ‘1 dig a,”
am an") 0211
0.7” an; dnn‘)‘

 

 

The determinant is equivalent to an nt'h

degree polynomial in A which
is called the characteristic polyomial of the matrix A. For a real
symmetric matrix all the eigenvalues are real.
Definition: If _A_ is an n x :1 complex matrix with eigenvalues )1,
1 g i 5 n, then

6 (£1) = max | A1 I
is the sEctral radius of _A_.

105
3. Properties of Matrices
Theorem.A.1: If g'is any arbitrary matrix then
Hill 2. ed). [Varga (1962) p.10]
Theorem.A.2: If §.is an n x.n matrix then

M = we?)

where A? is the conjugate tranSpose of g, [Varga (1962) p.11]

 

Definition: An arbitrary n x.n complex.nonsingular matrix 5 and an n-
dimensional vector §_can be combined as g? g_; to form.a homogeneous
second degree expression in terms of the components of §_which is called
a quadratic form. If the value of the quadratic form is positive for all
non-trivial g then the form and the matrix glare said to be pgsitixg
definite.

Definition: If §,is a n x.n complex.matrix such that any two elements

 

situated symmetrically with respect to the principal diagonal are complex
conjugates, aji ==§ij, then §_is called a Hermitian matggx. Notable
characteristics are:

1. Diagonal elements must be real.

2. Any symmetric matrix with real elements is a Hermitian matrix.

3. The eigenvalues of a Hermitian matrix are real.

4. The eigenvalues of a positive-definite Hermitian matrix are
positive.

Definition: A square matrix.g_is diagonally dominant if

“15?) 3 filau‘l

J=l,

Jae:
If the strict inequality holdsfor all i, §,is said to be strictly
diagonally dominant.
Theorem.A.3: If §.is a Hermitian n x.n strictly diagonally dominant

matrix with positive real diagonal entries, then §_is positive definite.

106
[Varga (1962) p. 23 1
Theorem Ll}: If A is an n x n Hermitian matrix, then
|| 5” = pg) [Varga (1962) p.11]
Theorem Lg: If A is an n x n complex matrix, then A is convergent
if and only if
(3(5) <1 [Verge (1962) p. 13]
Thus, we see that if the matrix A in Eq. (A-1) is symetric and
positive definite, then 5 is convergent if the spectral radius of A is
less than 1. In the next subsection the convergence rates of several

iterative methods will be considered.

A. Convergence Rates of Iterative Methods
The system of linear equations (A-1) can be written in the form
(1 - 1.4.) H. = E. (AL-3)
where g is an n x n complex matrix and I_[_ is the identity matrix. If
(:_L - g) is non-singular a unique solution vector 11 exists. Consider

an iterative method for the solution of Eq. (4-3)

 

2(”1)-!U(“)+£ Ian-1.2.3

For any vector iterate 9f“) the difference Um)

(A-h)
- Q - go“) is a measure
of the deviation of the vector iterate from the solution vector. The

(hit)

error vector _E_ can be written in terms of the preceding error vector

§(m) g g E-(m-1)
This is obtained by subtracting (A-B) from (A-h) and it can be readily
shown E(In) a E” §(0)
From Theorem A.5 we conclude that the error vectors g0“) will tend to
zero for an arbitrary E“) if and only if g has a spectral radius €(_Ig)< 1.
Theorem A.6: If A and g are n x n matrices, then

(1) l|£||>0 or i=9;

107

(2) if k is a scalar, “kg“ = [k l [15,”;

<3) Hue” .<. Hell + Hall3

m Mean s “an Hill:

(5) Hi!“ _<_, ”A“ ll _qll for all vectors y_ of n components.
Also, there is one nonzero vector _v in the n-dimensional vector space
such that

“A!” - “A“ Hill [Varsa (1962) m9]

Since the solution vector iterates g0“) and the associated error
vector iterates g“) are n-dimensional, Theorem 11.6 can be used as

justification for
New” - ”ram“ <uru lli(°)ll

 

ll ("fill

Hence, “am H as an upper bound estimate of the ratio of HEW) H to
II 33(0)” provides a basis for the comparison of different iterative
methods. If g is a Hermitian matrix then

urn - the)“

Definition: Given two :1 x n complex matrices g and g. If for some 111 > 0,

° ||f||<1 then
R(.!m)' ~41! [llgmllk’ ]- Jbgbfi-IL- (A-S)

is defined as the average 22 _of convergence £9; :_n_ iterations of the
matrix 3. [Verge (1962) p.62.J The convergence of g is said to be
iteratively faster than the convergence of 5 when Mg") > R (f).

The significance of Rm") as a measure of the average rate of convergence

may be seen by consideration of the average error’ reduction factor per

iteration 0: HEM)” k"
a” 113°)” J

108
For the case when “gull <1
0' s H!“ 11"" = ““3” (.-.)
where e is the base of the natural logarithm. Thus this) is the
exponential decay rate for the upper bound of the average error re-
duction 0‘, per iteration in an m-step iterative process.
Let NIn - 1/R(_I~f‘). Substituting into (A-6) we find
O’Nm .<. 6
from which we conclude NIn is a measure of the number of iterations
required to reduce the norm of Q“) by a factor e.
Theorem A.7: R(Am), the average rate of convergence for m iterations
of an n x n convergent matrix A has a limiting value of -In E(A) as
m increases without bound. [Varga (1962) p.67 ]
Definition: The asmtotic 22L! 2;; convergence Rx (A) is
Rag (A) = -1n 9(5)
Corollgz: Let A be a convergent n x n matrix, then
Roe (A) .>_ ME“)
for any positive integer m.
For Hermitian matrices A and g, the spectral radii may be used
for comparison of rates of convergence, since by Theorem [1.6,
9(a) <9(1.3)<1 implies ”an” <ugmu<1 and hence
R (Am) > My?) , so that of two convergent Hermitian matrices, the one
with the smaller spectral radius will have a faster average rate of
convergence for any 111.
It should be noted that though H g“ ||—-> 0 as m—boo for two
iterative schemes with matrices A and _B_, it is possible in general that
for a selected value m1, matrix A may be iteratively faster than g but

for a second value m2, 3 may be iteratively faster than A.

.109
Identification of the convergent iterative methods for biharmonic
difference equations is aided by consideration of the following three
theorems. The form of matrix A'in Eq. (A-1) is the determining factor.
Theorem A.8: If A_is a strictly diagonally dominant n x n complex
matrix, then the associated Richardson‘s and Gauss~Seidel matrices are
convergent and the corresponding methods are convergent for an arbitrary

initial vector 11(0). [

Varga (1962) p. 73. ]

The matrix ADassociated with the biharmonic difference equations
is not strictly diagonally dominant and Windsor (1957) has shown that
Richardson's method is not convergent for the biharmonic equation.
Theorem A.9: If A_= Q +-g;+-gf is an n.x n Hermitian matrix where Q
is Hermitian and positive definite and (Q +'§) is nonsingular, then
the Gauss-Seidel iterative method is convergent if and only if A,is
positive definite. (g? is the conjugate transpose of'g). [Varga p.78.]
Theorem A.10: If i - p + _q + (2* is an n x n Hermitian matrix and g is
Hermitian and positive definite, then the successive overrelaxation
method is convergent for any arbitrary'gflo) if and only if‘A is
positive definite and (Q +609.) is nonsingular for 0< (.0 < 2.

[Varga (1962) p. 80.]

5. Optimum relaxation factors

Definition: An n x n matrix g'which has zeroes and ones for elements
and only one non-zero element in each row and each column is called a
permutation matrix.

Definition: Given A_an n x.n complex matrix with n >.1, if there exists

an n x,n permutation matrix 2 such that

[*0

lb
'11
ll

 

 

L$2 £22 j
where A11 and A22 are square matrices of order k and (n-k) respectively,
then A is called reducible. Otherwise A is called irreducible.

Theorem A.11: If A is an irreducible n x n matrix, then: (1) A has a
positive real simple eigenvalue equal to €(A) the spectral radius;

(2) Increasing the value of any element of A will increase Q (A); (3)
Corresponding to the eigenvalue (MA) there is an eigenvector with all

its elements positive. [Theorem of Perron and Frobenius, Varga (1962) p.30]
Definition: If A is an n x n irreducible matrix with non-negative

elements which has a single eigenvalue of modulus €(A), it is said to

be primitive. If A has I: eigenvalues with modulus of e (A) then A is
gygl_:I._c 2f. _igggx A, k _>. 2. Each eigenvalue of modulus €(A) is a

simple eigenvalue. [Verge (1962) p.35 ]

Definition: A square matrix A of order n is said to be M EM

2A index A (k > 1) if an n x n permutation matrix A exists such that

 

 

2 9 2 5.,
A 0 O 0
'1' -2)1" ‘” "
252’s a .9 9 2
l 31' '
| \\1
| I
0 O A 0
L-' -' '-&ar- J

where the null diagonal submatrices are square. A may or may not be
reducible. A matrix can be simultaneously weekly cyclic of different
indices. [Varga (1962) p. 39 ]

Again consider the matrix equation

111
i932".

where A is an n x n complex matrix which can be partitioned

A“. 51,2 —’ "’ "" ALN
52,1 52,9. - - " £2,”

.4. a (A‘7)
_ AM! Am - - — Arm.

 

 

where the square diagonal submatrices are nonsingular. Let Q be formed

of the submatrices Ai’i,

 

 

5.1" Q 9.

9 i2), 9.
Q-

o o A ‘

L" " ”M;

then A is also nonsingular, and the matrix equation can be written
(i-mi+22-£

or _1 (Au-8)
u-a2+2 F

1

where A - _I_ - _B_- A is the iteration matrix for the block Jacobi iterative

method
2(m-1-1) ... 2 £011) + D-1

.E
The matrix A is called the block Jacobi matrix of A.

Definition: If the block Jacobi matrix A is weakly cyclic of index p,

 

then A is Ecyclic in the partitioned form (A-7). [Varga (1962) p.99 J

Definition: The p-cyclic matrix A is consistently ordered if all the

 

eigenvalues of the matrix

112
_B_(k) as kfl + R413") .3.
are independent of k for k 7‘ 0, where A is the block Jacobi matrix with
sero diagonal elements, A and A are respectively strictly lower and
upper triangular matrices such that A - A + A. [Varga (1962) p. 101]
If the partitioned form of A is block tri-diagonal, it is con-
sistently ordered and 2-cyclic. The optimum relaxation factor 60b
which maximises the asymptotic rate of convergence of the block successive
overrelaxation matrix for p = 2 is given by
(4)6: -__3___.
I + 1/1 - (”(5)
where A is the block Jacobi matrix. [Varga (1962) P. 110 1

For the cyclic Chebyshev semi-iterative method the acceleration

parameters are given 2 C .1 l/( ‘
(1),, = #12 , ‘ A .
e (5) Cm(/é(5))

where A is the block Jacobi matrix. [Varga (1962) p. 138]

 

The alternating-direction implicit method for the case of a fixed

acceleration parameter r is a slight variation of Eq. (67) p.33
2(n+1) _ B:— U(n)+ 4!.

If there is a block Jacobi matrix AR associated with Ar, the asymptotic
rate of convergence will be a function of 62(h).

Thus we find that the optimum relaxation factors for successive
overrelaxation and the cyclic Chebyshev semi-iterative method as well
as the acceleration parameter for the alternating direction implicit
method are functions of the spectral norm of an associated block Jacobi
matrix. In fact Verge (1962) shows that the three iterative methods

considered have the same asymptotic rates of convergence for Laplace's

equation solved for a square.

113

There appears to be a dearth of good approximations of (3 for
irregularly shaped regions. It is common practice to use approximations
based on the numerical results obtained in the iterative solution.

