Numerical Approximation of Partial Differential Equations, Selection of Papers Presented at the International Symposium on Numerical: Analysis held at the Polytechnic University of Madrid

411 Pages·1987·10.52 MB·English

by  Eduardo L. Ortiz (Eds.)

NU M E RlCAL APPROXI MAT10 N
OF PARTIAL DIFFERENTIAL EQUATIONS
Selection of Papers Presented at the
International Symposium on NumericalAnalysis
held at the Polytechnic University of Madrid,
September 17-13 1985
Edited by
Eduardo L. ORTIZ
Department of Mathematics
Imperial College of Science & Technology
London, United Kingdom
1987
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD .TOKYO
Elsevier Science Publishers B.V., 1987
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Library of Congress CataloginginPubliation Data
International Symposium on Numerical Esalysis (1985 :
Polytechnic University of Madrid)
Numerical approximation of partial differential
equations.
(North-Holland mathematics studies ; 133)
1. Differential equations--Nerical solutions--
Congresses. I. Ortiz, Eduardo L. 11. Title.
111. Series.
~370.1576 1985 515.3’5 86-24116
ISBN C-444-70140-0 (U.S. )
PRINTED IN THE NETHERLANDS
V
PREFACE
The International Symposium of Numerical Analysis
This volume contains a selection of papers on problems arising in the numerical solution
of differential equations. They were read at or contributed to ISNA, the International
Symposium of Numerical Analysis, held in Madrid on September 17-19, 1985. Papers
dealing with other topics and presented at that meeting will be published in a separate
collection.
ISNA was the initiative of computer scientists Rafael Portaencasa, Rector of the Poly-
technic University of Madrid and Carlos Vega, Vice-Dean of the Faculty of Informatics
at the same university. Over a period of years this dynamic university has run a fruitful
cooperation research agreement in the field of Numerical Analysis with Charles University,
Prague. It seemed then natural that ISNA should be developed as a joint effort of the
universities of Madrid and Prague.
After some preliminary discussions in 1983, a Scientific Committee met in Madrid at
the beginning of 1984 under the Chairmanship of Rector Portaencasa, with Professor Vega
acting as Secretary. His members were Professors Wolfgang Hackbusch, University of
Kiel; Oliver Pironneau, INRIA, Paris, and a distinguished group of Spanish academics:
Professors Enrique Alarcon, Alfredo Bermudez de Castro, Carlos Conde Lozano, Jose
Manuel Corrales, Covadonga Fernandez Baizan, Luis Ferragut Canals, Manuel Lopez
Quero, Francisco Michavila Pitarch, Jose Luis Morant Ramon, Pilar Perez Alonso, Libia
Perez Jimenez, Arturo Ribagorda Garnacho and Antonio Valle Sanchez. Professors
Ivo Marek, from Czechoslovakia, Yuri Kuznetsov, from the USSR and John Whiteman,
from England, were members of this Committee, but could not be present at the meeting.
I also had the privilege of being invited to participate.
The organizers of that meeting suggested the list of invited speakers, the topics to be
emphasized and possible dates. A second and equally live meeting took place in Prague,
where we were the guests of Professors Zdenek Ceska, Rector of Charles University and
numerical analyst Professor Ivo Marek, Fellow of the Czechoslovak Academy of Sciences.
A subgroup of that Committee discussed there the final details, and a call for papers for
the first ISNA meeting was sent out.
The symposium took place in September, a most pleasant time of the year to visit Madrid.
Our deliberations were held at the Campus of the University of Madrid, where the Poly-
technic University has its headquarters. It attracted over 120 active participants from
more than 20 different countries. Both the quality and quantity of papers contributed
to the symposium exceeded the most optimistic expectations of the Scientific Committee.
vi Preface
At the end of the Madrid Symposium it was agreed that ISNA II will take place in
Czechoslovakia in the Summer of 1987.
Papers submitted covered a broad spectrum of our discipline: Linear Algebra; Numerical
Methods of Approximation Theory; Computational Statistics; Analysis and Complexity
of Algorithms; Numerical Methods for Differential Equations; Optimization; Special
Problems of Science and Engineering; Inverse and Ill Posed Problems and Topics on the
Teaching of Numerical Mathematics.
