A mathematical model of coupled plates and its finite element method

Computer Methods in Applied Mechanics and Engineering 99 (1992) 43-59 North-Holland CMA 253

A mathematical model of coupled plates and its finite element method Wang Lie-heng Computing Center, Academia Sinica, Bei/ing, China Received 3 July 1991 This paper investigates the mathematical modeling of coupled plates and its finite element approximations folle~ving the approach proposed by Feng Kang and Shi Zhong-ci.

1. Introduction

The mathematical modeling of elastic composite structures, i.e., elastic structures that comprise substructures of the same or diffelent dimensions (three-dimensional substructures, plates, rods, etc., usually made of different elastic materials), is a problem of practical importance, since such elastic structures are very common in engineering problems. Nevertheless, it is only recently that this problem has been studied from a mathematical viewpoint and numerical analysis. About 10 years ago, mathematical modeling of such elastic composite structures was proposed by Feng and Shi [1, 2], from a mechanical and mathematical viewpoint. Very recently, mathematical modeling of such elastic composite structures, called elastic multb structures, has been investigated by Ciarlet et al. [3, 4], from a theoretical viewpoint using asymptotic analysis. In this paper, we consider the mathematical modeling of coupled plates and its finite element approximations following the approach of [1, 2]. The notation is as follows: (cf. Fig.

1)

O1

the mid-surface of the elastic plate

n'l' the elastic thin plate, with thickness 2t I (0 < tl << 1) f' = ( f l ) , f l = f l ( x l , x2), i= 1, 2, 3, the applied surface

force in ,(21 = (ul~), uli - ul(x I, x2), i = 1, 2, 3, the displacement vector in £J1, with u °i = 0, a = 1, 2, u 3I ~Ou~On = 0 on F0 the mid-surface of the elastic plate ~'~2 the elastic thin plate, with thickness 2t 2 (0 < t 2 << 1) 2 f , 2 _ (f2i), f~ =fi(x2, x3), i= 1,2, 3, the applied surface force in ~22 U 2 -u,2 = u~(x2, x3), i = 1, 2, 3, the displacement vector in n 2 HI

Correspondence to: Dr. Wang Lie-heng, Computing Center, Academia Sinica, P.O. Box 2719, 100080 Beijing, China. 0045-7825/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved

44

Wang Lie.heng, A mathematical model of coupled plates 3

~2t 1

i

/o

e2

el -r

///'

t/ 2t 2-

\& Fig. 1.

Assume that there exists a rigid junction on the part F between these two plates/~ and ,0 2, i.e. (cf. [4]), u Iffiu s

onF,

-o,.~=o3.~

(1.1)

on r.

(1.2)

The equality (1.1) means that the displacements u' and u s of plates D, and/'/2 are continuous on F, and the equality (1.2) means that the rotational angles of plates n t and/'J2 about F are continuous on F. Consider the total energy of the elastic composite structure consisting of these two plates under a virtual displacement ~ - (v ~, v=):

3(~)= ~D(~, ~)- F(~),

(I.3)

where (with ~ ffi (u', u=)),

= Dt(u', v') + Dz(u a, v2);

(1.4)

6.~(vI) = I,(o.v~ , + oBv:), ( 2E, t, ~((1- ,,,)~.~(v')+ ~,,~.(v')8.~) ; Qo~(v') = ~, 1 - v~/

(1.5)

Wang Lie-heng, A mathematicalmodel of coupledplates

gal3(v )---0a.OO3 1

D

[M.~(v')-

45

I

,

2E, t~ 3(~ = ~) {(1 - ,,,)K~(v') + v,K,,(v')~o,,),

a, fl, 3"= 1 , 2 ;

(1.6)

l ~o.~.(v~)= 1~(o~.v~. ~ + a~.v~.) | Qw#.(v 2) __ -2E2t2 - - ~ { ( 1 - v2)ew~,(v2) + [ 1 - v2

vz%.,.(v

2

)8,,.v } ;

(1.7)