Consider point successive overrelaxation. Forsythe and'Wasow
(1960) p. 250 Show that the eigenvalues }d_of the matrix of the simplest
iterative scheme, the point Jacobi method, are related to the eigenu
values 721 of the matrix of the successive overrelaxation method by
71. A: . This is applicable to the eigenvalue )(1 of maximum
modulus and hence to the spectral radius of the successive over-
relaxation matrix. The Optimum relaxation factor as derived by Forsythe

and wasow p. 253 is

2 2

L05 8 m : [fl (Aw-9)

The error vectors from one iteration to the next are related
Hamil 5. He II ll§(m")ll
where A,is the appropriate matrix for the iterative scheme being
considered. The dominant eigenvalue Th is the limit of II Am)" / [[§(m"1)“
as m increases without bound. Any vector norm of A,may be usede One
computational procedure for estimating Cub consists of starting the
fprcblem solution with w- 1, then after a number of iterations approxi—
mate ‘fi;
7(1 ” ”Em ll / “Em-1)“ (A-10)
and solve (A-9) for an approximate value of 00b.
Another approach to approximating cub consists of selecting

various values of Ce), running through several iterations for each
and then by comparison of results, select the best.

The procedure used in the ISOPEP code consists of computing a set

11b
of values for 6.) obtained by applying Equations (A—9) and (1-10) every
10 or 20 iterations. This has the advantage of being an automatic
procedure, and reasonably good results have been obtained for a number
of problems. Approximating (db numerically as the solution proceeds is
more computing art than science... Better methods for determining optimum
relaxation factors would contribute much, to the usefulness of iterative

‘thOd' e

APPENDIX B

ISOPEPBA FORTRAN PROGRAM FOR THE ITERATIVE SOLUTION

OF PLANE ELASTOSTATIC PROBLEMS

The system of computer programs named ISOPEP was written as a
general system for the analysis of plane elastostatic problems. It
includes a set of FORTRANwII subroutine subprograms and a main program
which provides linkage of a selected subset of these six subroutines
and additional boundary-value subroutines which must be provided for
each problem.

Thislappendix includes a brief description of the general sub-
routines, six examples of boundary-value subroutines, instructions for
preparing input data for the program, a sample of the output, a flow
diagram.and listings of the FORTRAN Source decks. I
lasts

This name is used for the system.of programs and also for the
main program which provides linkage between the subroutines. The
system was designed for use on an IBM 1620 with a‘20 K'main memory
and a CDC 3600. The number of subroutines linked by ISOPEP may be
reduced to increase the storage available for arrays. Only the Call
statements in ISOPEP and all Dimension statements need to be changed.
Only one of the three iterative method subroutines SORLX, ADI or CHEB
is normally used.
9.92m

OUTIN is the input and output subroutine. It provides for the

initialisation of arrays. When problems using a relatively coarse mesh
115

116
spacing have converged, the point values are stored for later use in
calculating an initial stress-function distribution for a finer mesh.
212119.

The initial stress function distribution for the second or sub-
sequent mesh refinements of a problem.can be computed by interpolating
between the values obtained from an earlier coarseemesh solution. The
problems may be solved consecutively, with the earlier solution saved
in memory, or the preceding solution may be read from.punched cards.
Execution of this option is controlled by the input of an appropriate
value of the control number MESH.
mass

The stress components 0;, 0'; and TV will be computed and
punched out on cards if specified by one of the options determined by
the input of a control number NSTRS. The difference equations used to

calculate the stress components at each interior point are

0;” —“'-'-’- [ U2;J'+1"2Ui,l +Ui,J-11/;.f- '
0}“. =5 [ U}+hj"'2Ui,J' + Ui¢bJ1/hl 3

”Km 5“ "'[-U2‘+1,J‘+T U2'+1,J-T Ui-1,J'+I+Ui‘bj"/Ih‘ '
£031.!

AQAAA is the subroutine for the point successive overrelaxation
iterative method. This has been adopted for general problems with
irregular boundaries. As written, it is limited to simplybconnected
regions for which all mesh lines,parallel to one of the Cartesian
coordinate axes, are continuous segments connecting two boundary points.

ADI
The alternating-direction implicit method has been written as two

117

subroutines ADI1 and ADIZ. The second is executed immediately after
the first. The specification of two subroutines makes more storage
available for arrays when the programs are run on the disk-oriented
IBM 1620. The solution domain is limited to rectangular regions.
gggg

The cyclic-Chebyshev semi-iterative method subroutine has been
run only on the CDC 3600. It is written in FORTRAN II and could be
divided into two or more subroutines for a computer with a limited
main memory. Only rectangular regions can be treated with this program.
It is necessary to specify an even number of interior mesh rows for
this subroutine.
PB N BD

The values of the stress function at points on the boundary must
be computed only at the beginning of the problem.aolution. This is
part of the initialization of the solution array. A subroutine of
this type must be provided for each problem. Examples are included
for 1 5 N 56.
PB N EX

Derivative boundary conditions must also be provided for each
problem. These may be treated by the introduction of exterior points
or with special finite-difference equations for each point on the
boundary. Both approaches are illustrated in the examples included
for 1 g N _f 6. This subroutine must be executed during each iteration.
lgpgt’Preparation

The input to the problem can be provided on a single 80 column
card unless point values of the stress function U13 are provided. The

deckeof stress function values would be placed immediately after the

118

ITERATIVE SOLUTION OF PLANE ELASTOSTATIC PROBLEMS

FLOW DIAGRAM 0F ISOPEP

 

START ;;.

 

 

I INITIALIZATION
l KPE-1 INOT-1

,fe

 

  
 

 

 

CUTIE ,
Read g CALL OUTIN .
Input Data ‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L

 

 

 

 

 

 

* PRINT Title of
Solution Method

 

 

 

 

 

 

 

Nbv \

PBNBD

Calculation

0f Boundary CALL PBNBD CHANG

Values U3 ::::::]< , Interpolation
Routine for

 

Initial 013

 

 

 

 

PBNEX

‘Calculation
,of Exterior

Values UExt

 

 

 

STRESS
Calculation
of Stress
Components

 

CHEB Routine for _,
Cyclic Chebyshev ‘
Semi-iterativelhth.

SORLI Successive
Overrelaxation
'Iterative Routips

 

 

 

 

 

 

 

 

ADI, Alternating - I
Direction Implicit INOT , 2

_lterative Routine

 

 

 

‘

 

 

 

 

 

 

 

 

 

 

Ies:__

CNSTRS 0 3L
4

Y3;

 

 

 

A
fl

OUTIN J
Print (or Punch) m CALI. OUTIN M Cznerlgledgn
Output }

jNo

 

 

‘1 ll. ( (I'll, all...

119

control card which includes identification, dimensions and control

numbers for input and output options. The stress function values must

be provided in agreement with the FORMAT (I5,7F10.7/(5X,7F10.7) ).

Output is punched in this form in order to simplify restart procedures.

The FORMAT specification (F10.3,F10.8,2F10.4,101A), is used for

the first card.

The following list gives the use and symbolic name

of each of the 1A numeric entries on the card.

 

 

 

Column Symbolic
Numbers Name Function
1-10 PRNO Problem.number for users identification.
11,20 DE Convergence criterion, usually in the range 1Om5=10°7.
21-30 REA Relaxation factor
314.0 SPYIJ H _E ”II Norm of the error vector in the preceding
iteration when restarting. Set to 1.0 initially.
h1-Lh HZ Number of mesh intervals along the xeaxis.
AS-AB MY Number of mesh intervals along the yaaxis.
49-52 IF (2) If y = 0 is an exterior boundary
(3) If y‘= 0 is a line of symmetry
53-56 JP (2) If x.= 0 is an exterior boundary
(3) If x - 0 is a line of symmetry
Note: Two mesh lines exterior to the domain are reserved
for derivative conditions if the boundary is a line
of symmetry otherwise only one line is reserved.
57-60 N N-so: Only one card of input is required. The initial
value of the stress function at all interior points
is set to 0.1.
N>0: Count of number of iterations completed. A deck of

 

Column
Numbers

Symbolic
Names

 

120

Function

 

 

61-61.

65-68

69-72

73-76

N<02

NOUT

ND ND

point values of the stress function must be provided
immediately following the first data card.

This is the control number which terminates the
processing of a sequence of data sets. A final

data card should always be provided with this entry.
Hhximmm.number of iterations permitted. This is the
choice of the user and provides for punched output for
restarting.

Output will be printed after NOUT iterations and

every subsequent set of NOUT iterations.

3,NT: The initial value of EPA will be used for all

iterations.

. ND'< NTz‘ The relaxation factor will be computed and

DESK (1)

(2)

(3)

 

changed at the end of every ND iterations.

No mesh refinement calculations

Read in values of the stress function from.a prior
problem and interpolate to find an initial stress
function distribution for the current problem.

Use the stress function stored in.memery from the
preceding problem.as the basis for interpolating te
find an initial stress function distribution for
the current problem. The user must be sure there is
at least one problem.apecified on input cards pre-
ceding this one.

 

121

Column Symbolic
Numbers, Names Function

A 'Am. i'.‘ -.,L-,_...i.- 14ae.-‘.uu .u‘.‘ “‘9‘ lL-~.—“n:—‘|’_u ‘Ammn 4.. “.-.-.. v I“.— Lr.W

 

 

_ _. ‘_.._" --.-‘_..-‘__.. ..—

 

 

77—80 NSTRS '< 0 Punch Specified stress components

INSTRsl = 1 Print. (and punch) 0.3.,
INSTRSI = 2 Print (and punch)O-Sf and o;

 

INSTRSI .. 3 Print (and punch) Txy9 o; and 0;
The second control card is used only if MESH = 2 on the first
card. The four integers read in with a (L15) FORMAT Specification pr0w
vide the dimensions of the problem solution prOVided on the cards

following this control card and the number cf mesh spaces along the yaaxis

of the problem to be solved.

 

 

 

 

Column Symbolic

Number Name Function ___
1-5 IP Number of mesh columns of input data

6~1O JP Number of mesh rows of input data

11-15 M1 MY for the input solution

16-20 M2 MY of the problem to be solved.

Example gf_Input Data
First half of the first card

Card columns

 

 

 

 

 

O 1 2 3 A
123456789012345678901234292829 M2 67§20
1.190 .00001 1.5 1.0

Second half of the first card
Card columns
5 6* 7 8
12345678291234567890123h56j8©01234567830

5 1O 3 2 1 200 200 10 1 -3

 

 

 

 

 

122

IBM 1620 OUTPUT OF SAMPLE PROBLEM

¢¢JOB 5
##FORXSB 1
0078“ CORES USED

SUCCESSIVE OVERRELAXATION

0 SOLUTION OF THE BIHARMONIC EQUATION C.L. DAVIS
PROBLEM CONVERG. MESH SPACES MESH
NUMBER CRITERION MX MY IF JF STRESS
1.190 .0000100 5 10 3 2 1 -3
RELAX. SUM OF ITERATIONS POINTS NOT
FACTOR ERRORS COUNT MAX.N0. CONVERGED
1.5000 1.0000 0 200 50
1.6352 1.7275 10 200 “5
1.6325 1.h1ko 20 zoo #5
1.5618 “.7068 30 200 “S
1.u7o7 .1865 #0 zoo #5
1.8761 .0372 50 200 #1
1.558 .0136 60 zoo ht
1.598 .0082 70 zoo #3
1.5832 .0085 80 200 #0
1.5579 .0019 90 zoo so
1.5258 .0006 100 200 28
1.8878 .0002 110 200 .9
1.190 .0000100 1.h878 .0001 5 10 3 2 116 200
200 10 1 -3.. - . ..
STRESS FUNCTION
2 3.6000000 3.0200000 2.8800000 1.9800000 .7200000 -.7200000
3 3.5072552 3.3395735 2.7961993 1.9099653 .696h0hh -.7200000
5 3.302709“ 3.1353795 2.6327760 1.798069“ .6555203 -.7200000
5 3.0285702 2.8790977 2. #277819 1.6699096 .6106627 -.7200000
6 2.6981876 2.5725967 2.1885523 1.5266087 .5626013- -.7200000
. 2.3065270 2.2096326 1.9063169 1.3587928 .5068087 -.7200000
1.8h37695 1.7788512 1. 5669582 1.1528863 .h365079-.7zooooo
9 1.3121758 1.2792279 1.1621542 .8963012 .3hk3551 -.7200000
10 .7h826h2 ..7h100h7 .705771h .5860380 .2259561 -.7zooooo
11 .2501268 .25h2h91 .2625820 .2h90h07 .0918619 -.7200000
12 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 -.7200000