We decided against a large volume of proceedings, which would have been over one
thousand pages in length and agreed to subdivide the papers accepted into sections. I was
given the task of editing and introducing those in the field of differential equations and
related techniques, which constitute the present volume.
The first Part contains papers concerned with some of the techniques of Approximation
Theory which are basic to the numerical treatment of Differential Equations. This last
topic is specifically considered in Parts Il-V. In the first of them, numerical techniques
based on discrete process such as Finite Differences, Finite Elements and the Method of
Lines are considered. Methods based on polynomial or rational approximation, such as
the Tau Method, Collocation, Pade and Spectral Techniques are discussed in Part II I. The
following section is devoted to Variational Inequalities, Conformal Transformation and
asymptotic technqiues. Finally, Part V contains a number of papers dealing with concrete
applications of differential equations to problems of Science and Engineering, where a
variety of techniques are used in an innovative way to produce desired numerical results.
This meeting would not have been possible without the help and assistance of several
leading scientific institutions and Government departments of Spain and Czechoslovakia.
The former provided finance and a number of facilities; the latter made possible the parti-
cipation of a large number of scientists from Czechoslovakia. Professor W. Hackbusch,
who could not attend the meeting due to other commitments, gave valuable advice to our
Committee in the preliminary meetings of Madrid and Prague. To all of them we wish to
transmit the gratitude of the Scientific Committee.
During the days we spent in Madrid Professors Portaencasa, Vega and their Spanish col-
leagues did everything possible to make our stay pleasanr. The program, concentrated
into three days, was tight but broken by a delightful dinner at a medieval castle, just out-
side Madrid. The organizers of ISNA deserve our warm thanks for their fine achievement.
Finally, I would like to express the appreciation of the Scientific Committee of ISNA to
its Patron, Juan Carlos I, King of Spain, for his firm support and consistent encouragement
of scientific research.
Eduardo L. Ortiz
Imperial College
London, 1986
vii
INTERNATIONAL SYMPOSIUM OF NUMERICAL ANALYSIS
under the Patronage of
His Majesty Juan Carlos I, King of Spain
and with the support of:
The Ministry of Education; Ministry of Culture; the Royal Academies of Arts and of
Sciences; the National Institute for Scientific Research; the Secretary of State for Univer-
sities and for Scientific Research; the Boards of Scientific Policy, of University Education
and of Technical and Scientific Cooperation, Spain and the Charles University, Prague,
Czechoslovakia.
Chairmen
Rafael PORTAENCASA, Rector of the Polytechnic University of Madrid
and
Zdenek CESKA, Rector of Charles University, Prague
Advisors
Jacques-Louis LIONS, College de France
and
Yuri MARCHUK, Academy of Sciences of the USSR
Executive Sectetaries
Carlos VEGA, Polytechnic University, Madrid
and
Ivo MAREK, Charles University, Prague
Organized by:
PO LYTECHN IC U N I V E RSIT Y 0 F MAD R ID
CHARLES UNIVERSITY OF PRAGUE
Numerical Approximation of Partial Differential Equations
E.L. Ortiz (Editor) 3
0 Elsevier Science Publishers B.V. (North-Holland), 1987
RECENT PROGRESS IN THE TWO-DIMENSIONAL APPROXIMATION OF
THREE -DIMENSIONAL PLATE MODELS IN NONLINEAR ELASTICITY
Philippe G. CLARLET
Universitg Pierre et Marie Curie, and
tcole Normale SupErieure, Paris, France.
The asymptotic expansion method, with the thickness as the para-
meter, is applied to the equilibriwn and constitutive equations of nonli-
near thee-dimensional elasticity. Then the leading term of the expansion
can be identified with ths solution of well-known two dimensional nonlinear
plate models, such as the van h 6 n e quations.
