~I"~,-j-~L

EL'iEVIER Compu!. Methods AppL Mech. Engrg. 132 (1996) 135-177

Computer methodsin applied

mechanics andengineering

A priori error estimations of hp-finite element approximationsfor hierarchical models of plate- and shell-like structures

l.R. eho, l. Tinsley Odcn*TexCls Instill/te for ('lImI'IIlCllio"al a"d AppliCtJ Mathematics. The U"ilwsity of Texas III AI/st;n. Allsti". TX 78712. USA

Received 7 October 1995: revised 16 November 1995

Abstract

A priori error estimales of hp-finite elemeIII approximations for hierarchical models of the thin elastic plate- and shell-likestructures are derived. The error in the approximations of a hierarchical model consists of the modelling error due to assumptionson displacement or stress fields and the finite element approximation crror of diseretizations of slleh a model. Also.non-uniformity with respect 10 the thickness owing to numerical locking is included in the presenl error estimates. Numericalresults are gi\'en supporting the theoretical results.

I. Introduction

In a series of recent works [11. 211. we have explored the concept of hierarchical models of plate- andshell-like clastic structures. In this paper. we consider the development of a priori error estimates forIrp-llnite element approximations of parameterized models of such structures.

The basic idea underlying hierarchical modeling is the parameterization of a family of mathematicalmodels of certain physical phenomena and to use an adaptive strategy to determine which model amongthe family best characterizes the particular problem under study. In the present investigation, the familyincludes models of clastic deformation of thin bodies subjected to external forces. The parameterlinking members of the family is the polynomial order q of the variations of the displacement field u in aso-called thickness direction. which we make precise in the subsequent analyses. The highest hierarchyin the present investigation is that furnished by the full three-dimensional elasticity equations. whichcorresponds to the parameter q -+ x. Obviously, the practitioner only wishes to analyze the simplestmodel with which he or she can obtain simulations of an acceptable accuracy. Rarely is it necessary touse the full three-dimensional theory at all points within a thin body to obtain good predictions ofstructural behavior.

We study the problem of constructing lip-finite element approximations of hierarchical models ofelastic plate- and shell-like structures, the geometry of which is characterized by a reference surface (rJand a thickness d. normal to w. which varies but is symmetric over the surface. The resultingdiscretizations arc legitimately called Izpq-models. since the thickness variation parameter q identifiesthe model level and can vary over w. It is known that such models can mimic classical plate and shelltheories when d is small. but that locking phenomena can occur for small enough J liz and p. We addressthis phenomenon in this paper as well. Our principal mission here is to derive a priori error estimatesfor Izp-finite element approximations of these classes of hierarchical models.

• Corresponding author. Director of TICAM. and Cockrell Faculty Regents' Chair in Engineering '2.

0045-7825/96/$15.00 © 1996 Elsevier Science S.A. All rights reservedSSD/0045-7825(95)00985-X

136 J.R. ClIO. J.1'. (Jtlen I Complll. Met/rods Appl. Mel'll. Engrg. 132 (JI)

l.R. ClIO. l. T. Oden I ComplII. Me/luxis Appl. Mech. Engrg. 132 (1996) 135-/77 137

an+

an-

o

Fig. I. Geometry of a shell·like body.

p

(6)

are prescribed. respectively (the Dirichlet and Newmann boundary conditions). For convenience. wedefine the projected boundarics aWl> and aWN by

aWD = {((1 1 • 82• 0) E 1~} = w n I;)

aWN = {(8 1. (I ~. 0) E wi (8 I .0'::. ±d/2) E I ~}Referring to Fig. I. the position vector r of a point (J = (0 I. 8'::.(3) is represented by a position vector

r0 and a unit normal vector /I of a point (8 I • /1'::.0) on the reference surface.123 11 +' 12r(O . /I . f} ) = ru(8 .8 .0) + 8 /1(0 .8 ) (7)

and is expressed in thc Cartesian coordinates x = (Xl. x.::. x3) with orthogonal unit vcctors el• e2 and e3•by

, ito I 0':: 03)r = x ei = x . . e, . i= 1.2.3 (8)Assuming the maps Xi to be smooth enough.

rj = '

138 J.R. ClIO. J. T. Odell I Comput. Methods Appl. Mech. Ellgrg. /32 (1996) /35-/77

gij = gj' gj = gj' gj = gjlThe contravariant basis vectors g' arc reciprocal to the covariant basis gi:

i "ig . g, = ()j

( 13)

(14 )

go = g,,{3g/3 . gO = g"/3gfJ . g"/3 = g" . g/3 (15)

Now we list basic coefficients defincd on a reference surface (J 3 = D. First. the coefficients of the firstfundamental form are defined by

(16)

and

a" = a"fJa/3 •And the coefficicnts of the second fundamental form describing the curvatures of a refcrcnce surfaceare given by

( 17)

The coefficients of the third fundamental form are defined by

( 18)

The normal curvatures of a reference surface w in the direction of the base vectors a" arcI 2

CII =bll =hl• c2:=bn=b: (19)

and the torsions arc defined by

(20)

(21 )

Then. the Gaussian curvature c and the mean curvature Iz of the surface ware defined by. respectively.

c = IbfJl = b1b2 - b2bla 1 2 ) 221z = b: = b: + b;

And the principal curvatures. CII and c:2• arc the extrcmal values of normal curvatures (onc ismaximum while the other is minimum). Then. the principal radii of thc surface ware dcnoted by

RI=-l/CII• R2=-lIC22

Returning to a surface 03 = constant. g" in (10) can be expressed in terms of b~g =a +03"a Q' .Q

= a - 03lJ a/3" oP

=µPa° fJwith the introduction of µ~ defined by

µ{3 =oP -03b/3"" "

To obtain similar forms for the contravariant basis vectors gO. we writc

" A""g = fJa"

then A; are determined using the relation gO . gp = 0; = A~µ~ and arc given byA" = jj" + 03b" + (f)')2bOb"f + ...

/3 /3 13 "f fJ

(22)

(23)

(24)

(25)

(26)

IN. Cleo. J. T. Odt'll I COli/pili. Methods App/. MeL"'- Ellgrg. /32 (/9 }J./J.y

r'; 3 ( yall=g . }J."U).)./J

= }J.;'bY/J

ra - b y "\,,313 - - /JyU . U 1\ Ii= -Aab"

" /J

where I;y are the quantities defined on the referencc surface. Note that r~"= 1';3 = r~3= o.Thc infinitesimal volume dv at any point of a body is

dv = dr3' (dr, x dr2)= g3 . (gt X gJ dO

=yg dOwith g = Igi;l. and the infinitesimal surface ill on a surface of 03 = constant is given by

ill = 11 • (dr I x dr J= Vii dw

wherc de = dO 1 dO 2 dO3 and dw = de 1 d02• Moreover. Vii is given byyg = g3' (gl x g2)

( " /J)= g3' }J.) a" x UfJ}J.2= 1}J.IVa

(29)

(30)

(31)

(32)

(33)

I I ).: 2'where }J. =}J.l}J.2 - }J..}J.2·Suppose that a material particle located at r = r(O) is displaced to a new location R by a displacement

vector V. Then.

and

dR2 = dN· dR= (R.i dOi). (R.i dOi) = Gi; dO

i dOi

Here. Gii = Gj • Gi and Gj = aR I ao i. Then. the strain measure is deli ned as follows:

(34)

(35)

140 l.R. ClIO. l. T. Odell I Complll. Melhods Appl. Mech. Ellgrg. 132 (1996) 135-/77

.., ') , .dW - dr- = (G. -gOo) dO' dO'

'I 'I

= 2£. dO; dOi'I

(36)

So2Eii = Gii - gii

= gi . Vi + gi . V.i + V.i . V.i (37)We shall confine our attention to the linear theories of kinematics in which assumption of infinitesimaldeformation is used to drop the non-linear terms Vi' V.i in the strain-displacement relations. Then.instcad of (37). we have

2Eii = gi . Vi + g, . V,jThe gradient of vector function V.; is expressed in the following form.