ISOPEP

15207 NEXT COMMON

C L DAVIS

 

'1'] I ll! f,.ll ._ -

NVgCDUDGFQChU1aflflhJ

19=5CMO<D\Joun4rung

Iva‘CHO€D\JOun4runo

and”

“36.0000
'35.5362
'33.h66h
-2908952
"'25 0111189
-19e3796
-12 o98‘1‘2
-6e5898
-1eh520
.823h
0.0000

'118 o5‘1'93
-11.1799
-6.959h
‘5.62#3
-60 u 277
-7.1097
'3.2317
6.5773
20.8010
50.0252

0.0000
1.0001

.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

STRESS DISTRIBUTION
PROBLEM NUMBER 1.190

123

D1MENS10N a

5 X 10

(THE X COMPONENTS OF STRESS)

-36.0000
_.35 05693
“33.5273
‘30.1843
"25 08,453
-200 6‘02G
-1ho697h
-8.4126
-2 o7973
1.9220
0.0000

'36.0000
“35.2859
“33.2108
“30.6556
'27.7899
~24.12©8
-20.2178
-0 “o8779
-8.9500
“2011879
0.0000

-36.0000

“18.0000
-20.2893
-23 o2976
“27.1915
‘31.8593
'37o11r816
-tt.0129
”51.2908

-58.587h

(THE Y COMPONENTS OF STRESS)

‘18.085’+
“10.3767
96.2087
-500219
‘5.6863
-6o78117
-6118811’“
“3.8599
5.1062
23.2516
50.8487

(THE

0.0000
1.2516
2.7566
1.007%
5.01h#
5.8206
6.2547
5.8579
h.0619
I] o 0623
0.0000

~16.7602 ~1h.0069 -h.7191
—7.9622 -h.1866 «1.7288
~h.1570 ~1.6253 ~.3973
-3011235 ‘1.51% _o3203
“QOS'DJOS -20115115 -07735
-5.7123 "3.8090 -1.hSOD
-6.5145 -5.0678 ~2.1856
-S.1578 -S.3678 -2.62h6

1.3193 «2.6731 ~1.569S
18.0607 8.7956 h.2232
52.5163 h9.8081 18.3723
XY SHEAR STRESS COMPONENTS)

0.0000 0.0000 0.0000

205571 b.5686 8.5h83

5.2605 7.0668 6.0013

7.2831 8.7826 6.7863

8.9587 18.8801 7.7779
10.5005 12.3875 9.3130
11.6978 11.5828 11.5622
11.7719 16.2658 18.1712
.9.LL30 16.1769 16.1815

3.87t1 11.9953 1h.6509

0.0000 “0.0000 0.0000

NOTE TITLES ENCLOSED IN PARENTHESHS ARE

-125
“72.0000 -1hh.

0.0000
-19071191
-12.8959
-21.8675
'31.h798
~h2.6391
*56.698&
‘75.1290
‘98.8088
.6276
0000

.0000
o OC‘C'D
.0000
.0000
.000

.0000
.0000
.0000
.0000
. 000

.0000

CJQHDGDGHQCDCDCHQGD

.0898
.0000
.0000
.0000
.0033
.0000
.0000
.0000
.0000
..0660
9h.7634

I
U1
a

speecumcecuacscwv

NOT PART OF OUTPUT

O O O O 0 0'0 0 n O O O O O O O O O O O O O O O O O n

124
LBSTING 0F ISOPEP EORTRAN SOURCE DECKS

PROGRAM PAGE
ISOPEP FOR THE 18M 1620 125
OUTIN ” " 125
STRESS 126
CHANG 129
SORLX 13o
ISOPEP FOR THE one 3600 131
A011 ‘ 135
A012 136
CHEB 137
PBIBD 128
PBTEX 128
PBZBD 110‘
PBZEX 111
PBBBD 112
PBBEX 113
93130 111
P31Ex 115
PBSBD 11s
PBSEX 116
PB6BD 132
PB6EX 135
soR3 116
SOR1 118
SORs 118

TABLE OF SYMBOLS 119

325

c ISOPEP FOR THE IBM I62O
C THE DECKS LISTED HAVE BEEN RUN UNDER THE MONITOR I
c SYSTEM ON AN IBM I62O NITH 2®K MEMORY AND A I3II DISK FILE.
c DECKS ARE LISTED IN APPROPIATE ORDER NITH MAIN DECK LAST.
shoooszoO7OI3600O32OO7O2R9O24O 25II9636II3O IOz
##JOB 5 .I OUTIN IOIII/65
##FOR 53
*LDISK
SUBROUTINE OUTIN
c INPUT AND OUTPUT SUBROUTINE
DIMENSION u(Ih,2h),IM(2O),IL(2OI,DIs) PII3.7 I.UI(I3)
COMMON U, IK, D ,Mx, IF IA MI HMY JF JA MJ N,NOUT,ND NT, NOL RFADRFIO
I SPYIJ, PRNO, DE ,DI, KPE INOT, MESH' NSTRS,NINC,IP,JP ,AMI, IL,P,UI

IO FORMAT 'gFIO.3. .EIO. 8, 2EIO.A IOIAI

II FORMAT F 8.3 FII. 7,I5.5I u/) .
I2 FORMAT sOHO SOLUTION OF THE BIHARMONIC EQUATION
II9H c. L. DAVIS/)
Ih FORMAT (45H RELAX. SUM OF ITERATIONS * POINTS NOT/
thFACTOR ERRORS COUNT MAx.NO. CONVEROED/)
Is FORMAT (uIH PROBLEM CONVERG. MESH SPACES MESH/

usHNUMBER CRITERION Mx MY IF JF STRESS/I
I6F MEMAT(2FI M05 2I8)
I7 FORMAT (I7H STRESS FUNCTION/)
:0 FORMAT I5, 7FIO.7/ 5x,7FIO. .I;;
22 FORMAT I5 7FIO. 7/8 5x, 7FIO. 7
IFII-INOT5I33, 8O8
so READ IO, PRNO, DE, RFA SPYIJ Mx MY IF JF,N,NT, NOUT, ND MESH, NSTRS
PRINT Iz'
. PRINT IS
PRINT II,PRNO,DE,Mx,MY,IF,JF,MESH,NSTRS
PRINT In
K-MX*MY
PRINT 16,RFA,SPY|J9N,NI,K
lA-MX-I-l F
JA-MY-I-JF
M|IlA+E
MJ-JA+I
NDL-NDH
NINc.NOUT
lF(MESH-3) 9I,9R.9R
91 DO 92 J-I ,I3 .
DO 92 III :7
92 P(J |)30. 0.
9H) OOH J-I MJ
DO IOO I-I ZMI

100
101

102
105

103

”(I "DI-9.1
IL J =3
IK J IIA
|F(N) 200

,303,fl02

DO “0h JaJF, JA

READ
NDL-N+ND
RETURN

20, K,(U(fl J),I mfiF F,IA)

 

 

I..- I I! II I l.

126

IF( KPE ) 90,90 119

IF (SENSE SWITCH 1 { 90,121

IF(N-NOUT) 123,126,127

IF(N-NT) 115,89,89

NOUT=NOUT+NINC

GO TO 121

lNOT-I .

GO TO 90

INOT-3 ‘

PRINT IO,PRNO,DE,RFA,SPY|J,MX,MY,IF,JF,N,NT,NOUT,ND,MESH,NSTRS

- PRINT 17

105

III
110
200

c Mx MUST BE LESS'

116
117
118
11h
112
113
115

##JOB
##FOR

00 105 J-JF,JA
PRINT

IF SENSE SWITCH I) 1
IF INOT-z) 112,110 1
IF KPE) 115,116,11
STOP

J,(U(l,J),l-IF,IA)
1 1

THA I IN THE DIMENSION OF P(J,I) IF THE
SOLUTION IS TO B SAVED FOR THE NEXT PROB.

lF(MX-6 ) 117,1I7, 1h

DO 118 J-JF,JAHM. NH,

K-JGJF+1

DO 118 l-IF,IA

LII~|F+I

P(K.L)-U(I.J)

IPhMX+1

JP-MY+I

AMI-I./DI

IF(NSTRS) 112,11

PUNCH IO,PRNO,OE

PUNCH 17

DO 113.J-JF JA ‘

PUNCH 22,J,(U(I,J;,l-IF,IA)

IF(SENSE SWITCH 200,115

RETURN .. .

END

3
N
E
1

5 115 I
,RFA,SPYIJ,Mx,MY,IF,JF,N,NT,NOUT,NO,MESH,NSTRS

'.‘2 I
.I I.

23 ' STRESS IO/17/65 CLD

*LDISK

C

27

#0
#2
805

1 SPYIJ,PRNO,D .
1h FORMAT 18H PROBLEM NUMBER,F6.3.5X.12H 5|MENS|0N -.|3.5H X.|3’I

SUBROUTINE STRESS
STRESS CALCULATION SUBROUTINE.3'I- .
DIMENSION Uélh 2A) IRI2AI,IL£241,O£s),FIIa.7 ).u1(13)
DIMENSION s IAI,Sx(Iu),Txv( ). .
COMMON U,IK,D MX,IF,IA,MI,MY,JF,JA,MJ,N,NOUT NO NT,NOL,RFA,RFI,
E,OI KPE,INOT,MESH NSTRS,NINC 1P JR AM1,IL,P U1

FORMAT ll)

30 FORMAT 21HO STRESS DISTRIBUTIONI)

FORMAT l5, 7FIO.h/E5X, 7FIO.4;;
FORMAT I5, ;F10.#/ 5X, 7FIO.#
IAI-I./(DI*DI . . . ”I .
PRINT 30

 

Ill Ill ll llll Ill ‘ '1
ll

I127

PRINT IA, PRNO ,MX,MY
DO 81A .JaJF, JA
820 KI=IK(J )
- K2=IL(J)
IF(J-IL(3)) 822,821, 822
821 K2=3
822 DO 812 I-IF, KI
_ IF (I-Kz) 811,81O,81O
811 S(l)=0.0 .,
w. GO TO 812 -
810 S(l)a(U(l+I,J)+U(I-I,J)-2.*U(fl,J))*AI
812 CONTINUE .. .. .
833 K-Kl
IF(NSTRS) 815 81A, 81A
815 PUNCH A2, J, (SIL) ,L-IF, K)
81A PRINT M(S(L) LaIF ,K)
.1 PRINT 27
, IF( ABSF(NSTRS)-2) 832, 833, 833
833 no 8AA J-JF, JA
KI-IK J
KZ-IL J
IF(J-IL(3)) 835. 83h9835
83A K2-
835 DO A1 l-IF, KI
IF I-KZ) 839, BAO, BAO
839 sx l)-O. O
60 To 8A1
8A0 SX(|)-(U(I ,J+1)+U(I,J—I)— 2. *U(I, J))*Afl
8A1 EOEIINUE
IF(NSTRS) 8A2, 8AA, 8AA
8A2 PUNCH A2, J, mi IL-IF, K
8AA PRINT AO, J, HL=IF K
.. PRINT 27
IF(ABSF(NSTRs)-3) 832, 8A5,8A5
8A5 DO 85A J-JF, JA
KIaIK J
K2-ILJ
IF(J-IL(3)) 8A7, 8A6 ,8A7
8A6 K2- 3
8A7 DO 850 I-IF K1
IF(I-KZ) 8A8 ,8A9, 8A9
8A8 TXY(I)=O. 0
GO TO 850
8A9 Txv(I)-(U(I+I, J+1)-U(I-1 ,J+1)-U(I+1, J- I)+U(I-1, J-1))*AI/A.
850 CONTINUE
, KaKI
IF(NSTRS) 852, 85A
852 PUNCH A2, J, TXY(L§,L
85A PRINT AO,J, TXY(L
832 EEEURN