"K,~.~.(v ~) = -o.,~,o~, 2E2t~

M~'#'(v2) = 3 ( ] - ~'2) { ( 1 -

v2)Ka.~.(v2)+ v2K~.v.(v2)6o.~.}, a',/3',

3,'=2,3; (1.8)

F(T')=/nf'vl dx, dx 2 + fo2f2v2dx2dx3;

(1.9)

Et > 0, E 2 > 0 denote Young's modulus of the plates •t and n 2, respectively, and 0 < vt < 1, 0 < v2 < 1 denote the associated Poisson ratios. Latin indices have the values {1, 2, 3}, Greek indices and primed Greek indices take the values {1,2} and {2,3}, respectively; the repeated index convention for summation is systematically used; and c, c,, c 2 denote generic constants which may have different values in different places.

2. Mathematical model

We now introduce some Sobolev spaces (cf. [5]):

H = { T ' - (v 1, v2); VIE Ht12(nl):-- Ht(•I) × HI(n1) × H2(nt), v 2 ~ n2~t(fJ2):= H2(n2) x n'(n2) x n ' ( n 2 ) } ,

(2.1)

with norm "~i/2

I1~11. :=

2 + Y~ Ilva'll~.n2 2 2 22 + IIv,112.~21 , IIv~ll,.o, + lIIl iv3112.n,

V={~ffi(vt, v2)EH; v ,1, = 0 , a f f i l , 2, on F0, v 3t=ov~/onffio on r0 and v 1= v 2 on F, - 0 i v ~ = 03v ~ on F } .

(2.2)

(2.3)

Then the mathematical model of the elastic composite structure of two coupled plates is:

Wang I ie.heng, A mathematical model of coupledplates

46 Find

= (u ~, u 2) E V ,

such that

D ( ~ , T') = F ( ~ )

V°//" E V .

(2.4)

To prove the existence and uniqueness of the solution of the problem (2.4), we need the following lemmas.

L E M M A 2.1. The subspace V is closed in H with the norm (2.2). The proof is easy and we omit it.

L E M M A 2.2. The bilinear form D(~ll, °F) defined in (1.4) is continuous and coercive on VxV. PROOF. The continuity of D(. ,. ) is obvious. We only need to prove that D(. ,. ) is coercive, i.e., there exists c = const > 0, such that

D('I/', )> cll ll

V

(2.5)

EV.

To do this, from (1.4), we firstly consider D,(v ~, v 2) as follows:

M~(v')K~(v')-

2 + 2K~2) + v,(K,t + K22)2} 3 (2Ett~ 1 - ~ ) { ( 1 - vt)(K~, + K22

2Ett~ 4 ~ 2 2K~2 ) 3(1 + v,) (X~, + X~, + 2K~2)- ~ I~,t,(K,, + K~2 + , with Lam6 constants A > 0 and It > 0. Similarly, we have

Thus we have

Dt(vt, v l ) ~ 4 1 ~ l t l ~ l [

t 2

5

12

•

In the same way, we have

D,(v 2, v 2) >>-4tt=t=~

+

3

32

(2.7)

1

Since oy E V, v s = 0, a = 1, 2, on r o. Then from Korn's inequality and Poincare's inequality,

we have

D,(v',

Ilu ll,,., + 110511,.., •

(2.8)

47

Wang Lie.heng, A mathematicalmodel of coupled plates

From (2.7), (2.8) and (1.4), we have

v.ll,.o,

+ Ilv~ll2.n,

+

c2

,

II~..,.(v )11o.02+ Io,1~.,~

,

(2.9)

with c~, c 2 = const > 0. If the inequality (2.5) is false, there exists a sequence (T'k)~ V, such that

I1~11. = 1 V k , as k-->oo.