(38)

- I" _ "Vi - (vk,; - Vi r ".)g - V"jigHence. we have

(39)

1Eii = 2' (viii + Viii) (40)

In this curvilinear coordinate system. the Sobolcv norm IIvlll.!J is given by

Ilvlli.J) = L (Ivvl2 + jvl2)yg dO= f {fdl2 (lgl/g#viliV"ltl + IgiiviV;I)Iµ./ d03} Va dw

w -,112(41 )

Wc shall refer to the body J.l as plate-like whcnever bop = 0 and shell-like when b"fJ ¥: O. We are onlyconcerned with shell-like bodies with a non-negative Gaussian curvature. and choose the orthogonalcurvilinear lines (J I . (12 such that they coincide with lines of principal curvature. 11/ the applications of0111' methods described larer. we focus 01/ plate-like structures and cylindrical shell-like structures.

In particular. for the surface of a right-circular cylinder (0 1= 0. 02 = Z. 03 = 1'). we have

(42)

g22 = g33 = 1 . othcrs = (), , I

hi = bl = b2 = 0: 2 2

µ: = 1 . µI = µ'I = 0

r:J = r~1= 1/1' . others = 0

c = 0 h = -21 R_ 2

gil - I' •

b:=-I/R.

µ.: = 1'1R .r' -II - -I'.

whcre R is the radius of middle surfacc of the cylinder. The componcnts of the strain tensor are then

Ell = V 1.1 + 1'V3•21012 = (VI: + V2.1) •

E =vJJ 3 ..1(43)

2Eu = (vu - ~.I + V).I)and the norm IIv 11../1 is

(44)

It is useful to introduce the notion of membrane- and bending-dominated behavior. for thin elastic

l.R. Clro. l. T. Oden I Comput. Methods Appl. Mech. Engrg. IJ2 (/9961 /35-177 141

(45)

(46)

bodies with thickness symmetric with respect to the middle surface. For plate-likc bodies, It IS notdifficult to define such terms. In the assumed displacement field expansion in the thickness direction.even order terms of IIx' ",. and odd order terms of "l produce the membrane mode: on the other hand.thc respcctive odd and even order terms produce the bel/dil/g mode. This is discussed morc thoroughlyin [26].

But for shell-like bodies this distinction docs not hold since thc cven and odd modes in thedisplaccment field are coupled in the energy inner product. In such cases. we define these terms usingthe even and odd modes in the strain field. Then for constant curvatures. the even mode is

(lIoIP)even = (lIo•p + lI'Yt~p + 113bop )even - (l3b~b'YP(1I3)odd(lI"j3)."en = (11".3 + lI'Yt:3)",cn

= (1I".J)even - {liX(IIY).""n + 03bX(II'Y)Odd + }b~(1I3Io)mn = (113.,,),,,en - {8~(II'Y)e",'" + 03b~(II'Y)odd + }b:(11313)e,en ::::;(113.3)",en

and vice versa for the odd mode. These two modes bccome decoupled in the cncrgy inncr product if ygis replaced with Va (e.g. as the thickness of the structurc tcnds to zero).

We then calculate the two portions of strain cnergies of the even and odd strain modes.

2(t:;i)e,.n = (ltil; + Itili)",en2(E';)Odd = (uili + "i1Jodd

We shall refer to the shell-like problcm as membrane-dominated whenever the even strain modesproducc a larger portion of the total strain energy than the odd modes. and bending-dominated whenvIce versa.

4. Preliminaries

4. 1. Separation of variables

Let It = u(O I .02• () 3) be a solution of the boundary valuc problemLu =f in n = fll x D'. (47)

with compatible boundary conditions applied on an. fl is a connected bounded domain in lR3. Here.f = f(O) is a source function and (0 I, (2) E DI' 03 E fl'.. It is assumed that the solution u can be written

u(O) = F(O I. (}2)Z(03) (48)

LEMMA 4.1. Suppose v(lJ)=V(Ol.(J2)Z(03). (OJ.02)E.!2" 03ED'.. Then, VE(H1(fl=Jl1xD2»3implies that

VE(HI([JI)/ al/d ZEH1(D2)

PROOF. First consider plate-like bodies (8 1= X. 0'. = y. and (13 = z). then

IIvll~.f1 = r

142 J.R. Cho. J. T. Odell I ComplII. Methods Appl. Meek ElIgrg. 132 (1996) 135-177

+ 2!VtV3.,Z'ZI + 2!VzVl.2z'zl + (V~ + V; + V;)ZZ} dz dx dy

Since VE(/lI(D»). each term in (50) belongs to LI(n). Applying Fubini's theoremfunctions. we have

(lI;)2. (v,...)2. V\.2V2.,' VIV3,1' VZV3.2ELI (D,)? V E (ll '(ll, »3Z2. (Z,)2. Z'Z E L '(D2)~Z E H'(D2)

Here,

IIVI17.D, = In {\V ... /lVp, .. 1 + IVIZ} dr dy•

Next. for cylindrical shell-like bodies (0 I = e. e 2 = z. and e 3 = ().

(50)

to such L'

(51)

(52)

(53)

(54)

By rewriting (R + {)" = R"( 1 + {/R)" and applying Fubini's theorem. we have (except for the first andlast terms in (53».

) ) ~ Z ,(VJ R)". (V2.2t· Vi. V 3' VI.2V2.1' V2V3.2E L (D])

Z2-. (1 + {IR)Zz. (l + {IR)(Z,)2. (l + ,IR)Z'Z E L \D2)

For dI2

J.R. Cho. J.1'. Oden I COli/pili. Alethods AI/pl. Mec-Jr. EngrJ.:. 132 (1996) 135-/77 143

(57)

(59)

The countably infinite set {CP/} ;:0 must be maximal I (complete or total) in HI([1J. Suppose a subspaccW C H' ([12) is spanned by such a countably maximal set {CP/} ;=0' and P be the orthogonal projectiononto Wand Q the orthogonal projection onto W.l. Then.

z=Pz+Qz. "f/zEH1([1Z)

and

(Qz, CPt) = O. "f/CPt implies Qz = (), "f/z E H'(flz) (58)Thus. z = pz for every z E HI([1Z) ,', W= HI(fl;,).

An orthonormal basis {ljit} ;=n (maximal orthonormal set) forms a dense subspace W" in II I(fl2),which can be generated using the method of Gram-Schmidt orthonorma/ization when a linearlyindependent set of iiI(fl2) is given. Let us take a set (Qt};=o' Qt(O~) = (OJ)'. and follow the method ofGram-Schmidlort/lOnorma/ization for flz = (-dll. dI2):

CPo= Qll = ](QI·4JO)1 J

CPt = Q I - (-I.. A.) CPo= Q I = fJ''I'll' '1'0 I , ,

(Qz· cPt)1 (Qz' cPn). d-. 3;' d-cP2=Q2- (4J

1.cP

l)1 cPl- (cf}O.CPO)' CPo=Q'-12=(O ) -12

, ,_ _ (Q3·4JZ)1 _ (Q3·CPI)1 _ (QJ'cPo). = OJ 3_ 3d-(d-+20) tJ"'

cPJ - Q3 (CPZ' CP2). cpz (CPl' CPt)1 CPI (CPu, CPu) I CPo () 20(d;' + 12)

and normalize

(60)

(61 )

Any set {Q/}~-O such that span{Qt};_o = span {,11t};.0 also has the density property even though (Qt};':..llmay not be orthonormal.

Next, we consider the structure shown in Fig. 2 with varying symmetric thickness d(1i I. 0 2) ~ dmmaxwd(OI.02). We define a function 2(OJ)EII'(-dnlI2.dn,l2) such that

2(03 = {Z(oJ). -d(O I. 0;') ~ 20J ~ d(O '. Oz)) 0 , otherwise

Note that, for plate-like bodies. (1I1(W»3XH'(-d",12.dmI2)Hl( -dI2. d/2).

