~

~ElSEVIER Compu\. Methods App!. Mech. Engrg. 136 (1996) 3]7-.,45

Computer methodsin applied

mechanics andengineering

Adaptive hpq-finite element methods of hierarchical modelsfor plate- and shell-like structures

J. Tinsley Odena.*, J.R. ChoJ

Texas IIISlilllle for COII/plllatiol/al tllld Applied ;\fa/hell/min. Th.' Universily of Tex(/.~ III AUSlil1. AIiSlill. TX 78712. USA

Received 7 October 1995: revised 16 November 1995

Ahstract

A four-step adaptive strategy for hpq-linite clement approximations of hierarchical models 01 lhc .hin plate- and shell-likestructures is derived based on a priori and a posteriori error estimations. This strategy controls three parameters: the model level(I. the mesh size h. and the approximation order p so as to produce 4uasi-optimal hierarchical models and tinite element meshes.Numerical results arc given supporting the theoretical results.

I. Introduction

Plate- and shell-like bodies arc three-dimensional material bodies with a rhickl/('ss dimension smallcompared to the other two. This particular gcometric feature permits more tractable two-dimensionalmodels to bc dcwJopcd such as the classical Kin;hhoff and the Reissner-Mindlin thcories whichprovide a dimcnsional reduction by restricting the order of polynomial approximations of displacementsor stresses through the thickness. It is known. however. that the classical theories arc frequentlyinadequate for modeling complex stress states. particularly at boundaries of members. or in bulkynon-prismatic bodies. or at junctions of two or more members.

Recently. the notion of hierarchical modeling has emerged. The idea of hierarchical modeling is toproduce a family of mathematical models which can be parametrically connected in some way togetherwith a strategy for selecting a member of the family that provides an appropriate \evel of sophisticationto provide a given accuracy. The lowest level of the hierarchy may possibly correspond to classical beamor membrane theory or to a Kirchhoff-type theory. while the highest sophistication may correspond tothe full three-dimensional theory of elasticity. Which theory is appropriate for a given application mustbe determined by an independent judgement of the quality of the solution. Fig. I illustrates the ideathat hierarchical modeling can be implemented locally within a mode\. AI K representing a hierarchicalmodel for a sub-region Ilk"

At least two types of errors prevail in approximating a hierarchical modeling: (I) the error inherentin the hierarchical models of plate- and shell-like structures due to assumptions on displacement fields.and (2) the numerical error in the tinite element approximation of the solution corresponding to aparticular hierarchical model. For certain classes of problems. three parameters, h. p and q are used tocontrol accuracy. hp-meshing of the reference surface and q-approximation through the thickness.

• Corresponding author. Director of TICAI\1. and Cockrell Faculty Regents' Chair in Engineering #2.I GRA. Texas Inslitute for Compulational and Applied I\'lalhemalies. Email: jin-rae@licam.llIexas.edu .

004S·782S/961s15.00 © 1996 Elsevier Science S.A. All rights reservedSSDI 0045-7825( 95 )Cl0986-S

318 J. T. Ot/ell. J.U. Clio I Compll/. Melhods Appl. Alccll. Engrg. 136 (J<)<)fi) 317-345

Conventional Modeling Hierarchical Modeling

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fig. I. Illustration of cOllventional and hierarchical modelings.

Estimated errors of the approximate solution corresponding to an initial q-distribution and to an initialmesh may give somc indication of the accuracy of the particular model Icvcl and of the approximationof the solution of a particular model.

But the proportion of the total error due to modeling or to numerical approximation is unknown andcan vary non-uniformly over the structurc. However. the two error components arc orthogonal in anenergy inner product in this particular class of hierarchical modeling. so control of modeling crror maynot guarantec control of approximation error. and vice versa.

To the best of our knowlcdge. few q- or hpq-adaptive schemes have heen proposed. A q-adaptivitymethod for selccting optimal hicrarchicalmodcls has been studied by Vogclius and Babuska [29] and byBabuska and Schwab [5] for two-dimensional scalar-field problems. A scheme for thc hierarchicaladaptive modeling of multilaycred compositc structures has been introduccd by Noor ct al. [14]. Steinand Ohnimlls have devcloped an "pq-adaptive schemc for elastic structures {24,25].