D(T"k , T k ) - . * O

(2.10)

From the first relation of (2.10) and by the Rellich-Kondrasov compact embedding theorem, there exists a subsequence ( F t ) C (1"k), such that v=,t"'> v~' , '

o = 1, 2 ,

in L2(/'~1),

•3.1 ' ---->v~ in H'(I'~,)

(2.11)

2 %,,t--> v wa2 ,

a t = 2, 3 ,

in L2(~'~2),

2 171.1-"> D21 in Hi(/'/2)

(2.12)

From the second relation of (2.10) and (2.11), (2.12), we have that 1

v~=O,

a =1,2,

1 u3=0, in H2(nn),

in H*(/'/I),

(2.13)

and that (v2~,,i), a ' = 2, 3, are Cauchy sequences in HI(/'~2), (vl,i) 2 is a Cauchy sequence in H2(/22). Since V is closed in H, then we can see that ~ - - , T' = (o, v 2) and

~.,~,(v~)=o,

in H

(2.14)

Iu 2,1~.,, =o.

(2.15)

From the first relation of (2.15), v~(x2, x3) -'- a2 + bx3 , 1

t~(x2, x3) = as - bx2 , 2

1=

and taking into account v~ = v 2 - 0, v 3 = v 3

(2.16)

0 on F(x, = x 3 = 0), we can see that

a2=a3 =b=O,

which means that 2

v 2 = V3 = 0 in/'/2. From the second relation of (2.15),

v~(x2, x3)

= a + bx2 + cx3 ,

(2.17)

48

Wang Lie-heng, A mathematical model of coupled plates

taking into account vl2 = v~! = 0 and a3v5= t _alv~ = 0 on F, we can see that

v~=0 ona~.

(2.18)

Thus T ' = 0 in H, but this contradicts the equalities I[T't[l~= 1, Vl. The proof is completed. [] By Lemmas 2.1 and 2.2, and Lax-Milgram's Theorem, it can be seen that the problem (2.4) has a unique solution. 3. Boundary value problems and junction conditions We now formulate boundary value problems and junction conditions for the problem (2.4), which will be useful in the error estimate of finite element approximations for the problem (2.4). Rewrite the problem (2.4) in detail: Find •I E H112(J'~l), u 2 E H2tI(~'~2); u ! - 0 , alu ~ - 0 on Fo, u I - u 2, -~lu3! -- a3u 2I o n F ,

such that

j~ Qo,(U')%(v')dx, dx~+ fn,Mo,(u')Ko~(vl)dx, dx2 +

f~ Qo,#,(u2)e.,#,(v 2)dxzdx3 + fnzM°'#'("~)/C°'# '(v2)dx2 dx3

(3.1)

= fo f v ' dx,dx, + f. f'v2 dx2dx,. vv' ~ II"2(n,), e ~ ¢ H2"(n2); v' =0, a,v~ •0 on ~, e' = e ~, -a,v~ = a,v~ on r. Let v ~= 0 in (3.1). Then

f~ Q~,~,(u2)e.,r(v2)dx2 dx~ + fn M.,~,(u2)Ko,D,(v')dx2 dx,

fO2 f2v2 dx 2 dx3

Vv 2 ~ H2tt(f~2); v 2 -- 0, O3v~= 0 on F .

By Green's formula, from (3.2) it can be seen that (cf. Fig. 2,) x3

3

~e P~

Fig. 2.

Irl' • P~

x~

(3.2)

WangLie.heng,A mathematicalmodel of coupledplates

49

2 dx 2 dx 3 Jr fa 2 )n#.vo. 2 ds O#.Qo, # , (u 2 )vo. a2,j-Q++,#.(II

fo

fn a,,o,M,,#.(uZ)v~ dx2 dx3- f+az,r Mo,#,(u2)no,n#, a,,v~ ds ÷ f~ o2\r{ao.Mo.#.(u2)n#. + as.M~.#.(u2 )n°.s#.}v2Ids F~ t 2 + [M°.#.(u2 )n=.s#.]ri(P3)v=(P 3F) + [M,.o.(u2)no.s#.]fi(P~)v2(p'4) ~- ffl2 f 2 v 2

dx2 dx3

VV 2 E H211(~'~2); v 2 = 0, {~302 = 0

on F ,

(3.3)

where n' -- (n2, n3) and s' = (s2, s3) are the outward unit normal and associated unit tangential vectors on M22, respectively, ~s, = so, a=,, a,. = n=, a o. and [+Ifj-