IS a cOl/vex hull of (HI(w»3 x

LEM M A 4.2. Let 2(03) be as defined in (61) and assume that the (Wo principal radii of CllrWllllre R 1and Rz are constants and that d",/max{R1, R;,} < L Then, for allY v E III '(fl = w x (-clI2, dI2»]J

(62)

I For every z E H '(/1,). (z.

144 l.R. ClIO. l. T. Odell I COII/prtt. Methods Appl. Alt·ch. Engrg. /32 (1996) /35-177

Fig. 2. A shell-like body with varying symmetric thickness.

PROOF. See the Appendix. 0

Using a maximal set {lpt};~o' we write uq E Vq = span(l/1}i=o in the following form

II = F(O I. 0 ~)Z(O 3)q

IIq = F(O 1(2) 2: atlp,(03)t=o

Then. Lemma 4.2 and the density argument lead to

lim Ilu -11"111.11 =Iim IIF(Z - ± atlpt)II«1-% q_:c '"WIn 1.11

II q II.s; C F lim Z - 2: 1I 11',IN II II J.w q __ 1 II '=0 I 0 1.( -d",12.dmJ 2)=()

SO. II" ~ U as q ~ x.A uq in Eq. (63) is also expressed as follows:

q

II" = 2: {atF(OI, (2)}cp,(03)t=oq

" I 2 3= Li Vt(O .0 )cpt(O )1=0

Here.

Ilv II J.w = Ilal'lIl.w.s; 11I11IIFIIl.w < +x

Therefore. we have

Here. the boundedness of constants a, is proved by the following arguments.(l) If {CPI} is all orthonormal basis, then Parsevaf's relation holds:

"Ilull~= 2: laJ

J.R. ClIO. J. T. Odell I Complll. ,\tetllOds Appl. Mf'ch. Ellgrg. 132 (1996) 135-177

"1I111l~ ~ L latl2

1;0

(3) If {'Pt} is complete but not orthonormal. then

II"II~= C~a;'Pi'j~ aj'Pf) I'"

= L L laillall('Pi' 'P/)I ~ L laiI2Ii'Pill~;=u i;II ;=0

Suppose we write an elemcnt uQEV'1 as in (65): then llqE(U!(D)f implies that(1) UtE (H'(w»3 . VI.s;;,q(2) Z=span{'Pt};=u is dense in H'(-dI2.tlI2).

4.2. Three-dimensional elasticity problems

145

(69)

(70)

We next recoru for future reference the cquations governing the static equilibrium of a three-dimensional elastic body. We considcr infintesimal ueformations of a material body occupying a regionJ1 E ~3. The classical equations govcrning equilibrium arc

(Tii(u)lf+/=O infl

1/ = () on fo. l.s;;, i. j .s;;, 3(Tif(U)lli = t' on f,...

and the strain-displacement relations and the constitutivc equations are

1eii(u) = '2 ("ill + "iii)

if ifkl(T (u)=£ eH(Il). l.s;;,k,I.s;;,3

Covariant derivativcs (Tiilk arc defincd by

ii I - if tirf klr;(T k - (T.k + CT kt + (T jl

(71)

(72)

(73)

Herc. u = t/g; = "i1:; is the displacement field defined on fl exprcssed in the curvilinear coordinates 0and (Tii(u) arc thc contravariant components of Cauchy stress tcnsor

(74)

In (72). £ijkl is the fourth-order tensor of elasticities (clastic moduli) satisfying

and the ellipticity condition

E;jkt~ii~kt~al~12. V ~ E~3x3. ~ij = ~ii

(75)

(76)

where a is a positive constant. In (71). 11 represents the unit extcrior normal at a point 8 E aD. frepresents the body force. and t reprcsents applied traction on the boundary TN of an: an =fN U l~.l~ n I~)= 0.

Throughout this work. wc assume that n is sufficiently smooth (Lipscltitzian or smoothcr) and thatthe data f and t are smooth: e.g. f E (L \(1»3. t E (L 2(an»3. Furthermore. wc suppose that thetraction t+ acts on the top surface an.. and t- acts on the bottom surface i/{L of the body. and that theDirichlet boundary l~ is restricted to the latcral boundary ani'

If V(n) is defined by

V(n) = {vex) E (HI(fl»).l: v = 0 on l~) (77)

146 J.R. Cho. J. T. Odell I Compll/. Me/hods Appl. Mecll. ElIgrg. 132 (1996) 135-177

(79)

(80)

then we arc guaranteed that v E V(n) implies that the strain encrgy U(v) < +::>0, i.e. the displacementfields in V(ll) have finite energy. The variational or weak formulation of the boundary value problem(71) is

Find II E V(n) such that V v E V(n)a(lI. v) = I(v) (78)

whcre the bilinear functional a: Yen) x V(n)-IR and the lincar functional I : V(,(l)~1R dcfIne thcinternal and external virtual work. respectively:

f Jd/~a(lI. v) = Ejikt(lIk,/ - IImr~~)(ui.i - uJ~)Vg de3 dww -d/~I(v) = r tujVg dO + r tiujVg 0 that thcre exist two positive constantsc1• c2 such that for any II. v E yen)

(i) la(lI. v)l.;:;c,llllllvllvllv(ii) a(v. v);;;!; czl/vl/t

Thus. problem (78) has a unique solution. If r = 0 or mcas U;» = 0. thcn the solution to (78) isunique up to an arbitrary rigid motion. Note that the strain energy is

1U(v) = '2 a(v. v). v E Veil)

5. Hierarchical models

(81 )

(82)

LEMMA 5.1. Let dm = maxw d(e' . ()2) and assllme the two prillcipal radii of CIIrvature. R I and R2• tobe cOl1stallts and that dm/max{Rl• R2} < 1. Furthermore. let Pj he as defined ill (61). then for allYvq E Vq(il) C 1R3.

q

Ilvqlll.lJ';:; Cm 2: IIPjIL.(-d, 12.dml2)llVilll.Wi=ll "

where Cm> 1 is a positive constallt dependent only on dm• RJ and R2.

(83)

(85)

PROOF.

Ilvqlll.li = IiiVieD J. 02 )Pi(03)11 with v~= 0 for j > qai=Cl 1.11

q

.;:;2: Ilvi(O'.o2)Pi(e3)IL.'J (84)i-OApplying Lemma 4.2 for each Vi Pj' we have

q

Ilvqlll.IJ';:; Cm L IIPill,.(-dm,z.dmmllVilll.w 0J=ll

l.R. ClIO. J. T. Ode" f Comp/{(o Methods API'/. ,Hech. Engrl;. 132 (1lJ96) 1J5-177 147

With vq E Y

148 l.R. ClIO. l.T. Odell I Compuc. iHethods ApI'/. Alec". ElIgrg. 132 (1996) 135-177

~I

z

Fig. 3. Parametrization of a cylindrical shell.

6.1.2. Geometry approximarionTo approximate the geometry for meshing. we shall use an isoparametric represcntation of the

geomctry in which a master cube i1 = [-1. 113 is mappcd into a curvilinear domain in which the ~3coordinate is stretched linearly in the rhickness direction. Such a strctching can he accomplished in twoways. First. the rop and hottom coordinates (.1',. Zi' 0101'(bllllnm) at node i can be specified and second.the coordinates (Si' Zi' ?i)mid of a node on the middle surface can be specificd togcther with a vectord;v3." where v3.; is a unit vector normal to the middle surface at node i. If ,pia •. ~Jare shape functionsdefining the geometry of the square (~= {( ~I' ~2' ~3) E i1 : ~3 = O}, then these two maps are of theform.

