Computing methods in applied sciences and engineering. International symposium, Versailles, December 17-21, 1973

Lecture Notes in Computer Science Edited by G. Goos, Karlsruhe and J. Hartmanis, Ithaca

10 Computing Methods in Applied Sciences and Engineering Part 1 International Symposium, Versailles, December 1 ?-21, 1973 IRIA LABORIA Institut de Recherche d'lnformatique et d'Automatique

Edited by R. Glowinski and J. L. Lions

Springer-Verlag Berlin-Heidelberg New York 1974

Editorial Board: P. Brinch Hansen • D. Gries C. Moler • G. Seegm011er • N. Wirth Dr. R. Glowinski Dr. J. L. Lions IRIA LABORIA Domaine de Voluceau - Rocquencourt F - 7 8 1 5 0 Le Chesnay/France

AMS Subject Classifications (1970): 65-02, 6 5 K 0 5 , 65Lxx, 65Mxx, 65Nxx, 65P05, ?6-04, 9 3 E 1 0 , 93E25 CR Subject Classifications (1974): 3.1, 3.2, 5.1

ISBN 3-540-06768-X Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06768-X Springer-Verlag New York • Heidelberg - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress Catalog Card Number 74-5712. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

textes des communications

Ce colloque est organis6 par I'IRIA sous le patronage de l'International Federation for Information Processing.

This symposium is organised by IRIA under the sponsorship of the International Federation for Information Processing. Organisateurs

Organizers R. Glowinski J.L. Lions

PREFACE

L,e present v o l u m e r a s s e m b l e les travaux l~rEsentEs au Colloque "International sur les MEthodes de Calcul Scientifique et Technique", organisE par I ' I R I A - L A B O R I A du 17 au Zl D 4 c e m b r e 1973, sous le patronage de I'IFIP. C e Colloque a rEuni h Versailles prhs de 400 chercheurs et ingEnieurs de t o u s l e s pays du m o n d e . L'originalitE des travaux pr~sentEs, la qualit4 de l'auditoire et des questions pos4es tout au long du Colloque, attestent de l'extr~me intEr~t scientifique et technique, qui s'attache k l'usage des ordinateurs pour le calcul scientifique. Les par tieuli~r ement

organisateurs

tiennent h r e m e r c i e r

Monsieur Monsieur

Andre

DANZIN,

Directeur de I'IRIA

MichelMONPJ~TIT,

Directeur Adjoint de I'IRIA

L e Service des Relations ExtErieures qui ont p e r m i s

tout

:

de l'IRIA

l'organisation de ce Colloque.

les conf4renciers

N o s r e m e r c i e m e n t s vont, 4galement, h tous et aux diff4rents pr4sidents de seance : MM.

J.H. A R G Y R I S A.V. BALAKRISHNAN P. BROUSSE J. DOUGLAS D. FEINGOLD B. F R A E I J S de V E U B E K E P. MOREL W. PRAGER E. ROUBINE O.C. ZIENKIEWICZ

qui ont anita4 d'int4ressantes

R.

discussions.

GLOWINSKI

et J.L.

~, Institut de R e c h e r c h e d'Informatique et d'Automatique Zaboratoire de R e c h e r c h e de I'IRIA.

LIONS

PREFACE

This book contains the lectures which have been presented during the " International Symposium on Computing Methods in Applied Sciences and Engineering " organised by I R I A - L A B O R I A • u n d e r t h e s p o n s o r s h i p of I F I P . ( D e c e m b e r 17, 2 1 , 1 9 7 3 ) 4 0 0 people~ s c i e n t i s t s a n d e n g i n e e r s c o m i n g f r o m many countries attended this meeting in Versailles. The originality of t h e w o r k p r e s e n t e d , t h e h i g h q u a l i t y of t h e a u d i e n c e a n d t h e pertinent questions raised during the symposium show how important is, at the present time, the scientific and technical interest for the u s e o f c o m p u t e r s in a p p l i e d s c i e n c e a n d e n g i n e e r i n g . The organisers Monsieur

w i s h to e x p r e s s

t h e i r g r a t i t u d e to :

Andr4 DANZIN, Director

Monsieur

of I R I A ,

Michel MO,~,~PETIT, Deputy Director of IRIA

The IRIA Public Relations Office who have contributed to the organisation of this Symposium. They also address a n d to t h e c h a i r m e n of s e s s i o n s : MM.

who have

directed

J.H. A.V. P. J. D. B. P. W. E. O.C.

t h e i r t h a n k s to a l l t h e s p e a k e r s

ARGYRIS BALAKRISHNAN BROUSSE DOUGLAS FEINGOLD F R A E I J S de V E U B E K E MOREL PRAGER ROUBINE ZIENKIEWICZ

interesting discussions.

R.

* Institut de Recherche d'Informatique Laboratoire de Recherche de I'IRIA

GLOWINSKI

et d'Automatique

J.L.

LIONS

TABLE TABLE

DES MATIERES OF CONTENTS

TOME PART

I I

GENERALITES GENERA LITIES Methods of stuctural optimization W. Prager .................................... Optimisation et propulsifs singularit

L.

des par

syst~mes la m4thode

portants des

~s -

Malavard

..................................

Some contributions t o non-linear solid mechanics J.H. Argyris, P.C. Dunne .......................... ELEMENTS FINITE

20

42

FINIS EILEMENTS

One-sided approximation and plate bending G. Strang ..................................

140

Quelques mdthodes d' 414ments finis pour le probl~me d' une plaque encastrde P.G. Ciarlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

156

Un nouvel 414ment de coques g4ndrales B.M. Irons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

Numerical solution of the stationary Navier-Stokes equations by finite element methods P. Jarnet, P.A. Raviart ..........................

193

Finite elements method engineering problems B. Fraeijs de Veubeke

224

in aerospace ...........................

Viseo-plasticity and plasticity An alternative for finite element solution of material nonlinearities O.C. Zienkiewicz ..............................

259

VIII

Som~ superconvergence results for an II-- Galerkin procedure for the heat equation J. Douglas, T. Dupont, M.F. Wheeler

..................

Application de la m6thode des dldments finis - Un procddd de sous-assemblage J.M. Boisserie .................................. PROBLEMES NON-LINEAR

288

312

NON-LINEAIRES PROBLEMS

Formulation and application of certain primal and mixed finite element models of finite deformations of elastic bodies J.T. Oden ..................................

334

M4thodes num4riques pour le projet d'appareillages industriels avane4s S. Albertoni ..................................

366

Etude num4rique du champ magn4tique dans un alternateur t4trapolaire par la m~thode des 414ments finis R. Glowinski, A. Marrocco ..........................

392

Une nouvelle m4thode d' analyse num4rique des probl~mes de filtration dans les mat~riaux poreux C. Baiocchi .....................................

410

CIRCUITS NETWORKS

ET

TRANSISTORS AND SEMI-CONDUCTORS

Numerical methods for stiff systems of differential equations related with transistors, tunnel diodes, etc.W. Miranker, F. Hoppensteadt .......................

416

Conception, simulation, optimisation d' un filtre ~ l' aide d" un ordinateur A. Guerard .....................................

433

Computing methods in semiconductor problems M . Reiser ....................................

441

Simulation num4rique de la fabrication et du comportement des dispositifs semiconducteurs D. Vandorpe ...................................

467

TABLE TABLE

TOME PART

DES MATIERES OF CONTENTS

II II

MECANIQUES DES FLUIDS MECHANICS

FLUIDES

Recent advances in computational fluid dynamics T.D. B u t l e r ................................. Mdthodes et techniques d' int4gration num4rique adapt4es ~ i' 4tude des 4coulements plan4taires R. Sadourny ....................................

22

Flow ,computations with accurate space derivative methods J. Gazdag .........................................

37

Numerical simulation of the TaylorGreen vortex S.A. Orszag .....................................

50

Probl~mes et m~thodes num~riques en physique des plasmas k tr~s haute temperature C. Mercier, J.C. Adam, Soubbaramayer, J.L. Soule ......................................

65

-

Probl~mes de contr~le optimal en physique des plasmas J.P. Boujot, J.P. Morera, R. Temam

...................

107

Probl~mes de stabilit4 num4rique pos4s par les syst~mes hyperboliques avec conditions aux limites J.J. Smolderen ......................................

135

R4solution num4rique des 4quations de Navier-Stokes pour les fluides compressibles R. Peyret ........................................

160

-

PROBLEMES D' ONDES WAVES PROBLEMS Three dimensional flows around airfoils with shocks A. Jameson .....................................

185

Lage amplitude wave propagation in arteries and veins Y. Kivity, R. Collins ..............................

213

Increase of accuracy of projectivedifference schemes G.I. Marchuk, V.V. Shaydourov ......................

240

M4thodes num4riques en 41ectromagn4tisme J. Ch. Bolomey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

261

CONTROLE OPTIMAL

OPTIMAL CONTROL

Time-optimal control synthesis for non-linear systems A flight dynamic example A.V. Balakrishnan ...............................

289

Numerical analysis of problems arising in biochemistry J.P. Kernevez ...................................

312

Sur I' approximation num4rique d" in4quations quasi-variationnelles stationnaires A. Bensoussan, J.L. Lions .......................

325

Gestion optimale des r~servoirs d' une vallde hydraulique A. Breton, F. Falgarone .........................

339

FILTRAGE FILTERING

ET IDENTIFICATION AND IDENTIFICATION

Algorithmes de calcul de modules markoviens pour fonctions al4atoires P. Faurre ..................................

352

Estimation de param~tres distribu4s dans les equations aux d~riv4es partielles O. Chavent ....................................

361

Adapdon de la m4thode du gradient un probl~me d' identification de domaine J. Cea, A. Gioan, J. Michel . . . . . . . . . . . . . . . . . . . . . .

391

Application de la m 4 t h o d e des 414ments finis ~ la r4solution d' un p r o b l ~ m e de d o m a i n e optimal D. B4gis, R. Glowinski ........................

403

GENERALI,TES GENERALITIES METHODS OF STRUCTURAL OPTIMIZATION William Prager Professor Emeritus, Brown University, Providence, R. I., USA ABSTRACT The paper is concerned with methods of optimal structural design.

Typical ingredi-

ents of structural optimization problems are discussed in Sect. i.

The basic prob-

lem is identified as one in mathematical programming, in general nonlinear programming, and the difficulties are indicated that are experienced in the application of standard methods of nonlinear programming.

The following three sections deal with

optimal plastic design, which is the most developed area of structural optimization because linear programming is applicable to it.

Sections 2 and 3 are respectively

concerned with the optimal plastic design of a truss of given layout and the determination of the optimal layout of a truss that has to transmit given loads to a given foundation. Sect. 4.

Optimal plastic design of beams and grillages is discussed in

Section 5 is devoted to optimal design of elastic trusses and beams.

Com-

putational aspects of structural optimization are discussed in Sect. 6, and some new ideas are mentioned in Sect. 7. 1.

INTRODUCTION

To be well-posed, a problem of optimal structural design requires specification of the purpose of the structure, the design constraints, and the design objective. general purpose of a structure is to carry given loads.

The

In general, a structure

will have to carry several alternative sets of loads, but it may happen that the design of the structure is governed by only one of them.

Design constraints may

concern the geometry of the structure or its behavior under the given loads.

Geo-

metric constraints specify at least the space that is available for the s%-~ucture, but may go much further in restricting the shape of the structure and the dimensions of its members.

Behaviomal constraints set bow~nds on quantities that characterize

the response of the structure to the loads for which it is being designed.

Examples

of behavioral constraints are upper bounds on stresses and deflections, and lower bounds on fundamental natural frequencies or on the ratio in which the given loads would have to be increased before they would cause failure by buckling or plastic flow.

The general design objective is minimization of the combined costs of the

manufacture of the structure and its operation over the expected lifetime.

It is

typical for aerospace structures that the cost of the fuel that would be needed to carry additional structural weight and the accompanying reduction in payload are much more important considerations than the manufacturing cost. e Research supported by the U. S. Army Research Office - Durham.

In these ¢ircum-

stances, minimization of structural weight becomes the design objective.

Since

structural optimization is particularly important in the aerospace industry, a vast majority of papers on structural optimization is concerned with design for minimal weight. Naturally discrete or artificially discretized problems of structural optimization are essentially problems of mathematical progra~nlng, usually nonlinear program,ninE. The development of powerful methods of nonlinear programming, and the availability of computers with large immediate-access memories therefore raised hopes for automated optimal design of practical structures by direct application of these programruing methods.

Except for optimal plastic design, which may be treated by linear

programming, these hopes have not so far been fulfilled.

Automated design proce-

dures were in fact developed that solve the basic nonlinear progra~ing problem by gradient or feasible directions methods, or treat it by a sequence of linear programs, or transform it into an unconstTained problem by the intToductlon of penalty f~nctions (for a survey, see Pope and Schmit, 1971). According to Gellatly and Berke, 1971, however, the use of these direct methods entails a prohibitive number of design iterations for structures with more than about 150 design variables.

Even

for much smaller numbers of design variables, the possibility is troublesome that the procedure may converge towards a local optimum.

To get some assurance that a

global optimum has been achieved, it may be necessary to start the procedure from quite different initial designs and choose from the corresponding final designs the one with the smallest weight. Sved and Ginos, 1968, have pointed out another reason why direct search techniques may not be satisfactory. truss for given loads.

Consider, for example, the minimum-weight design of a One way of approaching the problem of optimal layout is to

start with a lattice of possible nodes and consider the truss in which any two of these nodes are connected by a bar.

The bar forces of this highly redundant basic

truss must satisfy conditions of equilibrium and compatibility, which will be incorporated in the formulation of the problem, and the search will be conducted over "feasible" trusses for which these conditions are satisfied.

The optimal truss,

however, may be a subtruss of The basic truss obtained from the latter by The omission of certain bars.

In this case, the bar forces of the optimal truss together

with zero forces in the omitted bars will not satisfy all compatibility conditions of the basic truss.

Accordingly, the optimal truss is not a member of The set over

which the search is extended.

Sved and Ginos concluded that it would he necessary

to supplement the search described above by a similar search of the feasible designs of each statically stable subtruss of the basic truss.

Since each of these subtrus-

ses has a different set of compatibility conditions, this would be an enormous task. Sheu and £chmit, 1972, have shown how it can be reduced to manageable size, at least for the comparatively small example trusses they considered.

(The largest of Their

haslc trusses has eight nodes and Twenty-Two bars.) In recent years, attention has tumned from the direct application of the seamch techniques of nonlinear p~o~Tamming to the derivation of optimality conditions and their use in desiEn procedumes.

Some of this work is surveyed in the following.

Various problems of optimal plastic design and optlmal elastic design ape discussed in Sects. 2-4 and 5, mespectively.

Section 6 treats computational aspects of struc-

tumal optimization, and some new ideas ame mentioned in Sect. 7. 2.

OPTIMAL PLASTIC DESIGN OF TRUSS OF GIVEN LAYOUT

By far the most developed area of structumal optimization is optimal plastic design. Heme~ the structural material is tmeated as rigid , perfectly plastic, and the structure is to use the smallest possible amount of material subject to the condition that a given state of loading should r~pmesent the load-carTying capacity of the structure. It was recognized quite early (Foulkes, 1954) that the optimal plastic design of a structure of given layout may be formulated as a linear proETam.

Consider~ for in-

stance~ the minimum-volume design of a truss with the layout shown in Fig. i.

The

bars of the truss are to be made of a rigid, perfectly plastic material with the

Fig. i:

Optimization of truss of given layout

tensile and compmessive yield limits disPegamded in the analysis. to be rigid.

+a , and the possibility of buckling is %o he

The foundation, which supports the truss, is assumed

To exclude degenel-ate cases, in which some of the four bars are miss-

ing in the optimal truss, we prescribe a minimum cross-sectional area

Aio

for bar

i , (i=l,... ,4) ~ and write the actual c~oss-sectional area of this bar as Aio + Ai, where

A. >. 0 . 1

At the joint

0 , which is taken as the origin of the coordinates

X~ , (u -- 1,2) , a load with components be denoted by

Pa

is to act.

The length of bar

i

will

h i , and the direction cosines of the axis of this ham (oriented from

the foundation towards

0 ) by

c.

The following concepts will be useful in the discussion of this pmoblem.

Fom gener-

ality, they will be defined without reference to the partlcula~ly simple example in Fi E . i.

A system of axial fomces in the bars of a truss that equilibrates given

loads acting at the joints of the tmuss will be called "statically admissible" fom these loads.

Fom specified cross-sectional areas of the bars, a statically admis-

sible system of bar forces will be called "safe" if the cormesponding axial stress in each ham does not have an absolute value in excess of

~ .

A system of mates of

extension of the bars of the truss will be called "kinematically admissible" if it is derived from velocities of the joints of the tmuss that do not violate the kinematic constmaints at the supports. Let us now return to the problem in Fig. i.

According to the static theorem of

limit analysis (see, for instance, PmaEer , 1959), the load

P

will not exceed the

load-car~ying capacity of the tmuss if there exists a safe, statically admissible system of ham forces

F.

for this load.

AccoPdingly, an optimal design is charac-

terized by the following llnear program : Minimize ~

O

Z.£.A. 1

!

(2. la)

1

subject to EiciaF i

=

P

,

(2.1b)

F i + ~A i >. -OAio ,

(2.1c)

-F i + cA i >. -qAio .

(2.1d)

The sum in (2.1a) is the excess of the total volume of the bars of the truss over the minimal volume

?i£iAio .

Equation (2.1b) stipulates that the bar forces

should be statically admissible for ~he load

P

Fi

, and the inequalities (2.1c) and

(2.1d) stipulate that these forces should be safe. If the dual variables corresponding to the constmalnts (2.ib) thmough (2.1d) are denoted by

v

, ~i ' ~i ' the dual program has the following fomm:

Maximize ~aPavc~ - cZiAio(~i + ~i )

(2.2a)

subject to

+ -~i "< ~o £'~ "

(2.2c)

To facilitate the mechanical interpretation of the dual program, the factor ~e o has been introduced in the objective function, where ~ is a reference mate of extension, o

Note that the variables applies in (2.2c) if Fi/(Aio + A i) value

o .

v

are not restricted in sign, and that the equality sign

Ai > 0 .

has the value

Note further that

-c , and that

Accordingly, either

~i

or

Pi

~i > 0

Pi • 0

only if the stress

only if this stress has the

or both vanish.

The dual problem may be given a mechanical interpretation by identifying vs,(e=l,2), as the velocity components of the joint P~

and

~. • 0

~i - ~i

O

in a collapse mechanism under the load

as the corresponding rate of elongation

Xi

of bar

only if the equality sign holds in (2.1c), i.e. if the bar

i

i .

Now,

is at the

l

compressive yield limit; in this case,

"i + 6 1

=

The same r e l a t i o n

Ixil is

Pi = 0 , I i = ~i - ~i < 0 , and

.

(=3)

obtained

for

Pi • 0 , in which

case

~i

= 0

and

X'z • 0

m

Accordingly, the constraint (2.2c) may be written as fell "< So ' with equality sign for

Ai > 0 ,

(2.~)

where

¢. is the axial strain rate of bar i in the considered collapse mechanism. 1 Note that the signs of s i and F i are coupled by the relation F . c . >. 0 . 1 1

(2.5)

If a truss of the given layout is at all capable of carr~ylng the load

P

, there

exists at least one statically admissible system of bar forces, and the cross-sectional areas

A. + A. may be chosen sufficiently large $o fulfill the constraints io 1 (2.1c) and (2.1d). According to the existence theorem of linear programming, this

fact assures the existence of solutions of both the primal and the dual problems. There exists therefore a joint velocity that entails rates of extension satisfying (2.4) and (2.5). Note that the linear programming formulation of our problem is readily adapted to the case where none of several alternative loads load-carrying capacity of the truss.

P'

P"

is to exceed the

For each load, we then have equilibrium equa-

tions of the form (2.1b) and yield constraints of the forms (2.1c) and (2.1d).

The

mechanical interpretation of the dual yields the optimality condition

Iql where P'

~' P"

+ ~':11 + ~" ....

"'"

...

"< So ,

with

equality sign for

A, • 0 ,

(2.6)

are axial strain rates in collapse mechanisms under the loads

If one of these loads is too small to influence the optimal design,

the corresponding rates of extension vanish. 3.

OPTIMAL LAYOUT OF A TRUSS

The preceding discussion of the optimal design of the truss in Fig. 1 obviously remains valid if the joint

O

is connected to the foundation by more than four bars.

This remark enables us to attack the problem of optimal layout by starting from a "basic" layout that comprises, in addition to the joint

0 , a large number of po-

tentlal joints at the foundation, and setting

Aio = 0

for all bars of thls layout

To allow for the possibility that some of them may be omitted in the optimal truss. In The limiting case~ wheme any point of the homizontal foundation is a potential joint, the optimality condition (2.4) calls for a homogeneous strain mate field in the stmlp between the foundation and the horizontal Through mates

¢

0

that has strain

satisfying The following conditions:

Icl

=

0

for the homizontal direction,

(3.1a)

=

c°

for the direction of any bar of the optimal Truss,

(3.]30)

¢ o

fom any othem direction.

(3.1c)

It follows from (3.1) that the bars of the optimal truss are along the principal directions of the considemed stmain mate field.

Depending on whether one or both

principal mates of extension have the absolute value in Figs. 2a and b.

~o ' we have the cases shown

In the first case, the optimal truss is degenemate and consists

P O

A

P

O

// (a)

(b) (a) Degenerate optimal tmuss for load inclined by 45 ° or less against vertical

Fig. 2:

(b) Optimal truss for load inclined by more than 45° against vertical

of a single bam

OA

along the line of action of The load~ which forms an angle of

45° or less with The vertical. the bars

OA

and

OB

In the second case, the optimal truss consists of

that form angles of 45° with the vertical, while the angle

between load and vertical exceeds 45 ° . The conditions (3.//)) and (3.1c) quite generally gover~ the optimal layout of trusses fop a single state of loading (Michell, 1904). truss for the transmission of the load of the limited width

AB.

P

at

0

Figur~ 3a shows the optimal to a rigid, horizontal foundation

The hams of the Truss follow the principal lines of a

strain mate field with pmlncipa! strain rates

±¢

o

; along the foundation, this

field has vanishing strain rate. onal net in the region shown.

In the circular sectors

bars.

The bars of the optimal truss form a dense orthog-

0CDE ; in the figure, only a few of these curved bars are

Along the contour bars

ACD OCA

and and

BDE , there are dense systems of radial 0EB , the axial forces are constant, and

the curvature of these bars causes small axial forces in the bars of the other family.

Accordingly, the interior bars of the truss are light in comparison to the

contour bars.

There is a substantial body of literature on trusses of this kind,

which are known as Miehell trusses (see, for instance, Hemp, 1966).

While Michell

trusses wlth their infinity of bars are not practical, they provide the smallest

C

E

A

B

A

(b)

(a) Fig. 3:

(a) Optimal transmission of load

P

to foundation

(h) Optimal transmission of alternative loads to foundation AB

P'

AB and

p,,

possible structural weight, which is used in assessing the efficiency of more practical designs.

The Michell efficiency of a truss is defined as the inverse ratio

of the weights of this truss and the Michell truss for the same loading. To formulate an optimization problem that leads to more realistic designs, one might enforce a finite number of bars by including the weight of the connections at the joints in the structural weight that is to be minimized, for instance by adding, for each joint, a fixed weight corresponding to the average weight of a connection.

As

is well known, fixed cost problems of this kind create considerable algorithmic dif-

ficulties. An interesting super~ositlon principle that furnlshes the optimal truss for two alternative loadings is due to Hemp, 1968.

The illustrative problem in Fig. 3h con-

cerns the transmission of the alternative loads represented by the vectors OP"

to a rigid foundation of the limited width

segment

P'P" , the given loads are written as

AB .

With

Or'

and

as the center of the

OQ -+ QP' , and the optimal trusses

("component trusses") ape deterlnined for the single loads consists of the single bar

Q

OQ

and

QP' .

The first

0C , while the second has the layout in Fig. 3a.

The

strain rate fields for the collapse mechanisms of the component trusses ~ which satisfy the conditions (3.i), will he called "cor~onent fields".

The line elements of

the plane of the truss may be divided into three groups depending on whether they are along a bar of the first or second component truss or not along a bar of either II truss. If EI and c are the strain rates of the same line element in the two component fields,

¢oe°l

I¢I I ". 0 , and its curvature under the

~°: = 0.00

~

200 ^

'ii ' . -ioo'

i~

A

//,~

\I'll

1//1 I

r

1

I

/~ ,/'t~'~'°3

!\ ]I

I '° i

iv,

-'~°1'

\~

-2004

Fig .11-45

w

Free Oscillations oF Plane Net - Small Step and Large Step Algorithms

•

ReLI

Error 151

r3 rn,~" + r 3 = 0

Break-down ol Cond

t =0

r0 = 1

T-

'=0 = 0

i

163.

10_5.

\, ~

10.6

~

t0-7

~

lo'~

I,.

"

-ro

J

TO=7.4162987508 TO T = ~-

LargeStep Step

isplacernents

___Small

( 3rd Order )

0 4 8

Fig. 11-46

16

32

6~

Simple Cubic Oscillator Free Vibration, Non-linear Large and Small Step Algorithms Error in Maximum Displacement and Time after One Period

137

Coefficients

of ~lrlx

-TT I

~T

2

1

~T -~T

0T2

I

2

~2 ~-2

- ~2 T2

3 2

-~T

7tT

~'r 2

I 2 --~'r

ST2

~t2

2-Y° T,33

- ~ST 3

3

I

TfGT

~T

~T

~t

3

_ 31,2

~r

-~6r 2

2!52I-3

- &T

3

~-Et

~

I-~5 ~

" I-g~6

3-6T~~

- 3-~6~

Table tf - I - Family of Conditionally Stable Algorithms

138

Coefficients

of H-IX

l

.~T2

~ 2

_ • T 2_I~ ~0T2

~T2

2

~60T2

I

2

- ~IT 3

~3

i ~73 T

11

- i~T

1-~ T

-~1 3

- ~]

T~ T ~

- ~1

~

3

3-g-~T

Table II - 11

U n c o n d i t i o n a l l y Stable Algorithm

- 3-g-~T

n = 1 to 4

139

Order

Period elongahon

tan ~ / 2

n

@ for

1 4 ~2T2 12

I

1

Period elongation for small COT

(Newmark) 2 ( Re~ | )

720

I +

3

(d6 1"6

100800 1

f

W2T2 ~ / / /

Table II - III

~t,j2~2

~4T4 ~

Summary of Properties of Matrix

4

3~

ELEMENTS FINIS FINITE ELEMENTS

ONE-SIDED APPROXIMATION AND PLATE BENDING

Gilbert Strang Massachusetts Institute of Technology

We discuss a problem which arose in the theory of one-sided approximation, tion.

but has at the same time a natural physical interpreta-

Consider a triangular plate,

with all edges free, from a vertex. small

A.

supported at its vertices but

subject to a point load acting at a distance

The problem is to compute the deflection for

We shall describe how this computation promised to be

useful in estimating the distance from a nonnegative function to the set of nonnegative linear functions

v < u.

u

The latter es-

timate leads to an error bound for finite element approximation of continuous quadratic programming problems,

including the obstacle

problem and the deformation of elastic-perfectly plastic materials.

This paper was prepared for the International Symposium on Computing Methods in Applied Sciences and Engineering at IRIA, Eocquencourt, France, in December, 1973. It will be published in the Proceedings of the Symposium. I am grateful for the support of the National Science Foundation (GP 22928).

141 INTRODUCTION

This paper represents one step, and unfortunately only a small one, toward an understanding of the rate of conver~enge of Ritz methods for variational inequalities. a subclass of variational inequalities,

It applies most directly to which we might describe as

continuous quadratic programming problems--the problem is to minimize a quadratic functional of the usual "potential energy" type, but with the class of admissible trial functions subject to inequality constraints as well as equations.

The central problems of plasticity

theory fall naturally into this framework. The Ritz method chooses a family of trial functions depending only on a finite number of free parameters

E qj ~j,

qj, and

minimizes the potential energy under the given constraints. produces an ordinary

This

(discrete) quadratic programming problem.

Our

goal is not to solve this latter problem--a number of good algorithms already exist,

and this source of quadratic programming problems

ought to inspire new ones--but rather to estimate the distance between the true solution

u

and its Ritz approximation

u h.

First, we recall how such estimates are established in the classical case, when there are no inequality constraints and the problem is linear. l) solution

There are three steps to an error estimate:

To establish some smoothness, u.

or regularity,

for the true

Without this step we can still prove convergence

(this has already been done for many variational inequalities),

but

normally we cannot establish anything about its rate. 2)

To show that if such a smooth function

u

proximated by the given family of trial functions, trial function

uh

can be well apthen the particular

chosen by the Ritz method will be close to

u.

142

In the classical case,

this step is made simple by the fundamental

theorem of the Ritz method:

uh

is the p r o j e c t i o n

inner product natural to the problem, nonlinear problems, to

u;

uh

of

u,

onto the trial subspace.

In

will no longer be the closest trial function

we have somehow to find an a priori estimate for

given only the approximation 3)

in the

properties

of the trial subspa@e.

To study the approximation properties

and extract the information

required

u - u h,

of the trial subspace,

in Step 2.

At this stage the

given variational problem and the Ritz method are no longer involved; we are assured that the unknown function number of derivatives, element method,

is discussed

When the problem than an equation, Nevertheless,

of the data,

becomes

inactive.

u - u h,

h2

of

Lewy,

u--limited

inequalities

not

but rather by

Step 2 has been studied by have led to

in Step 3, and ! believe that But in this paper we

in [3], and prove the approximation TaKen together with

[3], this paper es-

estimate for the strain energy in the error

when piecewise

variational

Stampacchia,

The two approaches

may prove to be simpler.

theorem which it requires. an

rather

"free boundary" across which an active

different approximation problems

tablishes

inequality,

as in linear problems,

and by Mosco and the author.

adopt the approach described

/,

Brezls,

a limited regularity

of an internal

FalK's analysis

this ap-

each of these three steps becomes more difficult.

inequality constraint Falk,

For the finite

trial functions,

is governed by a variational

and others have established

the appearance

it.

a certain

in [4].

some results are Known.

by the smoothness

possesses

and asked to approximate

with piecewise polynomial

p r o x i m a t i o n problem

u

linear finite elements

are applied to

like the St. Venant torsion problem.

143 PIECEWISE LINEAR APPROXIMATION FROM BELOW

We shall describe two attempts at proving a theorem on onesided approximation. dependent

interest

The first attempt ended in a problem of inin the theory of plate bending--to

deflection of a plate under certain loads. originally

motivated

(The applications

the theorem were not to plate bending;

only that the approximation be restated

which

it is

theorem which was finally wanted could

in terms of plates.)

Unfortunately,

too great to prove the one-sided approximation fication of this unsuccessful note,

find the

argument,

the deflection was theorem.

But a modi-

described at the end of this

does yield a proof. The theorem

in question

THEOREM I:

Suppose

plane,

and let

P2 = (I, 0),

T

can be stated in the following way:

u(x,y)

be the triangle with vertices

P3 = (0,!).

(I)

v(x,y)

0SvSu

P! = (0,0), c,

there

such that

on

u(pi) - v(Pi) < clul2,T,

The last quantity

function on the

Then for some absolute constant

exists a linear function

(2)

is a nonnegative

is the standard

T i = 1,2,3.

~2

seminorm,

which must be

finite or the result is vacuous:

lu

2

12,T =

f~

2

2

(Uxx + 2U~y ÷ Uyy) dx dy.

This is not the only norm of interest.

One could prove the result

more simply,

variables,

priate

~

and admit more independent rather than

L2

by using appro-

norms of the second derivatives.

In

144 fact,

this weaker result would be sufficient

we have made, in quadratic

in a joint paper programming.

for those applications, First,

vj

like

satisfying

~2

norm is the optimal

the consequences Suppose

the plane

Within each triangle (1) and

(2).

Let

of Theorem

w

Tj

value

chosen from the triangles

w

0 ~ w ~ u

(4)

u(Pi)

The seminorm

of

u - w

is the interpolate determined

surrounding

that vertex.

elul2, i.

- w(Pi) J

over the triangles

u--the

ui(Pi) theory

continuous

= u(Pi)--then

< --

clul2. Tj ;

(6)

lu-uilo, Tj

clul2,T.;J

(7)

lu-Uzll, %

clul2,Tj.

(7) are

respectively.

at the vertices.

each triangle,

and therefore

u I - w--except

that the

luI2

re-

[4] that within each triangle

• a lu-uil

uI - w

If

linear

it is a central

Tj

The left sides of (6) and

Pi"

immediate.

piecewise

(5)

first derivatives,

surrounding

in other norms are almost

of

by

sult of approximation

of

is the smallest

over the whole plane

is also computed

Estimates

function

piecewise

must satisfy

(3)

uI

Pi

by tri-

is a linear function

be the continuous

whose value at every vertex

Then

one

1 for piece-

is covered

there

linear function vj(Pi),

to error estimates

so we shall stay with it here.

we describe

T.

[3] with Mosco,

But the

wise linear approximations. angles

for the applications

L2 From

norms of

u - w

(4), we already

But this is a linear the estimates

and of its know the size

function within

(5-7) apply also to

norms are taken over the union

~j

145

of all neighbors mates hold for

of

Tj.

u - w.

!ation of the plane,

By the triangle In fact,

maps each triangle Finally, longest

c

depending only on the

side--a simple affine t r a n s f o r m a t i o n

into the original triangle

T.

we rescale the independent variables

side is

h,

the same esti-

they remain valid for any triangu-

with a constant

smallest angle and the largest

inequality,

so that the

and loom to see how this scaling affects

each seminorm:

T H E O R E M 2. ous p iecewise

Given

u > 0

on the plane,

linear function

(8)

wh

there exists a continu-

such that

0 £ w h £ u;

(9)

max u-wh i c h]/21ul2,nj; Tj

(lO)

lU-~hlo,Tj i C h21ul2,~j, lU-~hlOi C h21~12;

(zl)

lU-Whlz, Tj ~ C ~lul2,nj, The second inequalities

lU-Whll ! c hlul2.

in (10-11) come from the first,

squaring and summing over all triangles; only on the smallest angle.

the constant

The underlying

C

by

depends

domain can be altered

from the whole plane to a polygon. The theorem leads to error bounds for the finite element approximation

of variational

first application ing

IZ

Igrad vl 2

constraint

v < X

inequalities.

of this Kind,

We describe

in [3] a

to the obstacle problem of minimiz-

under the constraint

v Z ~-

(An additional

could be handled by the same approach,

if the two obstacles

are separated:

~ < X.)

The argument

at least can be

146

extended

to St. Venant's

cyllnder--where hyperbolic

torsion problem for an elastic-plastic

there is an unknown boundary between elliptic

regions

(and,

if the cylinder

well as an upper obstacle).

which need not distinguish

between the elliptic and hyperbolic

it simply minimizes over those piecewlse satisfy the stress constraints. methods,

a lower as

This free boundary appears automatic-

ally in the discrete approximation, during the computation

is hollow,

and

domains:

linear functions which

For an exposition

we refer to Glowinski-Lions-Tr~moli~res

of these

Ill.