(3.4)

- +Irj •

Then we have

_

c3#.0=,#,(112)= f 2 , ,

Qo,#.(u2)n#, = O,

and

Or' = 2, 3,

a ' = 2,3,

in 0 2 ,

- O,..#.Mo,#,(u 2) -- f~

in ( / 2 ,

M=,#,(u2)no.n#.--0

on 0n2\/",

a,,.Mo,s,(u')n ~. + 8+.M+,#,(u+)n,..s#.•0

on 0 n 2 \ F ,

, r' i

t

(3.6a)

Mo,#,(u2)no.s#,Irri(P'4) O,

Mo,#,(.2 )no s# }r,(P3)=O ,

(3.5)

on On2\F ;

,

,-

which is equivalent to

M2,(,2)(P;)--0,

M2a(u2)(P,~)-0.

Substituting (3.5), (3.6) into (3.1), it can be seen that

fnt Qa#(llt)ea#(vl)dxl dx2 + fQ, M°#(uZ)K°#(vl)dxldx2 +fr

Qo,#,(u 2)n# ,Vo, 2 ds +

fr

Ma'#'(u2)n°'n¢' o3v~ ds

+ Jr {ao'M='a'(u2)n#'+ a2M~'#'(u2)n°'s#'}v~ds + [Mo,e,(u

2

Fi t 2 p F' w 2 t )n.,s#,]r,(P2)vt(P2) + [M.,#,(u 2 )n.,so,lr+(P,)v,(P,)

= In f~v~ dx~ dx 2 1

V v I E Hll2(,Ot); v 1 - 0, 0iv ~ = 0 on Fo,

(3.6b)

50

Wang Lie.heng,A mathematicalmodel of coupledplates 1

or equivalently, in our case (cf. Fig. 2), taking into account v 2 = v I,c330~ = -c3~v3 on F, we can

see that

fOI Qa#(ill)ea#(vl)dx1

dx2 + LI

Ma#(u|)K°#(vl)dxldx2

-- fF Qa'3(u2)Dla' dx2 - fF M33(I'/2) olv~ dX2 - fF (Oa'Ma°3(u2) d- t)2M32(~2)}vll dx 2 + 2M23(~2)(p~)vt!(P~)- 2M23(tt2)(P~)v11(P~) =fn I flVl dx I dx 2 V v I E HH2(J'21); V I -- 0, t310~ = 0 on

Fo .

(3.7)

By Green's formula, from (3.7), it can be seen that (cf. Fig. 3)

- fn ~+Q+,,(u')v:dx, dx2 + fan Q,.,(u')n,vt.ds

-fa :JoMo/j(u')vdx,dx2-foa M../j(u')nan/jO,,vds +

f~a,{rg°M°#(ut)n/++ c~'M<~#(ul)n°s/+)v~ds

+ 2Mn(u')(P2)v~(P2)-2M,2(u')(P,)v~(p,) Q~ .(u2)Vo.'dx:

-

. Msa(u')~.v~dx:

fr

dx.

s 1 p p I + 2M23(u2 )(P2)vI(P2) - 2M2s(u2 )(PI)v~(PI)

fn

/~e I dx~ dx2

Y v ~ E H~2(/}~); v ~ = 0, O~v3~ =0

l

r f~1

q

ro ~t

Fig. 3.

B X2

lq

on ro ,

(3.8)

Wang Lie-heng, A mathematical model of coupled plates

51

from which and taking into account that r = r', PI = P~,/'2 = P~, we have

-o#Qo#(ul)---f1~ ,

a=l,

Q~#(u l)n# = O,

a = 1,2,

on afl~\(r U Fo),

UI~=0,

a = 1,2,

onFo;

i

- Oo#M,,,(u')

2,

inn l,

(3.9)

in F l ,

= f~,

on

(3.10)

o ~ , \ ( r o Fo) ,

O~M~#(ut)n# + #~M~#(ul)nos# = O, on oa, x(r u to), o n ~o"

Mn(u')(P~)-O,

M23(u2)(P~)=O,

(3.11)

a = 1,2.