Recent progresses in the application of this method, such as the
extension to more general constitutive equations and boundary conditions,
the effect of the assumption of polyconvexity, the application to one-dimen-
sional rod models, etc ..., are presented. Various open problems, regarding
in particular the existence of corresponding three-dimensionaZ solutions
and the nature of ahissible three-dimensional boundary conditions, are aZ-
so discussed.
1. THE THREE-DIMENSIONAL CLAMPED PLATE MODEL. Latin indices :
i,j,p, ..., take their values in the set {1,2,31 ; Greek indices : cl,B,u,...,
take their values in the set {1,2). The repeated index convention is syste-
matically used in conjunction with the above rule. Let (e.) be an orthonor-
ma1 basis in R3, and let w be a bounded open subset of the "horizontal"
plane, spanned by (ea), with a sufficiently smooth boundary y. Given E>O,
we let
RE =O X 1 -€,EL, r: =y X [-€,El , r?e =W X {?El.
Because the thickness 2~ is thought as being "small" compared to
the dimensions of the set o, the set 2 is called a plate, with lateral
surface 'T and upper am? lower faces I'p and r! . We are concerned with the
problem of finding the displacement vector field U' = (u:) :$ -+ R3 and the
second Piola-Kirchhoff stress tensor field CT'= (a?.) :?-+S 3 (we let S3
111
denote the space of symmetric matrices of order 3) of a three-dimensional
4 P.G. Ciarlet
3
body which occupies the set in the absence of applied forces. The plate
is subjected to body forces, of density fE= (fz) :? +R3, and to surface
forces, of density gE = (g:) : r: U r: --+ R3, on the upper and lower faces.
For simplicity, we shall assume that both kind of applied forces are dead
Zoads, i.e., both densities are independent of the displacement, and that
the horizontal components of the applied forces vanish, i.e., fE=O and
g" =O. Finally, we assume that the plate is cZmped, in the sense that it
is subjected to the boundary condition of place u: =O on the lateral sur-
Eace rE. Then the unknowns and the data are related by the following equa-
tions of finite elastostatics, which express the elastic equilibrium of
the plate (we let a: = a/axE where xE = (x?) denotes the generic point of
J j' J
the set ?) :
(1.1)
(1.3) u4=0 on rE.
We assume that the material constituting the plate is elastic,
homogeneous, and isotropic. Then its constibutive equation takes the
form :
where M: denotes the set of matrices F or order 3 with det F>O,
(1 -5) CE = (I+Vu E) T(I +VuE) = I + 2EE
denotes the right Cauchy-Green deformation tensor, and yoE,y1E,y2E a re real-
valued functions of the three principal invariants of the tensor CE. If we
assume that the set 3 is a mturaZ state, i.e., that oE=O if uE=O (note
that uE =O implies CE =I), the Taylor expansion of (I .4) in terms of the
Green-St Vemnt strain tensor E" defined in (1.5) takes the form
2-0 Approximation of 3-0 Plate Models in Nonlinear Elasticity 5
where A'>O and uE>O are the Lame' constunts of the material constituting
the plate.
We may assume that the material is hyperelastic, in which case
'G
there exists 3 stored energy function :MZ +R such that
Then finding solutions of (l.l)-(l.6) formally amounts to finding the sta-
tionary points of the total energy
when vE span a space of R3-valued functions satisfying the boundary condi-
tion 'V =O on 'T and the orientation - preserving condition det(I+Vv') >O
in 2 (for details about the above model, see e.g. Ciarlet, 1985 ; Ciarlet,
1986 ; Germain, 1972 ; Gurtin, 1981 ; Hanyga, 1985 ; Marsden & Hughes, 1983 ;
Truesdell & Noll, 1955 ; Wang & Truesdell, 1973).