147

AN UNSUCCESSFUL

PROOF OF THEOREM

Suppose we choose any linear 0 < v < u

in the triangle

T,

function

and which

v

1

which

is tangent

satisfies to

u

at a

M

point

~

in

T.

(Everything

now takes place

this subscript

from the norms.)

(i.e.,

with

functions

differentiable, we mean that v(Pi) in

T.

about the word

without

violating

(2) of the theorem,

Since i_~s tangent

Ig12 = luI2" to zero at

we have on the function

then

class

is by way of the interpolates at the vertices; (12)

in

"tangent":

the constraint

is to show that i f

g(Pi ) A clgI2"

satisfies

g k 0

The best grasp

~,

in fact almost the only one,

gl"

We Know that

suppose we can evaluate

d(~) = max{f(~)

g = u - v

v ! u

by the argument which follows.

our problem

~,

u

and tha~ none of the nodal values

We hoped to prove that the difference

the estimate

and we drop

which are not pointwise

we must be more precise

can be increased

T,

Since there are functions

lu12 < ~)

v(~) = u(~),

in

g - gI

vanishes

the quantity

I f(Pi ) = O,

If12 = 1].

Then we will have

gz(~) : (gz-g)(~) 0

KluI2 v =

at the vertices.

KIul2,

then we take

If v = 0

(2) in the theorem holds. and

u(P2),

say, are less than

Then we construct

v ~ u

in

T.

was maximal,

KIuI2,

v = cy, choosing

At the vertices

is still less than

P3 = (0,1). and

v < 0

g(~) = 0.

KIuI2;

P1

and

and at some

We introduce the folare at

On this triangle we still have in the new part below the x-axis.

is certainly not near either of the two new vertices,

and even if it is near the old g(P3) ~ CIgI2.

- v = ~tu12

u

is not.

c

u

think of the larger triangle whose vertices

(2,-1),

g > 0, because

u

g = u - v

~, because

(0,-I),

and

for which

and

If

T, we may choose the approximation

u(P1)

= u(O, 1)

the largest

of

easy.

P3'

it would still follow that

So this case is settled.

153 Suppose finally that

u

is near zero at only one of the ver-

tices, for example the origin: origin by a vertex at T' = Q P2 P3" which

If we still choose

v < u

in

P2" because by hypothesis

for some absolute constant v

Q.

C.

g

[

and

P3

for

exceeds

is small at

P3:

If it happened that

be near

KIuI2. g(P3 ) ! CIul2 g(P2) ~ g(P3)

would establish the theorem; but it is more

g(P2) > g(P3), and we must look for another

P2

c

will again be on or

Nor will the point

If we carry out the same argument for roles of

~

g(P2) = u(P2)

Therefore, we can conclude that

likely that

v = cy, with the largest

T', a coincidence point

then this choice of

Then we replace the

Q = (-i,-I), and consider the larger triangle

above the x-axis, and not near to

u(P I) ! KluI2"

v

v.

of the form

cx, the

are reversed; either it happens that

g(P3) ! g(P2) ! clul2' or

else

g(P3) > g(P2)

and t h i s

v

is also

not satisfactory. Our last idea is this:

consider the family

V e = ce(x cos 0 + y sin e), For each v ~ u Q!). e.

e, c e

in

is chosen to be maximal under the constraint

T', so there is a coincidence point

The values Therefore

ge(Pi) = u(Pi) - ve(Pi)

corresponding Theorem i..

(not near to

depend continuously on

v

6 = a

for which

ga(P2) = g~(P3).

The

is the one-sided approximation we want in

It is zero at the origin, where by assumption

U(Pl) ~ KluI2. P2

~e

(unless the proof was settled by one of the previous

cases) there is a value

vertices

0 ~ e ~ ~/2.

Evidently, and

the point

P3" so that

other--and therefore at both.

~

ga ~ cIuI2

cannot be close to both at one vertex or the

154

This proof appears to extend to three dimensional Whether the theorem holds in all dimensions, order of one-sided approximation nomials of higher degree,

problems.

and whether a higher

is assured for piecewise poly-

is at this moment unknown.

155 REFERENCES

I.

R. GlowinsKi, Numerlque Dunod,

2.

.

des In4quations

Paris,

L. Mirsky, Press,

and R. Tr~mo!i~res, de la Mecanlque

Resolutlon" "

et de ! a Physique,

197~.

An Intrqductiom

to Linear Algebra,

Oxford University

1955.

U. Mosco and G. Strang, inequalities,

.

J. L. Lions,

0ne-sided approximation

Bull. Amer. Math.

Soc.,

G. Strang and G. Fix, An Analysis Prentice-Hall,

New York,

1973.

and variational

to appear.

of the Finite Element Method,

QUELqUES METHODES D'EL~IENTS FINIS POUR LE PROBLEME D'UNE PLAQUE ENCASTREE

P.G. CIARLET Analyse Num~rique, Tour 55 Universit~ de Paris VI II, Quai Saint-Bernard, 75230 PARIS CEDEX 05

Co-..unication pr~sent~e au Colloque International sur les M~thodes de Caleul Scientifique et Technique, I.R.I.A., Le Chesnay, 17-21 D~cembre 1973.

157

O. INTRODUCTION L'objet d e cet article est de pr@senter dlverses m~thodes d'~l~ments finis, effeetivement utilis~es par les Ing~nieurs, pour r~soudre num~riquement le probl~me d'une plaque eneastr~e. Nous n'avons pas consid~r~ toutes les m~thodes : C'est ainsi que nous ne dlsons rien des m~thodes "hybrides" ou "mixtes" (cf. Oden [ 31] , Pian [ 32]), qui sont associ~es ~ d'autres formulations variatlonnelles que eel!e que nous donnons ici. Nous renvoyons le lecteur int~ress~ par l'analyse num~rique de ces m~thodes Brezzi

[ lOI , Oden

[ 31]

et aux travaux de Johnson [ 23,2~] . De m~me, nous

renvoyons ~ Glowinski [ 21] et £ un article ~ para~tre de Ciarlet & Raviart [ 16]pour des m~thodes "par d~composition" qui, tout en ~tant des m~thodes non eonformes, correspondent ~ une fa$on de poser le probl~me discret diff~rente de eelle d~crite darts cet article. Le plan de l'artiele est le suivant : Au §I, on rappelle la formulation variationnelle du probl~me d'une plaque encastr~e. On donne ensuite au §2 des exemples varies d'&l~ments finis conformes utilis~s pour approcher la solution de ce probl~me. Enfin, nous examinons au §3 diverses m~thodes d'~l~ments finis non eonformes pour lesquelles un essai de Oustifieation a priori du "patch test" de B. Irons a ~t~ fait, suivant les travaux r~eents de M. Crouzeix, P. Lascaux, P. Lesaint, P.-A. Raviart, G. Strang, e t p a r l'interm~diaire d'un "lemme de Bramble-Hilbert sur les formes bilin~aires", introduit en [ 12] . On indique aussi une m~thode de p~nallsation, introduite par BabuShka & Zl~mal [ h] . Chaque fois qu'elle appara~t dans une in~galit~, la lettre C d~signe une constante ind~pendante des diverses fonetions intervenant dans l'in@galit~ en question, ainsi que du sous-espaee V h consider@. i. LE PROBL~ME CONTINU Dans tout c e qui suit, on d@signe par fl un ouvert born~ du plan. E t a n t

dorm@ un

entier m, les expressions

I.,o.o-(,

~

~

fo ;l" o,j

,~/2

• ,,o.o:

m

2

)i/2

repr6sentent les semi-normes et normes usuelles de Sobolev; on rappelle que sur l'espace de Sobolev ~o(~), la 5~i-nor~e I-Im,~ - ~

= e n o ~ e ~quiv~ente ~ la norme

11v Urn,~. Consid~rons le probl~me de l'~quillhre d'une plaque, tel qu'il est d~crit dans le livre de Landau et Lifchitz [ 26] , par exemple. En l'absenee de forces, la plaque est repr~sent~e par l'ensemble R du plan, suppos~ horizontal. On note respectivement

158

e, E, et u, l'~paisseur, le module d'Young, et le coefficient de Poisson de la plaque. La fonction inconnue u, qui repr~sente la cote de la plaque par rapport au plan horizontal lorsqu'une force verticale de densitg F est appliqu~e ~ la plaque, rend minimale l'~nergie de la plaque qui, pour une cote donn~e v, est donn~e par l'expression

E(v) =~.I'~-' {(~)2 ÷ ~(~-~~-k~.,~y ~2~ ~y~2~}d~dy-.[ F~xd~.

(L1)

n

n

12(i-u2) l' ~nergie de la plaque s'~crit (£ un facteur multisi l'on pose f = ~---~-T---F, plicatif constant pros)

(1.~)

j(v) =

1 a(~,v)

- (f,v)

oG (.,.) est le produit scalaire de l'espace L2(2), en supposant par consequent l'appartenance de f ~ L2(fi), et oG la forme bilin~aire a(.,.) est donn~e par (1.3)

a(u,v) = .r {Au Av + (i-~) f~ ~2u

B2v

B2u S2v

B2u B2V~}dxdy

n .

~r~2u ~2v

~2u B2v

_B2u

=I {~ ~u Av + (~-~JL~x--Z~--z+ ~y-~~-9z+ ~ y

B2v

~}a~dy.

n La forme bilin~aire ci-dessus est continue sum l'espace H2(fi) × H2(e) et, de plus, elle est H~(~)-elliptique; pour le volt, on remarque que

(~.~)

a(~,v) = ~I~I~,~ + (1-~)M~,~

et on utilise le fait que, physiquement le coefficient de Poisson ~ v~rifie les in~galit~s 0 < ~ < ~1 •Dans ces conditions, il existe une et une seule fonction u dans l'espace V = H2(~) telle que o

(1-5)

J(u) = min J(v), veV ou, de fa9on ~quivalente, qui v~rifie les ~quations (1.6)

a(u,v) = (f,v) pour tout v e V. Pour interpreter - au molns formellement - le probl~me variationnel (1.6), on

utilise les formules de Green suivantes : (1.7) (1.8)

.~ Au Av dxdy =.~ A2u v d x d y - ~ .r {2

S2U

B2V

_~n vdyBAu

S2U B2v

Au~dy,BV

B2U ~2vI~ ~

~xBy sx~y ~--z~-9~-~-97~ -zt~x~y = ~2u ~v ~

o G ~ n e t ~repr~sentent

+#

~2u ~v ~n~t

respectivement les d~riv~es normales et tangentielles le

long de la fronti~re ~fi de l'ouvert ~. Dans ces conditions, si la solution u du

159

probl~me variationnel (1.6) est suffisa~ment r~guli~re, elle est aussi la solution du probl~me

A2u = f dans n,

(1.9) (I.i0)

~u ~n = 0 S~r ~ ,

u =

qui est effectlvement le module "classique" le plus simple pour une plaque encastr~e, l'encastrement ~tant pris en compte par les conditions aux limites (i,i0). L'application de la seconde formule de Green (1.8) montre que la contribution ~2v ~2u ~2v ~2u ~2 du terme .r~(1-~){ ~ ~x~y ~x-~Z~-~2-~-~ ~x-~dxdy est nulle. En d'autres termes, on pourrait se contenter de la forme bilin@aire a' (u,v) --,~Au Av dxdy, et c'est d'ailleurs ce choix qui est fr~quemment fait, puisqu'il conduit ~ la formulation variationnelle la plus simple qu'on puisse attacher au probl~me (1.9)-(1.10). La forme bilin~aire a' est encore V-elliptique car la semi-norme IAVIo,~ est une norme sur l'espace v -- H2(~), ~quivalente ~ la norme Hv 112,~. En fait, la possibilit~ de remplacer la forme bilin~aire a par la forme bilin~aire a' tient ~ ce que l'on se place dans l'espace H2(G), et non dans un espace strictement plus grand, comme l'espace O

H2(~); autrement dit, cette possibilit~ serait supprim~e si l'on choisissait des conditions aux limites autres que celles d'une plaque eneastr~e. On volt aussi que la forme bilin@aire a' correspond ~ la valeur ~ = 1 dans l'expression (1.3) et qu'en fait, toute valeur W de l'intervalle [O,1] conduira ~galemerit au m~me probl~me (1.9)-(1.10), la V-ellipticit~ ~tant toujours assur~e; cf (l.h). De la m~me fagon, toute l'analyse falte au §2 pour les m~thodes conformes s'applique sans changement ~ toute valeur ~ de l'intervalle [ O,1]. Cependant, pour certaines des m~thodes non conformes du §3, certains r~sultats ne sont plus valables si W e s t ~gal ~ l, alors qu'ils restent vrais pour les valeurs physiquement admissibles de ~, i.e., dans l'intervalle ]0,~ [ . En ce qui concerne la r~gularit~ de la solution u du probl~me (1.5), on peut montrer (voir Kondrat'ev [ 25] ) que celle-ci appartient ~ l'espace H~(~) O H2(~) des que l'ensemble ~ est un polygone convexe, ce qui est effectivement souvent le cas des plaques. A cet ~gard, il est int~ressant de remarquer que (sauf pour l'~l~ment de Morley; cf §3), l'hypoth~se "u ~ H~(G) " est aussi l'hypoth~se minimale qu'on utilise pour obtenir la convergence des m&thodes d'~l~ments finis, aussi bien conformes que non conformes. 2. METHODE s CONFOHMES On suppose comme au §i clue f E L2(R). On notera ~. II l'expression 1.12,R, qui est une norme sur l'espace V = H2(R). Ii existe alors des constantes M e t ~ > 0 0

telles que (2.1) (2.2)

la(u,v) I ~ M flu " fly U pour tout u,v e V, ~ fly 112 ~ a(v,v) pour tout v e

V.

160

Etant donn~ un sous-espace de dimension flnie V h de V, le probl~me discret consiste ~ trouver une fonction u h E V h telle que

a(~,~- h) -- (f,vh) po~ vh e Vh,

(2.3)

et ee probl~me a une solution et une seule, d'apr~s (2.2). Utilisant les in~galit~s

Uu-~ , z ~ ~(u-u~, u-uh) = a(u-~h, u-v h) ~

M llu-uh B ,u-v h ,

v~rifi@es pour to=re fonction v h E Vh, on en d~dult l'in~gallt~

(2.~) ~u-uh~ ~ C

inf UU-Vh~,

o~ la constante C = M/ct est ind6pendante au sous-espace Vh, de sorte que le probl~me de l'@valuation de l'erreur

Hu-u h II est ramen~ ~ un probl~me de th~orie de l'appro-

ximation - l'~valuation de la quantit~

inf H u-v h H • Vh~-Vh Pour obtenir l'in~galit~ (2.4), on a utilis@ de fagon essentielle l'inclusion

V h C V. On dit alors que la m~thode d'approximation est interne et clue, dans le cas o~ l'espace V h correspond ~ une m~thode d'~l~ments finis (cf. les exemples d~crits plus loin), la m~thode, et les ~l~ments finis, sont conformes, ou encore compatibles. Pour construire un tel espace Vh, on se donne :

(a) Une triangulation

de l'ensemble ~ en ~l~ments finis, i.e., ~ = K E ~ K , o~ A

A

les ~16ments K sont des triangles, ou des ,~uadrilat~re,s, d'int~rieurs deux ~ deux disjoints, et tels que tout cSt~ d'un ~l~ment soit ou bien un cSt~ d'un autre ~l~ment, ou bien une pattie de ~ h =

on pose

max hK, avee h K = diam(K) pour tout K 6 ~ h ~

(b) Un espace P, de dimension finie, de fonctions r~elles tel que s i v h est une fonction quelconque de l'espaee Vh, alors restr'vh 1K ~ P pour tout K 6

~h •

(c) Un ensemble de de6r~s de iibert~ attaches £ un ~l~ment fini "co,rant" K qui, d'une part, d~finissent une base de l'espace P et qui, d'autre part, sont ehoisis de telle fa~on que !'inclusion V h C H2(~) ait lieu; comme les fonctlons de l'espace P sont le plus souvent tr~s r~gu!i~res dans la pratique, l'inelusion pr&c~dente sera une consequence de l'inclusion V h C CI(5), ce que l'on pourra v~rifier clans chacun des exemples dorm,s ci-apr~s. On v&rifiera aussi sum ees mSmes exemples que Its conditions aux limites (i.iO) peuvent ~tre satisfaites exactement dans Its sous-espaces V h eorrespondant s. Si u est une fonction suffisamment r&guli~re d~finie sum 5 (resp. sur un ~l~ment fini K), on notera HhU (resp. HKU) la fonetion de V h (resp. de P) qui interpole

161 l a f o n c t i o n u s u r ~ ( r e s p . sur K), i . e . ,

dont l e s d e ~ s

de l i b e r t ~ s u r ~ ( r e s p . sur

K) sont i d e n t i q u e s £ ceux de l a f o n c t i o n u . De l a s o r t e on v o i t que r e s t r , pour t o u t K e ~ h " On v a e / o r s majorer i a q u a n t i t ~ lit~

(2.4) pat" Jlu-IlhU tl, et comme

(2.5)

inf

nhU

Ilu-vh II de l ' i n ~ g a -

= ~Ku K

VhEVh

tu-nhul2, £ ~,

lu-,Ixul2,£/

,

le probl~me d'~valuation de l'erreur se trouve ramen~ ~ un probl~me de th~orie de l'interpolation "locale", i.e., Bur un ~l~ment fini "courant" K. Ce probl~me a ~t~ ~tudi~ par de nombreux auteurs ces derni~res ann~es; voir notsa~ent Babu~ka & Aziz [ 3] , Bramble & Hilbert [ 8] , Ciarlet [ ii] , Ciarlet & Raviart [ 14,15] , Raviart [ 33J , Strang [35] , Strang & Fix [37] , Zl~mal [40]. Le r~sultat fondamental est le suivant: on consid~re une "famille" d'~l~ments finis pour laquelle le param~tre h tend vers z~ro, et qui ne deviennent pas "plats" la limite (pour une d~finition precise de cette notion, voir Ciarlet & Raviart [ lh] ). L'hypoth~se fondamentale est l'inclusion

Pk C P,

(2.6)

o~ Pk d~signe l'espace vectoriel des polynSmes (ici : de deux variables) de degr~ ~ k. Alors pour tout entier m ~ k+l tel que P C Hm(K), on a

(2-7)

tu-IIKUlm,K g C

hk+l-m

K

u k+l,K

o~ la constance; C eat ind~pendante de la fonction u et de h K. Par application de (2.4), (2.5) et (2.7) avec m = 2, nous obtenons

(2.8)

Ilu-u h

, < c hk-llulk+l,£,

en supposant que l'inclusion (2.6) a lieu. En consequence, on obtient une conver6ence d'ordre O(h) d_~s que !~ solution u appartient ~ l'espace H3(£)

que l'incluslon

"minimale"

(2.9)

P2 C p

est satisfaite. Utiliaant les techniques de dualitY, d@velopp~es par Aubin [ 2] et Nitsche [ 30] , on peut montrer que si l'inclusion (2.6) eat v~rifi~e avec un entier k > 3~ on a Uu-uh

Ito,q

= O(hZ),

U-U h U 2,£ mais ce r~sultat est ~tabli en supposant que la fronti~re de £ est suffismment r~guli~re pour que la solution u soit dans l'espace H~(£) pour tout second membre f~

L2(~) et qu'il existe une in~alit~ du type ~u 114,.£ ~ C lJf ~0,£ pour tout

f ~ L2(£); voir l~e~as [ 29, Th~or~ma 2.2, page 216]. Examinons maintenant un certain nombre d'exemples. Dana les figures, on utilise les notations suivantes pour lea degr~s de libert~ :

162

•

u°

(9

I

" o,1.

d~riv~e normale au milieu du cSt~.

ax~y Exemple i ( c f . Figure 1). Cet ~l~ment, que nous appellerons "~l~ment ~ 21 de~r~s de l i b e r t Y " , est apparu en 1968, simultan~ment dans au molns s i x p u b l i c a t i o n s ; v o i r

Figure i ce sujet Zienkiewicz [ 38, page 209] , et Zl~mal [ hO] o~ la th~orie de l'interpolation pour cet ~l~ment a &t~ faite pour la premiere lois. L'espace P ~tant ici l'espace P5, de dimension 21, on obtient donc llu-uh II ~ C h~lui6,n , en supposant que u 6 H 6 (~). A partir de cet ~l~ment, on peut construire un "~l~ment ~ 18 de~r~s de libertY" (cf. Figure 2), qui est ~galement apparu en 1968 (voir £ ce sujet Zienkiewicz [38, page 209] ). L'espace P, de dimension 18, es% form~ par les polynSmes de degr~ 5 pour lesquels la d~riv~e normale le long de chaque cSt~ du triangle est un polynSme (d'une variable) de degr~ 3. Darts ce cas, on a donc les inclusions P~ c p c P5, de sorte que, si u E H5(~),

,u-h, ~ c h31uls,n

163

Fi6~r e 2 La thgorie de l'interpolation pour cet ~l~ment a ~t~ faite par Bramble & Zl~mal [ 9] = Exemple 2 (cf. Figure 3). Cet @l~ment a ~t@ introduit par Bogner, Fox &

Figure 3 Schmit [ 7] ; il ne s'applique qu'K des plaques dont les cSt~s sont parall~les aux axes de coordonn~es. L'espace P, de dimension 16, est form~ des polynSmes du type p(x,y) =

~ ~i~xiy j, Ogi ,j~3

c'est-~-dire qui sont de degr~ 3 par rapport ~ chacune des variables. De l'inclusion

164

P3 C P, on d~duit que, si u q H~(~), ~u-u h

les majorations locales pouvant ~tre d~duites par exemple des r~sultats de Ciarlet

& ~aviart [l~l. Exemple 3 ( o f . F i g u r e h ) . Cet ~ l ~ n e n t , a p p e l ~ ~l~mem~ de Clou6h e t T o c h e r , du nom de s e s i n v e n t e u r s [ 18] , e s t t r ~ s i n g ~ n i e u x . I 1 p r ~ s e n t e l ' a v a n t a g e

de r ~ d u i r e l a

Fi6ure d i m e n s i o n de l ' e s p a c e

P t o u t en r e s t a n t

eonforme ; c f . l a d i s c u s s i o n au d ~ b u t du §3.

L ' ~ l ~ m e n t f i n i K e s t e n c o r e un t r i a n g l e , triangles

Ki, et ltespace

Pest

sur K dont les restrictions

q u i e s t lui-m~me l a r ~ u n i o n de t r o i s

form~ des f o n c t i o n s une f o i s c o n t i n ~ m e n t d ~ r i v a b l e s

K. s o n t des polynSmes de 1 d e g r ~ 3. Une f o n c t i o n de P e s t done d ~ f l n i e p a r 30 p a r a m ~ t r e s q u i s o n t eux-m~mes obtenus ~ partlr

~ chacun des t r o i s

des 21 ~ q u a t i o n s r ~ s u l t a n t

e t des 9 ~ q u a t l o n s que l ' o n o b t i e n t ainsi

que s e s d ~ r i v ~ e s p a r t i e l l e s

m a t r i c e du syst~me l i n ~ a l r e du p o i n t a ~ l ' i n t ~ r i e u r t h ~ o r i e de t ' i n t e r p o l a t i o n si ue

triangles

de l a donn~e des 12 d e g r ~ s de l i b e r t ~

en ~ e r i v a n t que l a f o n c t i o n e s t c o n t i n u e s u r K,

p r e m i e r e s . On m o n t r e ( c f . C i a r l e t

correspondant est r~guli~re,

de l ' ~ l ~ m e n t , r~sulte

[ 13] ) que l a

q u e l l e que s o i t

la position

e t q u e , moyennant q u e l q u e s p r e c a u t i o n s ,

e n c o r e de l ' i n c l u s i o n

PS C p . On o b t i e n t

la

ainsi,

H~(n), ~u-u h L ' o r d r e a s y m p t o t i q u e de c o n v e r g e n c e e s t done i d e n t i q u e ~ e e l u i de l ' ~ l ~ m e n t de

i ' E x e m p l e 2, avec l ' a v a n t a g e

de p o u v o i r m a i n t e n a n t c o n s i d Q r e r des domaines p o l y g o n a u x

q u e l c onques. De m~me qu'on passe de l'~l~ment ~ 21 degr~s de libert~ ~ l'~l~ment ~ 18 degr~s

165

de libert~ (cf, Exemple i), de m~me peut-on r~duire de 3 le n ~ b r e de degr~s de libert~ de l'~l~ment de Clough et Tocher (ici encore, les d~riv~es normales aux milieux des cSt~.s) en assujetissant la d~riv~e normale le long de chaque cSt~ du triangle ~tre une fonctlon lin~aire (d'une variable), ce qui conduit ~ un espace P de dimension 9 v~rifiant l'inclusion P2 C p, et donc ~ une erreur en O(h) si u E HS(£).

•

Exemple 4 (cf. Figure 5). L'~l~me~t K, que nous appellercns ~l~ment de Fraei,~s de Veubeke et Sander [ 20,34] , est ici un quadrilat~re convexe, et sa conception

a l

a2

aB

~4

rel~ve de la re@me idle que celle de l'~l~ment de Clough et Tother. Suivant les notations de la Figure 5 on note K I le triangle de sommets ala 2 e t a 4 et K 2 le triangle de sommets ala 2 e t a 3. On pose

R~ -- {p ~ c1(K)~ p : 0 s = ~Kl, p ~ P3 sur K~}, R 2 = {p E CI(K); p = 0 sur CK2, p E PS sur K2} , les compl&nentaires

~tant pris par rapport ~ l'~l~ment K. A!ors l'espace P, de

dimension 16, est la s o ~ e directe de P3 et des deux espaces R I e t R 2. Ciavaldlni & N~d~lec [ 17] ont r~ce~aent fair l'analyse de cet ~l~ment : compte-tenu de l'inciusion PS C p, on obtient encore, si u e H~(9),

Liu-uh n ~ c h21u]~,a.

°

D'autres ~l~ments finis conformes sont ~galement employ~s: par exemple, on peut "ajouter" ~ des espaces de polynSmes des fonctions "singuli~res" judicieusement choisies. Ces fonctions sont singuli~res en ce sens que certaines de leurs d~riv@es, secondes par exemple, sont non born~es. Pour pouvoir appliquer les majorations du type (2.7) avec m = 2, ii faudra donc d'abord s'assurer que l'inclusion P C H2(K) est satisfalte. On trouvera des exemples de tels ~l~ments page 199 du livre de Zien-

166

klewicz [ 38] , ainsi que dans Birkhoff & Mansfield [ 6] o~ se trouve

de

sur-

croit faite une th~orie de 1'interpolation "locale". 3. METHODES NON CO~FORMES Les ~l~ments conformes d ~ e r i t s c a r l a d i m e n s i o n de l ' e s p a e e

Pest

dans l e §2 s o n t d i f f l c i l e s relativement

~ m e t t r e en o e u v r e

~lev~e et la structure

de l t e s p a e e P

est parfois compliqu~e (cf. Exemples 3 et 4). Naturellement, ces difficult~s r~sultent de la n~cessit~ d'avoir des d~riv~es partielles premieres continues lorsqu'on passe d'un ~l~ment ~ un ~l~ment adjacent. Par ailleurs, si les ~16~nents precedents conduisent ~ des ordres asymptotiques de convergence ~lev~, c'est en supposant que la solution est plus r~guli~re qu'elle ne iIest en pratique. Nous avons vu en effet ~ la fin du §I quton avait seulement "u fi H3(R) '' si ~ est un polygone convexe. Dans ces conditions, l'incluslon minimale P2 C p slgnal~e au §2 est aussi optimale et des espaces P plus grands que P2 ne condulront pas ~ une meilleure convergence. Or il

se

trouve

que si P = P2 (le cas

ideal l) la seule fonction de classe CI(~) v~rlfiant les conditions aux limltes y

(1.10) est la fonction nulle (cf. un article ~ paraltre de A. Zen~sek). Les considerations qui precedent conduisent donc naturellement ~ la conception de m~thodes pour lesquelles 1'inclusion Vh c CI(~) n'est pas satlsfaite : effectivement, dans les deux premiers exemples que nous donnerons, on a seulement I' inclusion V h C C°(£) et le dernier exemple est encore molns "conforme" puisque cette derni~re inclusion n'est m6me pas satisfaite. D'une fa~on g~n~rale, on dit que les ~l~ments finis sont non conform.es, ou incompatibles, d~s que l'inclusion V h C V = H2(~)o n'est pas v~rifi~e. On supposera que l'inclusion V h C L2(~) a lleu, ce qui permet d'~crlre les membres de droites des relations (3.1) ci-dessous. Puisque les fonctions de V h sont localement r~guli~res, la fagon la plus naturelle de d~finir le probl~me discret associ~ ~ un sous-espace Vh d'~l~ments finis non conformes conslste ~ chercher une fonctlon uh ~ Vh qui v~rlfie

(3.l)

%(u~,v h) -- (f,vh) pour

tout

Vh e

V h,

o~, par d~finition, --

(3.2)

ah(Uh 'Vh)

KE~h

K

l'expression { "''} ~tant la m~me que celle qui intervient dans la forme bilin~aire a(.,.) donn~e en (1.3). De la mSme fagon, il nous faut d~finir une norme sur l'espace Vh; I~ encore, il est naturel de poser

(3.3)

"Vh "h = ( E

2 hI/2 Ivht2,K/ ,

167

mais encore faut-il v~rifier qu'il s'agit effectivement d'une norme, la positivit~ n'~tant pas automatique; on d~montre que c'est effectivement une norme pour les trois exemples que nous d~crivons plus loin; cf. Lascaux & Lesaint [ 27]. Des expressions (3.2) et (3.3), on d~duit que (3.h)

a n v h II~ ~ ahCVh,Vh) pour tout vh 6 Vh,

o~ la constante e = (I-~) est ind~pendmute du sous-espace Vh, et c'est d'ailieurs cette "'uniformlt~" de la Vh-elllpticit~ qui permet d'obtenir la majoration fondamentale (3.6) de l'erreur. Slgnalons que si l'on avait choisi la valeur ~ = 1 darts la forme bilin~aire a(. ,.), valeur physiquement irr~aliste mais admissible pour les m~rhodes conformes, la Vh-ellipticit~ ne serait plus n~cessairement uniforme pour les exemples que nous considgrons plus loin. Dans ce qui suit, nous eonsid~rons que la forme bilin~aire ah(.,.) et la norme II. U h que nous venons de d~finir sur l'espace Vh sont ~galement d~finies sur l'espace V o~ elles sont respectivement ~gales ~ la forme a( .,. ) et £ la norme II. I]. Une simple application de l'in~galit~ de Cauchy-Schwarz montre pour commencer qu'il existe une constante M ind~pendante de l'espace Vh teile que (3.5)

[ah(Uh,Vh)I g

M Ilu h Ith ilvh IIh pour tout Uh, vh 6 V h.

Soit ensulte vh = uh-w h un glgment queleonque du sous-espace Vh. Utilisant les relations (1.6), (3.1), (3.4) et (3.5), on obtient

[twh

"~ < % ( W h , W h) =

ah(U-Vh,Uh-V h)

+

(f,w h)

-

ah(U,W h)

M Ilu-v h IIh llwh li h + I (f,wh)-ah(U,Wh)i , de

sorte

que Wh flh ~ M flU-Vh IIh +

i(f ,~h )-%( u ,Wh)l IIw h IIh

Cette derni~re in~galit~, jointe ~ I' in~galit~ triangulaire U u-uh ilh ~ ilu-v h flh + ilWh fib' conduit ~ l'in@galit~ I(f'wh)-ah(U'Wh) I ) (3.6)

Uu-u h U h ~ C ( inf flu-v IIh +

\Vh Vh

O~ la constante C = max{1 + M

h

sup

.h Vh

"'h "h

i} est ind~pendante du sous-espace V h. Cette in~ga-

lit~, fondamentale pour l'~tude des m~thodes non conformes, est due K Strang [ 36,37]. On remarque que l'in~galit~ (3.6) g~n~ralise l'in~galit& (2.4), puisque l'expression (3.7)

Eh(U,W h) = (f,wh) - ah(U'Wh)

est nulle pour tout w h E Vh, d~s lors que l'inclusion V h C V a lieu. Compte tenu de l'expression (3.1) de la forme bilin~aire ah( .,.), on~obtient en utilisant les formules de Green (1.7) et (1.8) sur chaque ~l~ment K {~n.. et 8tK

168

dgsignant respectivement les dgriv&es normales et tangentielles !e long de la fronti~re 8K de l'~l~ment K):

~(u,~) : ~(U,Wh ) + ~(U,Wh) ,

(3.8) a~e c

(3-9)

El~(U,Wh) =

~ ~.~K{~Au

wh + (l-u) 8nKS-----~K~tK }dy, ~Zu

~wa

~)2U ~)wh

o les formes bilin~aires Eh(U,W h) et ~ ( u , w h) ~tant respectivem~nt d~finies sur los espaces H4(~) x V h et H3(~) x V h. Observons clue si le sous-espace V h v~rifie l'inclusion V h C C°(~), ce qui est le (as du rectangle d'Adini et du triangle de Zienkiewicz, le terme ~(u,w h) est nul et l'hypoth~se "u q H3(~) '' est suffisante. Consid~rons le terme

inf llu-vh llh, qui intervient dans l'in~galit~ (3.6). VhEV h ~u supposant l'inclusion "minim&le" P2 C p v~rifi~e, et l'appsmtenance de la solution u ~ l'espace H3(~), on obtient, compte tenu de l'expression (3.3) de la norme I]. llh, (3.11)

inf

2vh

llu-vh H h < C hlul3,~.

Dans cos conditions, au vu de la majoration fondamentale (3.6), l'id~e naturelle est de d~montrer une in~galitg du type

(3.12)

IEh(U,Wh) ] < C h l u l 3 , n IIwh IIh pour t o u t u e ~ 3 ( n ) , Wh e Vh .

Or on a vu en (3.8), (3.9) et (3.10) que l'expression Eh(U,W h) est elle-m~me une somme du type (3.13)

~ ( u , w h) = ~

Eh,K(U,Wh),

chaque expression Eh,K(U,W h) ~tant associ~ h &u soul ~l~ment K; une telle d~composition n'est d'ailleurs pas unique et c'est pr~cis~ment du choix d'une d~composition Oudicieuse que r~sulteront les ma~orations d~sir~es; cf. l'expression (3.15). L'objectif est alors d'obtenir des majorations du type

(3.1h)

l~,K(U,Wh) I < C hluI3,K tWhl2,K pour tout u e H3(~), wh e Vh

l a majoration (3.12) en d~coulant imm~diatement. Nous allons pr~ciser sur un exemple la d~marche qui conduit ~ une majoration du type (3.14). On consid~re le terme ~ ( u , w h) correspondant au triangle de Morley (of. Exemple 7). Et~nt donn~ un triangle K et une fonction g d~finie sur la fronti~re du triangle, appelons HKg la fonction qui, sur chacun des trois cSt~s ~iK, i ~ i ~ 3, I du triangle K, est constante et ~gale ~ la valeur moyenne ~i '~SiKgaY de la fonction g sur ce cSt~, h i ~tant la longueur du cSt~ ~iK. On pout alors transformer l'expression ~ ( u , w h) de (3.10) en remarquant, suivant Lascaux et Lesaint [ 27] , que l'on

a aussi ~wh KE~h

~wh

~K

~wh

En effet, Bur un cBt~ BiK de milieu a i , la fonetion nK ~

~wh

vaut ~

(ai) puis-

~wh

que, le long de ce c~t~, la fonction ~

est un polynSme (d'une variable) de de@r~ I.

