Computational Mechanics: International Conference on Computational Methods in Nonlinear Mechanics, Austin, Texas, 1974

329 Pages·1975·4.214 MB·English

by  W. F. Ames, M. Ginsberg (auth.), Prof. J. Tinsley Oden (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
Series: Texas Institute for Computational Mechanics (TICOM)
Adviser: .J .T Oden
164
lanoitatupmoC Mechanics
International Conference on Computational Methods in
Nonlinear Mechanics, Austin, Texas, 1974
Edited by .J .T Oden
Springer-Verlag
Berlin.Heidelberg (cid:12)9 New York 1975
Editor
Prof. .J Tinsley Oden
Department of Aerospace
and Engineering Mechanics
University of Texas at Austin
Austin, Texas 78712
USA
Library of Congress Cataloging in Publication Data
Computational mechanics.
(Texas tracts in computational mechanics) (Lec-
ture notes in mathematics ; 461)
"With the exception of chapter 27 all of these
lectures were also delivered at the International
Conference on Computational Methods in Nonlinear
Mechanics held in Austin, Texas, in September, 1974."
Bibliography: p.
Includes index.
.i Mechanics, Analytic--Congresses. 2. Nonlin-
ear theories--Congresses. I. Oden, John Tinsley,
1936- II. International Conference on Com-
putational Methods in Nonlinear Mechanics, Austin,
Tex.~ 1974. III. Series. IV. Series: Lecture
notes in mathematics (Berlin) ; 461.
QA3.L28 no. 461 QASOI 510'.8s 551'.O1'515
75-14390
AMS Subject Classifications (1970): 62-02, 70-02, 73-02, 76-02
ISBN 3-540-07169-5 Springer-Verlag Berlin (cid:12)9 Heidelberg (cid:12)9 New York
ISBN 0-387-07169-5 Springer-Verlag New York (cid:12)9 Heidelberg (cid:12)9 Berlin
This work is subject to copyright. All rights are reserved, whether the whole
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Under w 54 of the German Copyright Law where copies are made for other
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be determined by agreement with the publisher.
(cid:14)9 by Springer-Verlag Berlin (cid:12)9 Heidelberg 1975. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
ECAFERP
This volume contains a collection of special invited lectures no computational
methods in nonlinear mechanics prepared by specialists from a wide collection of
disciplines. With the exception of Chapter 2, all of these lectures were also
delivered at the International Conference no Computational Methods in Nonlinear
Mechanics held in Austin, Texas, in September 1974, which saw held under the
sponsorship of the .U .S National Science Foundation. ehT original intention of
the meeting, dna for collecting this set of lectures, saw to bring together in one
place the latest results no computational methods for nonlinear problems from a
rebmun of diverse areas in the hope that techniques that dah been found successful
in eno area yam have some impact no problems in other areas. In addition, it saw
hoped that the state-of-the-art in certain areas of computational mechanics could
eb summarized. I believe that a reader ohw examines the contents will agree that
both of these objectives have been accomplished.
J. T. nedO
Austin, 1975
ELBAT FO STNETNOC
ECAFERP ................................................................ iii
LIST FO SROTUBIRTNOC ................................................... vii
SERUTCEL
Bilateral Algorithms and Their Applications
.W F. Ames and .M Ginsberg ....................................
A Plate Analogy for Plane Incompressible Viscous Flow
J. H. Argyris, P. C. Dunne, and B. Bichat ..................... 33
Finite Elements and Fluid Dynamics
.G J. Fix ...................................................... 47
Experience with Forward Marching Nonlinear Solutions of
the Navier Stokes Equations
J. E. Fromm .................................................... 17
Perturbation Procedures in Nonlinear Finite Element
Structural Analysis
.R H. Gallagher ................................................ 75
Computational Methods for Stress Wave Propagation in
Nonlinear Solid Mechanics
.W Herrmann, L. D. Bertholf, and S. L. Thompson ................ 19
nO Free Surface Problems: Methods of Variational and
Quasi Variational Inequalities
J. L. Lions .................................................... 129
Continuous and Discontinuous Finite Element Approximations
of Shock Waves in Nonlinear Elastic Solids
J. T. Oden and L. .C Wellford, Jr .............................. 149
nO the Solution of Large, Sparse Sets of Nonlinear Equations
.W .C Rheinboldt ............................................... 169
Vl
Recent Developments dna Problem Areas in Computational
Fluid Dynamics
.P J. Roache ...................................................... 591
Regularization in Celestial Mechanics
.V Szebehely ...................................................... 752
Iterative Solution of Linear dna Nonlinear Systems
Derived from Elliptic Partial Differential Equations
.D .M Young ....................................................... 562
Visco-Plasticity, Plasticity, Creep dna Visco-Plastic Flow
(Problems of small, large dna continuing deformation)
.O .C Zienkiewicz ................................................. 792
LIST FSOROTUBIRTNOC
.W .F senaP J. L. Lions
tnemtrapeD of Mechanics dna Hydraulics IRIA - Laboria
ehT University of lowa eniamoD ed uaeculoV
lowa City, lowa 04225 truocneuqcoR .B .P 5
05187 eL Chesnay, France
J. .H Argyris
Institute fur Statik dnu kimanyD red Luft J. .T nedO
dnu nenoitkurtsnoktrhafmuaR tnemtrapeD of ecapsoreA gnireenignE
Universitat Stuttgart dna Engineering scinahceM
Stuttgart, tseW ynamreG ehT University of saxeT at Austin
Austin, saxeT 21787
.L .D Bertholf
Sandia Laboratories .W .C Rheinboldt
Albuquerque, weN Mexico 51178 retupmoC Science Center
University of Maryland
.B Bichat College Park, Maryland 24702
Institut fur Statik dnu kimanyD red Luft
dnu nenoitkurtsnoktrhafmuaR .P J. ehcaoR
Universitat Stuttgart Science Applications, Inc.