(91 )

and

(92)

Thesc maps arc illustratcd in Fig. 4, and dcgrecs of frecdom of the gcomctry (Si' Zi' ?i) arc obtainedusing an H~ projection of the parametrization 113. 141. Thc parametcrs cli' V3.i in the map are evaluatedat the point (~I' ~2)i on the master reference surface using the parametrization.

1\

nMap:x= F(S)----------

S2

SJFig. 4. Geometry approximation of a shell-like body.

n

C,2

(S;. Zj .«;;) lOp(Sj. Zj .«;;J(Sj. Zj.

l.R. Cho. l. T. Oden I Complll . .\fetJlOds Appl. Mech. E/lgrg. 132 (1996) 135-177

6.2. hp-Finite elements

149

We next consider approximations of the hierarchical models using hp-finitc element models. Thisinvolves the construction of a family of partitions of Pl'" of il into finite clements ilK' cach clcmenthcing the imagc of a map FK defined, for examplc. by (91) and (92):

(93)

(see Fig. 5). The refercnce surface w is then a mesh of curvilinear quadrilateral clements WI-:' the pointsof which are the images f~UI'~2'O). On w=[-1,1r={g=(~I'~2'O)}, we introduce families ofhierarchical polynomials shape functions ljit( ~1' ~2) which arc characterized as follows (with ~1 = ~, ~2 =1/) :

• Vertex functions

IX;(~'1/)=4(1:tO(I:t1/). i=1.2.3.4

• Edge functions

1X;·I(~'1/)=2'(I -1/)CPk(O· k=2,3 'PI

X;·2(~, 1/) = ~ (l + OCPk(1/), k = 2. 3, ·1'21xf'3(r 1/)=2' (1 + T/)CPk(O, k = 2. 3.... ,1'31

X;\~'T/)=2(1-0

150 J.R. Cho. J. T. Odell I CompUl. Methods App/. Mech Engrg. /32 (1996) 135-/n

The space 9'P( w) of master element shape functions (on the plane gJ = 0) is then

{{"v} {' E...I} {' R}. 1 ~ ., ,.;;:4 2.::: k :S::: }CPP(") Xi . Xk I XI' .....,.u~, .....~p:;

v W = span. , 1~/~(Po-2)(Po-3)/2 (95)

For p-version approximations. we take PI = P2 = P3 = P4 = Po; otherwise. it is possible to assignpolynomials of differing degrees on each edge E of OJ and to the internal (bubble) node of the masterelement. This is essentially the same convention used in our earlier applications of hp methods (see e.g.[21]).

Further. let .u2 q(-1, 1) be the space of polynomials in g3 of degree I ~ q. Then. the space ~pq (Ii) =9'1' (w ) x .u2 q (-1. 1) contains functions of form

np

rJq(€) = L at(~3)cPt(~I' ~2)/-1

(96)

wherc "'tE9'P(w) are "I' polynomial shape functions of the type defined in (6.2) and the numher ofsuch functions. respectively. and

q

a/(gJ) = L v;tp;(g3);-0

The local lip-approximation of a q-hierarchical function on element K is then

v,/,/(x) = rJqoF~I(x)np tJKOI

h '" '" ·tv~~~(s. z. 0 = L... L... Vi.aP;(Of/1(s. z). I ~ a ~ 3t= 1;=(1

(97)

(98)

One example is illustrated in Fig. 5 in which the coordinate system x = (s. z. 0 with unit base vectorses• ez and e, is used.

Here. we introduce typical interpolation properties of lip-spaces.

THEOREM 6.1. Consider a partition gp; of W into N(gp~) fillite elemellts which are affine equivalent toa master element Cd witll

.'-(9' h)w= U wK'

K-I(99)

where FK is an invertible matrix and aK is a translarion vector for the reference surface WK' Let

PK = sup{ diam (B) I B = ball in WK}11K = diam(wK)

and suppose

hK- ~ (J = positive constant V WK E gp~PK

( 1(0)

(101 )

Then. for any fUllction II E HS(wK). there exists a seqllence of interpolants whp E gppw"(w,,.), the space of

polynomials of degree ~p w" defined on WK' P,." ;;?; 1, and a constant C indepelldellt of II. P K or h K' slIchthat for any r ~ s and WK'

( 102)

where µ. = min(p",,, + 1.s).

PROOF. See 171. 0

l.R_ Cho. l. T. Oden I Comput. Methods Appl. Mech. Ellgrg. 132 (1996) 135-177 151

( 103)

Next. we define the space of piecewise continuous polynomials:

Vhp(w) ~ HI(w) . eO = U eOK

wKEf1t;;

V"p(w) = {v E 11 of {l is introduced as construction of finite elements flK.1'(:1',,)

ii = U iiKK=l

( lO5)

( 1(6)

Let vq'''(n) k Vq(n) be the hp-finite element space defined by

vq·II(n) = V"p(fl) nV'i(12)Now. the restriction of the test functions vq E Vq(n) to ilK is denoted

vqloK = vk E V'i(ilK) = V1

where Vl is the space of Vq(il) functions restricted to {lK' Let Vk,II be a local finite elementapproximation of v'J.: involving polynomials of order PK' Since the global approximation vq·II ofv'l E Vq(fl) must be continuous across the interelement boundaries. the global approximation spaceV'i·I'(il) is a subspace of Vq(il) containing vector-valued test functions. V'i,h E «:gO(ii))3. vq'''lox = V,:;h.Clearly. each restriction V'J.:·hof a function in the finite element space Vq·I'(il) can have differcntapproximation polynomial orders within each element. Then. the discrete problem corresponding to(86) is

Find U'/·II E Vq·h(il) such that V vq.l/ E Vq·h(il)

a(uq·h• vq·II) = l(vq·h) (107)

7. A priori error estimation

Let u and uq be an exact and a dimensionalIy-reduccd approximate solution of (7R) and (86).respectively. Then, thc hierarchical modeling error e'i is defincd by

eq E V(il) , (108)

Let uq·II be a solulion of the finite element approximation (I07) for the q = (ql' (f2. q) hierarchical

152 l.R. ClIO. l. T. Otlell I Compl/l. Me/hods App/. Mech. EIIgrg. 132 (/996) 135-/77

~2

Fig. 6. Finitc shell clement of the hierarchical modeb.

model (Fig. 6): then the finite element approximation error e,,·h of the q hierarchical model is definedby

(109)

The total error is then

e = II - IIq·I, = II - uq + u

J.R. ClIO. J. T. Otic" / COlllpUl. Methods Appl. Mech. Engrg. /32 (/1)1)6) 135-177 153

(115)

Babuska 134] to plane elasticity problems such as beam-. arch-. plate- and shell-like structures withuniform symmetric thickness with respect to the refercnce surface w.): We just recall here the mainsteps of the proof (for simplicity. we consider a uniform thickness heam-like body (0' = X. 02 = y).).

• Step 1: I Pick the test function va in (82) ttl he of the form vuC-\'. y) = F,,(x)II'..(2yld). Fa E H(I)(w).11'" E H l-dI2. dl21 n Wa• a = 1. 2. The spaces W" define thc membranc and bending modesaccording to the symmetric or anti-symmetric properties of the displacement components. Write

tS4 l.R. Cho. l. T. Odell' Complll. Mer/rods Appl. Mech. ElIgrg. 132 (1996) 135-/77

q

~ C L (IIV i_V i·1I 1I .. w11 i',II ../-dm'2.dmI2l),=u (120)

where

(121 )

Using the standard interpolation theorem, Corollary 6.1. for each Ilvi _vi.hll ....., the proof iscomplete. 0

7.3. Locking effects in the finite element approximations

As the thickness d of the bending-dominated elastic bodies tends to zero. discretizations errors maybecome very large for certain class of finite element schemes owing to the presence of two well-knownphenomena: boundary layer effect and locking effect (shear locking in plate-like bodies and shear-membrane locking in shell-like bodies).