85:” ,K)
L:IF k)

I28

##JOB S PBIBD IO/I7/65 CLD
##FOR 53
*LDISK
SUBROUTINE PBIBD
C CALCULATION OF BOUNDARY VALUES PROBLEM I

DIMENSION u(IA 2A) IK(2A),IL(2A), D(5), P(13, 7 ) ,UI(13)
DIMENSION PH(1A,2 AI

COMMON U, IK, O ,Mx, IF, IA,HI,MY,JF,JA,MJ,N,NOUT,ND,NT,NDL,RFA,RFI,
1 SPYIJ, PRNO, DE ,DI KPE,INOT,MESH,NSTRS,NINC,IP,JP,AMI,IL,P,UI
EQUIVALENCE (U, 'PHI
A=MY

DI=1./A

x=0.0

DO 103 1:53, IA

PH I ,JA)=O. O

IF x-. A) 103,103.1OA

XBX+DI

KBI

DO 105 lax, IA
PH(I,JA)=(2.*x*x—1.6*x+.32)*(-36. )

dd
CH3
:4»

I05 X=X+DI .
106 DO I08 J82, JA
108 PH(IA, J): -.72
X80. 0
DO 110 I =3.”
2) =(.s*x*x-.1)*(-36. )
IF X-. h) IIO,IIO, II“
IIO X=X+DI . ,.
IIR Kal
DO IIS I=K, IA
PH(I,2 )=(. h*X-.I8)*(-36 )
IIS X=X+DI
RETURN
END
##JOB 5 PBIEX IO/I7/65 CLD
tiFOR 53
*LDISK
SUBROUTINE PBIEX
C CALCULATION OF EXTERIOR VALUES PROBLEM I

DIMENSION U(IlIL 2A) II<(2A), IL(2A), 0(5), P(13, 7 ), U1(13)
DIMENSION PHIIA 2A5
COMMON U, IK, D ,sz IF, IA, MI ,MY, JF, JA, MJ ,N, NOUT, ND ,NT, NDL, RFA, RFI,
1 SPYIJ, PRNO, DE, DI KPE, INOT, MESH, NSTRS ,NINC, IP, JP, AMI, IL,P,UI
EQUIVALENCE '(U, PHI
DO 117 J=2 JA
PH 2 ,J):PH$h, ,J;
PHI J)- PH 5 J
117 PH MI ,J)=PH(IA-I, J)-0. 8*DI*36.
DO 116 I=1 IA _
PH I, 1)=PH I 3)
116 PH fiMJ)=PH(I, JA-I)
RETURN
END

 

 

lll‘llllllllllli

*1JOB
*tFOR

E29

53 CHANG EG/E7I65 CLD
S .

*LDESK

C

EGG
EOE
E02
E03

20

E8
22

2h

10

ES

E SPWY J PRNO, DE, 0E9 KE

SUBROUTENE CHANG
CHANGE MESH SPACENG
DEMENSEON UEEE9 2EE9E

K( E(2LE95 (5)9P(E397 )9 UECEB)
COMMON U9 flK9 n 9Mx9 9EE9£A

24E9'E
9ME9M M'Y9 EE 9EE MJ9 N NOUT ND 9NT HNDL RFA9 RFE,
9EM9E9ME2M: MSERs9 ”NENC, IE 9JP, AME, EL, P, UE
FORMAEKEEs)
FORMAT E59 7FE®98)
FORMAT flS§7FE© 72E5X97 FE®97E)
FORMAE Es 7FE0997E929/E @9EEE
EE(ME$H—2 25 229 2E9 22 .

READ . EEE9EP9EP9ME9M2
AMEaME .. 9
AM2=EM2

DO E8 J=E9JP

READ M729 K,,( P
PRENT 9 E@39J9(
GO TO 2h _
MzaMY

MEBAME+900000E

AME-E42 .

ozaE./AM2

DSmE./AME

JA2wM2+E.

JmE

L=® '

Y2.@o

flmE ’

X“ BOO

XZB-DZ

E2=o

ovavz

JYBL+3

EzaEZ+E

flXaEZ+2

EF(E2—JP E79 79 EE

xzaxz+02 A 9

oxgxz-XE

EE(AB$F(Dx-03)-. EEEOEEE 99 99EE

EF(DX-03) 899

uE(E 2)-P(J9 E)+DX*((P(J9 E+EE-PEJ9EEEID3)+0Y*((P(J+E,E)-P(J9I))/o3)
u(Ex9 JYE=UE(E 2E .

EF(Ez-JP) 59 59 E©

flefl+E

XEBXE+DB

DXaXZ-XE

flF(fl-EP) 89 89 E0

LwL+E

PRENE EE39E9EUEEE2E9E2EE9JP)

BF(L-JA2) E59 EE9 EE

Y2wY2+DZ .

EE(ABSF(Y2-83E-. EEEEEEE E29 E29 E6

 

{II II 1‘ ll 1’ ll 1 1". ':..

330

I6 IF(YZ 03) 6,112,‘J2
12 JmJ+I

Y2=Y2~DB

IF(J -MI-I) 6969I4
Ih RETURN

END

$¢JOB 5 SORLx IO/I7/65 CLO
##FOR 53 A
*LDISK
SUBROUTINE SORLx
c POINT SUCCESSIVE OVERRELAXATION SORLx
DIMENSION u(Ih, 2A),IK(24), IL(2A), 0(5), P(13. 7 ). UI(I3)
COMMON U, IK, O ,Mx, IP, IA, MI ,MY, JF, JA, MJ, N, NOUT ,N6,NT,NOL,RPA,REI,
I SPYIJ PRNO, DE DI, KPE, INOT, MESH, 'NSTRS, 'NINC, IP ,JP,AMI,IL,P,UI
IO FORMATI2FIO. 4, is, 2I8)
SYIJ-O. O
KPE-O
KTBO
JBBJA-I
I3O DO IAO J- 3, JB
IBBIK J)'I
lC-IL J)
‘ DO 133 l-IC,IB
IMBI-I
IMMII—Z
13I YIJ a .05*RFA*(8. * U(l+l SJ)+ U(IM, J)+ u(I, J-I)+ U(I,J+1))
-2. *( u(I+I, J+I)+ U 1M, J+IS + U( (I+I J-I)+ U(IM, J-I ))
2 - U(l+2, J)-' U(IMM, J)- u(I, J+2)- U(I, J- 2))-REA* u(I, J)
Y2=ABSF(Y|J)
SYlJ-SYIJ+Y2
, U (I, J): U(I, J)+YIJ
IF (Y2-DE) 133,132,132
132 KPE=KPE+I H.
133 CONTINUE
C A ADO CARDS FOR SOR3, SORh OR SOR5 BETWEEN STATEMENTS 133 AND 160
tho CONTINUE _
. ‘ RFI=SYlJ/SPYIJ
SPYIJ=SYIJ
134 IF (N-NDL) I37. I36, I36
.136 NDL=NDL+ND . .
RPA=2. /(1.+SORTE(ADSF(I.-RF1)))

PRINT IO RFA, SPYIJ ,N,NT,KPE
137 RETURN A,
T 1 END ,
¢¢JOB 5 ISOPEP C L DAVIS IO/I7/6
##FORXSB 1
C PROGRAM ISOPEP
C ITERATIVE SOLUTION 0F PLANE ELASTOSTATIC PROBLEMS

DIMENSION U(1h,2h),lK(24),lL(2h),D(5),P(13,7 ). UI(I3 )

COMMON U, IK, D ,Mx,IF,IA,MI, MY, JF, JA, MJ, N, NOUT NO, NT, NDL, RFA, RFI,

I SPYIJ, PRNO' OE ,OI,KPE,INOT, MESH, NSTRS, NINC, 16, JP, AMI, IL, P ,U1
5 FORMAT '(28HI SUCCESSIVE OVERRELAXATIONl)

 

lIIIII‘I-‘l‘t‘l

IBI

50 INOT-I
RFI.I0
PRINT 5
KPE-II
CALL OUTIN
IF(MESH-z) 55.52.52

52 CALL CHANG

C NEXT INSTRUCTION ASSURES COMPATIBILITY 0F SUBROUTINES 0N IBM 1620

55 U(MI,MJ)-ABSF(SQRTF(I.O))

CALL PBIBD A

60 N-N+l

CALL PBIEX

CALL SORLX

INOT - 2

CALL OUTIN

IF(INOT-Z)‘50,68,60

68 :EEKPE) 80,80’60 .

80 ABSFINSTRS -I) 50.82.82
82 CALL STRESS A .- . .
ggDTO so
*LOCAL,OUTIN,SORLx CHANC,STRESS,PBIBD,PBIEx
1.190 .OOOéI 1.5 - . 1.0” 5 IO 3 2 O zoo zoo 10
1.198, .OOOOI 1.5 1.0 IO 20 3 2 O zoo zoo Io
. I.I90 .OOOOI I.67 .OOOI ”5 IO 3 2 -1 zoo zoo 1o

(LAST THREE CARDS ARE EXAMPLES OF INPUT DATA

, ISOPEP FOR THE CDC 3600
THE DECKS LISTED HAVE BEEN RUN UNDER THE SCOPE SYSTEM
THE MAIN DECK IS PLACED FIRST. THIS IS FOLLONED BY THE
SUBROUTINE SUBPROGRAMS OUTIN CHANCE, STRESS SORLx,
THE DATA.THE SAME SUBROUTINES AS LISTED FOR IHE IBM I6zo ARE
PBNBD PBNEx AND THEN AFTER NECESSARY SCOPE CONTROL CARDS

I USED EXCEPT ARRAY DIMENSIONS ARE CHANGED AND THE FIRST THREE
CARDSIIIJOB 5) (IIFOR 5) (*LDISK) ARE OMITTED. THE
SUBROUTINES PBéBD AND PBGEX ARE LISTED. NOTE PBBBD USES
COMPLEX TYPE VARIABLES NHICH ARE NOT AVAILABLE IN
FORTRAN II BUT ARE AVAILABLE IN 36oo FORTRAN.

ggog,331547,ISOPEP6, zo.DAVIS,C,L. 9/zI/65 GROUP C
' .
IROCRAM ISOPEP6
C ITERATIVE SOLUTION OF PLANE ELASTOSTATIC PROBLEMS.. .
DIMENSION u(6u 6A),IK 6h;,IL(6h),D(5).P(31.31),Ul(6h)
DIMENSION UXD(€A) UYD 6# w L .
COMMON U IK,D Mx,IF IA,HI,MY JF,JA MJ,N NOUT ND,NT,NDL,RFA,RFI,
1 SPYIJ,PRNO,DE,DI,KRE,INOT,MESH,NSIRS,N NC,IP,JP,AM1,IL,P,UI
ACOMMON uxo,UYD A
5 FORMAT (zBHI SUCCESSIVE OVERRELAXATION)
so INOT.I I
PRINT 5
KPE-l
CALLAOUTIN

00000000000 0

0000

I32

IF(MESH~2) 55,92,52
52 CALL CHANG
55 CALL PBBBD
60 NmN+I

CALL PB6EX

CALL SORLX

INOT a 2

CALL OUTIN

IF(INOT-z) 50,68,6O
68 IF(KPE) 8O,8O 60
80 IF(ABSF(NSTRSI~I> 53,82,82
82 CALL STRESS

GO TO 50

END

INSERT SUBROUTINE OUTIN - CHECK ARRAY DIMENSIONS
INSERT SUBROUTINE CHANGE - CHECK ARRAY DIMENSIONS
INSERT SUBROUTINE STRESS - CHECK ARRAY DIMENSIONS
INSERT SUBROUTINE SORLX - CHECK ARRAY DIMENSIONS