La contribution des deux int~grales curvilignes associ~s aux deux ~l~ments adjacents ce e~t~ est done nulle si ~.K est un cBt~ "int~rieur" ~ ~, et si ~.K est une pari i

~ h a i) = O, compte tenu de la deuxi~me condition aux limites, de tie de B~, alors ~--~" sorte que l'int~grale eorrespondante .F (...)dy est encore nulle. @.K 1

On est donc amen~ ~ ~valuer des expressions du type 2u

Bw.

@wh

Ah,i ,K(U,Wh) = .r~. K{-Au+(I-~) ~+--~)~t K (~nK - "K ~-~-}v.~d7 1

(3.16) On constate

alors que

(3.17)

Ah, i,K(u,wh ) = O pour tout u e H3(O), w h • PI,

(3.18)

Ah,i,K(U,Wh) = O pour tout u • e2, "h • Vh~w~

En effet, si w h • PI, la fonction ~

est constante le long d'un cSt~ ~iK du

triangle K et la relation (3.17) provient de l'invariance des fonctions constantes par l'application ~K" La v~riflcation de la relation (3.18) r~sulte de ce que ~.K(%-~K¢)d 7 = 0 s i ¢ • PI, d'apr~s la d~finition de l'application ~K' ce qui est I ~wh bien le cab pour la fonctlon ¢ = ~ , puisque W h • P2" On d~montre (cf. Ciarlet [ 12] ) le 1creme suivant, qui est la g~n~ralisatlon au cas des formes bilin~aires du Leone de Bramble-Hilbert | 8] , ~tabli pour des formes 1in,aires. l~mmg. Soit ~ un ouvert de R n de fronti~re suffisamsent r~6uli~re, soit k et deux cutlers, et soit V un espace de fonctions v~rifiant les inclusions _ p~ c V c H~+I(o). L'espace V ~tant norm~ par U • ~ + 1 , ~ ' _solt A :

H k+l (

~) x V -~ R

une forme bilin~aire continue, de norme ~IA ~ , et telle ~ue (3.19)

A(u,w) = 0 p.gur tout u • Hk+l(~), w ~ P£,

(3.20)

A(u,w) = 0 pour t0ut" u • Pk

, w • V.

Alors il existe une constante C, q u i n e d~pend que de l'ouvert ~, telle que (3.21)

IA(u,w)I ~ C HA ~ lUlk+l,R lwI£+l, ~, pour tout u • Hk+l(R), w • V.

Une lois ce lemme ~tabli, on l'applique aux expressions Ah,i, K de (3.16): on passe comme ~ l'accoutum~e de l'&l~ment "courant" K ~ un ~l&ment "de r~f~rence" sur lequel est appliqu~ le lemme, ce qui conduit ~ une majoration du type

170

(3.22)

I.'%,i,K(U,~h)l~ C h luI3,K J~hl2,K po= tout ~ ~ ZS(~), ~h ~ V h ,

le facteur h provenant des changements de variables dans les int~grales lorsqu'on passe de K K ~ , puis de ~ K int~grale sur

K. La dlfficult~ r~sultant de ce que l'on consid~re une

un cSt~ plutSt que sur tout le triangle

se traite ccmme dans

Crouzeix & Raviart [19, Lemme 3].

Les a u t r e s t e r m e s i n t e r v e n a n t darts l ' e x p r e s s i o n

~ ( u , w h) se t r a i t e n t

de faqon

a n a l o g u e , avec ~ v e n t u e l l e m e n t des m o d i f i c a t i o n s " t e c h n i q u e s " , e t c e l a a u s s i b i e n pour l e t r i a n g l e de Morley que pour l e t r i a n g l e

de Z i e n k i e w i c z ou l e r e c t a n g l e d ' A d i n i

(of. les exemples). Si nous raisons la somme de routes les relations (3.18) sur tousles ~l~ments finis K, nous end~duisons que (3.23)

Eh(u,wh) = 0 pour tout u e P2, w h E Vh,

or cette derni~re relation n'est autre que le c~l~bre "patch test" des Ing~nieurs, trouv~ empiriquement par Irons [ 22] comme une condition n~cessaire de "convergence" des m~thodes d'~l~ments finis non conformes. Nous nous sommes efforc~s ici pr~cis~merit de faire appara~tre cette n~cessit~ en consid~rant le patch test 'comme l'une des de ux invariances pol[nomi~es n~cessaires ~gur °btenir une majoration du t ~ e

(3.14),

une lois connu le lemme ~noncg plus haut. Examinons maintenant quelques exemples. Pour chacun des sous-espaces V h correspondant aux trois exemples qui suivent, les conditions aux limites (I.I0) sont prises en compte de la fagon la plus simple: tousles degr~s de libert~ sont nuxs lorsqu'ils correspondent ~ des noeuds situ~s sur la fronti~re. On constatera par ailleurs dans le cas du triangle de Zienkiewicz que le patch test se traduit par une restriction sur la g~om~trie des ~l~ments. Exemp1e 5 (cf. Figure 6). Cet ~l~ment, que 1'on appelle "trlangle de Zienkiewicz"

Figure 6

171

a ~t~ introduit dans Bazeley, Cheung, Irons & Zienkiewicz [ 5] • L'espace P e s t icl f o 1 ~ des polyn6mes p E PS pour lesquels

6p(a) - 2 I p(ai) ÷ I i=l O~ les points ai, 1 ~ i ~ 3, e t a

-- 0,

i=l sont respectivement les sommets et le barycentre du

triangle. L'espace P v~rifie les inclusions P2 C p C Ps,et sa dimension est 9. Lascaux et Lesaint { 27] ont montr~ que le patch test est v~rifi~ si et settlement si tousles c~t~s de tousles triangles d'une triangulation donn~e sont parall~les seulement trois direction, et qu'alors si u E HS(~),

,u-~ "h < C b lu13,~, ce qui est la solution du "probl~me de l'Union Jack" (cf, Zienkiewicz [ 38, pp. 188189] ). Par ailleurs, le fair que la valeur au centre de gravit~ ne soit plus un degr~ de libel~.' (cf. la relation (3.24)), condition ~galement trouv~e par les Ing~nleurs comme une condition empirique de convergence, volt ici sa justification d&ns le fait que la fonction de base correspondant au barycentre "ne passe pas" le

patch test.

•

E_xemple 6. (cf. Figure 7). Cet @l@ment, que l'on appelle "rectangle d'Adini",

()

x

Figure 7 a ~t~ introduit dans Adini & Clough [ i] • L'espace P consiste en tousles polynSmes du type "

p(x,y) = ~isxY 3 + ~slxSy +

~ijxlY J , O~i+j 0 est ehoisie de faqon ~ obtenir le meilleur ordre de convergence. Exemple 8 (cf. Figure 9). Le point a es% ie barycentre du triangle, et l'espaee Pest

l'espaee P3 , dont la dimension est i0. Cet ~l~ment a ~t~ consid~r~ par

Fi6ure

9

174

Babu~ka & Zl~mal [ 4]. Avec le choix , = 0 dans la forme billn~aire, ils ont montr~ que au-u h ilh ~ C h 1/2 gu Ji3,n

(3.30)

pour le ehoix optimal £(h) -- C h 2. Du fair de la p~nalisation, il n'y a plus de patch test ~ passer, ce qui permet de conserver la valeur au baryeentre comme degr~ de libertY, K l'inverse de ce qui se passait pour le triangle de Zienkiewicz (cf. Exemple 5). REF~ENCES [ i] Adini, A.; Clough, R.W. : Analysis of plate bending by the finite element method, NSF R e p o ~ G. 7337, 1961. | 2] Aubln, J.P. : Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by @alerkin's and finite difference methods, Ann. Scuo! a Norm. Sup. Pisa 21 (1967), 599-637. [ 3] Babu~ka, I.; Aziz, A.K. : Survey Lectures on the Mathematical Foundations of the Finite Element Method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differentiai Equations (A.K. Aziz, Editor), pp. 3-359, Academic Press, New York, 1972. [ ~] Babu~ka, I.; Zl~mal, M.: Nonconforming elements in the finite element method Technical Note BN-729, University of Maryland, College Park, 1972. [5] Bazeley, G.P.; Cheung, Y.K.; Irons, B.M.; Zienkiewicz, O.C. : Triangular elements in bending-conformlng and nonconforming solutions, Co._nferenee on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. [ 6] Birkhoff, G.; Mansfield, L. : Compatible triangular finite elements, J. Math. Anal. Appl., ~ para~tre. [ 7] Bogner, F.K.; Fox, R.L.; Schmit, L.A. : The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas, Conference on Matrix Methods in Structura I Mechanics, Wright Patterson A.F.B., Ohio 1965. [8] Bramble, J.H.; Hilbert, S.H. : Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1971), 362-369. [ 9] Bramble, J.H.; Zl~mal, M. : Triangular Elements in the finite element method, Math. Comp.2~ (1970), 809-820. [ i0] Brezzi, F. : Sur la m~thode des ~16~ments finis hybrides pour le probl~me biharmonique, ~ paraltre. | ii] Ciarlet, P.G. ; Orders of convergence in finite element methods, The Mathematics of Finite Elements and Applications (J.R. Whiteman, Editor~,pp. 113-129, Academic Press, London, 1973. | 12] Ciarlet, P.G. : Conforming and nonconforming finite element methods for solving the plate problem, Conference on the Numerical Solution of Differential Equ ations, University o-f Dundee, July 03-06~ 1973. | 13] Ciarlet, P.G.; Sur l'~l~ment de Clough et Tother, K para~tre. lh| Ciarlet, P.G. ; Raviart, P.-A. : General Lagrange and Hermite interpolation in R n with applications to finite element methods, Arch. Rational Mech. Anal.

~6 (1972), 177-199. | 15] Ciarlet, P.G. ; Raviart, P.-A. : Interpolation theory over curved elements, with applications to finite element methods, Computer Meth. in Appl. Mech. and Engnrg i (1972), 217-249.

175

[ 16] Ciarlet, P.G.; Raviart, P.-A. : A nonconforming method for the plate problem, para~tre. [17] Ciavaldini, J.F.; Ngd~lec, J.C. : ~ para~tre. [18] Clough, R.W.; Tocher, J.L. : Finite element stiffness matrices for analysis of plate in bending, Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. [ 19] Crouzeix, M. ; Raviart, P.-A. : Conforming and nonconforming finite element methods for solving the stationary Stokes equations, I, ~ para~tre. [ 20] Fraeijs de Veubeke, B. : Bending and stretching of plates, Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. [21] Glowinski, R. : Approximations externes, par ~l~ments finis de Lagrange d'ordre un et deux, du probl~me de Dirichlet pour l'op~rateur biharmonique. M~thode it~rative de r@solution des probl~mes approch~s, Conference on Numerical Ana].ysis, Royal Irish Academy, 1972. [ 22] Irons, B.M.; Razzaque, Ao : Experience with the patch test for convergence of finite elements, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, Editor ,~---pp. 557-587, Academic Press, New York, 1972. [ 23] Johnson, C. : On the convergence of some mixed finite element methods in plate bending problems, K para~tre. [ 24] Johnson, C. : Convergence of another mixed finite-element method for plate bending problems, Report No. 27, Department of Mathematics, Chalmers Institute 0f Techn°l°$7 and the Universit[ of GStebor6, 1972. [ 25] Eondrat'ev, V.A. : Boundary value problems for elliptic equations in domains with conical or angular points, Trud[ Mosk. Mat. Ob~6". 16 (1967), 209-292. [ 261 Landau, L.; Lifchitz, E. : Th~orie de l'Elasticit~, Mir, Moscou, 1967. [ 27] Lascaux, P.; Lesaint, P. : Convergence de certains ~l~ments finis non conformes pour le probl~me de la flexion des plaques minces, ~ para~tre. [ 28] Morley, L.S.D. : The triangular equilibrium element in the solution of plate bending problems, Aero. Quart. 19 (1968), 149-169. [ 29] Ne~as, J. : Les M~thodes Directes en Th~orie des Equations Elliptiques, Masson, Paris, 1967. [ 30] Nitsche, J. : Ein Kriterium f~r die quasi-optimalit~t des Ritzchen Verfahrens, Numer. Math. ii (1968), 346-348. [ 31] Oden, J.T., Some contributions to the mathematical theory of mixed finite element approximations, Tok_T_~Seminar on Finite Elements, 1973. [ 32] Plan, T.H.H. : Finite element formulation by variational principles with relaxed continuity requirements, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equat~0ns (A.K. Aziz,Editor), pp. 671-687, Academic Press, New York, 1972. [ 33] Raviart, P.-A. : M~thode des El~ments Finis, Universit~ de Paris Vl, Paris,1972. [3hl Sander, C. : Bornes sup~rieures et inf~rieures dans l'analyse matrlcielle des plaques en flexion-torsion, Bull. Soc. Ro)-. Sci. L i ~ e 33 (i96h), 456-hgh. [ 35] Strang, G. : Approximation in the finite element method, Numer. Math. 19 (1972), 81-98. [ 36] S~rang, G. : Variational Crimes in the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differentlal Equations (A.K. Aziz, Editor), pp. 689-710, Academic Press, New York, 1972. [ 37] Strang, G.; Fix, G. : An Anal[sis of the Finite Element Method, Prentice-Ball, Englewood Cliffs, 1973.

176

[ 38] Zienkiewicz, 0.C. : _The Finite 'Element Meth0d in En~ineerin~ Science, McGravHill, London~ 1971. [ 39] Zienkiewicz, 0.C. : Constrained variational principles and penalty function methods in finite element analysis, Conference on the Numerical Solution of Differential Ecr0~tions, University of Dundee, July 03-06, 1973. [ ~0] Zl~nal, M. : On the finite element method, Numer. Math. 12 (1968), 39~-~09.

S

B'r,~ce M. I r c ~

Reader, University of Wales, Swansea.

The general purpose thin shell element described here is essentially a nonconforming quadratic Ahmad element, with ~ o r d e r nodal parameters which accept multiple junctions.

integraticm, and with A historical introductiom

clarifle~ the develDpmemt and leads smoothly into the fcrmalati~, and the diacus~iem of patch-test convergence.

The geametry and the shell theory are new

and are ccmsistent with rigid body responses of a general patch.

~plementatlon

will be via a foolproof shape function routine. Introduction et ~oti~ation "Semiloof" est prohablement la derni~re phase de la recherche de l'auteur pour un ~l~ment de coques~)une recherche motiv~e par un m~contentement profon~ envers presque tousles ~l~ments de toques familiers.

La liste suivante des conditions

requisea abr~ge cette m~fiance: (i)

II f~ut un ~l~ment ~e raideur direct, car heauccup de millions de dollars

sent d~j~ perdus sur les progranm~s d'ordinateur qui admettent tels ~l~mauts meulement. (ii)

L'~l~ment dolt ~tre utilisable ("mixable" en anglais) - au s e ~ exact de

l'~preuve de rapi~gage (q) ("patch teat" an anglais) - avec lea ~l~ments voisins tris~agulai~s ou quadrilatSraux, ~

avee lea membranes isoparam~triques, et avee u,

cerZain ~L~ment eonvenable de poutre~ (iii)

Lea ~l~ments de toutes formes ne violent Jamais lea mouvemeats rig%des, J

J

~insl que taas lee assemblages de tels elements.

Sewiloof est le premier quadrilat~re de !a famille Ahmad qui a eatisfait *

.

J

nln~erlcuement aux epreuvea (ii) et (iii).

.J

I

J

Lee raplecagea d' elements

triangulaires et quadrllateraux Imbr!ques ont r~ussi aux essais d'ordinateur. Um ~l~.ment de poutre tres special real~se" " r~cemment _oar F. Albuquerque, de Louren~o Marques.

178

(iv) La plupart des coques technologiques ont des angles vifs et des embranchement~ ~Lultiples. Dc~e, un ~l~ent dolt modeler ceux-ci, sans exceptions pathologiques, et sans complications telles que transformaticas locales etc. (v)

Quelque soit le module choisi, on doit le traduire par l'interm~dialre d'une

routine de fonctions de forme nolre"

("shape function routine" en anglais), une "boite

('black hcx" en anglai~) de laquelle l'utilisateur ne dolt l~aS se

acueier et ~

laquelle se trouvent toutes los complications telles que

transformations et conventions de signe. De cette fa~on le chercheur peut garantlr fermement ~ e l'utillsateur pourra incorporer facilement l'~l~ment dane /

.

sea progran~ne, et que toutes les caracterlstiques matricielles exotiques de l'~l~ent, dcnt c~ aura besoin plus tard - In~vltahlement ~ ccttrt d~lal pou~t

Wo~e:

~tre aJout~es sans peine.

Si les ~l~ments iso-P sc~t touJours en vogue, e'e~ en grande pattie que

la routine des fcnctions de forme est cr~e, v~rifi~e et bien documaut~e~ et que tout le monde peut lui donner des t~ehes multiples.

Nous euvisageons maintenant

une routine plu~ ccmpliqu~e, mais conservant ces avantages. (vi)

La routine de~ fonctions de forme doit rester abordable au Programmeur de

maintenance grace ~ une co~e lucide, une documentation complete, et des ordres d' i~pression intermediairs facultatifs. (vii) Toute routine dolt %ire convenable et ~ r ~ l a b l e .

Par exemple le

proe~d~ se change souvent ~ la premiere rencontre avec un nouvel ~l~ment: ll faut que l'utilisateur ne doive jamais l'en avertir. (viii) Encore c'est un bienfait de fournlr ~ l'utilisateur des dia~lostics abondants~ L'endroit naturel pour bien d'entre eux est la routine dee fonctions de forme. (i~)

Ii faut que l'~l~ment ne montre auoun caprice en fonctionnement@

Par

°~

exBmple i! dolt ~iter tousles pleges tels que rang defectueux. (X)

Use pr&oision extreme n' est jamais requise, mais un mailla~e ~oseier

dolt donner des r~su!tats acceptables.

Solon le consensus general, lee

contraintes qui varient lin~airement satisfont le mieux lee besoins teehnologique s~

179 Di_~position des Noeuds Semiloof

II es% di~ficile de critiquer la disposition de la ~g.1 du poimt ae rue de l'utilisateur - elle a semBl@ la plus attirante parmi celles examlm~es ~ laref. q.

Aux coins et ~ mi-c~tes^ " nous avons u, v, w, les fl~che~ an directions

globales.

Cecl suffirera pour un~ membrane, mais ne peut pas emp~cher le

pivote~ent pour time coqme avec flexion.

Afin d'assurer une conformit~

approximative pour les pentes, nous Introdui~ns les rotations aux points de J

G~ss~ le long de chaque cSte.

Les 32 degr~s de libert~ devraient suffire

d~terminer lem champs de contraintes lin&aires, ~ la lois pour les actians de flexion et de membrane.

Pour la logique du programme nc~s groupons les 5

vaz@ables le long d'un cSt~ ccmme si elles agissaient en son milieu.

La

convention de signe - ~ laquelle l'utilisateur s'interesse rarement - d~pend des num~ros de noeuds des coins voisins, disons N I , N 2.

Ceux-ci ne sont

jamals ~gaux, car il me rapportent ~ des cordonn~es nodales.

Disons N2> Nq.

Le sens des rotations aux points c et d est fix~ par la r~gle de la main droite pour le segment orient~ Nq-~ N2, c pr~c~dant d.

wia ~es~?a2:::t~eco~ ~i~,1 Pr~curseurs de Semiloof

.

On peut resoudre~ une probl~me 2D avec noeuds de Loof

~ng1~is)s~tues .

("Loof nodes" en

. . . atux ~eu~x .points ae Gauss. sur chaque cote d'un el~ment, comm~

indiqu~ ~ la fig.2.

I1 faut ajouter un 9me noeud au centre de l'~l~ment: la

recherche de~ fonctions de forme est donn~e a la ref. 7 f ~

A cause des noeuds

manquants a1~x coins, on ne peut mSme pas assurer C ~0) entre lee elements. z

180

•

•

.

J

P~rtant, le dlscontl~11te tend vers P2(~), la polynSme de Legendre qui s'annule aux deux points de Gauss (ici ~ arpente le cote, -I~ ~ ~ I ) dent l'int~grale et premier moment sent nuls.

Nous examinons l'~preuve de rapid,age, les contraintes

~tantmaintenues constantes.

Touteperturbatic~de

cet ~tat unlforme provoquera

des disconti~uit~s de d~placement entre les ~l~ments. comme P2(~) le travail r~sultant est nul. suffise, I e ~ u v e

Si les sauts varient

Par consequent, po~u~u que Is rang

de rapze~age est satisfaite.

Not__._~e. Nous raisons honneur a Leer.

A la ref. ll il mentionna des points de

Lobatto mais il parla de son dessein d'ess~yer lee points de Gauss ouand l'occasion se pr~senterait.

~. ,.,

2.

.

3

(l';,~

L element de Visse

(fig.3), parmi d'autres, qui postulent une variation A

i

lin~aire de moment de flexion sur chaque cote.

Nous pouvons f ~ e r

sans

peine de tels ~l~nents en termes de moments localis~s: M = (moment/cm.). d(c~t~)/d

(I)

aux deux points de Gauss, en sorte que tout travail est l'addition de M fois la rotation aux deux neeuds de Leer.

Ensuite une inversion partielle (~) ("part-

invermion" en anglais) donnerait preclsement " " ~ la version "eq~ivalente en raideur directe avec rotations de Leer.

3.

La membrane isoparametrique de fig.~ adapte lea /

fonctions de forme N.(~,~) qui etabllssent une 1

correslx~enee (~,~)-~(x,y,z) en 3 dimensions.

Y

(2)

(z;i

Ctest tr~s simple.

.

p

p

Lea complex~tes se presentent

seulement en calculant les d~formations etc., qu'au doit exprimer en cordonn~es locales X, T darts le plan tangentiel de la membrane.

Nous arena

181

ou ~ar exemple X est le veoteur unit~ en direction X, ~ est le vecteur base covariant ~(position)/~, ainsi ~Ni/'~X,

BNi/~Y

et

.

est le produit scalaire.

Nu~s calculons

au point do~n~ ~, ~ . Les @~formations et autres

quantit~s de ce genre suivent facilement.

Par example, la contri~uti~

~W/~ X ~ cause d'un d~placement ~i au noeud i e s t •

4-

x

,_I~. toque feuillet~e d ' A ~ d

("membrane-stack" an ~

~

originairement con~ue quand les raisonnements Pi~. 5

vectoriels ~taient manifestement absents dans

Q,) Son

les manuels d'~l&ments finis, ~tait r~alis~e comme ~tude introductive. utilisation g~n~rale @tait inattendu.

Eous discutons seulement le quadrilat~re ~ 8 noeuds et 40 degr~s de liberteo Chacune des 8 lignes rigides de la fig.5 se soumet aux fleches u,v,w a la misurface du feuillet, et lie les membranes. l'intervalle

(~ , ~+d~ ),

Chaque feuille correspond

comme indiqu~ ~ la fig.5. G~om~tri~Aement noms

avons une brique iso-P 20-noeuds, mais~ x,y,z varient llnealrement aveo ~, en sorte que chaque membrane peut avoir une ~paisseur nonuniforme. Elastiquement nous n'avons pas encore une coque r@aliste.

Le~ membranes ressemblent ~ un livre

dont les pages Elissent sans contrainte.

ments

~

.

Ncus

nous proposons de les coller afin de leur donner une raideur de flexion. (i)

Ii y a deu_x techniques:

Fi~.6

Abroad ajouta l'~nergie de d~for~tion de

glissemsnt ~ mesure que ohaque membrane glisse contre les voisines. imag~ grossi~re: le glissement aux ~

s

ermines est faible ou ~

O'est une en

r~alit@, hors du cas rare o~ des charges tangentielles sensibles s'appliquent la surface. Toujc~rs nous p r ~ o n s

que le glisse~ent varie comme (~ - ~2), e%

il ~Ait que les normales courbent comme indiqu~ ~ la fig.6. Ahmad effectivement divise YXZ

et ~YX

par ~

&fin de rectifier le module dont les normales

sont rigides. Notons que ~XY' une d~formation d~ns la membrane, n'est pas modifi~e.

182

(ii)

Alternativement, pour une coque mince nous voulons annuler

YXZ

et

YyZ o

En ca~ de oertains ~l~ments d~llnquants mais comp~titifs ("delinquent elements" en anglais, avec de~ "discrete Kirchhoff assttmptions")

nc~s annulcr~

~'XZ et

/

~'YZ aux points isole~, cholsis avec soir~

Les contraintes nous permettent

d'~liminer des variables nodales convenables avant d'assembler lea ~l~ments. Note.

Les deux techniques ne ~'excluent pas.

Par exemple, en principe on pout

annuler toutes variations de glissement, afin @e rendre eelui-ci constant. Autrement, on pout le contraindre ~ varier lir~air~ment avec ~ et ~, etc.(~) 5.

,,feuillet~e

La coque

d'Ahmad avec int~Tation 2x2.

Par suite d'un examen des

r~gles d' int~gratica, Too remarqua que Gauss 2x2 dcane des r~sultats remarquable~nt amellores, pouvu qu'il pre.sente seulement los contra~ntes aux ^

ll~.

memos points 2x2.

t

/

Cette observatlon

est toujcurs inexpllquee, cepen~nt

ells a inspir~ au moins deux innovations importantes. Discutons pourquoi les r~sultats sages nous ~tonnent ~cialsment,

surtout

/

lorsque la plaque c~ la coque est mince. En tel cas l'energle de d~formaticn de glissement applique de fair 8 contralntes rigi~es, car on s~it que le~

I

o~--~

3 noeuds/element

2 glissements aux points de Gauss 2x2 sent o

reli~s lin~airement et ind~pendamment aux variables nodales. Donc examinons un mailla e raffin~ c o m e indlqu~ & la fig.7 p~ar une plaque d'Ahmad.

Chaque element ajoute 3 noeuds, et

~g.7

c h a ~ e noeud a 3 variables - une fl~che et deux pentes - de sorte que chaque J

J

element ajcute 9 variables.

Soustrayons les 8 contraintes.

Alors ncus

n'aJoutons qu'un soul degr~ de libert~ par ~l~ment de plaque suppl~mentaire. N@anmoins los rSsultats nous plaisent. 6.

Le premier ~l~ment d61inquant iso-P se sugg~ra d'une part par la sagesse

s Lmprevae de la coque feuillet~e avec •

in

~ tegration 2x2, et d'autre part par son

Inefflcacit~ technologlque intrinsiq-e.

Car le glissement entre les membranes

voisines ne connote que pour quelques pour cent de l'~nergie de d~formation I

totals.

.

s

En reallte, il imports peu qu'on l'inolue cu pas.

Pcurtant la toque

feuillet~e fournlt une variatica de glissement quadratique en ~, ~ , maim •

p

.

llnealre en d~formation de flexion. •

I

Jusq'ici nous avers trop de degr~s de liberte; mais quels 8 devons-nous omettre? La fig.8 nous donne un indice. lin~airesa

Imaginon~ que AB soit une poutre aux flexions

On pourra la d~terminer entmerement'~ ave~ la fl~che et la pente aux

deux bouts, sans los deux variables.

Nous nous demanderons sl cn pout aussi

183

%ter les variables semhlables en cam de plaque ou de coque.

• Les glissements

" ~

¢ £d

J

soit oms Fig. 8

Note.

Par consequent, pour un grand probl~me chaque ncuvel %l~ment ajoute 5

variables au lieu d'une.

Si nous averts an trouv~ 4 en sus o'est que la fl~che et

ls pente ~ ms-cote ~ " ne doivent plus se conformer entre les 61&merits. Los nouveaux degr&s de libert~ a~liorent les r~sultats, un petit peu. Razzaqae rut le premier K coder les &l@ments d&linquants iso-P, et rapporta que la precislon rut parmi los meilleures publi~es pour los elements de forme g&n~rale.

II se peut que la plaque d~linquante soit l'aboutissement de la

formulation isc-P. (')~@ )

Mais la toque d~linquante fsit d~faut en pratique. .

A

Deux variables, une fl~che

I

et une pente, disparaiss~nt &e chaque ram-cote. On dolt choisir avec soin leurs directions, et par consequent on doit choisir aussi les 3 variables restantes. ,,

+,

Razzaque a d~termin~ cos directions in~pendan~nent dans chaque element, sans ~e soucier des l~g~res inconsistanee~. On soup~onne qu'il s reussi ~ cause de la smmpllclte geometrlque de ses probl~mes.

Sans doute il n'aurai% pas obtenu de r~sultats utilisables pour des

toques ~ angles vifs et ~ embranohements ~altiples.

Note.

Los % l ~ e n t s d~iinouants, sent justement denornmes." I Dlrmges" " par !'intuition

physique plus que par route theorle de ooques admise, nous avons trouv~ un element competztlf.

Bien qu'il ne satisfasse a l'epreuve de r~aple~ge que pour

I i un maillage de parall~logr~mes (~ventuellement inegaux), los s elements difformes r~ussissent presque. (et il es%

D'ailleurs, quand nous commettons ce d~lit particulier .

J

d

diffieile de preclser en termes generaux ce o~e cola signifie

quand la glissement s'sr~uule aux points choisis), quel statut conserve-t-elle, I' ~preuve de rapid,age?

184

D&placements de Semiloof Libres de Contraintes Avant d'aVpliguer le~ contra~ntes, nous donnons le d~placement de tout point comme l'addition de trois classes: I.

• D~placements aux coins st ~

u,v,w oausent une translation des 8

" ^ •

m l - c o t e s

Nous comptons 2& degr~s de libert~

lignes rigides comme indiqu~ h la fig.5. de cette sorte. 2.

Rotations aux noeuds de loof et au centre, qui introduisent une difference de •

4

.

.

s

.

depiaoement entre les surfaces superleure et inferleure.

Nous con~ptons 18

degr~s de libert~ de plus. 3.

Une seule fonction bulle.(1-~2)(1-~ 2)

qui donue un d&placement dans la direction perpencliculaire a l' element au centre .%

(~ = V = 0), et qui s'annule aux front~eres. de satisfaire ~ l'~preuve de rapi&oage.

(fig.9)

•

Ce supplement permit

Nous devons fournir tout ~tat de

flexion constants dans un element, en general quadrilateral et plat, disons

ou Ni( ~ , ~ ) sont les fonctions de forme de coins et de m~-cotes.

~is

is

quadrilat~re n'exige pas touts la base, et par exemple x

=

A + B~

+ C?

+ D~ •

I

Si nous ajoutons le 43me degr~ de liberte, la fonction bulls, c'est cue 2 2 . A la ref.~ nous avons manqu~ cette w = x renferme le terms ~2 condition requise. Semiloof st les Contraintes de Glissement Nous pcursuivons en r~duisant les 43 degree de libert~ aux 32 de la fig.1. t " (~ la suite de plusieurs qui echouerent) se Les 11 contraintes qui ont reuss~ groupent naturellement en trois classes: I.

~

Les 8 pentes~ Indeslrees~ ~ i mlx noeuds de L o of. La rotation

de la normele octane indiqu~ ~ la fig.t0 engendre un glissement ~YZ"

~

~

~

Done il est naturel de contraindre la pente a v e c l a

condition ~YZ -- 0.

~ig.10

185

Note.

Lea 8 glissements aux noeuds de Loof sont reli6s h peu pros via la

matrice identlt6 aux 8 rotations ela~namees. dire¢:te, leg condition~

~XZ = ~YZ = 0

En outre, l'alternative la plus

aux points de Gauss 2x2 ne sont m~eme

paa ~md6pendant as. .p

I

L'~preuve cle rap~eoa~e. Consid~ons encore l'inflmence de l'epreuve sur la feu&lle au niveau ~

~

~+ d ~.

Perturhons les noeuds "int~rieurs"

(c-a-d cerne~ ~ completement • d' "elements ~ - se reporter ~ la ref.~po559,

583)

et demandons-nous si le travail virtual g'annule dang un champ de contraintes planes uniformes.

.J

Aux noeudg @e Loof A e t B ~ la rue en plan du raplecage, 16

fig.t1, lea fl&ches~ normalea a RS sont continues

~

.

.

]

A

.

.

.

entre lea ~l~mm2~ts voisins ~ cause des pentes nodales costumes.

Si leg fl~ohes dens la

directicm de AB se~t ~galement continues, c'est qua WR, Wset wll, lea fl~chea aux coitus et ~ mi-cSt6, moat communes., e~ qme le glissemen%

Fifo11

~(YZ s'annule ~ cause des 8 contraiutes

~anposees. Par ccms~quent nous devoms satisfaire l'&preuve de rapxegage quelques soiemt lea 3 derni~res contrai~teg qu'il nc~s plalse d'imposer. 2.

Lea 2 pentes au centre.

a lea glissemen%s lat~reux

Observation pr61imlnalre. ~(XZ' ~YZ"

Une plaque, en plan XY,

Or, poar Z fixe, e% X,Y tournant, la

quart%it~ bidimensionnelle

me %ransforme cOnm~ un vecteur. Lea 2 contraintes. et n o u a

Noua z~s~rvons les vecteurs unit~s au centre, X 9 et Yg'

fixtures x9 .

e(gupe

±cie)

employant lea points de Gauss 2x2.

2 points de Gauss. Exact. Note.

Pr~lable~ent une version plus simple

l'~preuve de rapid,age. con%raintes ncuuniformes~

S~

4

mi-cgte. Trop souple.

a ~chou~, malgr6 sa reussite

Ella se d6formait trop souplemen% d~ns leg champs de Au lieu des deux in%~Erale8 de (7), nous avons

contrain% lea deux glissemen%s au centre.

D'une mani~re semblable l'~l~ment

quadratique de poatre indiqu~ ~ la fig.12 est %rop souple avec une seule ccmtralnte.

{

186

3.

La font%ion bulle

es~ ~videmment fiche d'une cer%aine c~nbinaison ~e

glissement=:

est ~me e o n t r a ~ t e cr~dibleo

•

/

•

Nou~ ~ceferons transformer l ' ~ t ~ g ~ a l e a

(gllssemen% normal) d(arc)

=

0

(9)

frontiere e% oelle-oi es% la forme employ~o G~ometrzqaes~

S em~loqf

nSemiloof= est un module de coque mince, et nous faiscms les supposi%ions I

/

geome%ri~ues oz~dihles c o ~ e suit. (i)

Pour commemmer, nous ~enclons approximativement normales ~ la surface

moyenne du feuillet les lignes nodales rigides de la fig.5. J

Par conqueror, lea

I

el~ments ne m'imbriquent en general pas aux embranchemen%s. (ii)

I

J

La coque es± si mince que l'~lement de surface d(superficie) pour route

membrane peut ~tre confondu avec am valeur sur la mi-surface. (ill)

Le~ membranes sont presque parall~les.

L'&preuve de raple~age a aeja admis %ous les quadrilat~res pla%es.