The boundary layer effect limited within very thin region ncar boundary is caused by the boundarylayer terms varying exponentially in the normal direction to the boundary in the exact solution ofsingularly perturbed elliptic boundary value problems. This phenomenon is rcported to be different fordifferent types of loadings, boundary conditions and the models of the hierarchy. Due to the singularbehavior near the boundary, the regularity of solutions decreases. and this decrease is proportional tothe strength of the boundary layer. We will not discuss this here in detail; but the reader is referred to[3,27.30] and the references therein.

Locking phenomena occur if low order standard finite clement schemcs are used to approximatebending-dominated thin clastic bodies. Locking is manifested whenever the approximation error locksand does not decrease (or decreases very slowly) with a refinement in the mesh parameters hand p inthe pre-asymptotic convergencc ranges.

The mechanism leading to locking can be explained in the context of the cylindrical shell shown inFig. 5 which is subjected to a uniform normal traction t,(cv) = t,o The analysis is done using adegenerated three-dimensional shell model. This model has five displacement components (in right-circular cylindrical coordinates. sec Fig. 5): three in-plane displacements U = (II,. II z' II,) and tworotations 0 = (0,. OJ defined in the middle surface w. Then. we have the dimensionally-reducedvariational form

find Uc/ = (11.0) E Vd(w) such thatstJ(O, 1/1)+ r2b(uJ, vJ) = l(v). 'V vd = (v, 1/1)E VJ(w) (122)

( 123)

where VAcv) = {vd E [H '(w)f : v = 1/1= 0 on (lcvD} 0 The two bilinear functionals stJ(', .) and b(· .. ) are

.91(0,1/1) = E 2 f {(1 - V)K"I3(O)K"fl(l/1) + VK",,(O)K/3/3(I/1)}R dO dz12( I - v) ,..

b(ud. Vd) =, E~) f {(I - v)p"P(u)Pufl(v) + vp",,(Il)P/3fl(v) + (1- vh',,(lld h'" (vd)}R dO dz1 - v ...and, the linear functionall(v) is

l(v) = L t,v,R dO dz (124)where K",p. P"fJ and 'Y", arc the bending. membrane and transverse shear strains (a =5. f3 = z). and t,being a normal traction calibrated by d3.

l.R. ClIO. l. T. Oden I Comp/{(o !t/l·tllOds Appl. Mech. Eflgrg. 132 (1996) 135-177

- -2Knp(t/!) = if/nIP + if'13lu2PnP(v) = V"I13 + Villa2y',(v,t) = V,I" -1/1,.

155

(125)

(126)

Hcre. we recall that ii are quantities defined on the reference surface.As d-O. thc bilinear functional b(lId' vd) mllst tend to zero in order to have a finite strain energy

V vd E V.rCw). and accordingly two sets of constraints prevail (cr. /3 = 1. 2):au, II, auz

"II =0' 0 --+-={) --=0In ., as R : dZ

all. ", all all. ilu.p =o· -+- =--'-+-"-= -"-= 0

nil . as R c1z as iJz (127)

Eq. (126) are well known as the Kirchoff-Love constraints while Eq. (127) describes the membrancconstraints relatcd to thc assumption that the middle surface is inextensible. If the finite elementapproximate spaces arc not rich enough (low order coarse meshcs). then approximate solutionsdegenerate owing to these constraints.

REAfARK 7.1.(i) There is no membrane locking for plate-like bodies because of 11R = 0: therefore. shell· like

bodies may present morc serious problems than plate-like bodies with regard to locking.(ii) If the membrane strain energy dominatcs in (122). then :iJ( ... ) vanishes without creating any

constraints. Thus it is clear that membrane-dominated problems do not suffer this type of lockingphenomena. 0

We ncxt consider the convergencc results relcvant to our work. Above all. it should be emphasizedthat the pure analysis of locking is available if the boundary layer effect is excluded_ Since the boundarylayer may result in a decrease in the regularity of solutions. the analysis of locking becomescomplicated. However. an introduction of periodic boundary conditions can exclude this effect. whichfor a square plate with sides 2a arc of the form

u;(a.)')=II;(-a.)'). Iyl:;::a} .• t = 1. 2. 3

II;(X. a) = IIJt. -a). \xl ~ a (128)

We shall denote the spacc H'(Jl) with pcriodic boundary conditions by lI~cr(J2).First. we record the error bound for the Reissner-Mindlin plate model in which thc vertical

displacement IV and the two rotations 0 = (0 •. 02) are approximated by the orders Pw and PO'respectively. We define solution sets H,.d = {Ud = (w.O) E H~;:"(n) : (J - VIV = O}~ H~::(n).

TllEOREAJ 7.3. For The standard finiTe elemenT approximation (107) of the Reissner-Mindlin modelII" E H,.'I using the IIniform rectangular elemenTs with shape f"flctions ,cfP( OJ) in (95). the h-versionmethod exhibits The uniform convergence rate for all d of 0(hP-1) when s ;?: r + 1. and shows locking oforder O(h I) when s;?: P...+ 1 (e.g. there is one-order loss in h-convergence rates due to the presence oflocking).

Approximation orders Robustness order [29. 30J-(](h')

r=p-1

Locking order-O(ht)

1= 1

PROOF. See [29.30J. 0

[56 IR. Cho. J. T. Odell I Complll. Methods Appl. Mech. Ellgrg. 132 (1996) 135-177

The next theorem describcs the error bound for the degenerated cylindrical shcll model forbending-dominated problems clamped at two opposite sides parallel to the axis of the cylindrical shell(cf. Fig. 33).

TH EOREM 7.4. Let us consider the solurion u corresponding to the degenerated three-dimensional shellmodel for bending-dominllted cylindriml shells with cort/a angles a < 'TT. Then, standard finite elementschemes using qUllsiuniform rectangular elements with shape functions Y( w) hllve the following errorbound

( 12l)

where IJ = (s - 1). µ. = min(s - 1. pl. Moreover, if the mesh is aligned with the axis of cylinda and thesolurion u ¥: II(Z). then the following improved uniform error bound holds for p ~ 3

hminC.,-l.p-2)111I-1I'111£(Il)~C I' 11"llw1'p

where C is a conS/(l1/l dependellt of s and the lOt/ding.

PROOF. Sce [25J. 0

( 130)

Recently, Suri [29) and Suri et al. [31l] studicd shear locking for higher-order plate models (q ~ 2).Higher-order terms in the displacement field lead to higher-order terms in the thickness variable withinthe bilinear functionals d(·.·) and b(-, '). and they are constraincd to zero as d-Il. So. higher-ordcrterms do not create an additional source for locking. This argument holds for higher-order models ofcylindrical shell-like bodies.

Finally. we summarize thc relevant results and provide the uniform crror bound of finite elemcntapproximations for the hierarchical models of the bcnding-dominated thin plate- and cylindricalshell-like bodies.

COROLLARY 7.1. Consider the stllndard finite element scheme (107) for the hierarchical models ofbending-dominated thin elastic plate- and cylindrical shell-like bodies. For every (I ~ j ~ q, U I E[H;,c,(w) 13, and furtha11lore U i ¥: U i(z) for cylindrical shell-like problems. Then. for the lfuasiuniform hlind p, the following uniform error bound holds for rectangular (curved) elemellts defined in the spaceyP(w) and aligned with principal curvature lines.

(131 )

where v = s - 1. it = min(s - 1, P - 1) for plate-like bodies and min(s - I. P - 2) for cylindrical shell-like bodies.