SUBROUTINE PB6BD
BOUNDARY VALUES FOR REGION OF INFINITE PLATE WITH A SoUARE HOLE
TYPE COMPLEX PHI PSI,PDU OMEC,DOMG DOMGB,DPHI OPHIB PSIB,zET
DIMENSION U(64,6A),IK(6H ,lL(6h),DI5).P(31,BII,UI(6A)
DIMENSION UXD(6A),UYD(6A ,
DIMENSION DUX(62),DUY(62),X(6z),Y(62),UBx(3I),UBY(3I)
COMMON U,IK,D,MX,IF,IA,M|,MY,JF,JA,MJ,N,NOUT ND,NT,NDL,RFA,RF1,
I SPYIJ,PRNO,DE,DI,KPE,INOT,MESH,NSTRS,MINC,IP,JP,AMI,IL,P,UI
COMMON UXD,UYD
LAGRANGE INTERPOLATION FORMULA FOR UNEOUAL INTERVALS

GRANF (XA) =UA*(
5(fi§;¥2)*(XA-X3)*(XA-Xh)*(XA-X5)I/((X1-X2)*(XI-X3)*(XI-Xh)*(XI-X5))
+

iifié;XI)*(XA-x3)*(XA—xu)*IXA-x5))/((X2-XI)*(x2-x3)*(x2-XA)*(xz-xs))
2(XA;¥I)*(XA-XZ)*(XA~X4)*(XA-XS))/((X3-X1)*(X3-X2)*(X3-Xh)*(X3-X5))
+UD , '
g(Xé;XI)*(XA—X2)*IXA-X3I*IXA-xs))/((x4-XI)*(xu-x2)*(XA~X3)*(x4-x5))
+U _
9IXA-XI)*(XA-Xz)*(XA~X3)*(XA-X4))/((xs-XI)*(x5-x2)*(X5-x3)*(xs-XR))
chXF(RH,TH)A(COSF(TH)/RH-RH**3*COSF(3.*TH)/6.+RH**7*COSF(7.*TH)/
I5 . *R
chYFERHSTH)a(SINF(TH)/RH+RH**3*SINF(3.*TH)/6.-RH**7*SINF(7.*TH)/

15 . * -R
22 FORMAT (zOIA) -
50 FORMAT (ABHO INFINITE PLATE WITH A SQUARE HOLE C L OAVIS//)
6O FORMAT (IIHO THETA = ,FIO.5,6HRHO a ,F8.2,IsH NO BOUNORY PT./)

PRINT 5O 2

A=MX

DIaI./A

RaI.I796h523/3.

PEEBOBLI'E59265

THETA=2.*PI

LIM=37

I33

DANGaLIM-I
DANGauPI/(2.*DANGI
XMAXaI.0
YMAXfiII o O
DLROa-.05
DO 6hU lafloLIM
RHOaI.0
IF(THETA-7. *PI/h I 62D, 6IO, 6IO
6IO RIDX:XMAX-ABSF(CORXF(RHO, THETA))
IF(RIDX) 6I6, 63D, 6I2
6I2 RHO=RHO+DLRO
IF(.IO—RHO) 6IO ,6IO,6IA
6Ih PRINT 6O,THETA, RHO .
GO TO 630
6I6 Do 6I8 K=I 50
DRO=((CORXF(RHO, THETA)/XMAX-I )*RHO)/(R*(RHO**7*CO$F(7. *THETA)/7 -
Iz. *RHO**3*COSF(3. *THETA)/3. )IXMAx-I. )
IF(ABSF(DRO)-. OOOOOOI) 63D, 630,6I8
6I8 RHO=RHO- DRO . - VA
GO TO 6Ih
62D RIDYaYMAX—ABSF(CORYF(RHO, THETA))
IF(RIDY) 624 ,630,622
622 RHO .RHO+DLRO _
IF(O. I-RHo) 620, 62D, 6Ih
62h DO 626 K-I ,50
DRO=((CORYF(RHO, THETA)/YMAx-I )*RHO)/(R*(RHO**7*SINF(7. *THETA)/7. -
I2. *RHO**3*SINF(3 *THETA)/3. )/YMAX-I. )
IF(ABSF(DRO)-.OOOOOOI) 630,630,626
626 RHOsRHO-DRO .
GO TO 6Ih
630x I)=CORXF(RHO, THETA)
Yl)=CORYF(RHO, THETA)
ZET=CMPLX(RHO*COSF(THETAL RHO*SINF(THETA))
OMEGuR* I° /ZET-ZET**3/6 +ZET**7/56. )
DOMGaR* .I25*ZET**6- 5*zET**2-I IzET**2)
DOMGBa CMRLX(REAL(DOMG),'-AI MAGIDOMG)).
PH1=R*(. 25/2ET+ 426*2ET+.Oh6*zET**3+. OO8*2ET**5+. 00h*ZET**7)
DPHI=R*(. 028*ZET**6+. 0h*ZET**h+ U38*ZET**2+. h26-. zs/zET**2)
OPHIB: CMPLX(REAL(DPHI), -AI MAG( DPHI)) .
PSla-R*( 5/2ET+(.5&8*2ET- A57*ZET**3-. 026*ZET**5-. 029*ZET**7)/
I (I.+. 5*2ET**A-. IZS*ZET**8))
PSlBa CMPLX(REAL( PSI), -AI MAG(PSI ))
PDUaPHI+(0MEG*DPHfl B)/DOMGB+PSIB
DUX(I)aREAL(PDU)
DUY(I)=AIMAG(PDU)
640 THETA aTHETA + DANG
C BOUNDARY POINTS REGION OF AN INFINITE PLATE
READ 22, (IL(J), J33, JA)
UBYEI ;:—.Osh5259
UBX .323200z
LUMaLlM/Z A,
DO 650 Rafi, LUM
KaLIM-I
UBX(I+I)=UBX(I)-. S*(DUX(K)+DUX(K+I))*(X(K+I)- X(K))

USA

65D UBY(B+I)2UBY(E)+.5*(DUY(I)+DUY(fl+I))*(Y(I+I)-Y(l))
' BOUNDARY POINTS ON THE SQUARE
K=IL(3)
DO 652 l=3,K
UEK,I;=0.0
652 U I,K =0.0
YVAR=0.0
K=3
DO 662 J=JF JA
654 IF(YVAR-Y(K5) 66D,66O,656
656 lF(K-LUM+2) 658,658,660
658 K=K+fi
GO TO 65A
660 XI=Y(K-2)
x2=Y(K-I)
X3=Y(K )
Xh=Y(K+I)
XS=Y(K+2)
UA=UBY(K-2)
UB=UBY(K-l)
UC=UBY(K )
UD=UBYEK+I§
UEBUBY K+2
U(lA,J)-GRANF(YVAR)
UA-DUX(K-2)
UBaDUX(K-l
UC=DUX(K ‘
UD=DUX(K+I)
UEHDUX<K+2)
UXD(J)-GRANF(YVAR)
662 YVAR=YVAR+DI
XVAR=0.0
K=Ih
DO .672 I=IF,IA
66h KB=LUM+K
... KC=LUM+2~K
lF(XVAR—X(KB)) 670 670,666
666 IF(h-K ) 668,66 .670...
668 K=K-I .. .. ..W
. GO TO 664
670 XI=X(KB+2)
x2=x(KB+I)
X3.X(KB _)
XA=X(KO-I)
X5=X(KB-2)
UA=U8X(KC-2)
UB=UBX<Kc-I)
UC=UBX(KC )
UD=UBX(KC+3)
UE=UBX(KC+2)
U(I,JA) . GRANF(XVAR)
UA=DUY(KB+2)
UBaDUYéKB+i)
UC=DUY KB )

I35

UDaDUYéKB—I)

UEaDUY KB-z)

UYD(I) . ORANF(XYAR)
XVAR=XVAR+DI

RETURN

END

SUBROUTINE PBGEX

672

C BOUNDARY DERIVATIVE CONDITIONS

VALUES OF STRESS FUNCTION AT EXTERIOR POINTS WHICH SATISFY

DIMENSION U(61I, 610,1] K(61I),II.(6PI), 0(5), P(3I, BI) ,UI(61I)

DIMENSION UXDC6A) UYD(6h I

COMMON U, IK, D ,Mx, IF IA, MI ,MY, JF, JA MJ, N ,NOUT NO NT NDL, RFA, RFI,
I SPYIJ, PRNO, ”DE, DI, KPE, INOT, MESH, NSIRs, NINC, IP, JP, AMI, IL, P, UI

COMMON "UXD, UYD
KHIL(3)‘I
DO 685 JaJF, JA
IF(J-K) 682 68
683 U(2 J)-U( (K+2, J
oIo 685
682 U(K, J)aU(K+2, J)
GO TO 685 -.
68h ӣ60 ,J -U 5, J
MI

O J -UA
685 U ,J)-U(IA-I, J)+UXD(J)*2. *DI

DO 690 I-IF

IF( I-K) 686 686 688
686 U(l ,K)-U(I ,R+2I

GO TO '69O I
6880 U§|m ,I)-U 5)

Io
I I
,MJ)-U(I3
ETURN
END
SCOPE
9LOAD

9RUN, 20, 3080
I.7Ol+Ti

,68h

A-I)+UYD(I)*2.*DI

.OOOOI
I9 I9 I19 I9 I9
3 M3 '3 .3 ”3 .
3 ’3 3 3 3 3
I. 7OI .OOOOI . Io5
.LAST FIVE CARDS ARE EXAMPLES 00.

65 I.
,I9 I9 “I
3 3

“3 .3

mmmg.
muu$5
Emm$5

I.O I
INPUTTDATA

TO USE ADI,

3 3 025002500
I9 I9 I92 I9 I9
.3 v3 3 3 3
3 3
3 3

3 3 3
-I 200 200

REPLACE THE CALL SORLX CARD IN THE ISOPEP
CALL ADIZ

CHECK FOR AGREEMENT OF DIMENSION AND COMMON STATEMENTS

SUBROUTINE ADII

C

C

C , MAIN DECK WITH TWO CARDS, CALL ADII,
C

C ALTERNATING DIRECTION

IMPLICIT METHOD ROW SOLUTION

DIMENSION U(28, ,g2), ,UH(ZPI~, PEELHSOI, HCSO) ,G(50), 8(50)

DIMENSION IK(52
COMMON U,
I SPYIJ, PRNO, DE, DI,
COMMON oUH, F, G ,6,I,J, x, K

IL(52),

KPE, INOT, MESH: NSTRS ,NINC, IP, JP ,AMI,

IK, D, MX, IF, IA, MI ,MY, JF, JA, MJ, N, NOUT, ND, NT, NDL, RFA, RFI,

IL, P ,UI

I

I36

EQUIVALENCE (I ,HI,III‘ II ,RNR), (8(I), CJ ), (GII), RFA)
JB—JA-I
DO I66 J23,J8
IDAIA-I
SYIJa0.0
KaIB-I
LaA 2
Do I66 I23
F(I )38. *IUII J+II +UII LJ-I))+A *(U(I+I J)+U(I- I J))-U(I,J+2)-
2.*(U(I+I,J+II+UII ,J+I)+U(lwI,J-I)+UII+1,J-I)I-U(I,J-2)+
2(RNR-Iu. )*U(I, J)32
F(l— A) 150,15I,
EI)=I.0/CJ
I =-u.*C(I) I
(I =(F(l) u(I 2, JIIA,ID(I-I,J))*C(I)

I F(l)aF(l)-U(I 2, J)

2 DELN=4.+8(I-2)

WJaCJ- C(I-2)+DELN*8(I~I)

G l)=I.O/WJ

B l)=(- u. +DELN*G(I I))*G(l)
IF(K-I) I53, 15h, I55

F I)=F(I +h. *U(I+I, J)

F I)-F(I -U(I+2, J)

G I)=0. D

H I)-(F(l)-H(I-2)+DELN*H(I-I))/WJ
CONTINUE

IB-IB- 2

DO I66 l2-I,lB

l-IA-IZ

K-l-Z

YIJ=H(I

IF(I2- 2 I66,I62, I6I

I6I YIJaYIJ- -GEI)*UH(K+2, ,L)

I62 YIJ-YIJ- B I)*UH(K+I L)

I66 UH(K, L)=YIJ .
H RETURN
END

SUBROUTINE ADI2
c ALTERNATING DIRECTION IMPLICIT METHOD COLUMN SOLUTION
DIMENSION U(28,52), ,UHI2A.6h8;, ,F(50), H(SO), C(50) ,B(50)
DIMENSION IK(52), IL(52)D
COMMON U, IK ,D,Mx,IF, IA ,MI ,MY, JF, JA, MJ, N, NOUT, ND, NT, NDL, RFA ,RFI,
I SPYIJ,PRNO,DE,DI,KPE, INOT, MESH, NSTRS, 'NINC, IP, JP, AMI, IL,P,UI
COMMON UH,F,G,B, ,J,x, K
EQUIVALENCE (F,H ,(F(I )IRNR), (8(I), CJ ), (6(1), RFA)
SYIJ=0.0
KPEaO
I8=IA-I
DO I81 Ia3,I8
KaI-Z
JBHJA-I
DD I76 J=3,J8

ISO

and
mm

_._.
OU'I #W

qua—o
0“." “U1

V-

C5C’ run

I37
L-J-2
F(J)-RNR*UH(K L)+6.*U(I,J)-h.*(U(I,J+I)+U(l,J-1))+U(I,J+2)+U(I,L)
IF(J-u) I70, III,I72
‘70 SJ .300/CJ .MH ..