Un autre

o

prinoipe aomine ~os main%enan%: desormals nous exigerc~s qu'tm rapie~age d'elements de tou%es formes ne viole j ~ i s

les mouvements rigides,

%

nous avons fair les hypotheses sans assez de soins,

A Is ref.1

No%ons:

(a)

Les &l~men%s strictement iso-P pourvoien% les mouvements rigides,

(b)

T~/te ligne rigide - comme on la ooque feuillet~e d'Ahmad - es% tu~e

contr~mte d'accord avec un mouvement rigide. (C)

Un couple d'~lements fournlrait un mouvement rigide seulement sl les

variables nodales qui les llent ensemble (en particulier les pentes)

se

d&finissent ex~ctemen% pour %c~% mouvement rigide donn~. Il faut que route lizne rigide ~ ,~?= constante soit normale, ~ chaque noeud de Loof, ~ la tangente frontlere.

Autrement un mouvement de rotation ~

normal

la surface moyenne, con~ne lux~iqu~ ~ la fig.~3, provoquerait tune "rota%ion"

187 A

f a u s ~ autour de Sj.

Si nous devons

@,

/% K

~ ) ~ -" ccastante

compenser les ~paisseurs dorm's, c'est que eette ccr~/tion n' eat pas assuree automatiquement.

Quand nc~s rencontroms

um nou~el ~l~nent, nous devons remplir

FIR.i

plusieurs t~ches: Stade !.

Nous intez%oolcns les ~paisseur~ Ti, i = ~ a 8, donn~es aux coins e%

mi-c~te~, afin d'emgendrer los Tj, j -- ~ ~ 9, aux noeuds de Loof et au centre. Nous ~levons ~j (la premiere ~ apprccch~ation) perpendiculaire ~ la mi-surface. St ado 2"

A f ~ d'assurer que le~ Tj, j = I a 9, d~terminent une base

geometr~que darts l'espace de fonctions de coins et de m~-cotes, " " " ^ "

N~(

~, ~ ),

i -- ~ ~ 8, il faut imposer une condltion aux ~pai~seurs Tj, c'est ~ dire

• j : ~j

multiplions par (-I) j

(-~

(~-p.sjsj)

(9)

et l ' a d d i t i ~ o ~ :

i

I

- p.sj sj) = 0

(io)

Trmduisc~ en f o z ~ ~tricielle:

~e~ucc~p de techniques sont possibles: celle-ci emt imdereglable, c ~ 3x~ e~.~ d~inie positive,

la matrice

N o t o ~ que les ajusteme~ts f o r ~ s sont de m~eme ordre

~e grandeur que les variations ~' ~paisseur normale T d ~ s un ~l~ment iso-P dont lea T i sont co~.~t~ts aux c o i ~ et mi-cotes~ Puis nous construisons et conservons les d~placements relatifs

~. = ~. x ^

qui engendrent la pente ur~ite conm~_ue avec les el~nents voislns: aussi m.O : qul domnerait une pente approximativement unitaira le long du cote, J . . . une variable destin~e pour l' elLm2matlon. Stade ~.

Nous constrtLisons et r~luisons la matrice des contraintes.

Sta~e ~.

Nous nous sermons de cette matrice-ci pour creer les fonctions de

#

¢

I

force a l'endroit donne ~, 7. ~auf dans le cas d'un nouvel element, nous sauto~e directement au 4me ~tade.

0

188

Le~ ~esures @e D~formation pour une Coqae Peuillet~e Ninoe Ccmme dans~ les theories clasmiques de coques, notre but est simplifier le calcul ~anm introduire des inexaetit~des excesmives+

Nc~s employons les symboles

suivants: A

Z

=

normale unit~ au plan tangent ~ la surface moyenne.

/%

=

/ / S nermale dirig~e vers i t emtezaeur ~ana le plan tangent ~ I' element ~ la

X

.

fronti~re. =

d~placament ~ la surface moyemae. i

•

.

,p

•

deplacement relatif entre les surfaces superieure et inferzeure, tel que ~

rj Ni, Lj Ainsl T

=

= &

au meme point ~ i +V•

S j,

s

8 fonctio~s ae forme de coins et @e mi-cb~es, e% 9 foneti~ms ~e Loaf. = INid i

et

~

=ILj

j,

i= I ~ 8

et

j =I ~9.

vecteur d'~pais~eur

: ~ +~+__ :

~

+ s~ + ~ ,



-~

-w

->

~ a(v,~)+b(u)v,?)+b(v,u)T)-(x,div (4.20) has

a uniRue

solution

: P r o b l e m ( 4 . 2 0 ) is e q u i v a l e n t a function ~V such that

(4,21)

-+

->

.+

v)=(v,g)

for all v ¢ X

(~,X)EV x L2(O.)~

to the followinF~ one

~ a(v)?)+b(u)v)~)+b(v)u)~)=(v)g)

:

for all v ~ V .

Since a(v,v)+b(u,v,v)+b(v,u)v)= >~

~ a(v,v)+b(v)u)v)

( ~ -~]l~)II~] 2 >s v ~I]~ 2

e x i s t e n c e and u n i q u e n e s s L a x - M i l g r a m theorem.

of the

>p

for a l l - @ E V)

solution

~

V follow

from

the

We shall also need the f o l l o w i n g r e g u l a r i t y p r o p e r t y for the solution (~,X) of p r o b l e m (4.20) : there exists a c o n s t a n t A>O w h i c h does not depend on ~ such that :

ll~)r2)+ IXll, ~ ^ll~Ilo,~

(4,.22)

In fact) by u s i n g t e c h n i q u e s of K o n d r a t i e v [ 8 ] and G r i s v a r d [ 7], one can prove that this p r o p e r t y (4.22) holds p r o v i d e d t h e ~ p o l y h e d r a l d o m a i n Q is c o n E e x and the s o l u t i o n ~ is smooth enough (for example ~ K (CI(~))s). T h e o r e m 4 : Assume that H y p o t h e s i s H.I! h"~Id. Then| for h small enough, we h a v e (4~23)

II~h-~l[O,~ ~ Ch k+l

for

constant

some

Proof (4.24)

: Taking

V=Uh-U

in

which

is i n d e p e n d e n t

(4.20), we get

and

(4~=22)

of h.

:

(Uh-U,g)= v a ( u h - u ) ~ ) + b ( U , U h - U , ~ ) + b ( U h - U , U , @ ) - ( X , d i V ( U h - U ) )

On the other (4.,25)

C=C(u~p)

(2.22)) (4.9) the e s t i m a t e

hand,

using

(2.19)

and

(3.9), we may write

v a(uh-U,?h)+b(Uh,Uh)~h)-b(u,u)~h)+(p,div

for all ~ ~ V . Now, c o m b i n i n g +h n that d i v ~ =O, we obtain :

(4.24)

and

(4.25)

:

~h )=0 and n o t i c i n g

204

(Uh-U,~)= v a(uh-u,~-~h)+b(Uh-U,Uh,~-~h )+ + b(~,'+Uh-U"+ ,~ - - ~ h ) - b ( U h - U , ~ h - U , ~ ) - (×,diV(~h-~))-(p ,dlv(~O-? • -~ ÷ h)) Thus, we get for all ~h E V h l(Uh-U,g)I

(4.26) Next,

we

have

by

+Cl(Tlul+[lUhll)}llUh-URl~-~hll c I llUh-Ul2 l]~l+l ( × , d i V ( U h - U ) ) I + l

.< { ~

+

+

(p,div(~-~h))

(4.7)

1

:

lXll,n

(4.27)

] (×' dlV(Uh-U))I=I " "+ "+ "~ "+ "+ "+ (X-PhX'diV(Uh-U))l- -> -~ -+ ~Ph-Ph pjO,~ "< IIP-OhP~IO,~+ ~ Cl~Uh-U~+C2 ( ~ h ~ +~II) llUh-U~ and the desired 5. THE EFFECT

estimate

OF NUMERICAL

(4.34)

follows

INTEGRATION

at once.

: GENERALITIES

The practical application of the finite element method (3.8) requires the computation of various multiple integrals. Although most of these integrals involve only polynomials and can be computed exactly, it is easier and faster to use numerical integration techniques ; we shall show that it can be performed with no loss in the order of accuracy of the method. Note that these numerical integration techniques are essential when using curved isoparametric finite element methods. Let us describe the numerical quadrature method that we shall use. We proceed as in Ciarlet ~ Raviart [ 3 ] - L e t K be a fixed nondegenerate N-simplex of I~m. We choose- ~ certain quadrature formula over the reference set ~ : L (5.1) ~(~) = / ~(~)d~ is approximated ~X ~a(~)=£=~ I ~ ~(~£) K for some specified points ~ £ ~ K and weights ~£ which will be assumed to be positive. For each K 6 ~ h , let F K : ~ FK(~)=BK~+b K, ~K ~ ( s N ) ' hK~N, be an affine invertible mapping which maps K onto K. We may assume t h a t the Jacohian determinant JK=det(BK ) of the mapping F K is positive. Then, there corresponds to (5.].) the quadrature formula over K : L (5.2) IK(~) = K / ? ( x ) d x is approximated by IK,a(~)= ~-£=i ~£'K~(b£'K)

206

where (5.3) Then,

~,K

= ~£JK

any integral

(5.4)

I(~)

la(~)

over

' I~£~L.

the polyhedral

domain

Q

= /~p(x)dx

is a p p r o x i m a t e d (5.5)

' b£,K = FK(b£)

by

=K~

IK'a(~)" h

In all the sequel, we shall assume that the function ~ belongs to the space (C (~)) . Let a~(u~,v~), b h ( U ~ , V ~ , W h ) and (~,vh) ~ be the a p p r o x i m a t i o n s of a ( ~ , ~ b ) , - b ( ~ h ) and (~,~h) resulting from numerical integratioE': . . . . N ~h + ~ u~ h ah(Uh,V h) = la(i__~I ~ , ~ ), ..... = ~x i ~x i Note

that

(5.6)

the analogue

bh(Uh,Vh,Vh)

of property

= 0

(5.7)

functions

holds

:

for all ÷Uh,V+ h e X h

Now, we replace the discrete following ones : (i) Find

(2.18)

problems

(3.8)

~U h E V h and ~h e # h ~ such

and

(3.9) by the

that

~ % ÷ (~h,div ~h ) = (~,Vh) ~ ah(Uh,V+ h )+b h (Uh,Uh,Vh)÷ h for all Vh~ X h

(ii) Find (5.8)

a function

u h ~ V h such

that

v ah(Uh,Vh)+bh(Uh,Uh,Vh)~f,v

h) for all V h ~ V h.

Remark 3 : We shall ignore the p o s s i b i l i t y i n t e g r a t i o n in the d e f i n i t i o n of the space V h = {~hI~ h ~ X h , la(~hdiV

of using Vh :

~h ) = O

for all ~h e ~h }

and in the c o m p u t a t l o n of (ph,div vh). The interest p o s s i b i l i t y is limited because we shall prove in §6 error ~Uh-Ull is optimal when the q u a d r a t u r e formula exact for all p o l y n o m i a l s of degree 4 k+k'-2, which to an exact c o m p u t a t i o n of (~h,dlv ~h ). For

studying

problems

(5.7)

and

(5.8),

H y p o t h e s i s H.4 : The q u a d r a t u r e ' formula f o l l o w i n ~ p Koperties :

we

shall

(5.1)

(i) The....w...e..i...g..h..t...s. g.• are p o s i t i v e and the set P k , _ l - u n i s o l v e n t subset, i~e~ ....... (5.9)

P ~ Pk'-i

' P(bz)=O

' I~Z6L

===~ p=O

numerical

need

of this that the (5.I) is corresponds :

satisfles

the

{~£~L ~=I contains

;

a

207

(li) There exists an i n t e g e r s ~ k'-I such that the ~ u a d r a t u r e formula (5.1 ~) is exact for . . .all . . . p o l y n o m i a l s of desree ~< s0 By H y p o t h e s i s H.4 (i) and [3 , Theorem 3] for instance, exists a constant a>O i n d e p e n d e n t of h s~ch that (5.10)

ah(Vh,V h) >~ a~Vh] I

As a first

consequence

is a norm over us now

(5.10)

X h.

introduce

the following

quantities

:

sup Ibh (~h'Vh'Wh) I Bh = + + ÷ V + + + U h ' V h ' W h ~ h IlUh~hlIVhlIh~Wh~h

(5.12)

+

~2 (5.13)

_>

I (f ,Vh) h I

If h =

sup +

v

Vh~- h

Then,

for all V h ~ X h.

+ + + I/2 ~VhN h = ah (Vh,V h )

(5.11)

Let

of

there

we have

~I +

~ h h

the analogue

of Theorem

T h e o r e m 6+: Assume that H y p o t h e s i s function f satisfies

2.

H.4

(i) holds

and

that

the

(5.14) ~ ll~l[h < I Then,

there

exists

a unique

pair

of functions

solution of p r o b l e m (5.7). Moreover, ~'~aractgrized as the unique solution Proof

: The proof

By taking

Vh=U h in (5.8), 1

"+

(5.15)

follows

tl"Uh Ith

the same we get

(~h,~h) ~-V h x ~h/~

the function Uh~_V h can be o-T Drob--"-~e-~~5.8)

line

as that of Theorem

2.

the inequality

~

"< --

! ~11h

"

Before e s t a b l i s h i n g the convergence of B h and l]f|h as h tends to zero and before deriving error estimates, we need introduce some notations. ~qe set : (5.16)

E(~)=i(~)-ia(~),

EK(?)=IK(~)-IK,a(?),

E(?)=l(?)-la(?)-

Given a fixed K~=~,,, the m a p p i n g F K deteKmines o n e - t o - o n e c o r r e s p o n d e n c e s between the points ~ of K and the points x of K by X=FK (~) and between

and

the functions

1 We shall (5..17)

~=FKI (x) ,

make

and

constant

IK(?)=JK.~(~)

~ : K--~

and+ ~ : K - - ~

by

?° use of the above , IK,a(?)=JK.~a(~),

correspondences. EK(?)=JK.E(~).

Note

that

208

Let us introduce now m~O and any q~l : (5.18)

wm'q(~)

We provide

the Sobolev

= { v l v e Lq(~)

wm'q(~)

with

,~

~vllq

i~i~ m "

that

we need

Hypothesis

integer

, lal6m }

seminorm

)l/q

eq(~)

=

Era(~)

'

a slightly

H.~I : There

llv~m,Z, n = Ivl~,n ,

,

refined

exists

version

such

Ivlm,2,~

of H y p o t h e s i s

an operator

(Wh)N)/~ ~ ( ( H I ( ~ ) O C ° ( ~ ) ) N ; X h) o (i)

for any

:

wm'2(~)

Then,

wm'q(~)

v ~ Lq(~)

the norm and

(5.19) Nvlm,q,a = ( ~

Note

spaces

rh£~((Hl(~)

=

IVlm,

~

•

H.I. ~C°(~))N

;

that

(4.1) holds

+ + qs,q,K ) i/q ~ C h m + l - S l v÷l m + l , q , ~ ( ~-F Irhv-vl , O~ szm+l Ke~ h N for all ~ ~ (wm+l'q(~)) N and all m with O~mSk and m+l--- > O (3) q

(ii)

(5.21)

In practice,

Hypothesis

H.I holds

as soon as H y p o t h e s i s

H.I does.

Now, we want to study the convergence of B h and ~ f ~ as h tends zero. To this purpose, we need some technical lemmas.

to

Lemma 3 : Assume that H y p o t h e s i s H.4 (ii) holds for some i n t e g e r s=r+kr'/-I with r~O. Then, there exists a constant C>0 i n d e p e n d e n t of h and K such that

(5.22) IEK(2vi ~xj for all K ~

w ,

vI

l

I~xjlr+l,K

and all vi,w i

O,K

Wh•

Proof : This result is similar to L 3 , T h e o r e m 6J but it will be e s t a b l i s h e d in a slightly different way. Since H y p o t h e s i s H.4 (ii) holds with s=r+k'-l, we may write : ~(~vf __

~wi)

~xj ~xj

^ =

~v i

E((---~

~xj

~ v i ~w i r

)

3xj ~xj

)

,

where, for any integer m>O, ~m denotes the o r t h o g o n a l p r o j e c t i o n o p e r a t o r in L2(K) upon Pm" Since ~vi -~r ~vi and ~w. belong to i

~xj (3) so that wm+l'q(~) ~ C ° ( ~ )

by the Sobolev's

~xj

~xj

imbeddin~

theorem.

209

the finite-dimenslonal space Pk,_l ~ there exists a constant Cl=Cl(K~k'pr) such that

l[(('x......j _ .~r ~xj By the Bramble-Hilbert we get :

I .< c I

il~xI

-

~'

o,[

r ~xjllO,R

lemma [ i ] in the form given in [2 ,Lemma 7],

r ~xj~O,K

~ ~xj ~r+l,K

and therefore

t~(~xj ~xq I -< c3 l~xj r+l,K l~x111O,K Since (cf. [ 2 , f o r m u l a 0, We hive (5.25)

I~B .< lira infll~hllh h+O -w

Proof : Define l=limh-~oinf~VhlIh" First, we extract a subsequence yet denoted by {~h } such that lira I[~h~ h" h÷O

= ~.

Now~ sinceOC=X(~(H2(~)) N is dense in X, we can find a sequence {~n}n>.l of functions of O~ such that lim ~ n = ~ n÷~

strongly in X.

210

Then~

consider

(5.26)

the expression

:

ah (vh-rhVn, vh-rhVn ) = ah (Vh, Vh)-2 ah (rhVn, Vh) +ah (rhVn, rhVn).

By using

(5.24) with r=O and Hypothesis "~ + + + la(rhVn,Vh)-ah(rhVn,Vh)

H~.I (ii), we ~et

I ~< Clh( K ~

:

+ 2 K-) i / 2 ~ h l I IrhVnl2,

n

4

2hlVnI2,a I h

and therefore lim h-~O Similarly,

ah(rh~n,~ h) -- lim a(rh~n,~ h) = a(~n,~). h÷O

we obtain

lira lira a (rh~ n $ (~ ,v ) ÷ ) = h÷O h÷O a h (rh~ n ~rhVn ,r h n ) = a n ÷ n "

Hence h+olim a h ( v h - r h V n , v h - r h v n) = 12-2a(~n,V)+a(Vn,Vn) and lim lim ah($h-rhSn,~h-rh~n) n+= h÷O Since

the expression

(5.26)

= ~2-a($,~).

is >4 O, the conclusion

follows

at once.

Lemma 5 : Assume that Hypothesis H.4 (ii) holds and that the triangulatfon ~ h verifies the uniformity qondition (4.33). Then, we have : N (5.27)

Ib(Uh,Vh,Wh)-bh(Uh,Vh,Wh)

I .< Ch

tluh~ IlVhll IlWh/I

for all U h , V h , W h ~ X h. Proof

: First we note

that condition

(4.33)

implies

-N/q

(5.28)

~VhIIL~(~) ~< Clh

IlVhlILq(f~) for all V h ~ W h and all q>~l.

Let us prove this classical inequality for the reader's convenience. Let v, 6 W and let K ~ , ~ ; since ~ h E Pk,, there n h ^ exists a constant c2=c2(K,k') such that ll~h~L~(~ ) .< c2~hllLq(~ ) and therefore llVhllL~(K) ~< C2JKl/q~vhIILq(K ) But j-i = det(BK I ) ~< K

IIBKlll ~ t

where denotes the s p e c t r a l Lemma 2J and (4.33), we get

°atrix-noro.

hus, usin

r2,

211

j-I h~/ N K ~< ( ~ ) o(K)

-N "
.l (the case ~x.l r=O is left to the reader). We write : ~v

f~v

°,,"

•

l I By Hypothesis H.4 (ii), we get

i

1 and by the Bramble-Hilbert lemma (6.14)

i

i

~xi~V"(~j-#'o~j)) I ~< c5 1~i---l~xilr,KlWj^ ~v. I l~(~i---~J II,K

Similarly, we_get by H.4(ii) E(~ i ~x

9o~j) = E((~ i

~x

-f 1

r

(~iJ))~oWj) ~x i

and therefore~.. (6.15)

I~(¢~i~vj ~'o~J)I ~< c6 ~x.~

~Wj~o,4,K ~Xi r+I,4/3,~

Thu~, it follows from (5.23), (6.14) and (6.15) (6.16)

IEK(Vi~--xwj) I ~< c6hr+l{ v i lwj Ii, K + i l exi|r,K

~v_!l +

IVi xi[r+l,4/3,K

lwjlIO,4,K

216

Then,

(6.11)

is a consequence

of (6.16) and

v.-~J ~ c8¿Vi~r,4 l~x i r,K

iv

ax i r+l,4/3,K

K

laxi~r,4,K

,

.< e9 Ivillr+l, 4, K

Consider next

[

•

r+l~K

N

(f'wh)(~'~h)h = ~.=I ÷~

K~I

EK (f iwi )

For estimating A4, we shall use the following been proved in [ 3, Theorem 4].

lemma which has

Lemma 8 : Assume that Hypothesis H.4 (ii) holds for some integer ~ I with r~0. Assume, i'n addition, that the'fu'nction f.~.wr+l'q(~) .constant C>O (6.17)

for some q>~l with r+l-N>0. Then, q independent of h and K such that

IEK(fiwi)I

for all K ~ h

there exists

a

~< Chr+l~fi~r+l,q, K llWi~l,q, K

and all w i ~ Wh, ! + --1 =1. q q'

We are now able to prove Theorem 7 : We make. the followin$ (i) H.~p..o..thesis H.I holds

assumptions

;

(ii) Hx.~.qthesis .}!......4 holds for some inte$.er s=r+k'-I with O~.r~k.71 ; (iii) ~ ~wr+l'q(~)) N with q~2, r+l-~>O and satisfies (2.22) ; q

condition

(iv) The solutio.n. (u~p) of (2.19) verif.i.es the smqothness properties (.4.9) ; (v) T.h..e.triansulat, ion ~ h (4.33). Then,

for h small enough,

UhG Vh a n d we h a v e

(6.18)

satisfies

the

the. uniformit%, condition

pro.blem (5.8..)...has a unl.q.ue .solution

estimate

l~Uh-]~ ~< Clhk(lulk+l,a+Iplk,a) + C2 hr+l(lul

where C 1 .and C 2 a r e

positive

^ r+z,a

+

+lull2r +2, a+ll~Ilr+l,

constants

independent

2.-

Proof

: Taking vh=rh u in (6.6), we obtain A 1 ~< ( 1 + 2 ~ )

~u-rhUll+l[u-rhulm+llP-OhPllo,f~

q,~) of

h I u and

217

By applying (6.19) Next,

(2.24),

(4.7)

and (5.21),

we get

A I ~ c3hk(I~Ik+l,Q+Iplk,fl)+C4h2r+2 using

(5.24) with vh=rhu

and

:

I~2 r+2,~

(5.21),

we obtain

(6.20) Now,

A 2 ~ Cshr+ll~ I r+2,~ using Lemma 7, we get Ib(rh~,rh,u,wh)-bh(rhu,rhu,w

h) I -


angles

v21

a and B s h o u l d b e s m a l l .

TYPE~,,,,,7 -Reference

~ .....

~

v12 -The

---'~ u3

FRAHEELED~NT INCLUD,ING :

- A s y . u ~ _ t i o _ nE "

A.8 u-

DEFORMATION

(internal)

a I + a 2 x + a 3 X2 + a 4

¥ m" e + aI

SHEAR

t X + a2

x3

t X2 + a~ X3 a3

i!

2 t x - eI ÷ e2 x ÷ e3 x

%y m B1* ~z

"

+

~2t

X+

" + B~ el x +

(eventually 2rid d e g r e e )

e3!

X2

" X2

~3

the dlaplacements

up v t w a r e r e d u c e d

to

X

237

- D escr ip_~on

ThSs element is a straight prismatic member which deforms in extension. bending and torsion. The local axes are the inertia axes of the cross section. As the shear deformation is included in the theorY used for bendlng, the displacements u~ v~ w are represented by modes independent ~w of those used to describe the rotations. (~y is not equal to ~ , etc...). The difference between the elements of type i and type 7 is the same as between the plate bending elements of type 15 (Kirchhoff)

and type 8

(Hencky). This element should be used to represent beams reinforcing membrane elements and which have a bending stiffness in the plane of the membranes. The connection can be conforming with membrane elements of

2nd and 3rd degree (Type 3, 4 or 20). A11 t h e c h a r a c t e r i s t i c s

o f t h e c r o s s s e c t i o n can v a r y l i n e a r l y

along

the element neutral axis.

/x

,7/ J"

,,< ,/~_G~" .! +-,

/,

./

'~/"21

i

*~,.,=/J.~M[,._!y,~

e"

~ v " / " .,.-', ~ y

,,, 'Yl "i

e.:.

u--

"/~//

"/

.I jf

wl2

~

-21 -

"12

"

, /

,, "

!

n

'. -~

238 TYPE

8

CONFORMING }DDERATE

TRIANGULAR

TtilCK~ESS

PI~TF, BENDING

(TI!EORY OF

ELEMENT

OF

HKNCKY)

- Reference • G.SANDER,Dc.Thesis,Collection des Pub.Fac.Sc.Appl.L~ge,N°i5(1969) - _ A s i u ~ i o . . n . _ , u = s $x ( x , y)

v.,

sly

(x, y)

#x " al + a2 x + a3 y + a4 x~ + a5 xy + a6 72 ty

' ÷ ' X + ' y + ' X 2 + ' xy + , y2 " uI a2 a3 a4 a5 a6

w

= B I + B 2 x + B 3 y + B 4 x 2 + B 5 xy + S 6 y2

B7 -

x3 + B8 x2y + 89 xy2 +

810 Y

3

Descri£tlon Thls plate bending element is derived according to the theory of llencky for

plates

of moderate thickness. It differs from the Kirchhoff plate

theory by the independence of the rotations ~x' ~y of the cross section and the slopes ~w/~x, ~w/~y of the mean plane of the plate, It allows to take into account the shear deformation and the edge effect due to the transverse rotation around the axis normal to the mean i>lane. The element Is derived with fixed degree and constant thickness. The Young's modulus used to compute the bending energy and the shear modulus used to compute the transverse shear energy can be independan~. This allows ~o use the element to model non homogeneous materials llke sandwich, multi-layer 7

trusses etc.

/ '~

I t~

t2.

I / i

"-L

!t

/

~ !

A+

1[ Y23

"0

i

/ *

,

/

ft'l

•

U

I-

.~o ....

~

- +h

,

.,_

w

/

1

-'--~.~ u ~2-~

ii h=/ o/ •yA

i. i"~-.~ ~" L23

o.. "ix_-

L

239

- D e ~ r e e s of freedom The element can only be used in the X-Y p l a n e . The sequence of the gener a l i z e d displacements Is. in the global XY axes.

q'

= (wl gxl Syl

w2 @x2 @y2

w3 ~x3 ~y3

w12 4x12 ~y12 w21 w23 ~x23 gy23 w32 w31 ~x31 ~y31 w13)

The i n t e r n a l degree of freedom w necessary to d e r i v e the element i s e l i m i nated by condensation. To allow a correct representation of the boundary conditions, special axes can be defined at the 6 points where rotations are expressed. Only one set of special axes can be defined per element, by giving their direction cosines in the LOCAL,,axes. This transformation of coordinates does not necessarily affect all the rotations.

TYPE. 9

~UILIBRIUM OF

qU~)RILA%RAL

PLATE

BENDING .ELE,~f~NT

MODERATE ..THICKNESS (REISSNER ~{EORY.)

- Reference : G.SANDER,Dc.Thesis,Coll.Pub.Fac. Sc.Appl.Li~ge,NOl5 (1969) -Assumptions ; In e=ch triangular region, the bending moment field in tha oblique axes defined by the 2 internal interfaces of length a and b is : Mx

- B1.

S 2 a + 83 ~b

x =s4+ss~+~6 "

l b

÷ s9 Zb

÷ s8

A p a r t i c u l a r s o l u t i o n f o r c o n s t a n t d i s t r i b u t e d load p i s superimposed pslnn

l

" ~6

~lO

(n is the angle of she oblique axes)

240

This plate bending element is derived by the equllibrium theory of Relssner for plates of moderate thickness. It differs from the equilibrium formulatlou of the Kirchhoff theory by the form of ~he complementary stress energy which includes the contribution of the transversc shcar and by the continuity requirements for the surface tractions. In this element the £endlng and twls=ing moments are continuous across an inter-

face as well as the shear forces. There are no corner loads. Although the form of the bending ~oment field is the same as in the equil~rlum Kirchhoff plate bending element of ~pe 13, it is impossible to derive a triangular element free of additional constraints (or klnematie deformation modes)due to the increased number of interface generalized ~rces, The solution of building a super-element composed of 4 triangles allows to reduce the additional constra~ts ins~e the assemblage. The element is derived with fixed degree and constant thickness. The Youngts modulus can be ~dependent of the transverse shear modulus. -De~rees of freedom

The element can only be used In ~ e X-Y plane..The sequence of generalized tortes

is :

;' " ~ n l 2 M6n!2 V12 Mn21 Man21Hn23 Man23 V n23 M n32 M sn32 I~34M=34

V

n34 Mn43

M

an43

M

M

n41

V M M ) an41 n14 n14 snl4

/"

M

M

){In

Mn

3~34.~

In32

Pn

Jb II

/

y

./" /

./M

n.-.~..,,~.x

for ND - 3 on all interfaces

-i

I 12

The s e q u e n c e o f t h e g e n e r a l i z e d d i s p l a c e m e n t s f o l l o w s t h e same l o g i c as for the conforming membrane elements of types 3~4 or 20 : the displacements of the vertices are followed by those of points defined on the interfaces

taken sequentially

turnin E anticlockwlse.

q' = (d I d 2 d 3 d 4 d 5 d 6 d 7 , . . d12) w l t h d i - (u i v i wi ) The d i s p l a c e m e n t s o f t h e v e r t i c e s

a r e always e x p r e s s e d i n g l o b a l axes

while on each individual interface the special local axes defined for

the elements of tvves 3 or 4 can be selected by the phase I,

246

TYPE

15

,CONFp?~IING QUADRILATERAL

PLATE

BENDING,,,,ELLMENT

(KIRCHI{OFF THEORY) -Reference : de V~UB~K~.Int.JI.Sol.Str..4 (1968~ -~s~p!i~n

£ : In each triangular region 2 w m al + o2 x + a 3 y + a 4 x

÷

-

2 + n 5 xy + a 6 y

"7 x3 + °8 x2 Y + n9 xY 2 + ~i0

y3

£ezcai£t!o~ This plate bending super-element is obtained by assembling the 4 trlan8ular elements defined by the diagonals of the quadrilateral. In each of these triangles the deflection is represented by a complete cubic. The 40 corresponding parameters are reduced to 16 by expressing the continuity requirements of th~ deflection and normal slope along ~he internal interfaces of the quadrilateral. These 16 lhdependent parameters are finally expressed in terms of the 16 generalized displacements (w, ~x ' ~y ' at the 3 vertices and ~n along each interface) necessary to insure a strict continuity of the deflection and normal slope along the external interface of the quadrilateral. The element is programmed with the possibility of using a variable thickhess D anlsotroplc stress-strain relations and various special support options. The element can be used as an equilibrium membrane element by the stress function method. It is the Southwell analog of the element t y p e 16. In this case the thickness has t o be constant.

- ~e~rzes of freedom The element can only be used in the global X-Y plane. The sequence of t h e 16 generalized displacements is :

q'

~w

~w_~_ w2 ~w

(wl °~! ~Y2

~w

~x2 ~Y2

w3 ~w

~w

~x3 ~Y3

w4 ~w

~w

~w

~w

~w

~w

~ ~Y4 %n12~nz3 ~n34 ~n41

247

~,~'Y

Ij

w2//

/

%'

x3

n34 V

x4 w1

~.

n41

Y$

~X

g

0

Oxy

oblique l o c a l axes

u ~

Ox y

c a r t e s i a n l o c a l axes

Oxg y8

global axes

The local axes O~ is directed from 0 toward the middle of the edge 12. The normal slopes w

are expressed in the middle of each edge. nlJ

- S~ec!a! ~ i o n ~ I*/ Variable thickness : the thickness is constant unless the variable ICHOI - 1

In this case the thickness at point O is interpolated from the values defined alon~ the 2 diagonals at this point by a linear variation. The average of the 2 values is assumed.When the thickness is variable all the other special options are ineffective.

2"/ A n i s o t r ~

: it is controlled by the variable IANISO. IANISO = 0 corresponda to the Isotropic case, Fou~ anisotropic layers can be superimposed to the parent plate. Each of the~e layer can have an independent thickness tan i an4 I E tani 3 • the bending rigidity of each layer is ~-~

248 TYPE

16

E_qUILiBR!UM OUADRILATERAL (LINEAR

- References :

MEMBRANE

ELEMENT

STRESS FIELD)

deVVEUBEKE,AFFDL-TR-66-88 (1966)

- A ss_umpt_io_n~ : In each triangular region :

°x

" ~1 + ~2 x + B3 y

Oy

= B4 + 8 5 x + ~6 y

x

= B7 - B6 x - B2 y

- ~eEc~iEt!o ~

This e q u i l i b r i u m membrane is a super-element o~tained by assembling the 4 triangular elements defined by the diagonals of the quadrilateral. In each of these triangles the stress field is linear as indicated in the

assumptions. The 21 corresponding parameters are reduced to 13 by expressing the continuity of the normal and tangential surface tractions along the internal interfaces of the quadrilateral. The 16 generalized forces necessary to determine uniquely the linear variation of the surface tractlons along the external interfaces are expressed ~n terms of the 13 parameters. As the 16 forces satisfy 3 global equilibrium equations, the element is free of spurious kinematic modes. The generalized forces are the local values of the surface tractions Tsn . O n (times the thickness)

at each vertext along an interface. Note that although expressed at a

vertex,

they are interface

variables,

The e l e m e n t i s programmed w i t h t h e p o s s i b i l i t y

varLable t h l c k n e s 3 , -De~es

of freedom

of using a linearly

249

\ T

T43

3

,t, ~

4

"='=W T41

4 (local oolique

• ~N4I

axes)

Y8 (local axes for stresses) :X

(global axes)

0 The s e q u e n c e o f t h e g e n e r a l l z e d

f o r c e s i n t h e l o c a l axes Oxy i s

8' = (H12 TI2 N21 T21 N23 T23 N32 T32 N34 T34 N43 T43 N41 T41 NI4 TI4 ) where

face

Nij P Tij are cartesian normal and tangentlal components of the surtractions

multiplied

by t h e l e n g t h o f t h e i n t e r f a c e .