Stuttgart, tseW ynamreG 221 aL Veta .N .E
Albuquerque, weN Mexico 80178
.P .C ennuD
Institut fur Statik dnu kimanyD der Luft .V Szebehely
dnu Raumfahrtkonstruktionen Department of Aerospace Engineering
Universitat Stuttgart dna Engineering scinahceM
Stuttgart, tseW ynamreG ehT University of Texas at Austin
Austin, Texas 21787
.G J. Fix
tnemtrapeD of Mathematics .S .L nospmohT
ehT University of Michigan Sandia Laboratories
nnA Arbor, Michigan 40184 Albuquerque, weN Mexico 51178
J. .E mmorF .L .C Wellford, Jr.
I. .B .M Department of Civil Engineering
Monterey dna Cottle sdaoR University of Southern California
naS Jose, California 41159 soL Angeles, California
.R .H Gallagher .D .M gnuoY
tnemtrapeD of Structural Engineering Department of Mathematics
Cornell University ehT University of Texas at Austin
Ithaca, weN York 05841 Austin, Texas 21787
.M grebsniG .O .C Zienkiewicz
retupmoC noitarepO~secneicS hcraeseR tnemtrapeD of Civil Engineering
nrehtuoS Methodist University University of Wales, aesnawS
Dallas, saxeT aesnawS 2AS 8PP, United modgniK
.W nnamrreH
aidnaS Laboratories
,euqreuqublA weN ocixeM 87115
LARETALIB SMHTIROGLA DNA THEIR SNOITACILPPA
.W F. Ames and .M Ginsberg
I.I Introduction. In this paper, algorithms providing iterative improvable
upper and lower bounds are constructed for certain classes of nonlinear ordinary and
partial differential equations arising in transport phenomena. Developments are based
upon the construction of antitone functional operators which are oscillatory contrac-
tion mappings. Following the mathematical details, which include sketches of con-
vergence and uniqueness proofs, several examples will be presented. Application of
the general results will be made to nonlinear diffusion, non-Newtonian (power law)
boundary layer flow and to diffusion with a generalized radiation condition.
1.2 Preliminary Definition. Let R and *R be partially ordered metric spaces
and T be an operator whose domain D c R and range W c R*. Then T is said to be syn-
tone if v ~ w implies Tv ~ wT for every v, w c D and strictly syntone if v < w implies
Tv < .wT T is said to be antitone if v ~ w implies Tv ~Tw for every v, w E .D T is
monotone if it is either syntone or antitone.
A bilateral or error bounding algorithm for the solution of an operator
equation produces a sequence of approximate solutions such that the true solution is
bounded between every pair of successive approximations.
1.2.1 Introductory Example. sA a motivation for what follows consider the elementary
nonlinear equation
+C~T~ = -y(y - 2) , y(O) = 1 (1.1)
u~
the exact solution of which is
(1.2)
y(t) = 2/I + exp (-2t)
Problem (I.I) is convertible to the operator form y = TY, from which successive ap-
proximations nY = TYn-I can be developed, in a variety of ways. If the classical
Picard procedure is employed (I.I) becomes
Yd n
= -Yn-l(Yn-I - 2), Yn(O) = 1 (l.3a)
and the integral form is
t
nY = 1 - f Yn-l(Yn-I - 2)dt (l.3b)
W~
0
sA a first alternative a Newton-Picard approximation to (I.I) gives
nYd 2
dt + 2Yn-I - lYn = Yn-I ' nY (0) = 1 (I .4a)
with the associated integral form
1i f O -
nY = exp -2 (Yn 1 l)ds 2 2 (Yn 1 - l)dr ds + 1
- - Yn-I _ -
i i f exp- f i i
- 0
0
(l.4b)
Lastly an approximation which leads to a Bilateral algorithm, as will eb subsequently
demonstrated, carries (I.I) into
Yd n
dt = -Yn(Yn-I - 2) , Yn(O) = 1 (l.5a)
with the integral form
;i ex i
(l.5b)
= (Yn-I - 2)ds
Clearly, (l.3b), (l.4b) and (l.5b) all have the form nY = TYn-I but the properties
and complexities of the operators are vastly different. Since direct comparison is
not the present purpose only the properties of (l.5b) will eb briefly examined.