J-#.*G(J

J-(F(J)-U I .J-2)+#. *UII,J-I))*G(J)
GO TO I76 .

I7I F(J)-F(J)-U(I J-z)
I72 DELN-hJ+B(J-2

NJ-CJ-G(J-2)+DELN*B(J-I)

:53 1.3 00/ IWJ
.(-u.+D£LN *G(J-I))/NJ

IFJ (JB-I -J) I73! I7h, IZS
:J -F(J)+II...*.UI. lsJ-I-I
G J

H u(F(J)-H(J-2)+DELN*H(J-I))/WJ
CONTINUE
JB-JB-I
DO I8I J2-I,JB
J-JA-J2 . H
YIJ-H(J
IF(J2-2 I7 ‘1 8.177
g YIJ-YIJ-GJ U I, J+2
TIJ-YIJ-BJ *U I :J+I
9 Y2-ABSF(YIJ-U(I, J))
SYIJ-SYIJ+Y2
IF(YZ-DE) IBI.I80,I8O
I8. KPE-KPE+I .
I81 U(I J)-YIJ
SPYIJ-SYIJ
RETURN
END

TO USE CHEB INSERT THE FOLLOWING CARD AFTER THE
:éng CALL OUTIN STATEMENT IN THE MAIN ISOPEP DECK
REPLACE THE CALL SORLX CARD IN THE MAIN ISOPEP DECK WITH
THE FOLLOWING TWO CARDS

. CALL CHEB(NCHD)

;& NCHB I 2

‘ CHECK DIMENSION AND COMMON STATEMENTS

SUBROUTINE CHEB(NCHB)
CYCLIC CHEBYSHEV SEMI- ITERATIVE METHOD
DIMENSION u(ze g2), ,IK(52), 0(5) ,AKI3D).AL(3D,3D),5A(3o.3o),uz(3o)
DIMENSION IL (55
DDNNDN u IK, D Nx IF IA NI ,MY JF. JA NJ, N NOUT ND NT, NDL RFA ,RFI.
I SPYIJ, PNND’ DE, DI K5: INor,Nésu' NsIRs' NINC, II ,JN,ANI. IL, P .01
CONNDN 'AK 5A, AL ui, RHO
3o. IF(NcND-II 302. 3D: ans
302 NQ-(JA-3)/ 2 ..
L- JA-z /2
IF L-NQ 3Io,3Io.3Du

138

304 PRINT 3053
305 FORMAT (26H ERROR 000 NUMBER OF ROWS)

STOP IOIO
BIO NI-IA-3
NII-N|+NI
DO 312 K-I, NII

mg-O 0

K-0 0
DO 3I2 L-I, NII

fiéKh ,L :00 0o

3I2A
Do 3II K-I, NII
BIN AL K ,K)-20: 0
AL I ,2)-8. 0
AL I .3 -8. 0
AL 2, 'u -8. 0
AL 3, h)-—8. o
£5)-8. 0
AI: 1:“ .200
AL 2,3 =2.0
AL #,5 -2.0
AL 1,5 B‘00
AL 2,5):0. 0
DO 3‘5 K85 NII
AL K-3.K+I -AL(K-A,K I
AL K—Z K+I -AL(K-h,K-1
AL K-I, K+I -AL N-A,K- 2)
315 AL(K 'K+I)-AL K-u,K-3)
DO 316 L-I, NII
DO 316 KaL NII
3I6 AL(K L)=ALIL, K)
SA(I I)-$QRTF(AL(I I))
DO 320 L-25
320 EA$1,,L)-ALII,L)/SA(I,I)
32h DO 340 K=2, NII
. KI=K+I
KLRaK-h
IF(KLR) 325. 325 326
325 KLR=I .
326 KUPaK-I
- TEMP=0.0
Do 328 L-KLR KUP
328 TENP-IENP+$AIL K)*SA(L, K)
SA(K, K)-sIRTF(AL(K, K)-IENP)
IF(K-Nll) 329, 3&0, 340
329 LUP=K+N
IF(LUP-Nll) 332 332 33D
33D LUPaNII
332 Do 336 L-K1,LUP
TEMPao. o U
DO 33h L1=KLR KUP
334 TEMP-TEMP + SA(LI. K)*SA(LI ,L)
336 SA(K, L)-(AL(K, L)—IENP)/SA(K, K)

380
282

28k
285

385

337

' P2-8. *U I J+2 -2. *

3&8
350

352

. AK(h)-(P2-AK(I)*SA(I,h)-AK(2)*SA(2,Q -AK(3

35h
355

356
357

358

359
3609

I39

CONTINUE

IF (RFA) 282,282,288

A=MY

A-A*DI

B-MX

B-B*DI

B-(A/B)**2
RLAM-(5.Ihh*(I.0+B*B)+3.I 5*B‘WID /A)**h
RHO-I. 0/(I. 0+RLAM*DI**A/2.Q)
GO TO 285 .
RHO-SQRTF( 2.-2. /RFA)

RFZ-Z. 0/(2. 0-RHO*RHO)

RFIHI.

JC-JA-I

KPE-O ’

SYIJ-0.0

RFA-RFI

JB-3 I

?0 390 J-JB, JC, 2

3

Do 360 K-I, NII,
PI-8. *u I ,J-I -2.* EU él-I, J-I;+UEI+I,J-I;;$ I, J- 2 -u I,J+2

I+I J+2 +U I,J+2 I, J-I -U I,J+3
IF K-u 3L8.3su.3suu 2
IF K-Z 350.3509352
PI-PI+8. *u I-I.J)-2. *U(II I,J+I;:U(l- 2,J)
Pz.Pz+8. *u I-I J+I)- 2. *U(I- U(l~2,J+I)
AKI IgnPI/SAII I)

.(pz-AKIII*sA(I. 2))/SA(2, 2)
GO TO 360 .
PI-PI-U I-2, J)
P2-P2-U I-2, 'J+I)
AK(3)-(PI-AK(I)*$A(I.3)-AK(2)*SA(2.3 )ISA(3. 3)
)* SA(3.4))/SA(#.#)

GO TO 360 .
IF(K+3-Nll) 357. 355.355
PI=PI-U I+2,J J)
P2-P2-U n+2. J+I)
GO TO 357
PI:PI+8. *U I+I ,J)- 2 .*U(I+I, J+I)— UII+2 ,J)
P2-P2+8. *U I+I :J+I)- 2.  *U(I+I, J) ~UII+2, J+I)
KUPaK-I
KLR-K—A
TEMP-0.0
DO 358 KI-KLR KUP
TEMP-TEMP +AKIKI)*$A(KIK
AK(K ).(PI-TENP)/SA(K Kim
KLRaKLR+I
TEMPao. o
Do 359 KIuKLR, K
TEMP-TEMP+AK(KI)*SA(KI K+I)
AK€K+I)a(P2-TEMP)/SA(K+I, ,K+I)
II +fl

IhO

K=NII
U2(K) =AK(K)/SA(K K)
32(KTII7IAKIK-I) ~SA(K~I ,K)*u2(K))/SA(K-I, K-1)
00 380 La2 KI 2
K=NII -L
KUP=K+h .
IF (KUP-NII) 368,368,367
367 KUP=NII
368 TEMP=0.0
KLR=K+1
no 370 Ll-KLR KUP
37o TEMPaTEMP +5AIK, LI)*U2(LI
u2(K).(AK(K)-TEMP)/SA(K, K
KLR-KLR-I
IF(KUP-Nll) 372, 37h, 37k
372 KUPaKUP-I
37h TEMP .o. o
no 376 LI-KLR KUP
376 TEMP-TEMP+SA(K-I, LI)*uz(LI)
U2(K-II-(AK(K-l)-TEMP)/SA( K-l, K-I)
380 CONTINUE .
-3
no 390 K-I, NII 2
YlJl-RFA* 02 K -U(l (JII
YIJ2aRFA* U2 K+I)-uI I ,J+I))
Yl-ABSF(YIJI) I H
Y2=ABSF(YIJ2)
SYlJ-SYIJ+YI+Y2
WE =U(I, J +YIJI
I ,J+I -U(| J+I +YIJ2
IF(YI-DE) 38A, 382, 382
382 KPEBKPE+I
383 IF(YZ-DE) 390, 386, 386
386 KPE=KPE+I .
390 IBI+1 H
IF (JB-B) 392. 392, 39A
392 JB=5
. RFA=RF2
GO TO 3A7
39h RFI-I. 0/(I.o-RHO*RHO*RFA/h. 0)
RF2=I. 0/(I.o-RH0*RH0*RFI/h.o )
I SPYlJ-SYIJ
RETURN
END

TO RUN PROBLEM 2 REPLACE THE CORRESPONDING CARDS IN THE
MAIN ISOPEP DECK WITH THE FOLLOWING TWO CARDS

CALL PBZEX

CALL PBZBD

CHECK DIMENSION AND COMMON STATEMENTS

SUBROUTINE PBZBD

C

IAI

CALCULATION OF BOUNDARY VALUES OF PROBLEM 2

DIMENSION U(28, 52), IK(52), IL(52), D(s) ,P(25, I3) UI(25)

COMMON U, IK, O ,Mx, IP, IA MI :MY, JP, 3A, MJ, 'N, NOUT N6 NT, NDL, RFA ,RPI,

I SPYIJ, PRNO OE DI, KPE, INOI ,MESH, NsIRS, 'NINC, IP, JP ,AMI, IL, P ,UI
IOO RTPI-.5/3. III5927

A-Mx _ ..

DI.“ o/A

X-O. 0

DO I I93, IA

U(I 2 - - .S*X*X

IF X-. 25) IO5,IOS,I06
X-X+DI .

K-I

DO IO I-K, IA

U(I, 2 IO. 03I25-.25*X
107 X-X+DI - H

Y-I. E-IO
DO I08 J-2, JA
U(IA J;-((9../16. +Y*Y)*ATANF(. 75/Y)- (25. /I6. +Y*Y)*ATANF(I. 25/Y)
I - .547 *RTPI+. 03I25
108 Y-Y+DI

X-O. 0
00109 I-3 IA
U(I JA)-RTPI*( (( u.+(x -.25)**2)*ATANP((x-.25)/2. )- ((X+. 25)**2+A. )*
I ATANF((X+. 25 )l H.) I.)+. 03I2s
109 XIX+DI .
RETURN
END

SUBROUTINE PBZEX
DIMENSION u(za, 52}F IK(52), IL(52), DIS) ,P(25 I3) UI(25)
COMMON U IK, D Mx, IA MI 'MY JE, JA MJ, 'N NOUT N6 NT, NDL, RFA ,RFI,
I SPYIJ PRNO, DE, DI KPE INOI ,MESH, NSIRS' NINC, IP Jfi, AMI, IL, P, UI
"CALCULATION, OF ExIERIéR VALUES
POINTS OUSIDE OP RANGE x-O 0R GREATER AND x.I OR LESS
RTPI-.5/3. IAI5927
IIO Y-I. E-IO
w DO II2 J-2, JA
U2 ,J)-U(A, J
I'J)-U(5, J .
Y UMIéJ)-U(IA-I, J)+2. *DI*RTPI*(I. 5*ATANF( .75/Y)-2.5*ATANF(I.25/Y))
I12 - +

. POINTS OUTSIDE OF RANGE YaO OR GREATER AND Y92 OR LESS
X-O. O . , ‘ .