These forces

ere transformed in global forces S" -

F_

(x12 Eventually

F

F

F

F

F

is given a zero stiffness. along an interface

dinates,

.,. F

)

z14

t h e p h a s e i can d e c i d e t o keep t h e l o c a l n , t axes on an i n t e r -

face in which case the 3rd co~onent

wet

F

YI2 z12 x21 Y2I z21 x23

of t h e g l o b a l f o r c e s a t e a c h p o i n t

T h i s i s ~he c a s e when o n l y 2 membrane e l e m e n t s

and a~e c o p l a n a r w i t h o u t b e l n ~ i n a p l a n e o f c o o r -

250

TYPE

,EQUILIBRIUM BAR

17

VARYING

- Reference : - Ajs_u~p~io.n~

SH~

ELE!m~ AND

M.KIEFFER, SF-6 (internal) :

~(x)

CROSS

WITH.,,,,,,,,,,LTNE,ARLY SECTION

ARF~

(1969)

= S I + ~2 x

2 X

or S(x)

-

_Des_~ZZ~on

= SO + SI x + S 2 ~ - -

-

This b a r element i s i n t e n d e d to r e i n f o r c e the membrane element o f type 16. It is stresses by a linear axial shear and 2 tip loads. The cross

section can vary linearly

between t h e 2 e n d s .

(local)

- D e g r e e s o f freedom

r

X

/.

/x

7"

(global axes)

0 The sequence of the generalized forces in the local axes is

g' = {N 1 N 2 TI2 T21 ) whexe Ni i c a t i p 1cad and T i j t h e l o c a l v a l u e o f t h e s h e a r a t an end tfmes

the length of the bar.

Theae f o r c e s a r e e x p r e s s e d i n the g l o b a l axes i n t h e s e q u e n c e :

25t g' - (FXl F F F F F T T T T T T ) Yl Zl x2 Y2 z2 x12 YI2 z12 x21 Y21 z21 Eventually the forces TI2 and T21 are kept in local axes if this special

choice of axes has been decided by the phase I. In this case, zero stiffmess is given to the components T T and replaces T in the sequence given above, x12 z12 Tij YlJ TYPE

20

CONFORMING VARIABLE

- Reference : -_As~um_p~ion :

PARALLELOCRA}! DEGREE

A~D

ELE~N~

BUBBLE

WITH

FUNCTIONS

A.II (internal) u and v are of the form

2

2

PI (x, y) + (l - a ~ ) (1 - bye2) P2 (x, y) where both functions PI and P2 are P£ = (u 1 + a 2 x * ~3 x2 + "" + an+l n ) ( 8 1

+ B2 y + ~3 y2 + . . . ÷ g n + l

yn)

t~uncated at the degree requested by the user. - D es_cEiEtIo~ This element is a generalization of the classical rectangular membrane elements in this sense that it is extended to variable degree polynomials (from i to 9) and that it includes bubble displacement modes of degree independently variable (from O to 9). In this purpose the polynomials used to represent the displacements are split in two parts controlled by the functions PI and P2" PI describes the basic field and the interface modes Whi~e

P2

represents only bubble

displacement modes.

To allow easy comparison with other elements (type 4 or 14) the number of terms retained in the functions P1 (x, y) corresponds exactly to the number of generalized displacements necessary along the interfaces. This r

reduction of parameters is obtained by dropping in PI the terms of power greater or equal 2 in x ~ y

and the terms of power greater than n in

x O~R y, n being the degree of the displacements in the direction of the

edges. The complete form given above is always used for P2 (x, y). The thickness is constant and the material isotropic. The element can be used as a plate bending equilibrium element by the Southwell analogies. - ~e~rees of freedom

252

3

9/

s

/

//

/

10

4

/'

/

/

2

/ ~

x

/ II

,

! ! !

12

/

I

°

r

1

/

.

I

1

~ L14

~ ~14

n o d a l p o i n t s e q u e n c e f o r ND = 3 The g e n e r a l i z e d d i s p l a c e m e n t s a r e e x p r e s s e d a t e q u i d i s t a n t

points along

the i n t e r £ a c e , d e p e n d i n g o£ t h e d e g r e e s e l e c t e d . The s e q u e n c e o f t h e d i s p l a c e m e n t s , t h e c o n v e n t i o n s f o r r e d u c i n g t h e d e g r e e a l o n g an i n t e r f a c e , t h e o p t i o n o f l o c a l n o r m a l - t a n g e n t axes a r e e x a c t l y t h e same a s f o r e l e -

-

ments t y p e 4 o r 14. S t r e s s out~u~ The s t r e s s e s

a r e computed a t r e f e r e n c e p o i n t s which form a r e g u l a r mesh

d e f i n e d by ND-1 o r ITENS and o r d e r e d i n t h e s e q u e n c e i 1 1 u s t r a t e d The l o c a l c a r t e s i a n

a x e s £or s t r e s s e s

have t h e Ox a x i s p a r a l l e l

below. to the

!,

raise 4 - 1 .

! 10

1

.........

~o

Ii

12

t/

/

........ /I ............

1

~£erence

points for stress

2

1

3

! o u t p u t : 2nd d e g r e e i n x and 3rd i n y

253

Z

i

L h!2_. i"

,

-1

m

E,, 2.!0l°

~

v:o. t

=

~

.

0,01

TEST

CASE

NBK

1

:

CANTILEVER

U

BEAM

/,.

~"

/

\

'/

\

,

,, l ~ s

"

,

"/

.... "~"

~/,

'¢

,

1

2

',/

-

TEST

/'2

"

I/

"

: I I

i

/3

CASE

3

' /

•

/

2

l:6

CLAMPED

NBR

~

:

',~.',/

',

5

/s

CANTILEVER

,

SKEW

6

",/

"--,/6

PLATE

)

7

/

- 10 3

p

- 0.1 ,- 10 4

t p

E 2 ", 10 8

- 0.3

1.092

;

v

E1 -

.... V _ . . . . . . K

", /

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255

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256

REFERENCES I. FRAEIJS de VEUBEKE, B. "Displacement and equilibrium models in the finite element method" Stress Analysis, ed. O.C. Zienklewlcz and G. Hollster, Wiley, 145-197, (1965) 2. FRAEIJS de VEUBEKE, B. and ZIENKIEWICZ, O.C. "Strain energy bounds in finite element analysis by slab analogies" Jnl of StralnAnalysls, 2-4, 265-271, (1967) 3. FRAEIJS de VEUBEKE, B. "Basis of a well-condltloned force program via the Southwell slab analogies" USAF Report AFFDL-TR-67-80, (1967) 4. FRAEIJS de VEUBEKE, B. "Duality in structural analysis by finite elements, static-geometric analogies, the dual principles of elastodynamics" NATO Advanced Study Institute lectures on finite elements, Univ. of Alabmma Press in Huntsville, 299-377, (1971) 5. FRAEIJS de VEUBEKE, B., SANDER r G. and BECKERS I P. "Dual analysis by finite elements. Linear and non linear applications" Air Force Flight Dynamics Laboratory, Wrlght-Patterson AFB, Ohio, Technical Report AFFDL-TR-72-93, (1972) 6. FRAEIJS de VEUBEKEp B. and HOGGE, M. "Dual analysis for heat conduction problems by finite elements" Int. Jnl Num. Meth. Eng., 5, 65-82, (1972) 7. FRAEIJS de VEUBEKE, B., GERADIN, M. and }lUCK, A. "Structural Dynamics" CISM, Udine, (1973) 8. FRAEIJS de VEUBEKE, B. "Diffusive equilibrium models" University of Calgary lecture notes, (1973)

257

9. GERADIN. M. "Computational e f f i c i e n c y of e q u i l i b r i u m models in e i g e n v a l u e a n a l y s i s " P r o c e e d i n g s o f t h e IUTAM Symposium on High Speed Computing of E l a s t i c S t r u c t u r e s . Congr~s et Colloques de l ' U n i v e r s i t ~ de L i e g e . Place du XX AoQt. 16. 4000

LIEGE. 589-623. (1971)

10. GERADIN. M. "Analyse dynemique duale des s t r u c t u r e s p a r l a m~thode des ~l~ments f i n i s " C o l l e c t i o n des P u b l i c a t i o n s de l a Facult~ des Sciences Appliqu~es de l ' U n i v e r s i t ~ de Liege. 36. 1-173, (1973) II. IMBERT, J.F., GIRARD, A. and GERADIN, M. "Modal analysis of a satellite primary structure using a finite element procedure" Sympositnn on s t r u c t u r e s of space v e h i c l e s and s p a c e c r a f t . U n i v e r s i t y College London. (1973) 12. IRONS, B. and RAZZAQUE, A. " E x p e r i e n c e s w i t h t h e p a t c h t e s t f o r t h e convergence of f i n i t e e l e m e n t s " Conference on t h e m a t h e m a t i c a l f o u n d a t i o n s of t h e f i n i t e element t h e o r y " Univ. of Washington ( B a l t i m o r e ) , Academic P r e s s ,

(1972)

13. PLAN, T.M. and PIN TONG " B a s i s of f i n i t e element methods f o r s o l i d c o n t i n u g ' I n t . J n l Num. Meth. Eng., 1, 3-28, (1969) 14. SANDER, G., BECKERS, P. and NGUYEN, H.D. "Digital computation of stresses and deflexions in a box beam" C o l l e c t i o n des P u b l i c a t i o n s de l a Facult~ des Sciences Appliqu~es de l ' U n i v e r s i t ~ de Li~ge. 4, 87-137B (1967) 15. SANDER, G. "Dual a n a l y s i s of a multlweb swept back wing model" A i r c r a f t E n g i n e e r i n g . 6-16. (1968) 16. SANDER, G. " A p p l i c a t i o n of t h e dual a n a l y s i s p r i n c i p l e " P r o c e e d i n g s o f t h e IUTAH Symposium on High Speed Computing of E l a s t i c S t r u c t u r e s I Congr~s e t Colloques de l ' U n i v e r s i t ~ de Liege. Place du X.X AoGt, 16. 4000

Liege, 167-207. (1971)

258

17. SANDER, G., BON, C. and GERADIN, M. "Finite element analysis of supersonic panel flutter" In,, Jnl Numo Meth, eng., 7-2, (1973)

18. STRANG, G. "Variational crimes in the finite element method" Math, Foundations of the finite element method~ ed. A.K. AZIZ, Academic Press, (1972)

19. TABARROK, B, and SODHI, D.S. "The generalization of stress function procedure for dynamic analysis of plates" Int. Jnl Num. Meth, Eng., 5, 523-542, (1973)

20. TURNER, M.J., MARTIN, H.C. and WEIKEL, R.C. "Further development and applications of the direct stiffness method" Matrix methods of structural analysis. AGARDograph 72, 203-266, Pergamon Press, (1964)

VISCO-PLASTICITY AND PLASTICITY AN ALTERNATIVE FOR FINITE ELEMENT SOLUTION OF MATERIAL NONLINEARITIES by O. C. ZIENKIEWICZ Professor of Civil Engineering, University of Wales, Swansea and I. C. CORMEAU Aspirant F.N.R.S., Unlverslte Libre de Bruxelles (now at Swansea) •

.

J

Summary In this paper, authors present a formulation and some computational details dealing with a general elastic/visco-plastie material where nonlinear elasticity is admissible and the flow rule and yield condition need not be associated. If, in a visco-plastic solution method, stationary conditions are reached for the displacements, the solution to an equivalent plasticity problem is obtained. The visco-plastic approach thus provides an alternative technique to solve elastoplastic problems, and which is found to possess considerable merits vis ~ vis other iterative processes. In particular, non-associated flow rules and strain softening can be dealt with inageneral purpose program without requiring specific numerical artifices.

Further,

by providing always an equilibrating solution (within the approximations of the finite element discretisation) and, at displacementss~a~i~rmri~, ensuring a plastically admissible stress distribution, results always give a lower bound to collapse. The paper includes several examples to illustrate the application of the method to some problems of practical interest.

i.

INTRODUCTION

It is now customary to apply finite element techniques to obtain the solution of non-linear material problems (ZIENK!EWICZ27).

A great variety of specialised

numerical models were developed to deal with particular situations such as viscoelasticity (WHITE 19, ZIENKIEWICZ, WATSON and KING24), creep (GREENBAUM and RUBINSTEIN 9, SUTHERLAND 17, TREHARNE 18) or classical elasto-plastieity (ARGYRIS and SHARPF 2, MARCAL and KING II, NAYAK and ZIENKIEWICZ 12, ZIENKIEWICZ, VALLIAPPAN and KING25'26).

Not only metals but also polymers, cracking materials, rocks and soils

have been idealised by means of several ad hoc models (ZIENKIEWICZ27). The idea of a visco-plastic medium, involving both time and plastic effects is not new (BINGHAM4, FREUDENTHAL and GEIRINGER 8, REINERI6).

Surprisingly, however, it

280

arose only limited interest among structural analysts and engineers, despite its conceptual simplicity, generality and relative ease of implementation on digital computers. Early applications of visco-plasticity theory dealt with rigid/visco-perfectlyplastic plates and axisymmetric shells, under linearization assumptions, for static (APPLEBY and PRAGER I) and mainly dynamic loadings (WIERZBICK121, WIERZBICKI and FLORENCE22). The viscoplastic model does equally well in quasi static situations where the inelastic strains are of the same order as the elastic ones.

Closed form solutions

exist for simple geometries where the whole material is assumed to be above the static yield limit (WIERZBICK120) and numerical results are reported by CHABOCHE 5 for an elasto-visco-plastic structure subject to time varying thermal gradients. The formulation and experimental determination of viscoplastic constitutive relations were discussed by ZARKA 23 (microscopic approach - metals), LEMAITRE IO and PERZYNA 15 (macroscopic approach) among others;

these references contain

extensive bibliographies on the subject. Numerical methods for the solution of quasi static elastic/visco-plastic problems of arbitrary geometry were described by NGUYEN and ZARKA 14, ZIENKIEWICZ and CORMEAU 28, but so far only little numerical work has been published.

2.

FINITE ELEMENT FORMULATION OF QUASI STATIC SMALL STRAIN ELASTO/VISCO-PLASTICITY

Let a body ./~ bounded by a regular surface S under body forces ~

and surface tractions ~

be in quasi static equilibrium

, and subject to the boundary con-

ditions for displacements, velocities and surface forces

The material is supposed to be capable of transforming a mechanical energy input into both stored and dissipative forms, with possible interaction of externally prescribed thermal effects. It is well known (ZIENKIEWICZ 27) that a finite element approximation of a displacement (and velocity) field in _ ~ , by interpolation in terms of 'nodal' values .~ such as Z=o

c,~ 5=.

(2)

leads to a velocity strain distribution in /L (3)

261 where the matrices N imal s t r a i n s

and ~

depend on spatial coordinates

only if infinites-

are assumed.

With the introduction of stresses conjugate (in the sense of virtual work) to the velocity strains

T

and of consistent nodal forces .~

equivalent to ( ~

,~

), the principle of

virtual velocities yields the discretised equilibrium equations

(4a) (4b) as long as the changes of geometry are disregarded and assuming that rotations remain small to allow ~

to be simply taken as

(5) With regards to the reversible part of the total behaviour in view of isotropic cases, a specific stored-energy function, depending on the stress invariants, is supposed to represent the mechanical energy that can be released on instantaneous isothermal stress removal

where

The rate at which mechanical energy is stored can always be written as = c~-T M~ ~ where ~

(7)

is a symmetric, generally stress dependent, matrix given in Appendix I.

Further, it is assumed that the total stress power

~J = ~c~ ~-T~

can

be written as a sum of three independant terms

where

~

= (~_r~ 8

denotes the contribution of prescribed thermal effects and

~o : 0 "T ~~vF>.O ~-

designates the irrecoverable power of dissipation due to

viscous and/or creep phenomena.

Here,

~e

and

~vp

have dimensions of strains and

can be considered as internal state variables describing thermal and dissipative effects.

The balance equation for the stress power gives

T ( ~ _ M ~ _~o_ ~ ) : o

(9)

262 from which the stress velocities are obtained

Since dissipative and thermal effects are present, some heat supply per unit volume and unit time and heat flux will exist in ~ h principle of conservation of energy;

and on 5

to satisfy the

the quasi static hypotheses are therefore

extended to the consideration of processes where the retro-actlon of is negligible, such that

$o

onto ~ B

remains externally controlled everywhere in -f]- at

any time. Combining the local equation (I0), the general equilibrium equation (4b) and the strain displacement relation (3) one obtains

K T =J.o~.~r ~'J'~B

where matrix.

~.fl-

is a stress dependent tangential stiffness

Assuming instantaneous elastic stability, it is possible to solve (II)

for the velocities

A constitutive equation remains to be chosen for

~

; herein a generalized

form of the elastic/visco-plastic model proposed by PEI~IYNA15 is adopted since it covers several other models as particular cases.

PERZYNA's constitutive equations

state that the irrecoverable dissipation occurs only if the stresses exceed the static yield condition

)> where

a) b)

~

is a positive, possibly time dependent, fluidity coefficient

~(~-)

is a positive scalar-valued monotonic increasing function

in the range

~>~

such that

~-'f(~)

exists and possess similar

properties in the same range c)

the notation 4 >

stands for

d)

F= ~(o-~ ~vF')-7~< )

o

if

represents the yield function, being zero

when the static yield condition is satisfied. e)

y(~)

f)

q=Q(~,~.v~4-~

g)

~

is a static yield stress is the visco-plastic potential

is a hardening parameter, either state or more generally history

dependent and whose value is then given by the integration along the path of past states of an evolution equation

h)

~o

is a positive quantity introduced to make F / ~ o

dimensionless to

263

allow arbitrary forms of the function Specific forms for Fj ~

and ~

were derived by NAYAK and ZIENKIEWICZ 12 for

some popular isotropic yield criteria and plastic potentials, in a form convenient to computations and can be found in Appendix II. In isotropie situations F

and (~

depend on the stress invariants;

as shown

in Appendix I, it is also possible to write

where f

i s ' another s t a t e dependent symmetric m a t r i x which should be at l e a s t

positive semidefinite since

and that vanishes for any stresses below the current static yield surface ~--O The mathematical nature of &he final problem to be solved can be better appreciated after ~ •

-4

and ~ -4

*

have been eliminated between (3, IO, 12, 14) T

-I-9

T

-1. v F

(16)

~t thus g i v i n g r i s e to a n o n l i n e a r f i r s t

order system of p a r t i a l d i f f e r e n t i a l

f o r the stresses and hidden v a r i a b l e s ~ v p J k~,

equations

In the f o l l o w i n g sections,

simplifications will be achieved by considering linear elasticity and numerical integration. 3.

LINEAR ELASTIC/NONLINEAR VISCO-PLASTIC BEHAVlOUR

Isotropic linear elastic properties result from a particular form of the specific stored energy function:

B(4-=~) =

where E

0.,~2 i.

&E

3Z

(17)

E

and ~} are Young's modulus and Poisson's ratio respectively.

Equation (I0) becomes

-_ _D where D

is t:he usual elasticity matrix;

(18) integration in time gives O- I

~ / d e n o t i n g an arbitrary co-ordinate dependent stress distribution. The tangential stiffness matrix reduces to the ordinary stiffness matrix

264

which can now be assembled, part inverted and kept in tkis form once for all times ready for further operations involving

~-~ --

Initial conditions determining a solution to the initial value problem (16) may consist of an elastic set as follows: for

~

)~

~~ ~~

being given arbitrary values one has

- ~. ~,

(21)

i

,/0=¥C

o

eo

vPo

o)

4.

PARAMETRIC DISPLACEMENT FIELD AND NUMERICAL INTEGRATION

The diseretization of the velocity field (2) is generally not sufficient to eliminate fully the space variables ~ (~c.,,, ~ v p

and ~

which are still implicitly presence since

are allowed to vary within every finite element unless

simplex constant stress elements are used (ZIENKIEWICZ and CORMEAU28). Even if, at a given stage ~

of the calculations, the analytical form~[.~,~'~)

were known, the exact integration of

J-O.~T.~ ~vp ~-/I-

would remain

difficult since the instantaneous visco-plastic boundary crosses some elements; moreover

considering the possibly complex form of

~vP

, it is concluded that

numerical integration will be required, a context in which (iso)parametric elements find their natural justification.

These elements, fully described by ZIENKIEWICZ 27

have proved their capabilities in many non-linear problems. Displacements and velocities are interpolated over an element by means of parametric equations where the parameters

~

and ~

(2-D problems) are curvilinear

co-ordinates ranging from -i to +I: =~

~ (~,~ ~

I = ~ N ~ (~,~) ~

displacement interpolation

(22a)

co-ordinates transformation

(22b)

Introducing the Jacobian matrix

one has for planar cases,

and for axisymmetrie cases, where

$~.=o

is the axis of revolution

265

The matrix ~

depends on Cartesian derivatives of the displacement interpol-

ation functions NL

uonvergence,

and on the radial co-ordinate in axisymmetric cases

with

uniform mesh refinement, of the finite element approximations,

to the solution of the continuum problem, will occur if the following (sufficient) conditions are met: a)

the approximate displacement field can represent the exact one in the limit;

if the constant strain and interelement compatibility criteria are satisfied, this condition is fulfilled provided the exact strain field is bounded in-f~ . b)

nodal equilibrium equations are exactly satisfied when constant stresses

and body forces prevail in all elements, provided the exact stress field is bounded in YL . c)

Constitutive relationships are satisfied for all stresses and strains

appearing in the discretized equations (i6). Conditions (a) are always satisfied by isoparametric ( ~ by subparametric elements if linear relations

W~g= ~ N ~

~

elements and exist

(ZIENKIEWICZ27). Condition (b) requires an exact integration of

J~_ N ~

~-/k

for constant vectors o- ,

if body forces are present, ~v~

, ~ @

, -- .

Relevant 'square' Gaussian rules can

be found in Appendix III for various 2-D cases. Condition (c) is related to the convergence properties of the numerical method used to integrate (16) in time. In order to examine the effect of numerical integration on the discretization, integrals are approximated by the symbolic formula

266

where T

denotes the total number of integrating points in .O- and

This leads to a further diseretization since the approximate fields O~ , --~ vt , t~. depending on ~

and ~

are replaced by a finite number of time dependent

values at the integrating points only. Introducing 'structural' vectors grouping all integrating point quantities such

(27) it is possible to rewrite the diseretised equations in a compact form:

(28b)

where: B

~"

~

(28e)

(28d)

.

.

.

.

.

.

.

.

.

E

(28e)

is a constant structural strain displacement matrix made of elemental matrices computed at the integrating points is a constant diagonal matrix of -GA

D

coefficients defined in (26)

is a constant syrmetric matrix made of ~

matrices following each other

along the diagonal of ~) is a state dependent symmetric positive semidefinite matrix with structure analogous to that of and such that

C

and C

D = D~

~

=~-n C

In the case of linear elasticity, Eqns. (16) are replaced by (29a)

(29b)

(29c)

where

5=D S K

-4

T

D -D£

-4 (BOa)

267

is a constant, symmetric, negative semidefinite matrix and where

--

=

D B K- B c

~

'

(3Oh)

is a known, time varying, vector. For ideal visco-plasticity, Eq. (29c) does not exist and ~ '

depends solely on

so that finally the problem reduces to a nonlinear ordinary differential vector

iAl

equation

= where ~

=(~)(~

± ~-=

~)

(3l)

, though itself non-symmetric, is the product of two

symmetric matrices. Alternatively it is possible to derive a similar expression for weighted viscoplastic strains

I,,,

~

~

~-" ~..~

(32b) - T

but where the stresses are still implicitly present in

A'

and l '

It is seen that the size of the numerical problem is proportional to the number of integrating points in./')- ;

it is therefore of prime importance to use the lowest

possible order of numerical integration compatible with convergence conditions ( b ) mentioned earlier in this section if economy requirements are to be met. 5. 5.1

COMPUTATIONAL STRATEGY

Basic alg£rith~ The solution to (29) starting from initial conditions (21) can be obtained by a

time marching procedure. Let an equilibrium situation be known at a time t~a~

A vector of current pseudo loads is kept permanently up-to-date (33) Increments

interval

of hidden

variables

and pseudo

loads

are

calculated

over

a time

~ ~;~ = ~z,t~ - gzn ~_e . e (34a)

(34b)

268

(34c)

(34d)

v,,, +

(34e)

and the total displacements, strains and stresses are fully recalculated at time m~4

~

,,,v , ' 1 ' 1 4 4

(35c)

so that nodal equilibrium is maintained exactly at all stages of the computations. This procedure is slightly different from that presented earlier by ZIENKIEWICZ and CORMEAU 28 since it avoids the possible accumulation of errors due to the summation of incremental displacements;

the proposed process, however, would not apply in

the case of nonlinear elastic properties, where K T

is to be used with incremental

displacements only. 5.2

Time intervals selection Finite time intervals add new errors to the usual finite element space dis-

cretization errors.

Though a constant time step is the simplest procedure, it is

uneconomic near steady state if it is chosen as to give accurate results in early transient stages.

A variable interval is desirable, increasing when stresses

approach stationary and decreasing when the visco-plastic flow accelerates (when strain softening occurs in highly stressed regions or upon instantaneous load application). Two empirical criteria were used with success: Criterion I:

limitation of the incremental visco-plastic strains I

where

~

~

Z

~

Z

~

X

is the minimum taken over all the integrating points in.(]- and

269

where ~

is a time increment parameter specified by the user.

indicates the ranges

~4~O.~

Practical experience

for simple problems of contained plastic flow

o.O~ O=l~ "gil~F~/O F=I'~{~°'')-'=° ~ =lt~ °F=o= ~al~'O F o

Underthe additional assun~Qtionsthat f~(O~_]

is a homogeneous function of degree ~

~(O')-~=O

o

represents a non-concave surface in the space of .~°"

it is possible to prove two bound theorems:

272

lower bound theorem If, under the constant load~ ~ L ~ reached

in

s u c h that

, total stationarity conditions are

=o

o

then

where

the latter denotes the perfectly plastic collapse load parameter associated with-i~.~ and

~ This is an i~mediate application of the well known lower bound theorem of

DRUCKER, PRAGER and GREENBERG 7, since at tc~l stationarity, ~

is plastically

admissible and in equilibrium with ~ L ~ upper bound theorem If, under the constant loa~s ~v ~ reached in

such that ~ = O

,.partial stationarity conditions are and

= constant ~ O

The proof relies on the fact that, if ~

then ~ f ~ V

is an arbitrary plastically

admissible structural stress vector,

~ >O

, and follows lines

very similar to those of the lower bound theorem. These theorems bear out a very useful feature of the visco-plastic approach in limit analysis:

whatever constant loads act on a structure, the final viscoplastic

results will indicate their position with respect to the corresponding collapse loads.

Therefore, the safety of a structure designed to bear

specific maximum

service loads can be tested by a unique visco-plastic analysis under full loads; this contrasts with the incremental elasto-plastic approach where load increments must be specified ab initio according to some predicted value of

~ -

, an

estimate that is often not available with great accuracy. 7. As a first example,

APPLICATION

a rectangular plate of aspect ratio 2/1, unit thickness and

initially free of stresses, is clamped to a rigid support and instantaneously cooled; the restraint put on shrinkage strains near the fixed edge induces stresses which are assumed to violate the static yield condition of the plate material considered to be in a state of plane stress.

Symmetry permits to analyze only a half-plate, as

shown on Fig. i where the element subdivision is refined in the region of the expected stress concentration. Elastic and asymptotic elastic/visco-plastic displacement fields are almost identical, except for the stress peak region where a slight increase of displacements, parallel to the support, is observed.

The stress reduction is therefore

mainly due to relaxation at quasi constant total deformation. Elastic and asymptotic stress distributions (Fig. 2) show a significant stress variation along the entire bonded edge while the plastic straining -

~p=

~

~

remains localised at the corners.

(~

: strain hardening form, Appendix II)

273

RIGID

Plastic region

SUPPORT

~F///////~

T,.- . . . . .

+. . . . .

+ - - - ~ - - - ~ -

t-l--r**~,t

E=100 bars

~=0.52 Y=I bar

~T=-O.01

\

.

.

.

.

.

.

.

.

.

.

=0.03 time param. Perfect yon Mises pl ; Fo = Y ¢Ix)=x T =I seeG2 Gaussian integrat ~ l e r ' s rule

i.

.

'

,o

X

Deformed mesh(exaggeration ~ )

Mesh subdivision(96 elts.)

RECTANGULAR VISCO-PLASTIC PLATE. UNIFORM TE~Ph~ATURE DROP FIG. I Another typical application, concerned with limit analysis, elastic/visco-perfectly-plastic

is that of the

cantilever viewed on Fig. 3, and subject to either

an incremental prescribed displacement or an incremented concentrated load.

For

comparison with bending elasto-plastic beam theory it is useful to introduce the dimensionless variables

for which the theoretical load-deflection formula for the mid depth node at the free end is

5

~ 5

Fig. 3 also illustrates the time propagation of the viseo-plastic boundary when the beam is subject to a single step load increment

k = 1.5, for which no collapse

was encountered. Fig. 4 shows how various finite element results compare with beam theory. an incremented load, again, no collapse occurred for

k = 1.5;

With

EULER's rule failed

to converge under ~ = 1.51 after 50 additional time steps but HEUN's rule did reach a fully stationary state for ~

= 1.51 and ~

an even better lower bound to collapse, time steps were used. last results:

= 1.52.

~ L = 1.55;

Incremental displacements give in this case, however, smaller

There is no doubt about the lower bound character of these

stresses are in equilibrium with the concentrated reaction and no

longer violate the yield condition when steady state is reached.

274

IB

js

f

"

/"

c~ vp

~

O.Sm

o: ¢last |¢ ( ~[ ] BAUER & REISS

%

Ibar

]A RECTANGULAR PLATE. UNIFORM TEMP.DROP.STRESSES(section A-B) e:instantaneous elastic response vp. asymptotic state

B

is

o I o:eZasllc [~ l -,bar

0

"C'X~

A

I0

O.OS

~p

IA FIG. 2

The negligible difference between the deflections given by a unique algorithm (EULER here) for a load

~ = 1.5, applied either in a single step or incrementally,

is of prime interest and renews the suggestion that proportional incremental loading is superfluous in visco-plastic limit analysis (unless load-deflection characteristics must be fully determined). The comparatively poor deflections predicted by the fully integrated (G3) parabolic elements confirm previous tests in favour of the reduced integration technique in isoparametric elements, while the use of more complex cubic elements had no significant effect on the results, To finish with this problem, Fig. 5 represents the time variable stress distribution, under one step

~ = 1.5, in section AA of Fig. 3.

275

I^ I

time

Stcp~ I ° y ~__°-l~Ct~o o ~ ~ o

; / ~ /

Dimensions:l=36 cm ha6 cm

~

/

o

E=50000 bars 9=0.2 Perfect von Mises pl. F =Y=50 bars

~ ~

o.

~ ~-6"--o/

:il:o,

^

Linear flow rule4~(x)=x°

~ '\

o

(

kkQuadratic or cubic elements or G3 Gaussian integration

o

~h |

,

ELASTIC/VISCO-PERFECTLY PLASTIC CANTILEFER BEAM LOAD OR DISPLACEMENT BOUNDARY CONDITIONS FIG. 3

~.5

/

/

°/. (D

)

E

~

i< - -

NCREktENIAL OISPLACEMEt~*PARABOLIC **

O

• /F/

///

.

.

X

,/~/,--~ ~de~t~ca!

~/

//

not4,

"I

BEAM |HEORY

IINCREHENTAL

LOAD

;dem

idem

"dem

"dem

ONE STEP

LOAD

|or ~z~or 3x~

G2 ELTS*

EUN ~/~0~03

+PARABOLIC G2 £LTS+EULI[R

G3

.pARAgOLIC

2

HEUH

idem

I~em

~dem

G2ELTS.EUIER

GAu SSIAN INTEGRATION in "

T-0.~0

idem

space

3

ELASTIC/VISCO-PERFECTLY PLASTIC CANTILEVER BEAM NON-DIMENSIONAL LOAD-DEFLECTION CURVE AT FREE END. FIG. 4 The more realistic application which follows consists of an axi-symmetric pressurized thin shell for which experimental (DINNO and GILL 6) as well as numerical 15 elasto-plastie results (NAYAK and ZIENKIEWICZ) are available. Fig. 6 gives full details about problem specification, together with the localized deformation, at the sphere/branch junction.

Quadratic isoparametrie elements,

and EULER's rule were used once more. incremented.

2x2 Gaussian integration

The pressure was applied in a single step or

276

??

??

[! -

0.5-

-I i

O

i

L.ol .,4I ~, 1

(a) (h)

ELASTIC/VISCO-Ph~WFECTLY PLASTIC ~ CANTILEVER BEAM. STRESS REDISTRIBUTION IN SECTION A-A. (a)tShearing stress (b):Normal bending stress. FIG. 5 Two load/displacement curves (Fig. 7) at the branch end T and at junction A

show:

a)

a good agreement with the elasto-plastic analysis based on the same mesh pattern

b)

that an incremented or a one-step load give identical results

e)

an appreciable sensitivity to time interval size: overestimate the inelastic strains.

larger time steps

The number of time intervals needed in various analyses can be found in Table 2. When the incremental approach took 386 steps to reach stationary conditions under 1140 p.s.i, only 120 intervals were sufficient under full pressure. TABLE 2 Incremental

p (psi) 900 950 IOOO 1020 1040 1060

"6= O.15 time steps Incremental

= 0.03

14

31

5

16

20

p (psi) 900 920 940 960 980 iOOO 1020 1040 1060 1080 IiOO 1120 1140 1160

= 0.03 time steps One step

4

IO

6

p (psi)

1140

time steps

120

8

7

6

8

8

36

34

76

43

82

62 > IOO

1180 No convergence after 400 steps

As a last example, the relaxation of a rock mass around a lined tunnel is analyzed.

In this problem, described in Figs. 8 and 9, the rock behaves like an

277

SCALES

l

!

!

f'~

"°g'"'

/ l

, ,

\ •

/- //

~

displac,mcnt s [magnification

:20]

, / I~i//

L

I

N i

2s13

~ o 127

L, All materials

:E=2.912x107 pal ¢=0.3 Branch material:Y=38750 psi Weld material :Y=40540 psi Sphere material:Y=42340 psi Perfect von Mises plasticity Linear flow rule ~ (x)=x Mesh: 54 quadratic isoparametric elements. G2 Gaussian integration Loading: internal pressure applied incrementally or in one step.

AXISYMMETRIC ELASTIC/VISCO-P~ECTLY PLASTIC PRESSURE VESSEL.

FIG. 6

278 pressurcCpsl)

4000

~

...........

o- . . . . ~C

X incremental ¢iastoplastic F.E.~I.%| rl v.p.relaxotion under tt40pSi(~'=.03)

5oo

/

o ,nc~..,.,o, ..p.~.,o.o~io.(r..o3) A

. Id,~

__

steady state

....

continuing v.p. flow

(r=, s)

branah/sphere junotion disp.

VA

pr¢ ~surc(psi) expcrlm. [£]

-

~

........

13 v.p.rcloxotion under 144Opsi(T=.03)

0 Incrcmentol v.p.rmloxotlon (C=-03)

/

a /

--

, 0

. . . . . ~-----e-

.04"

id=m

(I:,4s)

Steady star"

branch top disPlacement .02x .OY

PRESSURE VESSEL: PRESSURE-DISPLACEmENT CURVES

FIG. 7 associated COULOMB visco-plastic material with a linear flow rule ~(~)=~: the sake of simplicity,

For

no displacements or relaxation effects are supposed to take

place until total completion of the tunnel. The instantaneous elastic response is depicted in Figs. 9, iO and 12(a).

The

yield condition is violated in separate zones (Figs. 8 and i0) and some tensions, less than

C c ~

, appear in areas distinct from the plastic zones.