We now formulate the coupled conditions on F. From (3.8)-(3.11), it can be seen that (cf. Figs. 2 and 3) f --iF

Q~l(u I ) V~I dx,. + fr Mll(U ') Olv~. dx2 - fr Q~,3(u 2 )vo,I dx2

-

fr

--

fl. ( 0 o . M a . 3 ( u

dx 2 - fr

{aoM~,(u')+

02M,2(u')}v ~ dx 2

2) -k t~2M32(u2)}Dl! d x 2 - 0

'qv t ~ Htt2(nl);

l•Oon

v z -- 0, 0iV 3

F0

(3.12)

from which we have

-Q,l(u I) - {Oo,Mo,3(u 2) +

a2M32(u2)} ffi 0 ,

- Q 2 1 ( u t) - Q 2 3 ( u 2 ) f f i 0 ,

-(O.MoI(u i) + 02M,2(ul)}

-

Q33(u2) =0,

(3.13)

,Ml1(U l) M33(u 2) --'-0, on F . -

Notice that conditions (3.13) and u I- u 2 ,

-01u ~ -- 03u ~

on r,

(3.14)

are the junction conditions arising from the rigid junction between two coupled plates, while the boundary value problems (3.5), (3.6), (3.9), (3.10) and (3.11) are normal formulas as in the usual single elastic plate.

Wang Lie.heng, A mathematicalmodel of coupled plates

52

4. Finite element approximation In this section, the finite element approximation to problem (2.4) is considered. Let ~'~, a = 1, 2, be regular rectangular subdivisions with the inverse hypotheses of O~ and K~z, respectively, which have the same nodes on F, and let Vh(O~) be the bi!inear finite element spaces associated with the subdivisions ~ : on/2.., a = 1, 2, Vh(12~) 2 be Adini's finite element spaces associated with the subdivisions ~'h on /2,~, a = 1, 2,

' ~ V h "-- { ~ "-- ( V 1 , Vh2); Yah

v~(n,), ~ = 1, 2, V3h1

2 ~ VI(~2), Va,h

OI ,

2 E = 2, 3, Vth

~.

v~(&)

,

V~(a2)

(4.1)

Vt,h(O)ffiO, V~h(Q)=O, 01v3h(Q ffi 0 Vnodes Q ~ Fo , and v~h(P) ffi v~h(P), --O~V~h(P) = 03V~h(P) V nodes P E F } .

Then the finite element approximation of the problem (2.4) is the following: Find ~h ~- Vh, such that Dh(°//h, ~ h ) = F(~h)

VT'h E Vh ,

(4.2)

where

t Q"o(uh)e'~(va) dx, dx 2 + ~.t , M.o(u3.)K.:

(v~,) dx, dx 2

+{f.~ Q., ,a,(u,)e,.,~,(¢~) 2 dx 2 dx 3 ~2

I

Let

frye, dxt dxa +

2

flush dx2 dx3 .

(4.4)

w' = (v ~, v ~) ¢ v~ 1 '/~1

+ ~ Iv,12.,, + )2 IIv.,ll,,,,. + ~ Iv,l~..2 "t a' v2

,

(4.5)

which is a norm on Vh. We have the following error estimate.

THEOREM 4.1 Assume that q/ffi (u l, u 2) /s the solution of the problem (2.4), with uoI H2(/2,), c~ ffi 1,2, u~ ~ H3(~']I) and u2,. e H2(a2), ¢' ffi 2,3, u~2 ~ H3(l"Jz). Let ~h =(uth, u~) be the solution of the problem (4.3). Then the following error estimate holds: .