DO IIA I-3, IA
Oil oI)-U(I3 '
U I MMJ)-U(I JA-I)+2. *DI*RTPI*( A.*(ATANF((x-.25)/2.)-
I ATANF((X+. 25)/2. ))> . A
IIh X-X+DI
RETURN
END

ALONG Y=O

...-g
O0
0“”

ALONG X-I

ALONG Y-2

C

SUBROUTINE Iasza
BOUNDARY CONDITIC-N: I
DIMENSION U(28 52), II
DIMENSION ARI7I
COMMON U, IK, D Mx,
1 SPYIJ, PRNo, DI, DII
EQUIVALENCE '(AR(II,

I)
I2
22 FORMAT (2014)
A-MX
DI-I./A
lB-lA-l
READ 22, I
PRINT 22,
JBuIK(3)+I
DO ZIO J-JB, JA
U(IA J)-. 125
Y-O. 6
DO 211 1-3, IB
U(I, JA)-. S*Y*Y
fi2)-Y

2IO

KrE,

U(I
EQUIVALENCE(D$3,U( I59)
EQUIVALENCE (DHI, UI I6

IhZ

IHE NOTCHED PLATE
I:EZI.D(5) ILISZ)

II MI ,MY, JF, JA MJ, N NOUT N
INOI MESH, 'NSIRS, NINC, IP,
29 )5. (D51, ,U(Ié7)) IIDS
3”(054, ,U( I60) (055 ,U(

IK J), J23, JA)
IK J), J=3, JA)

THE VALUE OF U ON THE CIRCULAR ARC

U(|03)-05*(Y-0 25)
Y=Y+DI

I U22, I#)-U 22 ,3 )
U 26,15)-U 26, 3 )
x- O. 6

JB-JB-I
DO 216 J-3, JB

THE VALUE OF Y CM TH
U(Ml, J)=,

211

E CIRCULAR ARC

5-SQRTF(. O625-X*X)

THE VALUE OF U ON THE CIRCULAR ARC

U(IA, J)=. S*(U(MI, J)-.

25)

0:
JPM
2.

I

PROB. NO. 3

NT, NDL ,RFA,RFI,
, m1 IL P,UI

U(

61

(156 )
)), (DS6,U( 162))

THE VALUE OF DELY/H FROM THE MESH POINT TO THE CIRCULAR ARC

212, 212
.25)/D1

IF(J- 3) 216
DEL=(U(MI J
IF(DEL- 1. I
DELBDEL’ i0
GO TO 213
U(IA-I, J)-DEL
X=X+DI
“4 ,h
UI6: 8 78
UI8,II =U I8; 3
U 19.12)=U 19.3
DO 2I7 K=2,11

KK-IA-K .
U(KK ,h)=U(lA-I, K+3
DSI-OS*UE'A-I'5)

I—

2I2
213
21h

215
216
)
)
)
)

2I7

DSZ-
053.2.*U(1A-I,7)-I..
Dsu-. S*U(IA-I.9) I

Dss.I. 5*U(IA~I.9)~.S

ZIS. ZIA, ZIh

)

lhB

DS6I.S+DSh
DHI=5.*DSS/3.

AR l)=;5+DSl ,

AR u =.25* 053+656+I.O)
AR S)-.5*DSS*DHI A
AR 6 .;5+u IA-I,9)-AR(5)
AR 7 -.5+O IA-I,IO) .
RETURN - ‘ n H-

END. , ‘

' SUBROUTINE PBBEX {
C EXTERIOR POINTS FOR NOTCNEO PLATE
DIMENSION U(28,52),IK(52),O(5).IL(52)
COMMON UIK,D NX,IF IA M|,MY JF,JA MJ,N,NOUT NO NT,NDL,RFA,RFI,
1lgPIAJiPfiNO,Dé,DI,K‘E,‘NOT,M§SH,NS§RS,NINC,I5,J5,AMI,IL,P,U1
220 no 223 l-3 IB '
U(I-MJ)-U(i,JA-I)«~
IF(i-IK(3)—I 222,222,223
222 u I,I;-uiI,5 ,. .,. ..
. u I,2 -u I,A
223 CONTINUE
c INTERIOR POINT ADJACENT TO CIRCULAR ARC ON HORIZONTAL LINE
- 7
JQI‘O
60 T0 22A
218 IQZO -.-

2| Jul-I if - '
22 .CA-I.-u(l.h) .
... CB-CA* CA-I. I2.
CC—CB* CA-2. I3.
CD-CC* CA-3. In.
. BB-CA-25 . ,
BC-i 3.*CA-6.)*CA+2.)/6. .
DAl-l.-CA+CBeCC+CD .. . ..
DAB-CB-3.*CC+6.*CD .
DAh-CC-h.*CD
DBI-BB+BD-BC-l; '
oaz-I.-2.*OB+3.*OC-A.*BO
O33-aB-3.*BC+6.*BO .
OBN-aC-4.*OO ‘
a .1./(DA2/DA1-DBZ/DBI)
u I J)-BE*(U(1 3)/DAI+(DBBIIBI-DA3/DAI)* U(I,J+I)
IT ?é?/DBl-DAh/6Al)*U(l,J+2)+(BD/DBI-CD/DA1)*U(I,J+3))
a + H . H . H H
IF (J-I ) 2I8,2I9 221 ,
221 IF(l-2h 225.22A,226H
225 [-23 _. .. .. ....

INA
J=I5
GO TO 222
226 IF(I-ZS) 221+, 2213, 227
227 OD 229 J=1,JA .
”.‘z'é ikfii'fi'i

IF(fi—IN(3)5 229, 229, 228

228 u(MI J)-U(IA-1, 3I+DI
229 CONTINUE
C INTERIOR POINT ADJACENT TO CIRCULAR ARC ON VERTICAL LINE
DUB. 5
l-IS

CC-CB* CA-2. ls;
CD‘BCC* CA-Bo ll}.
DB-CA-.s
BC.{§3. .*CA-6. g*CA+2. )/6.
BD- (h.*CA-12 .)*CA+22. )*CA-6. )/2u.

DA1-1.-CA+CB-CC+CD 1.

DAz-CA-z. *CB+3, *CC-A. *CD

DA3-CB-3.*CC+6.*CD

DAA—CC-h.*CD
, DBI-BB+DO-BC-I,

DBZ-I.-2.*BB+3.*BC~#.*BD

033-33-3.*BC+6.*BD .

DDAaDC-A. *BD

BE-I./(DA2/DA1-DBZ/DBI)

U I ,J)-BE*(U(IA, J)/DAI+DUD*DI/DBI+(D33/DBI-DA3/DAI)*U(I-I, J)
13 gé?/DBl-DAh/DAI)*U(I- -2 ,J)+(BD/DDI-CD/DAI)*U(I-3 )

II -|- .
IF(J-7) 230, 230, 232
2321 I.-I6

- IF(J-9) 230,230, 233
233 Ui'S'Iw -U 15.5 .-.

fig
2AA RETUfiN .
. END_

SUBROUTINE PBABD —
c BOUNDARY CONDITIONS FOR THE v-NDTCNED PLATE PROD. NO. 3
DIMENSION U(28, 52), IK(52), 0(5), P(25, 13) ,UI(25) IL(52)
COMMON U, IK, D ,Mx, IF, IA ,MI ,MY JF, JA MJ, N 'NOUT N6 ,NT,NDL,RPA,RPI,
I SPYIJ, PRNo' DE, DI, KPE, INOT,MESH, NS*RS, NINC, Ifi, JP ,AMI,IL,P,UI
22 FORMAT '(2OI&)' H
. A-Mx .-
Dl-1./A
lB-lA-l
READ I 22, ,(IKEJ), ,J-3, JA)
PRINT 22,(IN J), JuB, JA)
JB.IK(3)+1
DO 210 JaJD JA
210 U(IA,J)-.12§

J-s.
230 CA-l.-U(lA-‘ J)
CB-CA*§CA-1.;/2.

 

C

IRS

Y= ~O. 0
DO 215 |=3, IB
U(I JA)=. 5*Y*Y

IF(I-IK(3)) 215,215,212
BOUNDARY VALUES 0N NOTCH EDGE
212 KBI-M/N'
U(IoK)=os*(Y‘025)
215 Y=Y+DI
RETURN
END

SUBROUTINE PBAEX

EXTERIOR POINTS FOR THE V-NOTCHED PLATE
DIMENSION U(28, 52), IK(52), 0(5), P(25, 13) ,UI(25), IL(52)
COMMON U, IK, 0, MX, IF, IA, MI :MY, JF, JA, MJ, N ,NOUT, N0 NT, NDL, RFA, RFI,

PROB. NO. #

I SPYIJ ,PRNO,DE,0I, KPE, INOT, MESH, NSTRS, NINC, IP, JP, AMI,

220 OD 223' I=3,IA .
U(I MJ)=U(I,JA-I)
IF(I-IK(3)-I) 222,222, 223

222U I, WI)=UEI

I ,2)=U I

22A CONTINUE
DO 228 J= 1,JA
U(1,,J;=U 5,J)
U(Z, J 3U “J J)

IF(J-IK(3)I 228,228,226

226 U(MI J)-U(IA-I,J)+01...

228 CONTINUE .

. RETURN
END

SUBROUTINE PBSBD

BOUNDARY CONDITIONS FOR A PLATE WITH A SQUARE NOTCH PROB. NO.

DIMENSION U(28, 52), IK(52), 0(5), IL(52)
COMMON U IK, D MX, IF, IA, MI ,MY JF, JA, MJ, N ,NOUT N0 NT ,NDL, RFA ,RFI,

I SPYIJ, PPNO' DP, DI, KPE, INOT, MESH, NSTRS, 'NINC, IP, JP,m

22 FORMAT '(2OIL)'
IB-IA-I .-
A-Mx ,
DI-I./A
READ

JB=|K(3)+I

DO 205 Ja3, JD
205 U(JB, J)-O. O

00 210 J-JB, JA
210 U(IA, J)-.I25

Y: O. O

DO 215 I=3, IB
U(I JA)=. 5*Y*Y
IF(I- JB) 215,215,212
212 U(I, JB)-. 5*(Y-. 25)
215 Y=Y+DI .
. RETURN
END

2. (|K(J), J=3, JA)
PRINT 2, (IK(J), J=3, 'JA)

IL, P, U1

,IL,P,UI

C

d

Ih6
SUBROUTINE PBsEx
C EXTERIOR POINTS FOR A PLATE NITH A SQUARE NOTCH
DIMENSION U(28,52)§IK(52), 0(5), IL(52)

COMMON U IK, D Mx, IA MI 'MY JP, JA NJ, N, NOUT ND NT, NDL, RFA ,RFI,

I SPYIJ PRNO, DE ,DI,KPE, INOI ,MPSH, NsIRs' ,NINC,IP, JP ,AMI, IL, P ,UI
JB-IK (§)+l
IB.IA-
220 DO 225
_, U(I‘NJ

IF(
2220 UI'

(I, JA-I )
K )-I 222,222,223
,1 -U I,5
2 -U I ,h

GO ID 225
223 U(I JB—I)-U(| ,JB+I)
225 CONIINUE

. DO 228 J-I, JA
UEI ,Jg-Usfi J;
2 Jw

IF(J -JB) 22 226,226
226 U(MI J)-U(IA-I ,J)+DI.
228 CONTINUE H

RETURN
END

BOUNDRY CONDITIONS CAN BE INTRODUCED USING EQUATION (28)
AS INDICATED IN SECTION 6 - TREATMENT OF IRREGULAR
BOUNDARIES. THE ADDITIONAL INSTRUCTIONS REQUIRED

FOR SORLx FOR PROBLEMS 3 A AND 5 ARE LISTED UNDER

THE HEADINGS SOR3 SORh AND SOR5. RESPECTIVELY,

THE CARDS ARE INSPRTED.BETNEEN THE SUBROUTINE SORLx
STATEMENT NUMBERS 133 AND Inc.