If it is assumed that Yn-I < nY then it follows from (l.5b) that
exp S'1 s 2, SOPxe s 2' I
t t
= Yn+l
0 0
that is nY > Yn+l and (l.5b) is antitone. yB na analogous argument if it is assumed
that Yn-I > nY then nY < Yn+l" From (l.5b) it follows that each iterate is positive.
Thus, beginning with Yl = 0 it follows that Yl < Y2" yB application of (l.5b) the
following inequalities result:
Yl < 2Y ' 3Y < 2Y ' 3Y < 4Y ' 5Y < 4Y .... (1.6)
Since Yl 3Y the following inequalities result:
(I .7)
Yl < 3Y ' 4Y < 2Y ' 3Y < 5Y ' 6Y < 4Y ....
sA a consequence of the foregoing analysis the even subsequence {Y2n } is seen to form
a monotonic decreasing sequence of upper bounds to the exact solution. Since it is
bounded below the subsequence sah a limit "EY ehT ddo subsequence {Y2n+l } forms a
monotonic increasing sequence of lower bounds to the exact solution. It is bounded
above since every ddo term is less than all the even terms. Hence the ddo subse-
quence sah a limit "OY Upon proof (deleted here) of uniqueness dna convergence
to the solution of the original problem a bilateral algorithm sah been estaCDlished.
Moreover na estimate of the absolute error, at any step of the exact iteration, nac
eb made from ly n - Yn_ll. Further discussion of error analysis will eb given sub-
sequently.
ehT bilateral algorithm (l.5b), for problem (I.I), sah been carried out nu-
merically by Ginsberg I. Implementation for the computer saw accomplished by using
truncated Chebyshev polynomial approximations for the integral of (l.5b). Such ap-
proximations are found in practice to eb "close" to minimax approximations. Some of
the results dna comparisons with the exact solution (1.2) are shown in Table I.I.
ehT convergence is very rapid, requiring two iterates (after OY = )O at x = 0.25 dna
four at x = 1.00.
1.3 Literature dna Applications. In this section, some literature dna ap-
plications are reviewed.
1.3.1 Some Applications. Even though bilateral algorithms have not been readily
available a number of applications appear in the literature. nA early application is
eud to Weyl 2 (see also semA 3) ohw studied the Blasuius problem
f'" * ff" = 0 , f(O) = f'(O) = 0 , f'(~) = 2 (1.8)
yb means of what is won called a bilateral algorithm. Barnov 4 reports that non-
linear differential equations representing flexural dna torsional vibration of smaeb
or buckling of rods often cannot eb easily solved either analytically or yb existing
numerical methods in such a yaw that the positional relationship between the approxi-
mate dna exact solutions is accurately revealed. A bilateral approach is helpful
here; Baranov sah developed such a method for his specific equations. In naval war-
fare problems involving the esu of the Lanchester equations, Fabry 5 sah defined
a two-sided bounding scheme which offers a means of studying variations of certain
parameters before the equations are solved by some conventional (nonbounding) method.
TABLE I.I
BILATERAL SOLUTION ROF y' = - y(y - 2) , y(O) = 1
analytic solution lower bound upper bound
x = 0.25 505097149588391.1 025585902017846.1
y = 907304266819442.1 383958467203442.1 145489890390942.1
x = 0.50 764804281726941.1 167828858192817.2
y = 010062751711264.1 986659188979254.1 263466362558625.1
x = 0.75 105198159195687.0 4.481684580133818
y = 1.635148952387287 661194665515655.1 1.964470533182715
426776114294236.1 1.650773795468340
x = l.O0 0520600043748203.0 546039890650983.7
y = 567559551495167.1 063806498261724.1 510758329162138.2
649251773419737.1 661302671242268.1
525515168507067.1 840910946574667.1
:ETON esehT results are obtained from the program of Ginsberg I.
ehT halting criteria consisted of a check every two iterations
which used (n + l) (cid:141) 0.5 (cid:141) Ol -15 sa a bound no roundoff error
dna xam (fan_if, lanl, lan+ll) sa a bound no the truncation
error. For each iteration 9 terms (n = 8) are used in each
vehsybehC expansion.