II~t-%ll, ~ch

II.all2,.,+ll u3113.~, + ~ Ilu2,.Ih.,~ + Ilu, ll~.~2 •

(4.6)

PROOF. By the abstract error estimate for the nonconforming finite element approximation

53

Wang Lie-heng, A mathematical model of coupled plates

(of. [6]), we have

II°u- %lib ~ c [t v,,~vh inf I1~- %11~ + sup n~,',

I1~11~

(4.7)

J"

By using the usual interpolation error estimates, the first term on the right-hand side of (4.7) can be estimated as follows: inf ]l q / -

VhEVh

]~t'hl]h<<"ch{~-~ a ,a,lu"2,/~I + ]u3]3,fil ..]_1

Z.. ]ua,]2,fi2 -[-2

(4.8)

lu:[3,~2} .

It remains to estimate the second term on the right-hand side of (4.8). Using Green's formula repeatedly, and taking into account the boundary value problems and the junction conditions in Section 3, after some straightforward calculations, it can be seen that

E,,(ou, ~ ) = Oh(°//, W h ) - F(W'h)

={fa Q,,tj(u~)e,,o(w~)dx~dx2 + ~

f, M,,~(u')ror,(w~)dx,dx2

--~1~1 flwhdXl dx2}'l'(ftl2 Qa'[3'(il2)Ea'~l'(w~)dx2dx3

+ ~f,2g,,.,.(u2)K,.o.(w:)dx2dx3-~ f2w2dx2dx,}

fr Q,l(u '')wlh ds -- f~ Q~(u')w~h ds -- fr Q23(/,/ ~)W2h ds

=--

fr Qa3(u 2)w3h 2 d$--

fr

{OaMal(//l) + 02MI2(ul)} w3,~ ' ds

2 ds -- . {aa,Mo,3(u2) + O2Ma2(u2)}wI/~ -

-

¢1

f2

~'2

Mo,#,(u

)n.,n#,

O.,w m ds

= fr Q1'(ut)(w~h- w:,)dx z + fr Q33(u2)(w~h-w~h)dx2 -

-

r -

1

Ma,a,(u

) n a , n a, O . , w m d s

FzCF'

E Z fe M"tj(ut)n"nt~O'w~hds 'rI Fl~r I

i

FlY~aat -

Z Z

T2 F2EOr2

r2~'oa2

fF

t ra. O.,Wlh 2 ds, M.,a,(u2)n.,n

(4.9)

54

Wang Lie-heng,A mathematicalmodel of coupledplates

from the junction conditions (3.13), in particular in the last equality. Let v ~ denote the piecewise linear interpolation of o and Rt(o) - o - o ~. Then

~.,

-

FIeF

t

M~#(u')n~n~o~w~ ds - ~

= E [ M,,(u')a,W~h FCF JF

" M~,,tj,(u~)n,,,n#, a.,W~h ds '

F:~CI"'

M3,(u 2) O~w,. dx2

dx2 +

For 3F

-,,+ ~.f M,,(.').,(o,w~,)d.,+ Y_ f.,M.(.').,(o,~,)dx,. F'~f"

Since a~w~h(P)----OtW~h(P ) for all nodes P E r , which in turn imply that (a3w~h):! = -(atw3h)" on F. and taking into account the last equality of (3.13), then 8h = 0. Thus we have (cf. [71),

f,, Q,,(u')(w~ w,.)dx~ ' + frQ~(u)(w..

E,(~, ~ : . ) =

-

-

-2

w~.)dx~

M~/j(u )non#R1(O,,w~h)ds 2 f~, M.,.,(u')n.,a.,R,(O.,w~h)ds .

(4.10t

By the interpolation error estimates (cf. [6, 7]), and taking into account that wu, t andw~:, are the piecewlse linear interpolations of w~h and w~,, on F, respectively, we have

If,

2

Q33(u

(?

)(w3h-w~h)dxzi<~ch llu.,ll,,,,, 1

2

)(~ Iw~,,I,,.,)"' i

7.

.