1
IL?”
(3

00 0000000

. SOR3
IAI'IgITION TO SORLx FOR SEMI-CIRCULAR NOTCH
II 4.
IF(J-lK(3)-2) 230,230,130
230 OD 23I LI-I 5 . A
-LI/3-I 5* *(-I)**Li . .
L-J- (Ll-l) /3)*(-I)**( LI I- 12
231 2(5I.2.(U(K+I,L)+U( K-I, ,L).+U K ,L+I)+U(K ,L-l)-h».*U(K,L))
GO TO (233,235,236, 237,238 ,239.2AO,2AB, 25O,2sO,252,252,256, 260), K
233 Ifichoéo
23A D(2)-(TEMP+U(I,J)+.5*(U(|+I,J+I)+U(I+I,J-I)+DI)-2.*U(I+1,J))*ANV
GO TO 232
235 ANV-l../AR§12
TEMP-DSI* U I+I,J+I)-U(I+I,J))
GO TO 233 H H H
236 ANY-I. /AR(2)
TEMPI-DSZ*(U(I+I,J+1)-U(I+I,J))
TEMPu-TEMP-I-‘I'EMPI. ..
GO TO 233
237 ANV-I./AR(3)

I47

EEMIE-2§2Pn+' .S*(U(I+I, J+I)-U(I+I, J))

238 0(2g- U(I J)+U(I+I, J-I)+U(I+I, J+I)-3. *U(I+I. J)+(U(IA. J)-U(I+1, J))
0530(2)
GO TO 2A2

239 022);(U(I J)+. 8*U(I I+I J+I)-I. 8*U(I+I,J)+. 5*DI)/AR(4)

35-.(2§
GO TO 2&2
ZhO D(2)-U(I, J)+U(U+I, J+fl)+U(9+fl, J—fl)-3. *U(I+I, J)+(U(IA, J)-U(I+I, J))
1 /U(lA-I. J)
DS-D(2)
GO TO 2h2

E52)?§§(I+i, J+I)+U(B, J)- 2. *U(I+I, J)+DI /2. )/AR(7)
63.0 T0 2h2

DéZ):2. *(U(I+I, J+I)+U(I, J)- 2 .*U(I+I, J))+DI

os-D(2)

IF(J-13)21I2A ZSI ZSI

25I 0(2)-. *oslfla 05.

I M 05.0(2
GO TO 2&2

252 D(2)-(U(I+I J+I)+U(I, J +056*U(I+2 J) -(2. +056)*U(l+1 Jg+
lg?g;(U(I+I, ,§)-U(I+I,J) /U(I+I, h)+zI.-DSG)*DI* 5)/AR{6
GO TO 221:225

256 IF(I2

257 D E)

l

2h8D

250

(U I 2J +(U I—I J-I)+U(I+I J-I)-2. *U(l, J-I))*DSé+(U(I. 3)-
S/U(I,u u(I,J-I))/Dsé.

I'J-I/I
Io 2&2
258 D 5;-(U U(-l, J)+D$6*U(l-U LJ-fl)-(l.+056)*u(| J-fi)+. *DI*DS6)/AR(h)
l/Uglw 'IJ’*"("*“ J+“’*"(“*2 J)‘3 *U<'+1 J)+( U |+I.3)-U(I+1 J))

05.0(25-U(I, J)
KT .0 .
60 T0 252
260 IF(I-zh) 262 .263. 265
262 o(s)-os+u(I-I, J-I ) .
A KT-I .
GO 10 2A2

263 TEMPI-(U(I+I, J-I)-U(I J-I))*DSZ
TEMP-TEMPI+§U(I- I,J- -I5-u(I,J-I))*. 5+. 5*( .5-052)*DI
so T0268 - '

26h IF(l-26) 265,266, 252

265 TEMP-TEMPI .

. TEMPI-(U( I+IJ J-I)-u(I J-II)*DSI
IBIP.-IEMP+IEMPI.+ 5*(652—0511 )*DI
ANV-I./AR(2)
so T0268

(
1K0
G0
2

 

266
268
2&2

24h
2h3
245

230

231
232

234
235

£22

1&8

KT-O
TEMP--TEMPI+. 5*DSI*DI

ANV-1. lAR(1) .

D(5)-(TEMP+U(I J)-2.*U(l J-I)+.5*(U(I-I,J-I)+U(I+I, J-I)))*ANV
YlJ-(h. *o(3 -o(I)-o(2)-o(h)-o(s))*.05*RFA .
Y2-ABSF(Y|J . .
SYIJ-SYIJ+Y2

U(I ,J)=U(I, J)+Y|J
IF(YZ-DE) 2h3, 244, 2K4
KPEBKPE+1

IF (KT) 1h0,1h0, 2h5
I-|+1 _

GO TO 230

SORh********(SEE COMENTS on PAGE lhé)
Ao?érlou TO SORLX FOR A v NOTCH H
I +1
IF(J-IK(3)- I) 230 230,140
no 231 LI-I n..
l-éL1/3-15*(-1)**Li
L-J- (LI 1)/3)*(-l)**(L1-l
D(Ll)-(U(K+1,L)+U(K-1,L)+U(K,L+1)+U(K,L-I)-h.*U(K,L))
IF(J-h) 232 234, 234 .
g§2%?2)*(u(‘+1'J+1)+U(I'J)+U(|+I ,J-I)-3.*U(I+I.J)+Dl*.5)/3.
50 T0 2A2
0(5 )-05
IF(J-IK(3)-I) 235. 236 236
D(2)-2.*(U(| +1,J+1)+U(|, J)-2.*U(I+I,J))+DI
DS-D(2)
32128322. w. )5.“ I, W
O +1+O O
YIJ-(h. *D(3§- -D(I)- 0(2 )- D(h #5 -D(5);* 05*RFA
Y2-ABSF(YIJ .

'. SYIJ.SYIJ+Y2

2%

230

231

232
2k2

U(I ,J)-U(I,J)+YIJ
KPE-KPE+1 u.. .

SOR5********(SEE COMENTS ON PAGE 1&6)

?D?éT'°N T0 SORLX FOR A SQUARE NOTCH ..

I +1

IF(J-IK(3)-I) 230.230.1uo

no 231 Ll-155 2.-

K-l—-éL1/3-15*(-l)**Li

L-J- (L1-l)/3)*(-l)**(Ll- -Iz
D(L1)-(U(K+l L)+U(K-I, 'leu K, L+l)+U(K. L-l)-h. *U(K, L))
lF((J-lKE3)-15 232 2A2, 22

D(2)-(U I J *2 .+U(l+l J-I-I +U(l+l J- I)-lI. *U(l+l ,J))+DI
YIJ-(h. *o(3 -o(I)-o(25-o( )-o(5)$*. 05*RFA .
Y2-ABSF(YIJ . . .
SY'J-SYIJ+Y2
IF(YZ-DE) Iuo Zhh, zuu
u(I, J)-U(I, J)+YIJ' .

Zhh KPE-KPE+I

 

Ih9
TABLE OF SYMBOLS

PRNO, DE, RFA, SPYIJ, Mx, MY, IF, JP, N, NT, NOUT, NO, MESH,
NSTRS, IP, JP, Mu. M2. SEE INPUT PREPARATION FOR DEFINITION.

U(I,J)
I

J

MI

IA

- MJ

JA
P(J.I)

UI(J)
D1

KPE

INOT

IK(J)
IL(J)

DISCRETE VALUES 0F STRESS FUNCTIONS.
Row INDEX. '
COLUMN INDEX.

MAXIMUM l a MX+|F+I '
Ml-I'- LAST BOUNDARY RON INDEX 2
MAXIMUM J - MY+JE+I

MJ-I -LAST BOUNDARY COLUMN INDEX

STRESS FUNCTION DISTRIBUTION SAVED EOR GENERATION OF
INITIAL ESTIMATE OE U(I.J) IN NEXT PROBLEM (SEE

CHANG P. 129).

A COLUMN ARRAY USED IN CHANG EOR TEMPORARY STORAGE.
MESH INTERVAL H COMPUTED IN PBNBD; D1 DEPENDS ON
PHYSICAL DIMENSIONS OF THE SOLUTION DOMAIN. '
COUNT OF POINTS AT WHICH CONVERGENCE CRITERION WAS
NOT SATISFIED IN THE LAST ITERATION.

CONTROL SWITCH EOR SUBROUTINE OUTIN. INOT - I
SIGNALS INPUT OF NEXT DATA SET. INOT - 2 SIGNALS
USE OF THE OUTPUT PORTION OF THE SUBROUTINE.

LAST INTERIOR POINT ROW INDEX IN COLUMN J.

EIRST INTERIOR POINT Row INDEX IN COLUMN J.‘ _
IK(J) AND IL(J) ARE INTRODUCED IN PBNBD FOR SPECI-
FICATION 0F IRREGULAR BOUNDARIES.

 

[’1“!Ill 11 .1

150

SPECIFICATION 0F COLUMN TERMINAL POINTS FOR SORLX

A RECTANGULAR MESH WITH THE SPECIFIED DIMENSIONS OF THE
ARRAY U ENCOMPASSES THE PROBLEM SOLUTION DOMAIN. FOR A REC-
TANGULAR REGION THE ITERATIVE METHOD SUBROUTINE SORLX SNEEPS
THE MESH BY COLUMNS STARTING NITH J - 3, USING 35;I<IIA AND
CONTINUING UNTIL J - JA-I. IF SYMMETRY CONDITIONS ARE NOT
USED THE BOUNDARIES CORRESPOND TO THE RONS I - 2 AND I - IA
AND THE COLUMNS J - 2 AND J - JA. THE RONS I -'1, I'- Kl
AND THE COLUMNS J - 1 AND J - MJ ARE EXTERIOR TO THE SOL-
UTION DOMAIN AND CAN BE USED BY THE NRITER OF THE PBNEX‘.
SUBROUTINE TO SATISIFY BOUNDARY DERIVATIVE CONDITIONS. IF
I a 3 (OR J - 3) IS A LINE OF SYMMETRY THE VALUES IN THE
MESH LINE I - 2 (OR J - 2) ALSO MUST BE PROVIDED IN PBNEX
USINGDERIVATIVE CONDITIONS ALONG A LINE OF SYMMETRY."

A PROBLEM NITH IRREGULAR BOUNDARIES MAY SPECIFY A DIFF-
ERENT FIRST INTERIOR POINT, IL(J), AND LAST INTERIOR POINT,
IK(J), FOR EACH COLUMN. THESE VALUES MAY BE READ FROM CARDS
OR GENERATED IN THE SUBROUTINE PBNBD. IF THIS IS NOT DONE
ALL IL(J) ARE SET TO 3 AND ALL IK(J) TO IA-I BEFORE TRANSFER
TO PBNBD AND THE DOMAIN IS TREATED AS A RECTANGLE. THOUGH
-IL(J) AND IK(J) LIMIT THE SNEEP ALONG THE COLUMNS NITHIN
SORLX THEIR SPECIFICATION DEPENDS ON THE CHOICE OF THE NRITER
OF THE BOUNDARY CONDITION SUBROUTINES. IF HE ELECTS TO.COM-
PUTE SOME INTERIOR POINTS BY INTERPOLATION IN PBNEx THEN
HE CAN SET THE INTERIOR POINT RON INDEX RANGES ACCORDINGLY.
(SEE PB3EX P. 1A3)

I

MICHIGQN

STQTE UNIV. LIBRQRIES

 

31293003835869