PROOF. This follows immediately from Theorems 7.3. 74, Lemma 4.2 and the arguments onhigher-order models 131l]. 0

REMARK 7.2.(i) The p-version does not suffer locking: on the other hand the hp-version is locking free if the

h-version is free from locking.(ji) The intensity of locking depends upon the mesh density with respect to the thickness. 0

l.R. Cleo. l. T. Odell I COII/pl/t. Methods Appl. Mech ElIgrg. 132 (1996) /35-177

8. Numerical experiments

8.1. The d·. h- alld p-convergellce rates

t57

Fig. 7 shows an example of a plate-likc body. which is clamped at thc boundaries and loaded by auniformly distributed load gz. Due to symmetry. we consider only a quarter of the plate, and thethickness ratios arc chosen as follows: aid = 3. 10. 311and lOa. Fig. 8 summarizes thc computed globalrelative crror with respect to the hicrarchy q for each thickncss casc. Herc. the global relative error gGis dcfined by

[]]]]]] gz

a -x

a = 1.0 ind = ThicknessE = Young's Modulus

107psiPoisson's Ratio = 0.3

3 2gz = 4.0 x (dJa) Ibslin

Fig. 7. Clamped squarc plate-likc body subjectcd to a uniform normal traction g, on the top surface.

0.45

0.4

0.35

"- 0.3guu 0.25>..,'""E 0.2al

,Cl0

G 0.15

0.1

0.05

01 2

Hierarchy q3

ald= 3""-aid = toaid = 30 .-

158 l.R. C!w. l. T. CJelen I Compl/t. Methods Appl. Merh. Ellgrg. 132 (1Y96) 135-177

where U(u) and U(uq) represent. respectively, the total strain energies of the three-dimensionalelasticity and the q-hierarchieal model. In these calculations. U(u) is obtained approximately usingq = (9.9,9) and U(lIq) is estimated using q = (I. I. 0)*, (2.2.2). (3.3.3) and (4.4.4) with 16 uniformquartic clements.

The results show a decrease in the relativc error as the hierarchy q incrcases or thc thickness ddecreases. The variation in the relative error due to q (the modeling error) increases as the thicknessincreases. which implies that the variation in the actual displacement field through the thicknessdeviates from the classical thcories as d increases. On the other hand. the decrease in the crror withrespect to the thickness decreases as highcr hierarchies are used. as expected.

One remarkable aspect of the results occurs with the (1. 1, 0) model and is contrary to ourcxpectations. In Fig. 8. formulation for the (1. 1.0) model uscs thc modified material moduli E*. whichis denoted by the (1. 1.0)* model 110. 32]. But the formulation using this modified material moduli inthe non-thin region produces higher strain energy, and leads to lower relative error. However. it isdifficult to pick the limit thickness for thcse two formulations. Fig. 9 prescnts the results when theunmodified material moduli E for the (L 1.0) model is used. Contrary to the previous case. a strainenergy lower than the correct one in the thin region produces the higher relative errors.

From Figs. 8 and 9. we observe that results support Theorem 7.1. As the thickness decreases. slopesof the relative error become slower for any hierarchical order. and this is cxplained by the termell,.,.I/:}: and as the the hierarchy level incrcases, the relative error decreases for any thickness. Fig. 10prescnts the d-convergcncc rates for the hierarchical orders q = 1. 2. 3 and 4. where both results of the(1. 1. 0) model and the (1. 1. 0)* model arc presented.

log 1111 - u

l.R. ClIO. l. T. Oden I COli/pili. MClhods Appl. Mech. Engrg. 132 (11)96) 135-177 159

le-05

le-06

le-07

...0 Ie-OSt:OJ

-.;.D0a le-09

le-IO

le-Il

le·12I 10

Thickness ratio aid

Fig. 10. The d-cllllvergence rates of thc plate problcm.

-+--.o£} ....--.A-

100

L= 5.0 in

R = 1.0 int = Thickness

Body force (r0) = 4.0 IbsliJE = 10 7 psi

Poisson's Ratio = 0.3

Diaphragm (u r= U e= 0)

Fig. 1L Cylindrical roof with diaphragm boundaries.

problem. and the thickncss ratios chosen arc: Rtl = 3. Ill. 30 and 100 as in the casc of the plateproblem. rig. 12 presents the global relative errors with respect to hierarchical orders for eachthickncss. wherc the q = I model is the (1. 1. 1) model. For each thickness ratio. the global relativeerror decreascs with an incrcase in thc hierarchical model level. Also. the global relative error in eachhierarchical model decreases as the thickness ratio Rlt increases except thc (1. 1. 1) model.

In Fig. 12 (also in Fig. 14). the crror in the (1. 1. 1) model decreases as the thickness increasescontrary to our expectation. We used a calihration factor of the membrane-dominated problem (-() forloading. However. the rclative bending deformation becomes larger as thc thickness incrcascs for whichStich a calibration factor is incorrect (it must be between I and tJ). The loading calibrated by a factor oft produces a strain energy a little larger than that of a loading calibrated by a correct factor: however.this larger strain energy is closer by chance to the strain encrgy of three-dimensional elasticity. which

t60 l.R. Clw. l. T. Od"11 I Compl/t. Methods ;\ppl. Alec". ElIgrg. 132 (1996) /J5- /77