The state

of stress in the lining consists of high compressions in the wall and small stresses (some of which are tensile) at the crown and in the floor.

MOHR's diagram (Fig. II)

suggests that a considerable proportion of the loads are transmitted to the lining by shearing stresses distributed along the inclined parts of the rock/lining interface.

279

Mesh subdivision(63 quad.elsi Deformed mesh(exag.loo) Lining:E--4.32xS0|psf $ =0.15 Rock: E=0.72xS01psf $=0.20 C~14400 psf ~=~'-30 ° Mohr-Coulomb perfect plasticity(associated)7 Fo=C c o s ~ ( x ) = x Euler's rule; time increment parameter ~=0.05 ROCK RELAXATION AROUND A LINED TUNNEL

FIG. 8 Asymptotic displacements

~'~

Fig. 8 and their inelastic components

obtained after 27 time intervals are given in ~

-Go

show (Fig. i0) a shortening

of the upper and lower vaults and an elongation of the lining wall, thus causing a transfer of compressions from wall to vaults.

The mechanism of this transfer

appears from the orientation of the slip facets families

S~ and S~

in Fig. IO:

MOHR's diagram shows a reduction of the shearing stress and an increase of the normal stress acting upon these faeets3a family of which remains parallel to the rock /lining interface. Finally Fig. 12(b) depicts the asymptotic state of stress in the lining (suppression of lining tensions) as well as shrunken rock tensile zones.

Although the

present material properties did not cause pronounced plastic flow, the general conclusions drawn here remain~valid, with amplification, when the cohesion

was

reduced, resulting in the expansion of viseo-plastic zones and greater stress variations in the lining.

280

gO

80

70

60

50 70

80 90 LI~1ING Dt~"O~-ATI ON

~~0 ~

FIG. 9 8.

CONCLUDING REMARKS

Due to space limitation, only four examples have been presented and discussed. Problems involving non-linear flow rules, isotropic hardening or softening and nonassociated viscoplastic laws have been solved without creating any particular difficulty and bear out the versatility of the method in materially non-linear analyses. Though the present formulation deals with isotropic situations, it should be possible to extend it to anisotropic cases such as laminated and joint materials. The approximations related to the visco-plastic approach (finite element space diseretization and numerical integration in the time domain) stand out to be well defined;

this contrasts with the elasto-plastic approach, where several

computational

artifices such as corrections in plastic stress estimates must be used to restore the stresses to the correct yield surface and where equilibrium must be permanently checked. However the visco-plastic method was shown to lead ultimately to a non-linear system of first order differential equations, the numerical solution of which can, in principle, be obtained by a great variety of classical procedures.

281

LINED TUNNEL/PLASTIC COMPONENTS OF DISPLACEMENTS FIG. I0 In practical engineering applications, extreme accuracy is not the goal.

An

ideal scheme of integration should: - minimize the number of time intervals for a prescribed accuracy compatible with engineering practice -

prevent the appearance of numerical instabilities or, at least, detect them and take appropriate measures after their detection

- give an estimate of the truncation errors cumulated as the integration proceeds in the time domain -

-

remain reasonably simple to implement on computers (coding and especially storage requirements) be conceived to deal with systems of realistic size (at least 500 simultaneous differential equations)

To the authors' knowledge, such an ideal scheme has not yet been produced in relation to visco-plasticity applications and opens a wide field to future research.

282

z"I

'

~

~-

~

J

x

/

-

'

-

~

%

~'

'

l

:

- - . . . . . . i.i~io, .t~t.

.....

¢tosUc sta~,e msymptot,cs t a t y

~x1@thsltt 2

:

\\

~/-__~-

-2

/

STRESS C:KA.NGEAT I N T U I T I N G POINTS G1 ,G.2(de~±nec]. on P i g . 8 ) FIG. 1].

J

.

~o s

,b~I,~,,

,,

i

~I00bars

,o5 ,b,l,q,,

~

~OO bars

%erosion ELASTIC PRINCIPAL STRESSES

(a)

ASYMPT0r'2IO PRINCIPAL SrfR.I~SES (b)

\ FIG. 12(a)

FIG. 12(b)

283

APPENDIX I:

The derivatives

of

~

where

FUNCTIONS OF STRESS INVARIANTS

the

stress

invariants

~-~--

~

~

~ ~ o 3 ~j~

may be written

: -.

as

~-~

are symmetric matrices 0

0

symm.

0

0

9_ 0

o

o

o

X

-J¢~+9 5

symm.

one has

if

~'= ~4&

crTI~e T

~J

where

Similarly,

where

is a symmetric, stress dependent, matrix.

if

+~73

:

angle

"vl- ~t " v 2.

cohesion

friction angle

cohesion

~:

~:

friction

.

Z

uniaxial yield stress

uniaxial yield stress

Yield function

L~:

~(~'('):

~(~

V ysL~l(~

= ~(v)

(2.4)

and t s [O,t*], wherein

(2.5)

~Y) = (P~'Y) + I T'Y ds Moreover,

3n 2 u is also subject to the initial conditions, (z(.,o)-~u0(.),v)

:

0

;

-- 0

v

(2.6)

Let g~x) be a basis of L2(~) ; then each v in L2(~) is of the form v(x~=~vl(x)~x) vi(x) sL2(~) Let W01(~) be the Sobolev space of ~

~"

,

~

°

m

functions

whose derivatives of order m are in L2(~). We now construct of W01(~) a subspace S k(~) h m ' of finite dimension G, by identifying a collection of G linearly independent functions ¢l(X),~2(x), .... CG(X). The identification of these basis functions defines an L2(~)-orthogonal projection K h of W01(~)m into S~(~) such that the image of any ~(x) s W 0 1m( ~ ) ~ under Hhk is of the form ~h~(X)

where

=

(VllV2) O ~

The subspace S~(~)

G ~ ~ Z (~,~ ~ ) o~ (x) ~ V(X) e=l aVlV2da and ~ ( x ) ~ GZ ( ~ , ~ B ) ~i,6 (x). ~ ~ is assumed to have the following

(i) There exists a constant C I l~hV I Im _ 0, I~I - ~i,~2,a3

;

337

+

(ii) If p(x)

is a polynomial Hhp(~)

(iii)

of degree

< k,

= p(x)

(2.9)

For any r such that 0 < r < k, there exists

a constant

C > 0 such that If (I-Kh)Vl Is _< chk+l-s Iv Ik+l wherein

tv ik+12 =

~ 2da. f ~(I_~I--z Gk+iDav)

h is a real number,

generally

(2.10)

In all of these relations,

selected

so that 0 < h < 1 (i.e.,

h is a mesh parameter). Let E be the real numbers interval.

Let B[O,T]

be a linear

(generally we take B[O,T] S (~, [O,T]) Galerkin elements

= Sh(a)xB[O,T]

[O,T]-~E be a finite

space of functions

= cr[O,T],

r=0,1,2,or3).

which satisfy

3 G ~ ~ ~i i=l ~=I

=

(2.4)

defined

Here D~U = a and gIz,t)

(x,t) Aia(t)~

to seeking

among

(x)

V, > = £(V)~

the

(2.11)

¥ V ~ S kh (~)

= 0 ; (DtU(.,0)-Vo('), ~) = 0

= ZFmi(x,t)gm(X,0).. ~

Upon introducing

(2.9), we obtain a large system of nonlinear

tions in the coefficients

differential

+ <S( E giAi~(t)% i,~

)o,~6> j

(2.8) equa-

= FiB(t)

(Uo,gJ%8)o;

Z (gi+a,gJ+B)Al+(0) i,a

= (Vo,g3,B) o

Here M~

zat3 and F3~ denote the mass matrix and the generalized

M3+'le6= (pgi+~,gJ+6)O ; ~

(2.12)

Aid(t) :

Z (gi*a,g3}B)i l~ (0) = i,~

i,j = 1,2,3;

[O,T]

h (~) :

(Q(.,0)-Uo(.),V)

with

on

The space

for v ~ S k

(pD~U,V)°~ . . .+ . <S(U),VX ~_

Z M3 8Ale(t) i,a

time

is termed the space of semidiscrete

functions. Galerkin's method amounts k in Sh(~, [O,T]) the function(s) U(x,t)

into

and let

F!8(t) = £(gJ+8)O

a,B = 1,2,...,G.

(2.13) forces, (2.14)

338

Finite

Element

for a d i r e c t arbitrary systems ments)

Formulations.

and s y s t e m a t i c

domains

~ which,

of equations.

The f i n i t e - e l e m e n t application in general,

We p a r t i t i o n

concept

of the G a l e r k i n leads

provides method

to

to w e l l - c o n d i t i o n e d

~ into E subdomains

(finite

ele-

so that E e=l

On each e l e m e n t tions

X~(~)N

e

(2.15)

; ~e(~ ~f = ~, e @ f

e we i n t r o d u c e

a system of local

interpolat ion func-

such that

D X N~~~ e~ "xM") = 6MN 6~~ ; X e~ N- (x) = 0 for x ~ L here x M d e n o t e s Global

one of N e p r e s c r i b e d

"coordinate"

functions

N E e = U Z e=l N

%A(X)

(e) N ~ A is a B o o l e a n

where

coincides

functions

w i t h h = die ~e for the b a n d e d

[see

(2.10)

I~I = 0 in % (X) - U

for each

Z

defined

element

in

(2.16)

(e) N . ~ A=± if node N of L

element.

problems,

approximation

For example,

are p r e s c r i b e d

let gi(x,t)

are in sk([ e)

(2.16) 2 is r e s p o n s i b l e

in linear

a local G a l e r k i n

only values Also

generally

Property

of

independent

(i.e.,

~ N XN(X).

(e)(t) NZ m N(e) M ~iN

and of the

suppose

at nodes)

: ii(i[ ij : 6ij).

and

Then,

for

we have the local equations, + <S(Z (e) (t)ijXN(.)),~iXM(.)> e = feiN(t) ~ i NZ ajN



of the

(2.17)

transformation:

to f o r m u l a t e

(~}2)

each finite

where

(e~ e~ ~A XN ~ (x)

(2.6)-(2.8)]..

character

it p o s s i b l e

type

in ~ e ( M , N = I , 2 , . . . , N e ) .

}~ (x) are then given by

w i t h xAe~ and is zero if otherwise.

The local

makes

nodes

(2.16)

is the inner p r o d u c t e a r g u m e n t s t o ~ and e

in

(2.3)

obtained

using

(2.18)

restructions

(e) (e)_ mNM

(0 XN' e X~)

(e) = E a Na A l~ ; feiN = ~ ( ! i X ~ ) ; aiN

The g l o b a l

equations

are o b t a i n e d

-

by " c o n n e c t i n g

(2.19)

the e l e m e n t s

together" using (2.15) and the fact that ~ it), the global generalized force, is p r e c i s e l y E (e)N fe See [6,12] for a d d i t i o n a l E Z ~ e iN (t). details, e N

339

It is w e l l - k n o w n constructed h = ma~ [13],

so t h a t

{hl,h 2,

a . .

that

finite-element

(2.6) a n d

'he}' h e = d i a m e t e r

in e x t e n d i n g w o r k of F r i e d

class of e l e m e n t s

(2.8)

if Pe is the d i a m e t e r

scribed

in an e l e m e n t % ,

so t h a t

(2.ZO)

reduces

3.

[e

Indeed,

[14], h a v e

incompressible, the s t r a i n

p=m~n

to

{pe }.

sphere

Often

that can be in-

p=vh for v = c o n s t a n t > O ,

(2.10).

ACCURACY

AND CONVERGENCE

(2.9).

and prove

STUDIES

several

special

forms of

In some c a s e s we c o n s t r u c t

convergence.

homogeneous,

energy

and R a v i a r t

(2.20)

of the l a r g e s t

approximation

m a t e s of a c c u r a c y

Ciarlet

as

shown that for a w i d e

IVlk+l

In this a r t i c l e we s h a l l i n v e s t i g a t e the G a l k e r i n

can be e a s i l y

with h defined

can be r e p l a c e d by

hk+l < C -p- s I I (I-Kk)Vl Is __ where

interpolants

(2.8) a r e s a t i s f i e d ,

isotropic,

In p a r t i c u l a r , hyperelastic

esti-

we c o n s i d e r

bodies

for w h i c h

f u n c t i o n W is of the f o r m

W = W ( I I , I 2)

I3=I

(3.1)

where

I. are the p r i n c i p a l i n v a r i a n t s of the d e f o r m a t i o n t e n s o r 1 G = I + 2~ [see (2.2)]; i.e., u s i n g the s u m m a t i o n c o n v e n t i o n r r + 4, r s r s. I 1 = 3 + 27 r ; 12 = 3 + 47 r Z tYrYs-Tsy r)

; 13 =

16r+27rls s (3.2)

where

ysr = g r m y m s ,

are the c o v a r i a n t x i at t=0. Among

¥ms are the c o v a r i a n t components

components

of the m e t r i c

of ¥, and g r m

tensor associated

forms of W ( l l , I 2) in use in the c h a r a c t e r i z a t i o n

and s y n t h e t i c

rubbers,

we m e n t i o n

W = M l(I1-3) the H a r t - S m i t h

as e x a m p l e s

the M o o n e y

with

of n a t u r a l

form,

(3.3) 1

+ M 2 (I2-3)

form

W = C{ fe x p

[k!(Ii-3)2]dIl

+

k 2 1 n ~I2 }

(3-3) 2

and the B iderman form W = BI(II-3)

+ B2(II-3) 2 + B3(II-3) 3 + B4(I2-3)

(3-3) 3

340

H e r e M1,

M 2, C, kl,

examples

are c i t e d

Owing

(2.2)

= ~ • F

in 013

called

; Z = ~ W ~ ~

a static length

force

S(1)

Denote

this

D ~ d/dx.

We

even

problems

(S(Du),

(S(1) by s e t t i n g

v=V

Assuming S(1)

in

(3.6)

= S(A)

simple rod of

fixed

at xl=0,

h is

case,

subjected

normal

stresses

can be e l i m i n a t e d stress

we

to

per

unit

are

ab i n i t i o

is f o u n d

to be

i.e.,

if u(x)

ratio; then

S(1)

= S(I+Du).

type,



=

find

S(I)

a n d S(Du)

Likewise, (A,B) o.

u such

for one-

The w e a k

problem

Z(V),

f o r m of

that

V v ~ HI(o,L)

Galerkin

is the

i = 1 + u'.

use the n o t a t i o n

is then:

(3.6)

is f i n d

k U s Sh

(O,L)

such

¥ V s S~(O,L)

arrive

DV) ° = 0 , and

multiplier

end of p, and to a force

extension

Dv) O = £(v),

S(I)

(3.4)

~W

shall

we e a s i l y

~I + h 3 ~Yij

As a f i r s t

and u' = du/dx,

problem

- S(A),

the s t r e s s

cylindrical

transverse

~W

though

(S(DU),DV) o =

A=I+DU,

BVP.

pressure

of this

the a s s o c i a t e d

Denoting

free

the

longitudinal

boundary-value

whereas

its

~I ~ ~Yij

thin

Piola-Kirchhoff

displacement

interchangeably, dimensional

other

[6,pp.236-242]).

material,

that

2(1_i-3)

i is the

longitudinal

(3.1)2,

the L a g r a n g e

(see

Two-Point

the h y d r o s t a t i c

longitudinal

Z and

of a long,

at

Assuming

negligible,

of

hyperelastic

tensile

constraint,

2 ~ ~W ~=l~I~

=

pressure

Quasistatic

of f.

and the

; ij

the s t r e t c h i n g

incompressible,

Numerous

by,

are c o m p o n e n t s

3.1 N o n l i n e a r

where

is g i v e n

the h y d r o s t a t i c

consider

constants.

[6].

to the i n c o m p r e s s i b i l i t y

t e n s o r S of

where

k 2, and B i are m a t e r i a l in

3.7)

at the o r t h o g o n a l i t y

condition

V V e S~(O,L)

subtracting

that

(3.8)

(3.7).

~ C I, we h a v e

+ sl(~)el

;

~ =

I +

8(~-i)

;

eX =

i-

A

(3.9)

341 with 0 < ¢ < i, S' (~) =- dS([)/dl.

Hence

(3.10)

(S(1) - S(A),I-A) ° = (S' (~)el,e l) For most hyperelastic materials,

there are numbers Po' ~l > 0

such that ~o ! S ' ( ~ ) ! Ul for every I > Ic, where I c is some critical extension ratio > 0. Assuming this is the case,

~o I [et tl 2 !

(s' (~-)e t , t - A + ~ h t - t I h t ) = (S' (1)el,El) < pll leiIl O I IEll 10

where E l = I- Hhl

= I+u'=KhI-HhU'=(I-~h)DU=Eu,

and el=l-A=D(u-U)=De u.

Consequently

~_!

( 3. ii)

i lDeul I° o exist so that 0 < Po -< dS(1) dl V I s [Ic,~].

< Pl --

(3.12)

Let U denote the Galerkin approximation

of the solution

u of

(3.6) and e = u-U be the error. Then (3.11) holds. U k if U s Sh(O,L)_ for which (2.6)-(2.8) hold, then I IDeul Io ~ khklulk+l

;

Moreover,

(3.13)

I [eull O ~ khk+iIulk+l

where k is a constant independent of h. The results of

(3.13) follow from

with the observation that

(2.8) and

(3.11) together

I leul ll ~ el IDeulIo and

IIeulIo ~ hell IDeul Io

for the simple case under consideration. Thus, if polynomials of k Sh(O,L), the approximation is con-

order k are used in constructing vergent and I IDeul Io = o(hk), Remark:

I Ieul 10 = o(hk+l).

Observe that u - ~1 DS(u x) = u

C(Du)

/

1 dS dX

1p dl dS Uxx"

Hence

(3.14)

342

is the intrinsic wave speed in the material. S' (Du) exists V Du and

(3.12) holds,

with real and finite wave speeds.

is such that

in the body travel

9~en this occurs, we shall say that

S(Du)

satisfies

3.2.

Stretchin 9 of a Rod of Mooney Material.

the wave condition.

can be obtained illustrate,

When S(Du)

disturbances

if specific forms of W(II,I2)

suppose the material

More precise results are identified.

is of the Mooney type

[see

To (3.3),].

Then it can be shown that S(1) = 2(1

-

I-3) (IMI + M2);

h

=

-

1 -!

[M 1

+

I-IM2

(i

+

13 )

(3.15)

so that dS

dl = 2MI(I + 21-3) + 6M21-4 For stretching of the rod,

I > 1 --

condition and,

c

(3.16)

= i; hence,

S satisfies

the wave

in fact, dS 0 < 2M 1 k.

Ortega and Rhein-

[19] prove a version of the following theorem T h e o r e m 4.1

(Cf[19]).

where f: D~___En x

C o n s i d e r the system of equations

[0,i] ÷ E n.

to x and let its derivative J(x,p) singular on Dr[0,1] [or x(p(s))]

= ~f/3x be continuous and non-

for all s ~ [0,I] and assume a solution x(s)

exists.

integers kl,k2,..,

(2.1)

Let f be d i f f e r e n t i a b l e with respect

Then there exists a p a r t i t i o n of such that the sequence

remains in D and, after N load increments,

[0,i] and

{ r,m}, m = 0 , . . . , k r _ l lim x N'm = x(1). m÷~

In applications,

the major p r o b l e m w i t h this m e t h o d is its

inability to handle, w i t h o u t m a j o r modifications, J(x,p)

is d i s c o n t i n u o u s or

singular.

cases in w h i c h

Such cases are e n c o u n t e r e d

frequently in n o n l i n e a r e l a s t i c i t y in the form of bifurcations and limit points on the e q u i l i b r i u m path F: 4.2.

Stability,

Bifurcations,

x = x(p).

and Limit Points.

We consider a

m o d i f i c a t i o n of the procedure d e s c r i b e d above w h i c h can be used to determine limit points and points of b i f u r c a t i o n and to carry the solution beyond these along stable e q u i l i b r i u m paths. Ideally,

at critical points ~c such that det J(Xc,Pc ) = 0, we

introcuce a change of v a r i a b l e s x = Ay such that the m a t r i x H = ATj(xc,P)A

(4.7)

is diagonal and of rank r < n, w i t h zeros in the last n-r entries. Let y =

(YI'[2)T' [I being the first r rows of ~,

arbitrary r - v e c t o r of constants. constant.

and let ~o be an

We set ~ = A(Zo,~2 ) , holding p

This moves the solution off an e q u i l i b r i u m path F but

in a d i r e c t i o n tangent to F at the critical point ~c" constant, we iterate on p until f(x,p*)

Holding

= 0 (approximately).

P* > Pc' the p o s t c r i t i c a l e q u i l i b r i u m path is stable.

If

If p* < Pc'

it is unstable, whereas if p*-p, the test fails, a new ~i is selected (IZll

> IZol) and the process is repeated.

e q u i l i b r i u m path is reached,

Once a postcritical

the incremental loading process is

c o n t i n u e d w i t h p r e s c r i b e d load increments Ap such that Ap > o if

349

P* < Pc and Ap < 0 if p* < Pc" To determine x with sufficient accuracy, we employ a procedure ~c described by Gallagher I22], and evaluate the sign of the determinant of J(x,p)

at each load increment.

undertaking.

This is a numerically

Generally J must be appropriately

is only estimated by linear or quadratic the test is necessary by-passed

in the incremental

loading process.

point

for post-critical

too slow and expensive

scale nonlinear problems. effectively

for the practical

An alternative

is described in

[9].

analysis

just described

study of large

that has been used

In this process,

each bifurcation

(i.e., each critical point through which two or more stable

equilibrium paths cross)

is interpreted

critical point involving

only one equilibrium path)

system obtained by introducing system of equations. of perturbations of loads, ~.

and x ~c However,

interpolation.

since bifurcation points can be inadvertently

The elaborate procedure is generally

scaled,

sensitive

as a limit point

imperfections

These imperfections

in either the stiffness

(i.e., a

of a perturbed

into the original

are generally coefficients,

in the form the location

or both, and are represented by an imperfection

The incremental

scheme is based on the observation

, the post-critical approaches

equilibrium path of the imperfect

asymptotically

tion points

that of the "perfect"

parameter

that for small system

system and bifurca-

in the perfect system are reduced to limit points in the

imperfect system.

Thus, we proceed with the usual incremental

loading

Newton-Raphson

technique, checking det J as described previously, r r-i . until a critical point x =x c is reached. The system at x is

perturbed,

and the e q u i l i b r i u m path of the perturbed

beyond x c.

Newton-Raphson

iterations

system is traced

(with ~=0) then return the

system to the correct e q u i l i b r i u m path and the incremental process

is re-initiated.

Examples

of postbuckling

problems

loading solved

in this way are given in the next section° 4.3

Explicit

grating

Integration

Procedures.

(4.3), w h i c h r e q u i r e s

construct

the system of differential % + Cf(x,p)

where C is a damping matrix, x* of

(4.1)

lected so that

system.

(4.8)

integration

inte-

scheme, we can

equations

= 0

and ~(~)

is the steady-state

stable dynamical

Instead of numerically

an implicit

(4.8)

is usually 2-

solution of

Then the solution

(4.8), provided

(4.8)

is a

The damping matrix C can generally be se-

is stable.

Taking C to be the diagonal matrix

350

CI, one choice of an explicit integration scheme for

(4.8)

is

x r+l = x r + As c f ( r , p )

(4.9)

~

AS = Sr+l-Sr,

s ~ [0,i].

Since p is new held constant, x(s) may not

intersect with an e q u i l i b r i u m path at any points other than s=0 and s=l.

We cite an example p r o b l e m solved using this m e t h o d in the

next section. 4.4

Simple E x p l i c i t Scheme for T r a n s i e n t Response.

to the e l a s t o d y n a m i c s p r o b l e m

(2.9), which,

We now turn

for the present,

can be

thought of as a system of s e c o n d - o r d e r n o n l i n e a r differential equations of the form

~

+

x(0)

f(x,p) =

=

0

(4.10)

; ~(o)

x °

-

v°

Here M is the mass m a t r i x d e s c r i b e d in

(2.14).

~

We describe some

results in the next secion o b t a i n e d using the following scheme: The m a t r i x M is replaced by a "lumped" mass matrix mI.

It can be

~

shown that this does not deteriorate the accuracy of the approxim a t i o n for sufficiently smooth x(t) tion of wave fronts. solve

and it leads to a better defini-

This step also makes it p a r t i c u l a r l y easy to

(4.10) using explicit schemes.

Next, we replace

(4.10) by the

equivalent system = -!fm~(x,p)~ ,

v~(0) = Yo

(4 .ii) =

v

,

x(0)

=

x

~O

w h i c h we approximate using the divided central d i f f e r e n c e scheme v r + } = v r - 21 - mA_~t f~ (xr,p) ~ r+l x

r = x

1 + At v r + 2

(4.12)

This scheme is easily p r o g r a m m e d and has yielded surprisingly good results for some large problems. 4.4. ....A... F i n i t e - E l e m e n t Based L a x - W e n d r 0 f f - T y p e Scheme for Shock Waves in Elastic Materials. R~(g~) w i t h bases Suppose

C o n s i d e r two f i n i t e - e l e m e n t subspaces S~n(~) and

{ ~ ( x ) } G = 1 and {~h(x) }A=I' H respectively,

(~ =

[O,L]).

351

G ~ = (¢~, ¢~) A (x,t)

;

HA~ = (~A' ~r )

[Ae(t)~(x)

;

If A and Q are finite-element finite-element/Lax-Wendroff into

Q(x,t)

= ~BA(t)~A(X) -

approximations

of X and q of

(3.41), a

scheme is obtained by introducing

(3.41) and equating the projections

R~(~)

(4.13)

(4.13) k(~) and in S h

of the residuals

to zero:

y~HAFBF (n+l) =

~HArBr (n+l) = ~{HAF r - 2pAt-~2 (DS' (An) @ F'D@A) } By (n) At ~(S' (An ) - 7D~,

_ G 8A$(n+l ) = ~ { G B _ _At2(s , (in)D%~ 6

P

D~ A

)A~(n)

n + ~2

+ __ At ! (~A' ~ )BA (n)

D%B)}Ae(n) '

P

n + z2 Here A e(n)

~ A~(nAt),

generalized

forces.

"lumped" matrices

etc.

We conclude

5.1.

we replace G 8 and HAF in

so as to produce an explicit SOME NUMERICAL

this investigation

tained by applying representative

n n and £i' £2 are terms contributed

In general,

5.

integration

by

(known)

(4.14) by scheme.

RESULTS

by citing numerical

the methods described previously

results

ob-

to a number of

problems.

Large Deformation

comment on the numerical

of an Elastic Frame.

jected to a vertical

load P.

sionalized horizontal

As a first example,

analysis of large deformations

buckling behaviour of a Hookean two-bar

and P

(4 14)

and post-

frame shown in Fig.

i, sub-

If u = u/b and v = ~/b are non-dimen-

and vertical displacements

pd3/aoEb 3 is a non-dimensional

area and E the modulus,

we

of the center node

load, a ° being the initial bar

then the system is described by the equations

[2O ] (v-~) (V2-2~V + u 2) = P ; where ~ = c/b.

For ~ < /~, limit-point

since u = O V P. exist,

u[u 2 - (2~v - 2(2-v2))] behaviour

For ~ > /2, bifurcations

as indicated

in Fig.

i.

Figure

= 0

(5.1)

is encountered

in the equilibrium path

2 shows numerical

obtained using the u n c r e m e n t a l - l o a d i n g / N e w t o n

solutions

Raphson procedure

scribed in Section 4.1, together with a postbuckling

de-

analysis of the

type m e n t i o n e d

in Section 4.2, for the cases ~ = 1.0 and ~ = 1.5=

Good agreement

is obtained.

352

5.2.

Biaxial

has b e c o m e nonlinear using

Strip-Problem,

a standard elasticity.

the e x p l i c i t

on the finite

in.,

Another

0.05

5.3.

version

bodies in.,

solution

behavior

considered

degree

of AP = 0.4 ibs. from a positive (P = 2.8 ibs.)

Hence, 2.60

The

value

technique

failed

load pC was

ibs.

ibs.,

ated in the d e f o r m e d

placements

center

were

state

linear.

0.51 in. or 6.4 p e r c e n t

equations,

&P = 0.05

isolated

if the modulus of inertia

vertical,

the column

was applied at the top initial

deformation.

the v e r t i c a l

load was

(x2=L)

for which

of e l a s t i c i t y

within

case

the

ibs.

load w h i c h is approxi-

load.

Before

evalu-

buckling,

and the v e r t i c a l

dis-

at b u c k l i n g was

with a maximum

of the original width. the d e f o r m e d

shape.

After load

of the column to give the h o r i z o n t a l

increased

of load

at a load of

the critical

a small h o r i z o n t a l

While h o l d i n g again

restarted

at state of zero load,

and the p o s t b u c k l e d

determined,

changed

p o i n t occurred

height,

4

load was

matrix

displacement

Figure

critical

loading

in increments

I and length L were

near the critical

of the original

are sixth-

at a load of 2.65

well with

The v e r t i c a l

x2 = 0

to the range

of 0.07 in. or 3.5 p e r c e n t

load,

which

increment

ibs,

to converge

in w i d t h shows

The

is sub-

of incremental

at the seventh

change

at the critical

and

element m o d e l

load was applied

solution was

line r e m a i n e d

strain.

nodal displacements.

of the stability

This agrees

is pC = ~2EI/L 2 = 2.35

[24].

we cite

the surfaces

The finite

that a b i f u r c a t i o n

incremental

m a t e d by E = 6C 1 and the m o m e n t

the v e r t i c a l

The

load increments

the critical

with

and 72 unknown

equilibrium

corrections.

in

[9] on s t a b i l i t y

at finite

solved by the m e t h o d

indicating

< pC < 2.65

in

bodies

rotation.

The d e t e r m i n a n t

P = 2.4 ibs, with Newton-Raphson

were

.000[.

4, a 2.0 x 8.0 in. body

elements

to n e g a t i v e

the increment.

initially

(C 2 = 0) with C 1 = 24 ibs. per sq.

In Figure

72 n o n l i n e a r

with N e w t o n - R a p h s o n

obtained

in §4.2

For completeness,

load in the x 2 - d i r e c t i o n

polynomials,

strip in

obtained.

o b t a in e d

are n e o - H o o k e a n

of 64 t r i a n g u l a r

The resulting

c = .001 to

Bghavior.

results

fixed and x 2 = L fixed against consists

results

described

the same p r o b l e m was d e s c r i b e d

times were

of h y p e r e l a s t i c

thick.

to an axial

technique

of damping:

and P o s t b u c k l i n H

all 1/3 in.

jected

3 we i l l u s t r a t e d

methods

thick, M 1 = 24.0 and M 2 = 1.5 ibs. per

values

4-6 some recent

postbuckling

of a r e c t a n g u l a r

for c o m p u t a t i o n a l

of a strip of M o o n e y material,

of b a s i c a l l y

Stability

in Figures

In Figure

in.

for various

Agai n very rapid

stretching

time-integration

e l e m e n t model

8.0 in. square, sq.

The

test p r o b l e m

shape

the first

(p=0.15

ibs.)

the system an load constant,

from zero to the critical

by

353

the i n c r e m e n t a l technique.

For this case the e q u i l i b r i u m path of the

imperfect s y s t e m was stable to a load of 3.60 Ibs. where l i m i t - p o i n t type b u c k l i n g was indicated.

Holding the v e r t i c a l load c o n s t a n t at

3.0 ibs., the h o r i z o n t a l load was incremented to zero to p r o j e c t to the p o s t b u c k l e d path of the perfect system.

The m o t i o n followed by

the structure in removal of the horizontalload is shown by line segment I-J in Figure 5.

With the system now p r e s u m a b l y on the p o s t b u c k l e d

path of the perfect system,

the v e r t i c a l load was i n c r e a s e d from

3.0 lbs. to the second critical load of 3.75 ibs. < pc < 3.80 Ibs. w h e r e l i m i t - p o i n t type b u c k l i n g was experienced. creasing load increments from p o i n t J

The first two in-

(Figure 5) indicates that the

system has not r e t u r n e d to a point of complete relative m i n i m u m of the total p o t e n t i a l energy. c o n d i t i o n of the system.

This action is a t t r i b u t e d to the weak

The p o s t b u c k l e d path of the perfect system

from p o i n t K to the second c r i t i c a l point at point L does appear to be of c o r r e c t form in that it follows nearly p a r a l l e l to the imperfect system. A second example is indicated in Figure 6.

Here we see an arch-

type structure m o d e l l e d w i t h 72 triangular elements and ii0 degreesof-freedom.

A load was applied along the axis of symmetry in incre-

ments of 0.2 ibs., and the b i f u r c a t i o n point was passed in the increment from 1.4 to 1.6 ibs.

This is shown g r a p h i c a l l y in Figure 6.

The i n c r e m e n t a l s o l u t i o n was r e s t a r t e d at a load of P = 1.4 ibs. with the load i n c r e m e n t ~P = 0.02 ibso

In this p a r t i c u l a r case,

the

incremental solution did actually pick up the p o s t b u c k l e d path at a load of P = 1.50 lbs. Geometric deformationswere measuredrap~dlyfromthis point

(Figure 6),

and the critical load pC was isolated to the inter-

val 1.515 ibs < pc < 1.520 ibs. structure w i t h increasing load

S u c c e s s i v e plots of the deformed (Figure 7) are i n t e r e s t i n g in that they

show the step-wise t r a n s i t i o n to a b u c k l e d mode.

It is noted that one

of the members reverses curvature, which is typical of this type structure.

A small couple was applied at the vertex of the structure

and the load was increased to the critical.

The critical load of the

imperfect system was found to be 1.00 < p C
O}

dans 0

st, & partir d'ici et des r@sultats obtient

dx dy + ~O v dx dy

w, on remonte & ~,u par les #ormules

e O I w[x,y)

= y-Wy[X,y]

1

~ f~O

un Th@or~me d'existence

connus

sur les in@quations

st unicit@ pour le probl@me

du type

& fronti@re

[10) on fibre

413

(pour les d@tails Cf toujours [ I] ).

Successivement, [16]]

dans une s@rie de travaux

(Of [2], [3], [4], [5], [6], [8],

oe r@sultat a 6t@ adapt@ ~ des prebl@mes de filtration plus compliqu6s

[perm6abilit6 immiscibles

variable,

g6om@trie plus g@n@rale,

,..) et ~ des probl@mes & frontiers

lement subsonique autour d'un profil,

problems de Stefan

Tout en restant dans les probl~mes g@n@ral,

pr@sence de plusieurs

de filtration

aussi un int~r@t physique ligne d'@mergencs,

ii faut remarquer

: par exemple le d@bit et 3'abscisse simple pr@o@dent

qua, en

(tells qua (10)]

d@pendante d'un ou plusieurs param@tres

qui, dens l e c a s

(@eou-

...; Cf [4],[6],[11]],

on dolt @tudier non plus une in@quation variationnelle

mais une #amille d'in6quetions

liquides

libra de nature dill@rents

(ayant eux

initials de la

sent connus a priori].