(4.11)

By the error estimate for Adini's element [7], we have

FIEBr !

i

1

(4.12)

I~ ,,~,,. f,: M..#.(u:)n,.%.R.(O,.w~,)ds} ~ chl],:]l,.,(~ Iw~,l:.,,) ',2 "

Wang Lie-heng, A mathematical model of coupled plates

55

Finally it turns out that

,.:11 2..,

, + II u~ll,.,,, + E~. Ilu:,ll=.,,. + Ilu,2 ll.,_. )11~",,11,,. (4.13)

and the proof is completed.

[]

I. In fact, we can apply other nonconforming finite element spaces for VhZ(/'],,).

REMARK

For the case of the space Vh2(Oa)C H~(O,), such as Zienkiewicz's element space, the conclusion of Theorem 4.1 holds with the same proof as that of Theorem 4.1. For the case of the space V h2 ( D , , ) ~ H (/2,), such as Morely's and De Veubeke's element space, the conclusion of Theorem 4.1 holds essentially. The proof is given in Remark 2. 2. As an example, we consider Morley's element space V~(O~). Let ~rj~, a = 1, 2, be regular triangulations with the inverse hypotheses of/2, and~']2, respectively, which have I the same nodes on F. Let Vh(O,) be the linear finite element spaces associated with the triangulations ffh" on O,, a -" 1,2, and let V h2( O ~ ) be Morley's finite element spaces associated with the triangulations J'h, on n,,, ot = 1, 2, and REMARK

v,,={~,, = (v,',.V h~.) , V.h' E v'.(a,)..~ 2 v~.l, E

V},(a~), a'

= 1,2,

I 2 o~,,~v.(a,).

2 = 2, 3, v,, E Vh(O2),

V~th(Q) = 0, Vnodes Q E F0'

, Otvsl , ds = O, Vedges F C F0 ,

and vll,(P) = v ,", ( P ) , Vnodes P E F,

.,o:,....

(4.15)

The finite element approximation of (2.4) for Morley's element is the same as above. Then the following error estimate holds. THEOREM holds:

4.2. Under the same assumptions as in Theorem 4.1, the following error estimate

I1~ - %11,, ~

ch{~ Ilu~ll=.., + IIu~l13..,+ E

fllw

Ilu~,ll=..~= +

+ h(ll f~ll0.,~, + IIf2,11,,..~)} •

I1,~11~.,,~ (4.16)

Similar to the proof of Theorem 4.1, it is sufficient to note that, for Morely's element w h E V ~ H I ( O ) , by Green's formula, we have PROOF.

Wang Lie.heng, A mathematical model of coupled plates

56

a=~ f M.#(u)K.~(w,,)dx-fa fwhdx = - ~ f M.~(u)O.pWh dX- fa fWh dX

-~f Ool~o,(u)O,w, dx-~f~ ~o,
(4.17)

Let o t denote the piecewise interpolation of o, since vI E H~(O). Then

+ ~ f. O.M.o(u ) Oo(wh wl) dx + fa ffw'~- w,,)dx. -

(4.18)

Let

eth=~ f O°M.a(u)Oa(w,-w~)dx, e,,= f. y(w'-w,)dx, by the interpolate error estimate, le,,I ~<

chlul.~.o

Iw,12..

,

le=nl<~ch21lfllo,a(~

Iw,,I;,,/

(4.19)

Since w~ E Hi(/2), and by Green's formula, then

- f,,o~M.(.)O,< d~- f,, r
-z f~. M.~(u)n.n~ O,,wh ds + L~ OsM"o(u)n"s#w~ ds f~. M,,~(u)n.s# Os(w~- wh) ds.

(4.20)

Wang Lie-heng, A mathematicalmodel of coupledplates

57

We row estimate the last term on the right-hand side of (4.20):

~.,o~ f. Mo,(.:, o:~;- .,,)~,. Let

Pgw= N

wds.

IFl=

lds.

!

Since wh = wn at all nodes of 3 h, then we have

=

- eo[M,~#(u)])n,:# FEar

from which by the interpolate error estimates (cf. [7]),

le3,1~
(~

12 xl12

W,h..)