0.3

0.25

.. 0.2guu>

'OJ

'OJ-il 0.15..Oi.0o6 0.1

0.05

oI

~~~~~-..........--'-----

2Hierarchy q

3

Rlt= 3-+--Rlt= 10 nRlI= 30 .-0 ..RlI= 100 -tf--

4

Fig. t3. Global relative crror as a function of the hierarchy q when modified material moduli arc used for 'I = t. i.e.1=(1.1.11)*.

Icads to the smaller error. Results of the case in which we replaced thc (I. I. I) model with the(1. 1, 0)* model arc shown in Fig. 13.

Fig. 14 shows the d-convergence rates of the membrane-dominated shell-like body. There. results ofthe (1, 1.0)* model. which is a modified model as introduced in Section 5. arc included. Thed-convergence rate of this model. which is not given by the a priori modeling error estimate (114). is

l.R. ClIO. l. T. Odell I Compl/l. Metltods Appl. Alec". Ellgrg. 132 (/9%) 135-177 lot

0.01

0.001

O.()()() I..gu(;i le-05.D0(3

le-06

le-07

le-08

....... -........"',.......

B- ..

q=(I.I,I) -+-q=(I.I.O)* -.-q=2 .-1) ....

i::3 ---- q----.-- ---+---.

···D••.•

10Thickness Rlt

Fig. 14. The d.convergence rates of the membrane-dominated shell problem.

100

.. 0.10t:uu.~~](;i.D0(3 0.01

0.00110 100 1000

Degree of freedom

p= I -+-p =2 -+--p=3 .-f) ....p=4 ----

10000

Fig. IS. The It-convergence with respect to the degree of frccdom for thc thin plate (tlld = lOO).

observed to be a value between those of the (I. 1. 1) model and the (2. 2. 2) model as the thickncsstends to zero.

Fig. 15 rcpresents thc "·convcrgence rates of the (1. 1.0)* hierarchical plate modcl for the thin case(aId = lOll). Here. the refcrence solution is obtained using p = X with 36 uniform clemcnts. Whcnp = I. the crror curve is parallel and this is well known as the shcar locking phenomenon. Fig. 16 showsthe p-convcrgence rates. where exponential convergcnce rates are observed. Figs. 17 and 18 showresults of the 11- and the p-convergcnccs. respectively. for the (I, 1.0)* model for the thickness ratio

t62 J.R. Clw. J.T. Odell / CampUl. iHet/wds Appl. Alech. Ellgrg. 132 (1996) /35-177

guu>

';;J

'""E0;.co5

0.1

0.01

0.00110 100 1000

.Degree of frccdom

Fig. 16. The l'-convergence of the plate prohlem.

h=ln -h = 1/3 -+-h = 1/4 .-6 ....h= 1/5 -h = 1/6 -4-h= In ......

10000

p=1 -p=2 -+-P =3 ..D ....p=4 --

.. 0.1guu>co"E0;.c0

5 0.01

0.00110 100 1000

Degree of freedom10000

Fig. 17. The "·convergence with respect to the degree of freedom for the thick plate (aid = 3).

aId of 3. Numerical locking is not experienccd. but slower h-convcrgence rates are ohscrved whencompared with the thin case.

Figs. 19 and 20 represent the II-convergence rates of several hicrarchical models for the thin and thethick cases. rcspectively. Numerical locking appears for cvery hierarchical model at p = I for the thincase. For higher-order mouels (q ~2). the h-convcrgence rates bccome slower as the thickness becomessmaller. This is due to the decreasc in the regularity of the solution duc to the boundary layer effect [3.30]. But, the (I. I. 0)* model provides opposite results and these are explained by the fact that the

J.R. ClIO. J. T. Odfll I Complll. Metleods Appl. Afecle. En!;rg. 132 (fl)

164 J.R. Clro. J.T. Odell I COli/pitt. Mt,tllOds Appl. Mecll. Ellgrg. 132 (199(» 135-/77

.. 0.10t:QQ.~~](;j.&>0

6 0.01

--+-..-.-1') ....-...- ..--+--

..•....

0.00110

Mesh size In!Fig. 20. nle II-convcrgencc variations with rcspect 10 thc hierarchical levcl CJ for the thick plate.

0.1

om

h=1I5,q=l* -q=2 -+---q=3 .-c ....

0.00110

Order p

Fig. 21. The IJ-convergence variations with respect to thc hicrarehieal level If for lhe thin plate.

We next present results of the (I. I. I) hicrarchical shell model. Fig. 23 rcprescnts the II·convergenccrates whilc Fig. 24 shows the p-convergencc rales. The thickncss ratio R/t is 100 and numerical lockingis not obscrvcd. It is observcd that the p-convergence ratcs arc not exponential but algebraic.

Figs. 25 and 26 show rcsults of variations in thc II-convergcnce of several hierarchical shell models forthe thin and the thick (R / t = 3) cases. The II-convergcnce rates of the (1. 1. 1) model are almostunchanged with variations of the thickness. Furthcrmore. there is no incrcase in the regularity as the

J.R. ClIO. J. T. Od/'I/ / Complll. Metleods Appl. Mecle. EI/grg. /32 (/996) /35- /77

h= 1/5.q= 1- -+-q=2 _.uq=3 ..., ....

165

...guu>

'.::l

'"1!C;.0o6

0.1

0.01

0.00110

p= I -+-p = 2 -+---p=3 ...,....p=4 -)(-

Order p

Fig. 22. The p·convergence variations with respect to the hierarchical level q for the .hick plate.

F i i a

~

0.1..0t:uu.::~ 0.011!C;.00

60.001

100 1000Degree of freedom

0.000110

. ~10000

Fig. 23. The Ie-convergence with respect to the degree of freedom for the thin shell (R/t = 1(0).

thickness decreases because the Kirchhoff constraints do not prevail for membrane-dominatedstructures. Since the boundary layer cffcct dccrcases as thc model level decreascs. thc variations ofregularity of solutions decreases. For higher-order modcls (q ~ 2). the h-convcrgence rates decrcasesslightly as the thickness hecomes smaller due to the boundary laycr effect. Due to this effect. theh-convergcnce rates hecomes very slow after a certain error level. as shown in Fig. 25 (p = 4. q =(2.2.2), (3.3.3».

166 l.R. ClIO. l.T. Odell I Complll. Methods Appl. Mecll. ElIgrg. 132 (/996) 135-177

0.1..0t:uu>

'z:l

'" 0.Ql11C;.D0

60.001

h = 112h=1I3h = 1/4h = 1/5h = 116h=ln

-+--+-.-0....--4-

0.000110 100 1000

Dcgr~ of frccdom

Fig. 24. The p·convergenee of the shell problem.

1()()()()

-9--

.-......

.-+--

-w·-··--.- ..

.-0 ....--..--

~"""..............--....-.........11'-.

-·- ....------p=l.q=1--~. q=2

---....._~ ~ q=3•.---J'=2,q=1

q=2q=3

p=3.q=1'--. q=2

~

"'---. q=3. ....p=4,q=1

q=2..'...... "'~. q=3

"G. K

0.1..gllJU.~

l.R. Clio. l. T. Odell I Complll. Methods AppJ. Mech. Engr!(. 132 (1

168 l.R. ClIO. l.T. Odell I COlllpllt. Methods Appl. Alecll. ElIgrg. 132 (1996) 135-177

h=lIS,q=1 -q=2 ~-q=3 .-0 ....

0.1

0.01

0.001

0.000\10

Order pFig. 28. The p-convergence variations with respect to the hierarchical level If for the thick shell.

.... 0.1guu>

'::J

l.R. Cho. l. T. Oden I Complll. Methods ApI'/. Meeh. l:'ngrg. 132 (1996) 135-177 169

0.1gvv.::0;]OJ.0o(3 0.01

0.001

0. ..

'0 •.

a/d= I 000, p=1p=2

ald= 5, p=1p=2

)c slope = 2.0

10Mesh size I/h

---+--.-0 ....-

Fig. JO. The iI·convergence rales of the (1. I.0)' model for the thick (aid = 5) and the lhin (aid = WOO) cases.

"- 0.1gv

">.;::'"]

OJ.00(3 0.01

0.001100 ]000

Degree of freedom

h= liS -h = 1/10 -....--

~

10000

Fig. 31. The p·convergence rates of the (I. 1.0)' model for the thickness ratio aid = 1000.

Next, we consider the bending-dominated quad-cylindrical shcll shown in Fig. 33. We consider thchalf of the shell labellcd by mesh using the symmetry.

Fig. 34 represents thc II-convergence rates. where bilinear element suffers complete locking on thecoarse mesh while quadratic and cubic elements have too-order loss in the h-convergence rates.However. thc II-convergence rates arc improved by onc-order for p = 2. 3 on our final refined mesh.

170 l.R. ClIO. l. T. Odl'll I COlliI'm. Metlrods ApI'/. Meclr. Ellgrg. /32 (1996) /35-177

..gvv.~(;j

~(;j.0o6