V - RESOLUTION NUMERIQUE

La r@solution

num@rique d'in~quat/on

on dolt ajouter un algorithms de dlfficult@s

: on connait

tisation parfaitement de vue pratique programmation

pour la d@termination

des param~tres)

(en g@n~ral,

quadratique),

apr~s discr6tisation~on Par example

(Of [ 9],[2]

finies on remplacera

de disor@tisation]

n'offre pas

(Of par example [12]) de nombreux proc6d@s de discr~-

justifi@s du point de vue th@orique

tisant en diff@rences (h pas

telles qua [10) [dens le oas g@n@ral

et efflc£ent

du point

pervient ~ des probl@mes de pour 3es d@tails]

en discr@-

Ze conve×e K par une famille {Kh} h

de aonvexes constitu6s par fonctions constantes

par mor-

ceaux, qui sent non n@gatives et qui, sur le "bard discret" de 0 prennent des valeurs gh discr@tisantes tisation de (7],(8]

wh > 0

la fonction g; on tombe sur le s~st@me

~ Ahest

j

la discr6tisation

Ah Wh < 1

;

[qui est le discr@-

~ 5 points de A} :

Wh(1-A h whI = 0

dens ~' "int6rieur discret" de O Wh = gh

sur le "bard discret" de O ,

syst@me qul peut, par exemple, On remarquera finalement

@tre r@solu en adaptant que les r6sultats

essurent qua w h m w dans H I discret, non pas w mais l'ensemble ~

Oh = { [ x , y )

l'algorithme de S.O.R.

olassiques

or si l'on pose (analoguement

e O I Wh(X,Y)

sur les in@quations

mais qua la "vraie" inconnue Ou probl~me est

> O}

Q (12]]

:

414

la c o n v e r g e n c e de w h., k w dens H 1 dlscret n'assure p a s l e En e f f e t

on p e u t m o n t r e r

[ C f [ 4] ) q u e

= int~rieur

C. BAIOCCHI

,

Su un problema di frontiera

di idraulica. 107-127;

[ 2]

C. BAIOCCHI,

de ~h ~ ~"

[lim' ~h ] . h+O

B I BL I O G R A P H

[1]

"convergence"

~h c o n v e r g e v e r s ~ au sens s u i v a n t

I E

libera connesso a questioni

Ann. di Mat. pure e eppl.

(IV] vol. XCII,[1972),

note aux C,R. Acad. Sc. Paris, t.273

V.COMINCIOLI,

L.GUERRI,

(1971), 1215-1217.

G.VOLPI.

Free boundary problems in the theory o4 fluid flow through perous

[ 3]

C. BAIOCCHI,

media

: a numerical approach.

V.COMINCIOLI,

E,MAGENES,

Calcolo X[1973)

1-86.

G.A,POZZI;

Free boundary problems in the theory of fluid flow through porous media (1973)

[4]

C. BAIOCCHI,

: existence and uniqueness

E.MAGENES, Problemi di frontiera

libera in idraulica.

"Atti del Convegno Internazionale magematica"

[ 5]

theorems,Al1~, di Nat. XCVII

1.82.

V. BENCI,

: Metodi valutativi

A paraitre aux helle fisica

15-20 dicembre 1972. Academia Nazionale dei Lincei, Roma

Su un problema di filtrezlone

attraverso un mezzo poroso.

A praraitre aux Annli dl Mat. pure e appl.

[ s]

H. BREZIS,

G.DUVAUT, Ecoulement

incidence.

[ 71

H. BREZIS,

avec sillage autour d'un profil sym~trique

sans

C.R. Ac. Sc. Paris 276 [1973)

G.STAMPACCHIA, Une nouvelle m@thode pour 1'@tude d'@coulements

res. C.R. Acad. Sc. Paris 276(1973).

stationnai-

415

[8]

V. COMINCIOLI,

A theoretical

problems.

[ 9]

V.CDMINCIOLI

and numerical approach to some free boundary

A paraitre aux Annell di Mat. Pure e appl.

, L,GUERRI,

G.VOLPI,

Analisi numerica di un prohlema di frontiera

libera connes-

so col moto di un fluido attreverso un mezzo poroso. Publication du Laboratorio

[10] C,W. CRYER,

di Analisi Numerica C.N.R. Pavia

On the approximate

finite differences.

[11]

6, DUVAUT,

[12]

R, GLOWINSKI, J . L .

[1971].

solution of free boundary problems using

J. Assoc. Comput. Mech.,

R@solution

z@ro degree),

I?

d'un probZ@me de St@fan

17, N. 311970),397-411.

(fusion d'un bloc

C.R. Ac, Sc. Paris 276 [1973).

LIONS, R. TREMOLIERES, R@solution num@rique des in@quations de l a M@canique et

de l a Physique,

[13]

M,E, HARR,

[14] E. MAGENES,

A paraitre,

6rounwater

Ounod P a r i s ,

and seepage. New York ; Mc Graw-Hi11

Su alcuni problemi ellittlci di frontiera

con il comportamento

1962

libera connessi

dei fluidi nei mezzi porosi. Symposia Math,

Rome, 1972,

[15]

M. MUSKAT,

The flow of homogenous fluids through porous media.

York : Mo Graw-Hi11

[16] A. TORELLI,

New

1937

Su un problema di flltramione

da un canale. A pa~aitre,

CIRCUITS ET TRANSISTORS NETWORKS AND SEMI-CONDUCTORS NUMERICAL METHODS FOR EQUATIONS RELATED WITH

S T I F F S Y S T E M S OF D I F F E R E N T I A L TRANSISTORS, TUNNEL DIODES, ETC.

by

W i l l a r d L. M i r a n k e r IBM Research Center and Frank Hoppensteadt New York University

1.

INTRODUCTION

T h e m o d e l s of c i r c u i t s w h i c h c o n t a i n e l e m e n t s s u c h as t r a n s i s t o r s or t u n n e l d i o d e s a r e d i f f e r e n t i a l equations as is w e l l k n o w n . Because of the h i g h s p e e d of p e r f o r m a n c e of t h e s e c i r c u i t e l e m e n t s , the range of t h e p a r a m e t e r v a l u e s a n d of t h e d e v i c e c h a r a c t e r i s t i c s of c u r r e n t interest in t h e c o r r e s p o n d i n g differential equations r e s u l t in s o l u t i o n s w i t h an e x t r e m e r a n g e of b e h a v i o r . These solutions m a y be c o m p o s e d of s l o w l y v a r y i n g components, highly damped components, highly oscillatory components, and combinations of s o m e or all of t h e s e . This variation in b e h a v i o r of the s o l u t i o n s is c h a r a c t e r i z e d b y the term stiffness. In S e c t i o n 2 we w i l l d e s c r i b e a simple circuit model for a t u n n e l d i o d e a n d we s h o w h o w t h i s r a n g e of s o l u t i o n s does arise. The numerical s o l u t i o n of s t i f f d i f f e r e n t i a l equations meets with difficulties because of the e x t r e m e s in t h e r a n g e of b e h a v i o r of the solutions. In r e c e n t y e a r s a n u m b e r of n u m e r i c a l methods for stiff differential equations have been devised. (cf. G. G. B j u r e l (1970).) S o m e of t h e s e m e t h o d s have been applied successfully to c e r t a i n c l a s s e s of s t i f f d i f f e r e n t i a l equations. Nevertheless it is s t i l l n o t k n o w n h o w to d e a l e f f e c t i v e l y with this computational problem in g e n e r a l . Especially difficult is t h e c l a s s of d i f f e r e n t i a l equations which contain highly oscillatory components in its s o l u t i o n s . In p r e v i o u s studies (cf. W. L. M i r a n k e r (1973) and W. L. Miranker and J. P. ~rree~,~ (1973))one of us h a s p o i n t e d o u t t h e r e l a t i o n s h i p between stiff differential equations and differential equations subj e c t to s i n g u l a r p e r t u r b a t i o n s , and has exploited this relationship to d e v e l o p e numerical methods f o r t h e s o l u t i o n of s t i f f e q u a t i o n s . In t h i s s t u d y we w i l l e n l a r g e on t h i s p o i n t of v i e w to d e v e l o p e numerical m e t h o d s w h i c h c a n d e a l w i t h t h e f u l l r a n g e of b e h a v i o r in s o l u t i o n s just described. To d o t h i s r e q u i r e s t h a t a u n i f o r m m e t h o d be d e v e l o p e d for c h a r a c t e r i z i n g the s o l u t i o n s of s i n g u l a r p e r t u r b a t i o n problems throughout the f u l l r a n g e of i n d i c a t e d solution behavior. As is w e l l known (cf. J. C o l e (1968) a n d A. If. N a y ~ e h ( 1 9 7 3 ) ) t h e c h o i c e of asymptotic techniques u s e d to o b t a i n d e s c r i p t i o n s of s o l u t i o n s of d i f ferential equations d e p e n d s on t h e n a t u r e of t h e s o l u t i o n (e.g. s t r o n g ly d a m p e d or h i g h l y o s c i l l a t o r y ) . In S e c t i o n 3 we s h o w h o w the m u l t i t i m e technique of a s y m p t o t i c expansions m a y be c o m b i n e d w i t h t h e m e t h o d of a v e r a g i n g of B o g o l i u b o v

417

to p r o d u c e a procedure for deriving the asymptotic f o r m of s o l u t i o n s of s i n g u l a r l y perturbed differential equations o v e r the f u l l i n d i c a t e d r a n g e of s o l u t i o n b e h a v i o r . We do t h i s for the s i m p l e m o d e l p r o b l e m du e d-~

=

(A0

+ cA1)u

(l.l)

a n d we i n c l u d e t h e p r o o f of v a l i d i t y of the e x p a n s i o n to w i t h i n O ( e 2 ) . We m a k e t h e s e l i m i t a t i o n s f o r t h e s a k e of c l a r i t y of p r e s e n t a t i o n and because t h i s m u c h of t h e a s y m p t o t i c development is a d e q u a t e for the s p e c i f i c a t i o n of t h e n u m e r i c a l method. We e m p h a s i z e that this asymptotic theory (and t h e n u m e r i c a l m e t h o d ) m a y be c a r r i e d o v e r to non-linear problems a n d to a l l o r d e r s , b u t we d e f e r t h i s t r e a t m e n t to a n o t h e r study. In t h e s i n g u l a r limit, highly oscillatory solutions may conv e r g e to i n v a r i a n t manifolds of d i m e n s i o n greater than one. Thus the meaningfulness of d e s c r i b i n g such a trajectory b y a s e t of its v a l u e s on the p o i n t s of a t i m e m e s h is l o s t . Indeed this suggests the reasons f o r t h e l a c k of e f f e c t i v e n e s s of e x i s t i n g numerical methods for such problems. To r e m e d y t h i s d i f f i c u l t y we i n t r o d u c e in S e c t i o n 4 a n e w n u m e r i c a l solution concept for d i f f e r e n t i a l equations. We accept a quantity as an a p p r o x i m a t i o n to the s o l u t i o n at a p o i n t in t i m e if the q u a n t i t y approximates a n y v a l u e w h i c h the solution a s s u m e s on a n e i g h b o r h o o d of t h a t p o i n t in t i m e , t h e s i z e of t h e n e i g h b o r h o o d being arbitrary but positive. With this numerical solution c o n c e p t we t h e n c o n s t r u c t a numerical m e t h o d b a s e d on the asymptotic theory introduced in S e c t i o n 3. We i l l u s t r a t e the numerical m e t h o d b y m e a n s of c a l c u l a t i o n s b a s e d on s e v e r a l s a m p l e problems. 2.

and

In t h i s show how

A CIRCUIT

section various

MODEL

AND

STIFF

BEHAVIOR

we w i l l d i s c u s s a model for a tunnel diode circuit solution classes with extreme behavior arise.

A simple circuit representing tically in f i g u r e 2 . 1 a

L

a tunnel

diode

is

given

schema-

R

(a) E

C

v V

Figure

2.1

The current through the non-linear element is g i v e n by ! = f ( v ) , where the tunnel diode characteristic, f ( v ) , is the S shaped graph as indicated in f i g u r e 2 . 1 b . The differential equations describing this circuit are dv C ~-~ =

i-f (v) (2.1)

di L ~

= E-Ri-v

418 For certain ranges of values of the parameters, the simple model (2.1) is a s t i f f s y s t e m w h o s e s o l u t i o n s exhibit a variety of extreme behavior. We will now give an indication of this behavior. Introduce

the

new

R x = ~ t, in

(2.1).

We

variables CR 2 L

s =

'

I =

Ri,

F(v)

(2.2)

= Rf(v)

get

dv e -dx

=

I - F (v) (2.3)

dI -dx

= E - v - I

W h e n £ is s m a l l t h i s s y s t e m is s t i f f a n d s o l u t i o n s m o v e a l t e r nately through regions of slow change and rapid change. A typical family of solutions in t h i s c a s e is s c h e m a t i z e d in f i g u r e 2 . 2 a .

(o)

/if\

/\V

,:oh

Alternatively

, ~ o ^ \ \\ i ,-eo^\ \ \

~y

we

may

Figure

2.2

introduce

the

t z = -RC in

terms

of

which dv dz

--

X //8=0

(b)

DECREASING

6 =

(2.3) =

i/£

/

/ i

variables

,

(2.4)

becomes

I-F

(v)

(2.5) dI ~z W h e n ~ is schematized in

= E-I-v.

small the solutions behave figure 2.2b.

in

A different form of extreme behavior of certain other ranges of parameter values may the variable y : and

writing

the

an

extreme

as

solutions of (2.3) f o r be seen by introducing

t//~

system

manner

(2.6) (2.3)

as

a single

equation

419

d2v

dy

2'

2 +

~

dv v

=

:..~

~.~ )

du

434

~ la fonction

d~signant le produit de matrices complexes carr4es d'ordre 2, C est de ~n+l_~

~t8 d~finie par:

C(s,~ ) = ~J. k=l

ck(s, ~0)

m d4signe le nombre de quadripOles 414mentaires sont suppos4s @tre associ~s en cascade.

3°

qui

Pour une fr4quence donn4e, posons A = C(s, ~ ) =

Alors l'amplitude a e t le d~phasage ~ / ta relation ci-dessous : ea+j~ =

R0

constituent

et R

n

1 2 V RoRn

4rant les r4sistances On en d~duit

C11

F'z2

!:~21

12r"1

de la r~ponse sont donn4s par

(CIIRn + r12 + [~21

d'entr4e

le filtre : ils

RoRn + ['~22 R0)

et de sortie appliqu~es

au quadrip61e.

: ......

4°

Remarque :

Une application lin~aire de ~ 2.__)(~2 , repr~sent4e par un syst~me lin~aire complexe d'ordre 2 :

~1 + J ~ 2

I ~I +j ~ 2

~i

+J

6"1 +J

~2

~2

=

71 + j

~2

peut @tre consid4r4e comme une application lin~aire de t4e par le s "st&me ci-dessous :

L =

ol I

_4 2

~2

°~I

~I

•

4~

-~2 g2

71

72

"C2

et repr4sen-

435

B

- PROBLEME

On eherche dans le domaine K de variation des pararn&tres s, domaine qui a @t@ d~termin@ par exemple par un programme de simulation de fagon clue le gabarit en amplitude y soit toujours respectS, le point pour lequel la phase est "la plus lin@aire possible" dans la bande passante. Sur un ensemble dans la bande passante, cient de corr41ation.

de fr~quenees la lin4arit4

(U,~l) I = 1 & L convenablement

du d4phasage

est earaet4ris4e

r4parties

par le coeffi-

L

L (~IJ ~ 5--_ i=i Ce coeffieient

contraintes

Ce probl~me :

x

L (~o i_ ~)2 i=I

doit etre le plus proehe consiste

done

& minirniser

C

I.

pour s 6 K, sous les

L

C (s, ~ i )

- RESOLUTION

On introduit les multiplicateurs

R t , R 2 ...... et

I- p

de la valeur 2

!

Vl ~ ~" 1,2 A1 =

possible

le Hamiltonnien

:

Rt

:

~Is ;fl,r2 ..... ~L' L

~I'R2 ..... RLI = L

RIX (AI-C(s,W

I)

436

Remarque

: On note

(AI-C(s,~I))

R 1 ± (A 1 - C

eonsiddr~s

Minirniser pour s ~ K. On dolt ~

Mais

(s~ cO i )) le produit des

vecteurs

de

i- ~ 2 sous les contraintes

donc

chercher

(s ~) = 0

= _

~

cornme

~

point

s~

et

~2

+

~2

/I/1

scalaire

indiqu~es

point s ± on doit donc

D'autre s , on peut 4erire

part,

avoir

si on se trouve

revient

& minimiser

K tel clue : (s ~) = 0,

~I

et l

Au

de R 1 et de

~8.

=

I

:

en un point s relativement

proche

de

: n

i=l

~i 1

Al'optimum,

~

= O, et :

~) i

1 D

- ALGORITHME

1° - Choisir s (0), point de d4part. 2 ° - A l'@tape k,

s (k) @rant suppos~ connu, L analyses du syst~me

pour les diff~rentes valeurs (~ 1)1=1 & L fournissent 11 et A I, pour 1 = 1 & L.

437

O n en d~duit

:

et

3 ° - L e p o i n t s ( k + l ) e s t l a p r o j e c t i o n s u r l e d o m a i n e a d m i s s i b l e K du point s '(k+l) obtenu en modifiant une seule des composantes

~.'i(k+l) _ffi(k) _

.

d e s {k) :

~ (k)

~/~¢i l ' i n d i c e i d e l a c o m p o s a n t e modifi4}e e s t a u g m e n t 6 , it4ration.

m o d u l o n, de t ~ c h a q u e

4 ° - L e p r o c e s s u s e s t i t 4 r 6 j u s q u ' ~ c e clue s o i t a t t e i n t e u n e p r 4 c i s i o n r e l a t i v e r a i s o n n a b l e p o u r t e s v a l e u r s d e s c o m p o s a n t s , s u i v a n t l a n a t u r e du p r o b l a m e t r a i t G ou b i e n l o r s q u e l e p o i n t c o u r a n t s (k) s e t r o u v e ~ u n s o m m e t du d o m a i n e K. Note

:

Au cours des premieres it4rations, et si le domaine K est vaste, c o n v i e n t d ' a p p l i q u e r un c o e f f i c i e n t de s o u s - r e l a x a t i o n : 6"' ( k + l ) i

ff (k) i

_ ~

~(k)

:

p o u r 6 v i t e r un 6 1 o i g n e m e n t du p o i n t o p t i m u m .

, ,~+(hn n

=

e

(s)

-~+¢p p = e

(9)

445

with # called pseudo Fermi-potentials.

Equation

(7) becomes

~n = PnnV~ n = ~n e

and similarly for ~p.

(lO)

V~ n

It is also customary to introduce the logarithmic variables

~n = exp(~n) and ~p = exp(~p) yielding the form

~n = ~ne~?~n

In the zero-current or equilibrium case it is ¢ = 0 and equations

(Ii)

(8) and (9) revert

to the well known relations

n = e~

,

P = e-~

(12)

(13)

Combining Poisson's equation (5) with (12) and (13) gives the so-called Shockley-Poisson

equation

-V2~ = N + e -~ - e ~

(14)

which describes the zero-current case.

2.2

Limits of the Diffusion Model The diffusion model can be derived from the Boltzmann's equation by a set of

assumptions,

namely:

(i) The distribution can be fully characterized by the mean velocity v (or equivalently the second and all higher moments have negligible relaxation time). The problem can then be described by the first two moment equations. (2) Neglect of

~$/~t and

momentum balance) to get

of the Bernoulli term in the first moment equation (or

446 ->

nv : -J

= -n~

- DVn

(15)

n

(3) Neglect of magnetic forces and static treatment of the electric fields. Most serious is assumption (i) and its consequences, i.e., no thermal forces and no explicit energy balance equation. The electrical field-strength is not a good description of the physical state and equation (7) may lead to serious contradictions.

This becomes apparent in

regions where large density gradients are balanced by correspondingly high fields so that no current flow results. unphysical "hot-electron" effects.

In such areas, equation (7) would produce This problem has led to serious difficulties

in the analysis of the insulated-gate field-effect transistor.

A better form based

on the assumption of addltivity of the forces is [Reiser 71]

:

n

nV(F)~

(16)

with ~ = ~ + V log (n) and ~(F) the same form as ~(E) in equation (7).

2.3

Special Aspects of Various Devices Not all terms and equations of the diffusion model (i) to (6) are always

relevant and simpler models which are computationally less demanding may be adequate.

The following is a short discussion of the properties of various

important devices: (i) The bipolar transistor is based on injection of minority carriers and thus the full set of equations is relevant.

The solution depends critically on the

recombination and diffusion term and a density-dependent diffusivity may even be required to account for degeneracy in high injection levels [Hachtel 72]. A one-dimensional analysis is sufficient in most cases. (2) The field-effect transistor is a majority-carrier device and therefore neglect or approxi=mtive treatment of the minority carriers are usually justified. Recombination effects can be neglected, even if minority carriers are taken into account.

The solution is insensitive to changes in the diffusion term.

447

The proper nonlinear drift term, however, is important.

The shortcomings of

drift equation (7) are especially apparent in the current channel of the insulated gate field-effect transistor and use of the more appropriate form (16) is strongly indicated.

The fleld-effect transistor requires a

two-dimensional solution. (3) Gunn-effect devices.

In the case of "two-valley" semiconductors such as GaAs,

the solution is totally dominated by the nonlinear drift law which has a range of negative differential conductance. neglected.

Minority carriers (i.e., holes) can be

The proper form of the state dependent diffusion term is unknown

and many authors use a constant diffusivity.

Often no steady-state solution

exists and therefore a transient analysis is always required.

3.

}~R~ERICAL SOLUTION

The set of equations known as diffusion model are nonlinear partial differential equations of the parabolic type or of the elliptic type in the case of the d.c. problem.

All numerical solutions of these equations described so far in the

literature fall into the class of finite-difference methods on rectangular grids with low order centered difference approximations (so-called five-point formulas). The problem then is one of: (i) choosing optimal finite difference expressions, and (2) solving the resultant system of algebraic equations efficiently. The need for efficiency is absolutely predominant since especially in the two-dimensional case the demand on computing and the resulting costs may well be prohibitive for any extensive use of numerical analysis. The following well known properties of the solution are typical for an ill-conditioned system and make a numerical solution accordingly difficult: (i) Over short distances, the density variables may vary over several orders of magnitude. (2) The solution, especially the location of space charge areas is very sensitive to changes in the boundary conditions (i.e., the applied voltages).

448

(3) Important macroscopic quantities such as the device current are defined over differentials of the solution.

3.1

Mesh and Finite-Difference Approximation Regular rectangular grids are a preferred choice for the solution of

two-dimensional plane problems as they lead to simple finite-difference formulas and to sparse matrices with only few non-zero diagonals (a regular mesh is one where no grid line ends in the interior of the domain).

To save mesh points in

the uninteresting neutral areas nonuniform mesh spacings are often used.

Regular

nonuniform grids, however, also have drawbacks which in certain cases may even offset their advantages, i.e., (i) reduced order of accuracy, (2) numerically degenerate long-shaped rectangles in certain areas, and (3) computational overhead in the equation setup. There is a great deal of freedom in choosing finite-difference approximations to the original equations.

Two requirements are important, viz.,

(i) Consistency, i.e., the finite difference equation converges to the differential equation if the mesh is refined. (2) The conservation property of the continuity equation has its discrete analogue. The second condition turns out to be necessary in order to obtain a uniquely defined device current [Reiser 71]. class satisfying both conditions. in the literature.

We subsequently give several examples as found

For notational convenience a one-dlmensional uniform grid with

spacings h is assumed. straightforward.

Centered difference formulas are the simplest

The generalization to two space-dimensions is usually

Moreover, we give the examples for the electron current only and

omit the subscript n.

The basic centered difference approximation of the divergence

term is

V']Ix=ih = [~i+I/2 - ~i-I/2 ]/h + O(h2)

(17)

The various methods published differ in the approximations ~ of the current density, i.e., in basic variables n [Katakoa 70, Reiser 71, Suzuki 72]

449

~i+I/2

ni+l + ni ~i - ~i+l ni+l - ni 2 Vi+i/2 h + Di+i/2 h

(18)

or in logarithmic variables ~ [Slotboom 69]

$I+I12 $i+I - ~i ~i+i12 = ~i+i/2 e h

(19)

where exp($i+i/2) may he approximated by grid values by ll2[exp(~i) + exp(~i+l)] or exp(½ ~i + $i+I ) or better by [Mock 73] [exp($i+l) - exp($i)]/h.

Some

position-dependent scaling is required as ~ may become very large and, in fact, even exceed the range of floating point numbers of many present-day computers.

A

particularly simple form is obtained if the density function itself is used for scaling, in which case equation (19) becomes [Vandorpe 71]:

[P -~i+l- n i e -*i]

^ (~i + ~i+l)/21ni+l e Ji+i/2 = ~i+i/2 e -

h

(20)

"

As first observed by D. L. Scharfetter and H. K. Gummel all the above formulas may lead to gross errors in case of large voltage drops IV~I = I$i+1 - $i I over a single mesh cell.

This can easily be seen in equation (18) which in case of zero-current

condition, cannot be satisfied by both n i and ni+ I positive whenever IV~I > which is obviously an unphysical situation.

2,

As reported recently, the same type

of error is associated with equations (19) and (20) [Mock 73].

A serious

restriction of the mesh size may result in case of high bias voltages.

A scheme

which avoids this problem may be obtained by assuming J and E constant over the mesh cell and integrating the transport equation j -i = nE + dn/dx yielding

A~ ~i+i/2 = -~i+I/2 h

For

IA~!

ni+l ni l i - exp(A~) + i - exp(-A~)"

> i a pure drift current of the form

(21)

450

Vi+i/2 hA--~ni+l

if

A~ > 0 (22)

~i+I/2 = ~i+i/2 hA--~ni

is obtained,

if

A~ < 0

The superiority of this Seharfetter-Gu~el scheme was recently

demonstrated theoretically [Mock, to appear]. The discretization of Poisson's equation offers no problems. five-point formulas [Verge 62] are most widely used.

Standard

Care has to be taken that

the discretization of the boundary conditions does not violate the current conservation property.

Techniques for treating boundary conditions are well known

(viz., the so-called phantom-point method) and need not be discussed in detail.

3.2

D.C. Problem In case of the d.c. problem, the time derivatives in equations (I) and (2) are

put to zero.

In discretized form a set of nonlinear algebraic equations results

which may be symbolically written as

A(~)n

= -R

(23)

~(~)m

=

(24)

L~ = N + ~ -

where ~, ~, ~, R and N are the vectors of grid values, and A(~), B(~) and L are band matrices (three-diagonal in the one-dimensional case, five-diagonal in the two-dimensional case).

The proper form of A(~) and B(~) is defined by the

finite-difference formulas used, i.e., in the case of the standard centered difference formula (18) A(~) becomes

(25)

451

i Di+i/2 + ~i+i/2 ~(~i - ~i+l )

h2A(~) = h2[aij(~)] =

if j=i+l

1 -(Di+i/2 + Di_i/2) + ~- ~i+i/2(~)i - ~)i+l)

i - 2 ~i-1/2(~i-1 - ~i )

Di_i/2 - ~i_i/2(~i_l - ~i )

0

(26)

if j=i

if j=i-i

otherwise

where in case of field-dependent ~ and D it is ~i+i/2 = ~(~i - ~i+l ) etc. Two methods have been in use to solve this system, namely: (i) sequential iteration or Gummel's method, (2) Newton's method. Methods of the first kind are simple but have only a first order rate of convergence whereas Newton's method requires a more complicated equation setup but promise second-order convergence. Sequential iteration is the most widely used method.

It is based on the fact

that the discretized continuity equation is linear in the density variables n and p or ~n and ~p (but not in ~n and ~p).

This suggests the following algorithm:

Step i: Compute mobility and recombination rate for current values of ~k, k

k = V(~k)

and k

, R k = R(nk,p k)

Step 2: For ~k fixed compute n#+I and pk+l such that the continuity equations are satisfied, i.e.

452

k)nk+l .

o

_Rk _

,

k+l =

Step 3: Compute updated potential values ~k+l as a function of the newly computed k + l and k + l Steps 1 to 3 are repeated until the desired convergence is achieved (or divergence becomes apparent).

Note that both Step 2 and Step 3 lead to systems of

linear equations which in the two-dimensional case are of five-diagonal form and may be solved conveniently by standard iterative methods [Varga 62]. density variables ~

n

Logarithmic

and # may substitute for n and p. p

Many variations of this algorithm have been used.

They differ mainly in the

following: (I) The density state variables, i.e., basic variables n and p [Dubock 70, Vandorpe 71], logarithmic varlables ~

n

and ~

p

[Slotboom 69, Heydeman 71, Heimeier 73]

or other [Mock 73]. (2) The finite-difference formulas (see section 31). (3) The methods used in Step 3 (see discussion below). (4) The methods for solving the systems of linear equations in Steps 2 and 3, i.e., point relaxation (SOR) [Slotboom 69, Dubock 70], line relaxation (LSOR) [Vandorpe 71, Heydemann 71, Mock 73], Douglas Racheford methods (ADI) [Vandorpe 72, Mock 73], or Stone's method (SIP) [Heimeier 73, Mock 73].

LSOR is

particularly well suited for the insulated gate field-effect transistor whereas SIP is best if different device geometries are to be analyzed. The simplest way to treat Step 3 would be simply to solve the Poisson's equation _?2~k+l = N + p - n.

This formula, however, was never actually used as it would

lead to slow convergence.

Most widely employed is the method of the original

publication [Gummel 64], i.e.,

V2~ k+l + (n + p)[~k+l _ ~k] + N + p - n = 0

(27)

453

This formula is derived by linearization of the nonlinear potential equation -V2~ + ~p exp(-~) - #n exp(~) = 0 around the current estimate ~ k

A different

approach to accelerating is the following two-stage iterative process:

~k+l = ~k + k [ ~ + l

where -~"'i is defined by -V2~ -+I--~

=

_ ~k] + 8k[~k _ ~k-l]

N + p

-

n and k

(28)

and ~k are acceleration

parameters for which appropriate Chebychev sequences can be constructed [Mock 73]. A mathematical analysis of the asymptotic rate of convergence was recently achieved under some simplifying assumptions [Mock 72].

Convergence problems are

predicted for forward bias condition and high recombination rates.

Some variations

of the sequential iteration procedure avoiding these convergence problems were subsequently described [Seidmann 72].

The idea is to treat a linearized part of

the recombination rate computation in Step 2 rather than in Step i.

Note, however,

that the analysis which is based on a perturbation argument says nothing about the global convergence behavior.

Unfortunately, the informmtion about this most

important question is only fragmentary. are found in the literature.

Figures from 20 to 200 overall iterations

Generally, convergence is slower:

(i) the more charge is stored in the device (i.e., the higher N); (2) the higher the applied voltages; (3) the higher the recombination rate. Arriving at a suitable initial guess of the solution is a nontrivial problem.

In

case of large voltages~ the full voltage drop may have to be applied in small steps at a time.

It may even be necessary to build up the doping level slowly in order

to avoid convergence problems. Newton's method has so far been used in one-dimensional problems only [Scharfetter 69, Gaughey 69, Hachtel 72].

The difficulty with two-dimensional

problems is the fact that now the Poisson and continuity equation have to be solved simultaneously, thus leading to a larger system of equations with more than five non-zero diagonals. form

If for simplicity holes are neglected, the equations take the

454

(29)

where x is the combined vector of mesh values, i.e., x identity matrix.

T

= (n,~) and I is the

Then, Newton's method is:

x

k+l

=x

k

-

j-i (x k) F ~k) ~

(30)

or more conveniently for numerical computations:

j k+l = j( k)xk _ F ( k )

where J(x) is the Jacobian (i.e., Jij = ~Fi/~xj)'

(31)

Since A and L are five-diagonal

matrices, J has the structure

(32)

where A, A' = ~A(~)/~ and L are all five-diagonal submatrices.

Classical methods,

such as SOR, are not suitable for solving this system of linear equations since J has more than five non-zero diagonals.

Stone's method [Stone 68], however, is

exactly tailored to such "coupled systems" and allows for a efficient solution. Promising results with this Newton-Stone's method have already been achieved [Hachtel, private con.].

3.3

The Transient Problem Finite difference schemas of the Crank-Nicholson type [Richtmeyer 57] are the

simplest method for solving the transient problem.

In case of electrons in the

absence of holes, the discretlzed continuity equation takes the form: k+l At

k .,,k+l, k+l mt~ )~ + (i - ~)A(~k)n k

(33)

455

with an error term of the order 0(~t2, h2) in case of a = 1/2, O(At,h 2) otherwise. In case of e = 0 one speaks of an explicit scheme, e > 0 are implicit schemes.

As

is well known, the explicit scheme, which is computationally very simple (no systems of equations) has a strong tendency towards instability and may require prohibitively small time steps, limited by the following stability condition [Reiser 7Z]:

h2D '

At ~ min

where v

max

= max(~)

2D 1 Vma x

(34)

is the maximum drift velocity which in the case of

field-dependent mobility is always finite and well known.

Implicit schemes, which

avoid this stability problem, lead to nonlinear systems of equations similar to those of the d.c. problem and which have to be solved for every time step.

This

is feasible in the one-dimensional case but may lead to an excessive amount of computing in the two-dimensional case.

Therefore, despite the small At, explicit

schemes were used for Gunn diode analysis.

A compromise between the speed of the

explicit method and the stability of the implicit method is the half-implicit method of M. Reiser.

This method is based on the same principle as the sequential

iteration, i.e., the fact that the continuity equation is linear in n, provided is held fixed.

This leads to the following algorithm for carrying out one time

step :

Compute ~k for given n k according to e~ k = N - n k. Step 2: Compute new density from the system of linear equations k+l n

k - n

At

i ~. k. k+l i k-nk) ~n + y A~ 2 ~ ~

-- - A ~

Both the above steps require the solution of five-diagonal systems of linear equations.

Very efficient direct methods exist for the Poisson problem of Step i

(35)

456

[Hockney 70] which should be utilized whenever possible.

Standard iterative methods

are appropriate for Step 2. The instability giving rise to condition (34) is totally avoided by the half-implicit method.

It was, however, found that decoupling of continuity and

Poisson's equation introduced stability problems of a different kind, leading to the following restriction on At [Reiser 73]:

At < -

i

(36)

~N

Note that unlike equation (34), the above stability condition is independent of the space increment h.

Also the behavior of the unstable solution is quite

different from the exponential growth pattern of the diffusion (or linear) instability, i.e., it exhibits stationary standing waves in the density function. These oscillations preferably in neutral areas do not grow as time goes on but prevent the solution from reaching a steady-state.

The stability condition (36)

restricts the applicability of the half-implicit method to small doping levels N < !ol7cm-3, as they are typical in Schottky-gate field-effect transistors and Gunn-effect devices.

Where applicable, however, it is an efficient method and,

owing to the increased stability, faster than the explicit method. The transient solution is the most general one and a computer program for a transient analysis provides answers to the following problem areas: (i) large signal responses; (2) small signal analysis by means of Fourier analysis of step responses; (3) steady-state or d.c. analysis. To compute d.c. solutions one starts from a suitable initial condition and proceeds until a steady-state is reached.

Unlike in the case of sequential iteration, this

initial solution is completely uncritical and o

= N has proven to be a good choice.