The following existence result holds @f3 denotes the space of
matrices of order 3 ; Cof FEM3 is the cofactor matrix of FEB3) :
THEOREM 1 (Ball, 1977). Assume that the materia2 is hyperelastic
and that its stored energy function satisfies the fo'ozZowing asswnptions :
There exists a convex function W-E :M3x EI 3 x]O,+m[ such that (poZyconvexityl
(1.9) ?(F) = ?(F,Cof F,det F) for all FEB+3 ;
(1.10) ?(F) -+ +m as det F -+ O+;
the linear form
(1.12) vEEW19P(nE;R3) 4
is continuous. Then there exists at least one displacement uE satisfying
6 P. G. Ciarlet
(I .13) J" (u") = inf { J" (v') ; V" E }'U
where the functional JE is defined in (1.8) and the set U" of admissible
displacements is given by
(1.14) u" = {v" E w1y P(nE;R3) ; Cof (I+VvE) E Lq(n' ;M3), det(I+Vv") E Lr(QE),
det(I+Vv")>O a.e. in QE, vE=O on r:}.
It is always possible (Ciarlet & Geymonat, 1982) to construct a
class of simple stored energy functions satisfying assumptions (1.9)-(1.11),
and whose expansion in terms of the Green-St Venant strain tensor E" agrees
with the expansion (1.6) of an elastic material with arbitrary Lam6 cons-
tants AE >O and yE >O. Unilateral boundary conditions (Ciarlet & Nexas,
1985a) and conditions guaranteeing almost everywhere injectivity (Ciarlet
& Nexas, 1985b) can also be included in the definition of the set U" of
(1.14).
2. APPLICATION OF THE ASYMPTOTIC EXPANSION METHOD. For ease of
exposition, we assume that the constitutive equation takes the simplified
form
(2.1) 0" =AE(tr EE)I+ 2yEE'.
Although the associated stored energy function is not polyconvex (Raoult,
1985b), we mention that the present approach can be extended, at the ex-
pense of various refined arguments however, to stored energy functions
satisfying all the assumptions of Theorem 1 (cf. Sect. 4). A variational
formulation of equations (l.l)-(l.3) and (2.1) then consists in expressing
that the pair (uE,uE) satisfies
XCE,
(2.2) (U",UE) EVE
where VE={~EEW174(RE;R3); vE=O on r"}, xE=L2(RE;S3),
-5
JE{(%)
(2.3) osj u;pSij} rEijdxE -[E~:jE Fjdx" =O
R exE,
for all TE = (T:j)
2-0 Approximation of 3-0P late Models in Nonlinear Elasticity 7
=/cf:v:dxE + JE gE3vE3d aE for all vE = (v?) EVE ,
E
R r+ur-
"
where EE ~ pE(3AE+2pE) and yE - are respectively the Young rnodutus
XE + LIE 2 (AE+p €)
and the Poisson ratio of the material.
Our first task consists in defining a problem equivalent to the
variational problem (2.2)-(2.4), but now posed over a domain which does
not depend on E. We let
n=~xl-1,1[, r =YX [-1,11, r k =wx{?I~.
With each point xE=( x:)c?, we associate the point x= (x.)Ec by letting
c"
xa=xE, x3 =x:/E, and with the spaces VE, of (2.2), we associate the
spaces
c=
(2.5) v = { ~ E w " ~ ( ~ ;;R V~=O) on roj, L~(Q;s~).
With the unknown functions (uE,uE) EVE xc", we associate functions
(u(E),u(E)) E V x r defined by
(2.6) uE = E%,(E), u: = Eu3(€),
(2.7) DaE0 =E2U aB (E), UZ3 =E3Ua3(E), UE3 =E4U33(E),
XZ"
and with the "tria2 functions" (v~,TE~V)E appearing in (2.3)-(2.4),
XZ
we associate functions (v,T) E V defined by
(2.8) VE = E2V a' v3E = &" 3'
(2.9) TaEB =E2T aB' TaE3 =E3T a3' .c:3 =E4T 33'
where :u = E'U~(E) means that uE(xE)= E'u,(E)(x) for all corresponding
points ~ " €and2 ~ € 5e,tc ... . As regards the data, we assume that there
exist functions f3 : n -+R, g3 : T+UI'- --f R, and constants A,p, independent
of E, such that
(2.10) f5=s3f3 , g;=2g3, AE=A, yE=p
(as we shall indicate in Sect. 3, these are not the only possible assump-