0.1

q=(I,I,O)*, p=J -p=2 -+-

q=(2,2,2), p=1 ...,....p=2 --

toMesh size lilt

Fig. 32. The "·convergenee rates of the higher·order model for the thickness ratio aId = IlK)(l.

L=2.5 InR = 1.0 in

d = Thicknessg(8,x)=4.0 x (tIR/lbs/in

2

7 .E = 10 pStPoisson's Ratio = 0.3

Fig. JJ. Quad-cylindrical shell·like body with uniform normal tractiolls.

Fig. 35 shows the comparison between the thick and thin shells for p ~ 3. where the II-convergencc ratesof the thick shell are hounded by the regularity of the solution. The /H;onvergence rates arerepresented in Fig. 36. whcre the numerical locking is not observed.

From the numerical results. complete locking occurs for bilinear clcments of bending-dominated thinplate- and shell-like bodies. This is because every hierarchical model approaches the Kirchoff theory.the biharmonic equation. as the thickness becomcs thinner. For thc case of bilincar clements. rotationsof the normal to the middle surface are continuous and constant owing to the Kirchhoff constraints, andthis leads to a zero rotation function for the problems with built-in boundary conditions.

J.R. ClIO. 1. T. Odell I Compll/. Methods Appl. Mech. Ellgrg. 132 (/996) 135-177 171

".

....~slope = 0.0 .................

slope = 1.0....p=1 -+--p=2 ---P = 3 .-0 ....

"Q,.

0.1

'Q,.

·lI ...slope = 1.0·e.

·ct.·tI.

·tI.'1iI

'1!J.D

'1!J'[;j

1!J slope = 2.0

0.01 ..L10

Mesh size I/h

rig. 3-1. The h·convergence rales of the (t. t. 0)' model for tht: thickness ralio Rlt = 1000.

••• _u

--+--..g ....

-M--

'""',..... '"''...... ' ....'.. ''''...... '.,.... ....' ...

.-.:::: slope = 1.0

-~ .......slope = 0.0 ............Jltin ,p=1

slope = 1.0...• p=2p=3

Thick,p=1p=2

. p=3"1].. ~ slope = 1.0

·'D ..;Slope= 1.0·El.

li

tI.·tI.

'Q

'1!J.'0

'1!J

'Q1!J slope = 2.0

'01.

"

•,\ ,,,

~,,-, ~4

0.1

0.0110

Mesh size I/h

Fig. 35. The h-eonwrgence rates of the (1. 1. 0)' model for the thick (Rlt = 5) and the thin (Rlt = 10(0) cases.

On the olher hand. higher·order clements (p ~ 2) just lose their unlocked II-convergence rates oncoarse mcshes by one-order for plate-like bodies and two-orders for shcll-like bodics. Fortunatcly. it isobserved that they rccover their II-convcrgence rates as the mesh is refined. as mentioned in Remark7.2. Fig. 37 shows thc locking of the (2. 2. 2) model.

172 J.R. ClIO. J. T. Ot/I'II I Campl/t. Methods Appl. Alec/I. EI/grg. /32 (1996) 135-/77

0.1

0.01100 1000

Dcgrcc of freedom

h = 1/5h = 1/10

\~

--+-

10000

Fig. 30. The p·convergence rates of the (t. t. 0)' mollel for the thickness ratio Rlt = 1000 .

...ol:oo.~(;j]til.co5

0.1

0.01

·- ............n=I*,p=1~ p=2

p=3q=2,p=I

p=2p=3

10Mesh size l/h

-+ ..--

··0····-)(--

Fig. 37. The lI·convergence rates of the higher-order model for the thickness [alio RII = 1000.

9. Conclusions

An a priori error estimate of lip-finite element approximations of hierarchical Illodels for plate- andshell-like bodies is derived. The hierarchical model is an approximate theory within the three-dimensional theory of elasticity obtained by restricting the order of polynomial variations through thethickness in the displacement or stress fields. Therefore. the hierarchical model itself has an inherentmodeling error when compared to the full three-dimensional theory.

l.R. ClIO. l. T. Dden I Comp/{(o Ml'llIods Appl. Mecll. ElIgrg. /32 (/9

174 J.R. ClIO. J. T. OdE'1l I COlllplll. Mt,tllOds IIppl. Alech. Ellgrg. 132 (1996) /35-177

( 1) V V ap - (V vrj )(U V I'm ) ik "Il.n· .pg - '.a - j ia Yk.p - /PI k/3 g g= {(Vy.a - \tjr~

l.R. Cho. 1.7'. Oden I Complll. Methods Appl. Meeh. Engrg. 132 (1996) 135-177 175

Since A~ = (8~ + (}Jb~ + ... ).

(2.1) IV"A;b~glll~ = lV"b;gll + vob:(e3b~ + ... )gJ311

:S;I~.b;glll~ + 21~.b~gtJll~.b~«(J3b~ + ... )glll + lV"b~(O·lb~ + ... )gill1~2W.b~glll~ + 2IVob~(e3b; + ... )g/3l~ (144)

Substituting (144) into (143).

I(VZ) .31~~ 4IZI~lVob;gf.T + 4IZI21V"b~(e3b; + ... )g1l12+ 2IZ.3121Y;gr (145)With the results, (141) and (145). we have

IIv(e I, (1)Z(8.l)il~.n:S; r {IZI~(21V,1"~lllg'/gO/l1 + 1V11+ 4IV.b;g/312J1I+ 61V"b~(o3b; + ... )gjll~ + 21V3bp/3(83b:)gP@gilI2) + 21V1~IZ31~}ygdO

( 146)

Now. let us examine bounds of terms in (146).

(3.1) IV.b;g/l12 ~ 12V.li;brna.g/I12

= (2brna.,)21V12. brna. =rnax abs(h~)u.p

(3.2) IA;I ~ {I + (d",brna.l2) + 2(d",brna,/2)2 + }~ {I + 2(dmhma./2) + 22(dmbma.l2)2 + }= {I - (d",bma.)} -1 = Am

183b; + ···1 ~ {(dmbma.l2) + 2(dmhmax/2)2 + ... }

(d",bmax ) -= =bI - dmbmax

(3.3) lV"b~«(J'lb~ + ... )gIl12:s; IV.(2bma.8;)gill~l(e.lbma. + ... W,- ,

= (2hrn .. tbIVI"

(3.4) 1V3bpp(O .lb~ )gP @gP12 :s; (d",h V3 )2Ibl'/lby"gIlY gPU I= (d",hV.l)2Iali/lawb:b~g/lYgPal= (d",hV.l)2Iali/lafOb;b~A~A;(r A ~A;aAJI~ (d",hV3 )2(2hrna.)2IA ~A~A~A: 1

~ (2Am)~(2bma.dn/l)2IVI2

Therefore. with b* = 1 + 8(bmaY' (2 + 3b + 16h2d~JA~,)

IIV(o 1. 0 ~)z(03)1I~.n ~ L (f~~2 {IZI1(21 V;lu ~1/lg'/g"/l1 + b*1V 12)+ 21V12IZ ..J}Iµ.1 d03)v'll dw

(147)

( 14H)

(149)

(ISO)

(151 )

Next. we express the metric gil in terms of ail for separating intcgral ovcr n into one over wand thcothcr ovcr (-dI2. dI2). In order for this separation, let liS define

with

gil = A' Ai -kmk ma (152)

176 l.R. ClIO. l. T. Oden I ('ompUl. Melhods App/. Mech. Engrg. 132 (1996) t35-/77

{

k",-k", (l • k. m = 1 ?a = .-

kmg . k orm =3{'i'l "k -I?-i - i = 1\ k 1\ ",. I. J. . 11/ - .-

A kA In i iBkB m' others

then with a definition of Am = ( I - d",h rn",) -I ~ 1.

(153)

g"µ ".; (2A )2 "13111 Cl ( 154)

(155)

Therefore.

J ( f

l.R. ClIO. l. T. Odell I CompUl. Methods Appl. Mech. Ellgrg. 132 (1996) J.l5-177 177

(9) L Babuska and t.,1. Suri. Locking effects in the finite element approximation of elasticity problems. Numer. ~Iath. 62 (1992)439-463.

IIOJ 1. R. Cho. A study or hierarchical models or structures using adaptive hp·finite elemenl methods. Maslers Thesis. TheUniversity of Tcxas at Austin. 1993.

III1 1.R. eho and 1.T. Oden. A priori modeling crror cstimates of hierarchical models for elasticity problems for platc- andshell-like structures. Math. Compul. /llodel .. to appear.

1121 P.G. Ciarlel. The Finite Element Method for Elliptic Problems (North-Holland. Amsterdam. 1978).1131 1.. Demkowicz. A. Karafiat and 1.T. Oden. Solution of elastic scallering prohlems in linear acoustics using h·p boundary

clement mcthod. Comput. Methods Appl. Mcch. Engrg. 101 (1992) 251-282.114] 1.. Demkowicz. A. Bajcr and K. Banas. Geometrical modeling package. TICOM Report lJ2-06. The University of Texas at

Austin. 1992.1151 L Demkowiez. 1. T. aden. w. Rachowicz and D. Hardy. Towards a universal I1-p adaptivc finite clement strategy. Part 1.

Constrained approximation and data structure. Comput. Methods AppL Mech. Engrg. 77 (1989) 79-112.116) Ph. Dcstuynder and M. Salaun, Approximation of the geometry of a she \I for non-linear analysis, Preprinl. 1994.117) W. Fliigge. Tensor Analysis and Continuum Mechanics (Springer-Verlag. New York. 1972).It8) M.D. Greenberg. Foundations of Applied Mathematics (Prentice-Hall. Englewood Cliffs. Nl. 1978).[19J H. Kraus. Thin Elaslic Shells (Wiley. Ncw York. 1967).[20J L Li. Discretization of the Timoshenko bcam prohlem hy the p and hop versions of the finit