Furthermore, the method is very insensitive to large voltage steps and viewed as an iterative d.c. method, has a linear rate of convergence.

It is therefore

comparable in efficiency to the sequential iteration method but has the important

457

advantage of giving additional information about important time-constants of the device as a by-product of each d.c. computation.

3.4

On the Problem of Accuracy In many publications the adjective "accurate" appears in the title but they

generally fall short of convincingly demonstrating its justification.

The

mathematical theory is still far from producing error bounds for a given mesh and even the somewhat weaker question of convergence of a given finite-difference scheme (i.e., whether the solution of the difference converges to the solution of the partial differential equation) generally remains unsolved.

Some progress towards

a mathematical treatment of the convergence problem has been made recently [Mock, to appear].

Bounds on the error in the total device current, introduced by an

imperfect potential solution ~ were found.

The assumptions are (i) D = ~ =

constant, (2) ~ and p such that the continuity equations are fulfilled exactly [i.e., V-(V~ - ~V~) = 0], and (3) the current i is computed as a suitable average of the currents through the ohmic contacts.

Then the error II - ~i is found to be

bounded by

iI - ~i ~ C l ~ -

where C I is a suitable constant, the potential residuals.

(37)

li~resll

l i" I I is a norm and ~res = V2~ + N + p - ~ are

Note that this result is obtained by purely analytical

means for the general two-dimensional case with mixed boundary conditions.

In one

space dimension, equation (37) may be sharpened and applied to particular finite difference schemes.

It is found that standard centered difference formulas for

the Poisson problem together with Scharfetter-Gummel type formulas for the continuity equation yield an error O(h 2) in the computed device current. schemes are shown to have larger error, i.e., of the form O(h).

Other

This important

theoretical result supports the earlier finding that th Scharfetter-Gummel formula is better than other common discretization schemes.

458

In the absence of a rigorous mathematical theory, numerical experiments are the only means of investigating the properties of numerical methods.

The following

should always be performed and the results disclosed in the publications: (I) observation of the solution as the mesh is refined; (2) convergence properties of the iterative equation solvers; (3) check on the independence of the device current of the integration paths; (4) single and double precision computations. With regard to point 3, for example, it is generally found that a large portion of the required iterations is spent to slow and minute changes in the field variables in order to get a path independent current. Round-off errors are an additional source of inaccuracy.

Although there is

generally no error build-up in the Iterative equation solvers for the systems of linear equations arising, the finite precision arithmetic may lead to serious accuracy problems, especially in the case of fine meshes. computations are advisable when economically feasible.

Double precision

In the case of

Scharfetter-Gummel formulas, special attention should be paid to the evaluation of exponential terms in the denominators.

It is an advantage to test the value of

IA~I and revert to a standard flnite-difference formula in case of IA~I 0) De m~me de

CA

et C D

les coefficients de diffusion D A

et D D

varient en fonction

d'apr~s les lois : Z +~I

(9)

DD

= Do D

n. cn 1 Z +

(i0)

DA

= D°A

2 +~' Z +

÷ 13

CA '

~- ni

DOA, DOD sont les v a l e u r s des coefficients pour le s e m i - c o n d u c t e u r intrins~que,

@ et ~ '

sont des coefficients qui c a r a c t 4 r i s e n t le

p r o c e s s u s de

diffusion de l'impuret4 consid~r4e. Toutes ces qualit4s sont strictement positives de m ~ m e tions inconnues

CA

et C D

.

que les fonc-

471

I. 2.

- Conditions aux limites

L a fabrication des dispositifs se fait en plusieurs phases successives, les unes dites de pr4d4pSt ou k faible temp4rature,

on fair circuler un gaz porteur

d'impuret4s qui restent en surface du dispositif, les autres dites de redistribution qui se font h temp4rature 41ev4e pour permettre aux impuret~s de diffuser dans le volume. Les phases de redistribution sont a c c o m p a g n 4 e s

d'une oxydation en surfa-

ce ce qui va poser un probl~me particulier dfl ~ la pr4sence de deux milieux diff4rents (oxyde et silicium) les ~quations dans l'oxyde 4rant 14g~rement diff4rentes. Par ailleurs,, la position de la fronti~re g4om4trique ainsi que l'4paisseur totale varient au cours du temps. Les conditions aux limites ~ prendre en compte sont alors les suivantes : I. 2. i.

- ModUle unidirnensionnel

FLUX entrant

FIGURE

~

~

FLUX sortant (nul)

Si

j,,

;

t

0

A

B

..~

x

1 -. Diffusion unidimensionnelle (pr4d4p6t)

Dans le cas d'un pr4d4pSt, on suppose que le flux sortant est nul, soit

:

co (11)

x

CA (B)

=

~x

(B)

=

0

O n connaft les flux entrant en surface (IZ)

JD

= ~(Co

C D (A))

:

472

ou

sa

= o¢(Co

-c A(A))

On ne p r d d d p o s e en effet q u ' u n e i m p u r e t g ~ la fois. ~'

e t Co

sont des constantes positives.

On p o u r r a i t b i e n stir p r e n d r e

en consid@ration dans le m o d M e route

a u t r e c o n d i t i o n a u x l i m l t e s q u ' e l l e s o i t de t y p e D i r i c h l e t ou N e u m a n n s a n s a v o i r modifier sensiblement les programmes

num@riques.

La solution i n i t i a l e u t i l i s d e s e r a soit le r g s u l t a t d ' u n e p h a s e a n t 6 r i e u re,

s o i t u n e r g p a r t i t i o n u n i f o r m e ( c a s du p r e m i e r

p r g - d @ p S t ) p o u r la r e d i s t r i b u t i o n ,

o n n d g l i g e l ' i n f l u e n c e du c h a m p g l e c t r i q u e d a n s l ' o x y d e e t l e s v a l e u r s d e s c o e f f i c i e n t s de d i f f u s i o n sont d i f f 6 r e n t e s .

1

entrant

FIGURE

Z -

Si

sortant (nut)

,L

I

I

I

0

A

C

B

X

Diffusion unidimensionnelle (redistribution)

L e s c o n d i t i o n s a u x l i m i t e s en A (t)

et

B

sont les m ~ m e s que pour

l e p r g d @ p g t (aux v a l e u r s n u m g r i q u e s p r o s ) . Les gquations ~ l'interface

Si

SiO 2

sont

:

CSi =

m

m

CSi O2

(13)

: ~tant une constante positive (coefficient de s ~ g r g g a t i o n ) .

Csioz (~'- ±) v (t) -- Dsioz m

~ CSiOz

~

x

DSi

~ CSi x

~tant le rapport entre l'@paisseur de silicium oxyd@ et celle de silice cr@de V (t)

la vitesse d'oxydation. On tient compte par ailleurs du d~placement gdom@trique des points

473

et

A

C,

B

4rant s u p p o s 4 fixe. L a solution initiale est le rdsultat d'une p h a s e ant4rieure.

- Miod~le b i d i m e n s i o n n e l

I. 2. 2.

G

C

D

G

C

D

SiO Si

Si 0 2

YI

F

Si

B

l

E

A

0

F

E

A

x

FIGURE

3

- M o d h l e bidimensionnel diffusion (pr4d4pSt)

de

FIGURE

4

- M o d U l e bidimensionnel de diffusion (redistribution)

Nous raisons les m~mes hypotheses que pour le mod~te unidirnensionnel.

C e p e n d a n t d a n s c e t t e 4 t u d e , on d e v r a i t t e n i r c o m p t e de l a p r 4 s e n c e

d'oxyde

m ~ m e p o u r l e p r 4 d 4 p S t ( e n d e h o r s de l a " f e n ~ t r e " p a r l a q u e l l e s ' e f f e c t u e l a d i f f u sion). Cette pr4sence n'ayant qu'une influence relativement pas tenu cornpte rigoureusement e n s u p p o s a n t q u e , l e l o n g de On a donc

B C

:

B C

et D E

@c ~x

sur

C D

et A E

~y

sur

AB

J

-

~c

J

n

0

- 0

=

f(C)

e s t la c o m p o s a n t e n o r m a l e du f l u x d ' i m p u r e t 4 s . n

On a p r i s p o u r

J

la m ~ n

me fonction f que pr4c4demment

(~4)

A C D E

, il n ' y a p a s d ' ~ c h a n g e s a v e c l ' e x t 4 r i e u r .

sur

oh

faible nous n'en avons

et 4tudions u n i q u e m e n t le r e c t a n g l e

f(c)

=o~(Co

soit

:

- c)

P o u r la redistribution les conditions a u x limites sont identiques celles du p r 4 d 4 p S t (aux valeurs des constantes pros) et on ne tient plus c o m p t e

de

474

l'existence de la fen~tre de diffusion en supposant que la vitesse de croissance d'oxyde est inddpendante de se de redistribution.

Y

et que l'4paisseur d'oxyde est nulle au d4but de la pha-

L'~quation d'interface est la m ~ m e

que celle 4crite p r d c d d e m -

m e n t sur le m o d u l e unidimensionnel. I, 3.

- Position du p r o b l ~ m e m a t h g m a t i q u e

Nous sommes

donc a m e n d s

~ rechercher la solution, si elle existe,

d'un syst~me de deux 4quations paraboliques non lindaires. Ii ne nous a pas 4t4 possible de d d m o n t r e r l'existence et l'unicit4 de la solution de ce p r o b l ~ m e en raison des non lindarit4s figurant dans les 4quations. O n adrnet l'existence et l'unicit4 de la solution,

par analogie avec le probl~.~._e de la chaleur auquel on se r a m h n e

exactement s i l'on ndglige : les interactions entre les deux types d'impuretds,

-

-

la variation des "coefficients de diffusion",

- les effets du c h a m p 41ectrique. Ce syst~me dtant impossible ~ intdgrer formellement,

on remplace

le p r o b l ~ m e continu par un p r o b l ~ m e discret en utilisant pour ce faire une m 4 t h o d e de diffdrences finies. II conviendrait alors de d d m o n t r e r que la limite, quand les grandeurs des pas de discrdtisation tendent vers zdro, de la solution du p r o b l ~ m e discret tend vers la solution du p r o b l ~ m e continu. faite. N o u s s o m m e s

Cette ddmonstration n'a pu ~tre

conduits ~ admettre ce rdsultat, apr~s avoir vdrifid en faisant

plusieurs essais n u m 4 r i q u e s

sur le m ~ m e

cas, avec des valeurs de pas diff4rentes,

que la solution trouvde ne ddpend pas du maillage choisi. I. 4.

M d t h o d e s n u m d r i q u e s de traitement

P o u r tenir c o m p t e correctement des variations des p h d n o m ~ n e s

de

diffusion dans l'espace et le temps, nous somrnes a m e n d s ~ utiliser des pas de discrdtisation variables en X , Y , T. O n dcrit chacune des deux dquations A traiter sous la f o r m e (en bidimensionnel)

:

~u (15)

oh

L (u) =~)t

a

et b

C) x

(a

sont des fonctions de x

~u ~

)

~9

(a ~-~U~) Y

b

et t dans lesquelles on fair entrer tousles

t e r m e s non lindaires et ceux ddpendant de la seconde impuretd, syst~me complet en deux sous-syst~mes tion inconnue

CA

ou

CD .

= 0

sdparant ainsi le

oh l'on consid~re qu'il n'y a qu'une fonc-

475

O n approxirne les d~riv~es de

u

par une m ~ t h o d e

classique de diffe-

r e n c e s finies (7, 29). P o u r des raisons de stabilit~ (l i) n o u s a v o n s choisi une rn~thode purernent irnplicite. Cette fa~on de faire, n o u s arn~ne alors ~ utiliser un p r o c e s s u s de calcul it~ratif bas~ sur la m ~ t h o d e

du point fixe. Ceci n'est pas un inconvenient car, en

raison de la n o n lin~arit~ du problhrne, ment.

cette solution se serait i m p o s ~ e

naturelle-

P a r contre, un avantage i m p o r t a n t de ces choix r~side dans le fair que les

m a t r i c e s des syst%rnes d'~quations lin~aires rencontres dans le p r o c e s s u s de calcul sont

:

- tridiagonales pour le rnod~le unidimensionnel, - tridiagonales par bloc p o u r le m o d u l e bidirnensionnel, ce qui sirnplifie grandernent les calculs n ~ c e s s a i r e s h la r~solution des syst~rnes. O n pr~cisera ult~rieurernent les rn~thodes de calcul utilisdes. I. 5.

Algorithrne et prograrnrnes

L a rn~thode e x p o s ~ e ci-dessus a ~t~ prograrnrn~e pour les d e u x m o d U les uni et bidimensionnels. I)

L'algorithrne g~n~ral est identique dans les deux cas

:

E n t r e e des d o n n ~ e s et initialisation.

2) - Calcul du pas en temps.

Calcul ~ventuel de l'~paisseur d'oxyde c r ~ e .

3) - Calcul des coefficients de diffusion. 4) - R~solution des d e u x syst~rnes d'~quations. 5) - Test c o n v e r g e n c e et retour ~ventuel en 6) - Test final, retour ~ventuel en Les prograrnmes IBM

3.

2.

sont ~crits en F o r t r a n

IV

et utilis~s sur materiel

360 50 et 67 . Ils sont relativernent courts (1000 cartes environ pour le m o d u l e

bidirnensionnel sous syst~rne t e m p s n~cessite

CP/CMS)

et rapides. L e calcul pour un intervalle de

90 ~ 30 iterations p o u r le m o d u l e bidirnensionnel et de 15 ~ 3 it~-

rations p o u r le m o d u l e unidirnensionnel. Un module

de tracd autornatique fournit ~ l'utilisateur une representa-

tion graphique des r~sultats. I. 6.

- R~sultats obtenus

O n d o n n e ~ titre d'exernple sur la figure 5 le rgsultat de la sirnulation bidimensionnelle de la fabrication d'un transistor bipolaire sur du silicium. L ' a n a lyse physique de tels r~sultats a dgj& gig entreprise (I l) et les conclusions que l'on en tire sont trhs int~ressantes pour les ~lectroniciens, rant p o u r la c o m p r e h e n s i o n physique des p h g n o r n ~ n e s que p o u r l'arnglioration de la technologie et l'optirnisation

3 -I

J

J

Z

J

I D

J

i

i

~n

/

to

DoZ~

.....

-_.-

->

: l igne d '~quiconcentration

: jonction

~metteur

base

[ 40 zo

~(X,y)

404s

4 0 40

477

des dispositifs. Signalons s i m p l e m e n t

N ce propos que les p r o g r a m m e s

DIFFUSI

sont

utilis4s de fa~on courante dans plusieurs laboratoires. II - E T U D E

NUMERIQUE

DU

FONCTION-NEMENT

DE

CERTAINS

DISPOSITIFS

SEMI CONDU CTEURS L'4tude num4rique semiconducteurs solide.

pr6sent4e

On peut montrer

v4rifi4es

des ph4nom~nes

i c i r 4 s u l t e de c o n s i d 4 r a t i o n s

que sous certaines

on peut 4crire

classiques

hypothhses

raisonnables,

d e p h y s i q u e du g4n4ralement

le syst~me d'4quations.

(16)

J n

= q ~n

n

e + q dn

grad (n)

(iv)

j p

= q •p

p

e

g r a d (p)

-

q dp

(18)

-~t-~P- =

- R

- ~ q

div

(19)

¢)n ~-~-- =

- R

1 +-q

-'~ div On )

(20)

de c o n d u c t i o n d a n s l e s d i s p o s i t i f s

div (e) = ~

(n - p

(jp)

- Hop)

L e s s y m b o l e s utilis4s 4rant d4finis dens la table ci-dessous On transforme me (21)

ce s y s t ~ m e d'4quations en d4finisant un n o u v e a u syst~-

d'unit4s pour obtenir : J

-

-

(grad (P) + P

grad

(~)

grad

(~))

)

p

(22)

"~ J

1

-

~ (grad

(N)

- div

-') ( Jp )

N

n

(23)

-

p T N T

(24)

(25)

-

V

=

- U

U +

=

z~¢

= N

div

- P

--) (Jn)

hop

L e s inconnues principales seront les expressions analytiques de riv4es. L a fonction

DOP

variables g 4 o m 4 t r i q u e s.

~n' ~'p , U

est connue

N, P ,

~

, on supposera c o n n a ~ r e

en fonction de

N, P, ~2et de leurs dE-

(an g4n6ral n u m @ r i q u e m e n t )

en fonction des

478

TABLE

1

Syrnbole non normalisE

x, y

LISTE DES SYMBOLES

G r a n d e u r reprEsentEe normalisE

X, Y

Variables g4om4triques

T

temps

N

concentration d'41ectrons libres

P

P

concentration de trous

Jt

J

Jn

J

Jp

J

densit4 de courant externe

t

densit4 de courant d'Electrons n

densit~ de courant de trous P

--)

champ

E

e

41ectrique

potentiel 41e ctrostatique -1 -1

~p D

d

mobilitE des trous coefficient de diffusion des Electrons

n

n

coefficient de diffusion des trous

D

d

mobilit4 des Electrons

P

P

dop

DOP

dopage du semiconducteur

R

U

bilan g4nEration - recombinaison

TAUP

dur4e de vie des trous

TAUN

durEe de vie des Electrons

~P

q

charge de l'41ectron

479

N et

~p

et

P

sont des fonctions strictement

positives

de m ~ m e

que

~'n

. Z. 1.

Conditions aux limites

O n a 4tudi4 ce p r o b l ~ m e sur des m o d u l e s g4om4triques ~ une puis deux dimensions. r 4 g i m e dynamique,

L'analyse unidimensionnelle a @t4 faite ell r 4 g i m e statique et en en tenant c o m p t e 6ventuellement de p h 4 n o m ~ n e s

particuliers tels

la g4n4ration par effet d'avalanche (13) ou par effets des r a y o n n e m e n t s (18) l'analyse bidimensionnelle a dr6 r4alisde uniquement en r 4 g i m e statique. Z. i.I.

Conditions aux limites sur le rn0d~l e unidi!nensionnel statique Sur ce m o d u l e on ne peut 4tudier que des dispositifs ou les effets de

surface sont n4gligeables tels la jonction

PN

et le transistor bipolaire. base

emetteur

L

I I

Aux deux limites

m4talliques

sur lesquels

A et B

bipolaire

des potentiels

Aet

ModUle unidimensionnel de t r a n s i s t o r NPN

B

repr4sentent

les contacts

l e s c o n d i t i o n s de n e u t r a l i t 4

41ectrique.

il t a u t t e n i r c o m p t e du c o n t a c t de b a s e c e On peut r4soudre

partiellement

cette diffi-

(qui s o n t d 4 f i n i s h u n e c o n s t a n t e a d d i t i v e

pros) par une hypoth~se physique c o m p l 4 m e n t a i r e 2. i. 2.

-

B

on a a l o r s d e s c o n d i t i o n s de type D i r i c h l e t

on suppose v4rifier

de f a ~ o n r i g o u r e u s e .

cult4 en fixant l'origine

FIGURE 7

N, P , ~2 c a r l e s p o i n t s

Pour le transistor qui est impossible

de

i L

I 0

I

A

ModUle unidimensionnel jonction PN

sur les trois inconnues

i

I

x

B

A

-

!

®

® ®

®

FIGURE 6

eollecteur

donnant la valeur de

(o)

- C__qonditionsaux limites sur le m o d u l e unidimensionnel d y n a m i Q u e P o u r l'4tude d y n a m i q u e de ce p r o b l A m e il s'av~re que le choix des

variables principales dolt ~tre modifi4 et que l'on a int4r~t ~ 4tudier les fonctions N, P, E

les conditions aux limites sur

type Dirichlet,

N

et P

les conditions aux limites sur

E

4tant c o m m e

pr4c4demment

de

4tant par contre de type N e u m a n n .

480

(Z6)

~ E (A, T)

Ox

:

0E

(B, T)

e) x

V

=0

T

L e s conditions initiales 4tant alors soit le r4sultat d'un calcul en statique, soit le r4sultat d'un p r ~ c 4 d e n t calcul en d y n a m i q u e q u a n d on d4sire poursuivre une simulation. O n doit p a r ailleurs i m p o s e r rieur sous la f o r m e

(zT) 2 I. 3.

en outre la contrainte due au circuit ext@-

:

f ((~B(T)

~ A (T))'

Jr(T)) = 0

- Conditions aux limites sur le m o d u l e bidimensionnel statique N o u s ne d o n n o n s ici que les conditions a u x limites sur le m o d h l e

jonction

PN

par le n o m b r e le dispositif.

de

represent4 en figure 8. E n effet les divers dispositifs ne diffhrent que de contacts m4talliques et la fonction

DOP

(X, Y) qui caract4rise

P a r contre, on 4tudiera toujours les dispositifs en supposant l'exis-

fence d'une couche d'oxyde en surface, cette c o u c h e 4tant s u r m o n t 4 e d'une Electrode m4tallique (anneau de garde).

D

D'

C

~2

o~ ...... ! I !

A

FIGURE

8

- ModUle

A'

B

g 6 o m 4 t r i q u e de jonction

Sur le m o d u l e

PN

les parties h a c h u r 6 e s c o r r e s p o n d e n t k l ' e m p l a c e m e n t

des contacts m4talliques.

O n 4tudie les d e u x fonctions

A' B C D' et la fonction

sur le rectangle

Net

P

sur le rectangle

A B C D. C'est-~-dire que pour c h a q u e

481

dispositif les fonctions semi-conducteur

aans l'oxyde

Net

P

ne sont dgfinies que sur le rectangle repr~sentant le

(~'~z) alors que le potentiel est 4tudi4 dans le s e m i - c o n d u c t e u r

([l 1 Ufl2). ?Pour l e s c o n t a c t s

et sur

et

mgtalliques

Aa'o¢0/ on a

tels

7 ( X , Y) = 7 1

A'~'

D D'~'

et sur

D

et

on a

P (XI' Y)

= P1

N (X,

= N I

~I)(X,

'

Y) Y)

P (Xl'

Y)

= ~!12 = P2

N ( x 1, Y) : N z

sur

A' B

P o u r l'41ectrode de garde (X ~

on a

Pour

on a

et

Ies fonctions

n'

C

du d i s p o s i t i f

,)~'

ON --

0y sur

~

B C

0

et sur une interface 1

'

~'n

(zg)

c

3¥0 x

N

0N elx

0¥+ O x g

~x

--

,)y

0

~P -

~'p

0

:

1

) =

--

clx

relations

01~

( t) x

f et

=

:

,) P

-

ely

on ales

0N

(zS)

oh les fonctions

tellx'~

~

-

(Xo, Y) = %u3

z)- -P

( -~x

+ p

O x

) = f (N, P )

= g (~)

repr4sentent

les phgnom~nes

d'interface

et

C

est une cons-

tante. Dans les essais

num4riques

on a pris

:

A

(30)

f (N, P )

(31)

~(~) = B ( ~ -

oh A, B, ~ o

effectu4s

= p +--------~ ( P N - 1)

%0 o)

sont des constantes positives.

O n a donc u n p r o b l ~ m e

m i x t e ~ r4sou-

dre. Z.Z.

Existence et unicit4 de la solution

II ne nous a pas 4t4 possible de d 4 m o n t r e r la solution de ce p r o b l ~ m e

l'existence et l'unicit4 de

n o n lin4aire.

O n a pu alors obtenir quelques r4sultats tr~s parcellaires

d'une telle

d 4 m o n s t r a t i o n (Z5) m a i s les h y p o t h e s e s qu'il est alors n 4 c e s s a i r e de faire sont si

482

restrictives qae la port4e de ces d4monstrations reste tr~s limit4e. N o u s supposons donc que la solution du p r o b l ~ m e existe et est unique avant de mettre en oeuvre une m ~ t h o d e n u m 4 r i q u e pour en trouver une solution approch4e discrete. Z. 3.

M4thodes n u m 6 r i q a e s de traitement

P o u r chacun des m o d u l e s r4alis4s on a employ4 une m 4 t h o d e de point fixe de type it4ratif apr~s avoir discr4tis4 le p r o b l ~ m e par une m 4 t h o d e de diff4rences finies appropri4e.

L e choix de la m 4 t h o d e de point fixe it4ratif s'imposait en ef-

let assez naturellement en raison des non lin4arit4s du p r o b l ~ m e k traiter. P a r ailleurs on peut remarquer, ~

sur le syst~me k int4grer, que si l'on suppose connaftre

, alors on peut 4tudier s @ p a r 4 m e n t les p r o b l ~ m e s en

Net P

d4finis par les 4qua-

tions (22) et (24) d'une part, (Zl) et (Z3) d'autre part. Par contre si l'on suppose connaftre

Net P

on peut alors r4soudre le p r o b l ~ m e en ~7 d4fini par l'4quation (25).

Cette s4paration en trois p r o b l ~ m e s plus simples ~ traiter simplifiera notablement le traitement et s'ins6rera tr~s facilement dans la m 4 t h o d e de point fixe choi~ie 4galement en raison de ce d4coupage. Z. 3. i.

- M o d U l e unidimensi.onne ~ statique O n a repris pour ce p r o b l ~ m e l'algorithme propos4 par G u m m e l

(9) et

am41ior4 par D e M a r l (5) en lui apportant des simplifications importantes t24).'" L e syst~me d'4quations s'4crit alors :

(32)

J

(X)

:

1 ~[p

p

(33)

$

(X)-

P (X)

d ~) dX

N (X)

d dX

(X) n

:

U

=

-U

d X d J

(X)

P d X

(35)

+

dX

~'n

d J

(34)

,d,,N (X)

i

n

d P (X) dX

az (36)

- N (x)

P (x)

DOP

d X2 oh l'on c o n n a ~

~n (X)

,

~p (X) et D O P

(X).

U n e des principales difficult6s n u m 4 r i q u e s de ce p r o b l ~ m e provient du fair que les expressions (3Z) et (33) font intervenir les diff6rences de quantit4s tr~s grandes devant

J

et J c o m m e une 4valuation grossi~rement approch4e le montre. p n O n a alors int4r~t, pour faire apparaftre un t e r m e calculable de fagon pr6cise h in-

483 troduire

les variables

interm4diaires

(37)

N

=

h

e

(38)

P

--

f# e

A et

de variables

(39)

(X)

=

-

et de l'4quation

(40)

g

n

Y n (t)

XB

Jn (t)

e-

4tant calculable

(34), o n a

/Ax U

(X) =

les 4quations

(32) e t (33) t r a n s f o r m d e s

et l'on a pour une fonction

Cette expression limites

:

~' _~'

On peut ensuite int4grer changement

~l.d4finies par

PN , par exemple

~'(t)

car,

par ce

:

(B)

dt + ~

compte-tenu

des conditions

aux

:

A

(t) dt +

B

- A (A)

(8)

~,n(t)

e- ~l~(t)

dt

A

/.°

~n(t)

e-

U (y) dy

L"A

~B

~'n (t) e-

]

at

~(t)dt

O n voit appara~re dans ces expressions une autre difficult4 due h la prdsence des termes tels e

ou

e

. E n effet pour des polarisations appliqudes

de l'ordre de i0 volts ~Jvarie de 0 h 4000 environ. Par ailleurs les ordres de grandeur des quantit4s N e t

P

que les quantit4s Ai@et

varient de l'ordre de e~ou

e -~

I0 -I0 h 106 ce qui m o n t r e bien

sont impossibles ~ calculer sur ordlnateur.

O n r4soud ici ce probl~me en exprimant les 4quations de type (39) l'aide des quantit4s N e t P

(40

N (x) = e -~(x)

et on 4valuera fonctions

pour obtenir

telles

les expressions

:

N (8) e-

~'n :n (t) e-Y(t)

de type

~

~n

e

~(t)

dt

en introduisant

:

B (4z)

des

~

(x) : n

/B ~'n

X

e tf(X)

t~(t) dt : e (X)

e-q'(t)dt X

n

484 N (X)

ce qui permettra ensuite de faire a p p a r a ~ r e dans les expressions donnant des p r o d u i t s t e l s

e ~(X)

e- ~ ( X )

:

i.

O n aura done cette lois des quantitgs calculables sur ordinateur de fa9on simple, ceci quelque soit la valeur de la polarisation appliqu4e. E n outre, en reportant les valeurs obtenues dans les expressions type 41

on obtient i m m 4 d i a t e m e n t

la valeur de N e t

P

ou plus pr6cis6ment Aet ~ q u i

sont les inconnues que l'on cherche 5 d~ter1~niner . L a d4termination de ensuite ~ partir de l'4quation

d-----T - =

7

se fera

:

- ~£

- DOP

x

que l'on discr6tisera par une m 4 t h o d e de diff4rences finies apr~s avoir utilis~ un processus de quasi lin4arisation en posant, ~ la k + li~me it4ration :

(44)

"~k + 1 :

pour obtenir

I~k + O ( k

: d

z~ Z

(45)

_ O~k (Ae ~ k + /~e

Vk)

=

A e~k

-

d z ~k p-e "~k - DOP -----T

dx

dx

O n utilise alors l'algorithme g4n4ral suivant i) - D4termination d'une fonction

q2 ° (x) de d4part

:

(~ partir de consid4rations phy-

siques).

z) - Cal~ul ae A ot [~ ~ partir de ~y on en tire N et P. 3) - C a l e u l de

~

~ partir

de

i

et ~-

.

4) - Test convergence et retour ~ventuel en Z. 5) - Edition des r~sultats. Des programmes partir de cette m4thode,

correspondant aux divers dispositifs ont 4t4 4crits

lls sont 4crits en F O R T R A N

IV et comportent 5 ~ 600

cartes. Ils sont tr~s rapides et d'un emploi ais4. Ils sont d'ailleurs utilis4s de fa9on courante dans plusieurs laboratoires de consfructeurs et d'universit4s. Cependant, peuvent appara~re.

sous certaines conditions,

des difficult4s de convergence

U n e condition de convergence a pu ~tre m l s e en 4vidence par

M o c k (14) et des solutions k ce p r o b l ~ m e qui avait 4t4 constat4 exp~rimentalement ont 4t4 propos4s ce qui p e r m e t d'obtenir la solution dans tous les cas (21, Z7),

485

Z. 3. Z,

- ModUle unidimensionnel dynamique D a n s le cas d'une 4tude d y n a m i q u e

on se r e n d c o m p t e i m m 4 d i a t e m e n t

q u e le choix de la variable ~2 se r4v~le tr~s maladroit. de tenir c o m p L e de l'4quation d'4volution du c h a m p

(46)

-~)E ~

(X, T) T

:

J

(T)

_ (J n (X, T) + Jp (X, T) )

E comme fonction inconnue,

crivant en ce cas

:

(4s)

:

f) T

-Jx

:

4lectrique (4quation de Maxwell).

t

et donc de c h o i s i r

(47)

E n effet il est indispensable

les dquations donnant

s'4-

u

- ~ x

-

U Jp

Jn E x p r e s s i o n s clans lesquelles on r e m p l a c e U

Net P

par leurs valeurs en fonction de

N, P e t

~x

•

,)X

, J

n

, J

p

et

E.

D a n s ce m o d u l e unidimensionnel d y n a m i q u e

on a pu tenir c o m p t e de ph4-

nom~ne"s particuliers et fort importants de par leurs applications pratiques tels les effets de g4n4ration p a r avalanche (13) et effets de r a y o n n e m e n t s nom~nes

(8) qui sont des ph4-

essentiellement transitoires qul modifient e n t i ~ r e m e n t le c o m p o r t e m e n t

du

dispositlf en des intervalles de t e m p s tr~s brefs (quelques nanosecondes). C e m o d u l e utilise les m ~ m e s @voqu4es pr4c4demment par rapport a u t e m p s ,

techniques de discr4tisation que celles

et nous e m p l o y o n s toujours une m 4 t h o d e purement

de discr@tisation,

implicite ainsi qu'une quasi-lin4arisation syst~mati-

q u e p o u r 41iminer les non-lin4arit4s qui apparaissent dans les diverses expressions. C e qui nous a:m~ne de n o u v e a u ~ devoir r 4 s o u d r e des s y s t ~ m e s d'4quations lin4aires dont la m a t r i c e est tridiagonale en utilisant e n c o r e la s4paration du p r o b l ~ m e

sous-probl~mes

li4s, l'un donnant

E , l'autre N et P.

L ' a l g o r i t h m e g4n4ral est alors le suivant i) - Initialisation ~ partir du m o d u l e 2) - Evaluation du pas en t e m p s

en d e u x

:

statique.

et incr4mentation du temps.

3) - Evaluation 4ventuelle du courant ext4rieur. 4) - Calcul de

U

en tenant c o m p t e 4ventuellement de la g6n4ration par avalanche et

p a r effets ionisants. 5) - Calcul des corrections sur

E, N , P.

6) - Test c o n v e r g e n c e et retour 4ventuel en 4 ou 3 selon le signal appliqu@.

486

7) - Test fin et retour 6ventuel en 2. 8) - Edition des rdsultats. U n certain n o m b r e

de p r o b l ~ m e s restant ~ pr4ciser et en particulier

la gdn4ration automatique des pas de discr4tishtion en

X

et en t e m p s qui se fait

en fonction des r4sultats obtenus dans le calcul (13) pour r4aliser le meilleur cornp r o m i s possible entre l'accroissement de l'erreur, due ~ des pas trop grands et la dur4e du t e m p s de calcul ndcessaire, Des programmes

due ~ des pas trop petits.

de simulation ont 4t4 4crits h partir de cette m 6 t h o -

de. O n m o n t r e ~ titre d'exemple (figure 9) les r4sultats obtenus dans la simulation d'une diode hyperfr4quence en fonctionnement L'emploi de ces p r o g r a m m e s d~pend par contre 4 n o r m 4 m e n t

TRAPATT

.

est tr~s simple. L e cofit de la simulation

des p h 4 n o m ~ n e s

simul4s variant de quelques secon-

des C. P. U. pour un fonctionnement n o r m a l ~ plus de 20 minutes pour simuler une p4riode de fonctionnement hyperfr4quence en raison de l'extr~me rapidit4 des ph4nom~nes

(le t e m p s correspondant ~ une exploitation sur mat4riel C D C

6600).

Cependant la richesse des informations fournies p e r m e t d'expliquer des p h 4 n o m ~ n e s jusqu'alors peu clairs et justifie le coGt de telles simulations (8). 2.3.3.

- M o d U l e bidimensionnel statique .... C o m p t e - t e n u du c h a n g e m e n t de variables donn~ par les 4quations (37)

et (38) le syst~me d'4quations ~ int4grer d'4crit

(4:9)

(--"-~ e - ~ x ) + ~

~)x

4"

(51)

~)x

+

z

"~ ~'-~x

~

= ~e ~

~e-

-

~) z y

:

( T

e

1

1

"~y ) = U -~/ ~ " ) = U

- DOP

O n discr&ise chacune des 4quations par une m 4 t h o d e de diff4rences

f i n i e s s p 4 c l a l e m e n t adapt4e h c e type d ' 4 q u a t i o n e l l i p t i q u e a u t o - a d j o i n t e (29), soit si l ' o n appelIe r .1. j le d o m a i n e r e c t a n g u l a i r e e n t o u r a n t le point i, j d4fini p a r h. h. ij

l-i

x.i

Yj

2 m. j-i +

z

<x

/.. xi +

z

i

m.