(4.21)

•

Thus by the second equation of (3.13),

fr Q,,(u' )w,,' dx,- fr Q,,(u')w;,. dx2

E,.(~t. ~ r , ) - -

fr Q23(u , )w2hdx22 fr Q33(u 2)w3,2 dx2

fr {O"M*#(ul)n#+ O~Ma#(ut)n"sO}(w~h)'dx2 +fr (O*'M~'o'(u2)n#' + O"M*'#'(u2)n"s#'}(w~h)t +

dx2

- v~rf~ Mo#(U')non# O,,w~nds 3

-2

f. Mo,o,(u2)n.,n,,O,,,w~hds- I ~ - I~ + 2 (',;+ e,~)

F2cF av2

1~1

- - f~ o.(,')(~:~-(~)' dx~)- f~ O33(u2)(w~,

(w~h)')dx2

(o~,M,,,~(u2) + 02M32(u2) }(w~h )I dx2 -

fr Q.(u')(w~,)'

dx 2

- -

fr (OoMo,(,')+ a2M|2(ui)}(w~h) I dx2

+ e~rfi~ {Mll(ul)Olw;h + Maa(u2 ) OaWlh} 2 dx2 3

- Ith- I~ + X (ei~ + e,2h), i=1

(4.22)

58

Wang Lie.heng, A mathematicalmodel of coupled plates

where

'~=~ ,,~.,,fFM"' (4.23)

rtgto£1 t

12 =

Ma,/3,(l/2)/'/a,/'//3, On,W1hds.

E

F2¢a~2

By the first and third equations of (3.13), and taking into account that 2 1 I W3h = (W3h) on F, we have Eh(~, ?4/'h)=

1

2

Wlh = (Wlh)

I

,

{M,i(U')O,W3h + M,3(u2) O3w,h}dx2 3

(4.24) i=l

Let R~w = w - P~w. By the last equation of (3.13) we have [r~crfp

{M,,(U')atW~h + M33(uZ)a3W~h}dX21

=

Ro[M,i(u )lRo[O,w,hldx2 +

Ro[M,,(u )]Ro[O3w,h]dx2l

r

(4.25) where we have used the interpolate error estimates (of. [7]) in the last inequality. By the error estimate for Morley's element (cf. [7]) it can be seen that

(~

, : ),,2 (4.26)

i~ ~ chllu~ll~.n,

2 2 Iw,,12.,J ,

Finally we have

IE.( °~, Wh)l ~ ch{ll u3113.m ' 2 + Ilu~ll3,o~ + h(llf~ll0.o,) + IIf,ll0.o,)}ll ~hll~,

(4.27)

and the proof is completed !-1, Acknowledgment This work was done during the author's stay in Istituto per ie Applicazioni dei Calcolo "Mauro Picone" of Roma and Istituto di Analisi Numerica of Pavia, del Consiglio Nazionale

Wang Lie-heng, A mathematical model of coupled plates

59

delle Ricerche, of Italy as a visiting professor. The author wishes tO thank the Directors, Professor E. Magenes and Professor A. Tesei for their invitation.

References [1] Feng Kang, Elliptic equations on composite manifold and composite elastic structures, Math. Numer. Sinica 1 (1979) 199-208 (in Chinese). [2] Feng Kang and Shi Zhong-ci, Mathematical Theory of Elastic Structures (Science Press, 1981); also to be published by Springer. [3] P.G. Ciarlet, A new class of variational problems arising in the modeling of elastic multi-structures, Numer. Math. 57 (1990) 547-560. [4] P.O. Ciarlet, H. Le Dret and R. Nzengwa, Junctions between three-dimensional and two-dimensional linearly elastic structures, J. Math. Pure Appl. 68 (1989) 261-295. [5] J.L. Lions and E. Magenes, Nonhomogeneous boundary value problems and Applications, I (Springer, New York, 1972). [6] P.O. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978). [7] F. Stummei, The generalized patch test, SIAM J. Numer. Anal. 16 (1979) 449-471.