In this paper. a four-step strategy for "pcl-adaptive finite c1cment mcthods is introduced. in which qand h. p controlthc hierarchicalmodcls and thc finite clement meshes. respcctivcly. A parametcr a isintroduced as a measurc of thc relative proportions of modcling and approximation errors and a schcmeto cstimatc (¥ and the total error in an cncrgy norm is proposed. Basically, this algorithm is dcrivcd onthe assumption of equidistributing the error. and on the treatment of inequalities of a priori- and apostcriori crror cstimates as equalitics. The scheme builds upon the thrce-stcp adaptive strategy for"p-adaptivity described in [17.181 and adds a fourth step to measure and control modeling crror. takingadvantage of the orthogonality property of the error components.

The application of the adaptive algorithm to represcntative platc- and shell-like structures isdescribed. Examples arc provided in which an initial modcl of a cylindrical shell-like or a flat plate-likebody is adopted into a model of non-uniform q- and hp-clements. Estimates of the accuracy of suchcalculations are also given.

2. Notations

We let n E [R3 denote an open bounded Lipschitzian domain with piecewise smooth boundary an.The Sobolcv space H'"(fJ). m E 7L+. is a Hilbert space defined as the completion of {u E<'(5"'(fJ): lIull.".11 <x} in the Sobolev norm

Ilull~,.JI= L (ID"uI2 drlal.;m JlI

where multi-index notation is used: 0'= (al• a2• 0'3)' a; E 7L'. lal = at + a2 + 0'3 and

(2)

are weak or distributional partial derivatives and dx = dr I d\"2 dx 3' The inner product on the spaceH"'(fJ) is dcfined hy

J.T. Udell. J.R. ClIO I Comp/ll. Methods AI'pl. Mech. E/lgrg. 136 (IW6) 317-345

Fig. 2. A shell-like structure.

(Il.u)",.o= L f D"Il'D"ud\'la<l"'" If

319

(3)

Furthermorc. H~'(n) is dcfined as the closure of ~~(n) in the spacc lJ"'(n).Fig. 2 depicts a shell-like structure n which is characterized by the refcrence surface wand thc

thickness d. Here. the reference surface is the middle surfacc and the thickncss is varying andsymmetric with respcct to this surface. We let w E 1R2, n E 1R3 denote open hounded regions withpiccewise smooth boundaries aw and an. respectivcly.

(4)

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(0 I. (J 2) are orthogonal curvilincar coordinates in thc refercnce surface w which coincide withorthogonal lines of principal curvaturcs and 03 is a coordinate normal to w.

In Fig. 3, the position vector r of a point P; (81• 82

, 03) is described by a position vector r0 and a unitnormal vcctor fl of a point Po: (01.82.0):

123 12· 312reO . (J .0 ) = rll(/! ,0 .0) + (J fl(O .0 )

In the Cartesian coordinates x = (Xl. x2• x3

) with orthogonal unit vectors e •. e2 and e3,

, '( I 2 - 3)r = x e,.= x (/ . H . fj e,. i = l.2. 3 (6)

and. r.i = arl aoi arc (assuming thc maps x' to be smooth enough)

r.i = x'fe,. j = I.2. 3

p

o

Fig. 3. Position and normal vectors.

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3211 l. T. (hIell. l.R. Clio I Campul. Alelllods Appl. Mech. I:·ngrg. 136 (1996) 317-3-15

Two base vectors g" tangent to the surface fJ .I = constant arc dctined by.I

go = ro = '0.0 + fJ n"

- + 0.1- a" ",,, . a=1.2 (R)

with a" = r O.a = g,,(fJ I. fJ ~. 0). To such a tangent plane. the unit normal vector. " = g.l = rJ' is defined by

gi = giigi

/1=(gl xg2)/lgl xg~I=(al xaJ/laJ Xa~1

The contravariant basis vectors gi are rcciprocal to the covariant basis g;:i <oig . gi = 0 i

gi = giigi .

The coefficients of the metric tensor are defined by

gii = gj . gi = gi . gi = gil

gii = gi . gi = gi . g' = gii

(9)

( 10)

(II)

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I t tl t ".I 3" () I .13 Iall(. no e HI g,,3 = K.I"= g = g = an( g.l.l = g = .Now, we list basic coefficients defined on a reference

fundamcntal form arcsurface 03 = O. The coefficients of the first

(13)

The contravariant components are related to the covariant components according to

aU. a = Sa{3 {3

au = ll"pa{3

The coefficients of the second fundamcntal form describing the curvatures of a refcrcncc surface are

(14)

Thc coefficients of thc third fundamental form arc

(I5)

The normal curvatures of a reference surface w in thc directions of the base vcctors a" arc

(16)

and. the Gaussian curvature c of thc surface ware defined by

c = Ib~1= b:b; - b~b~

For a general surface 03 = constant. ga in (8) can be cxprcssed in terms of b~

g" = u" + 0.111."_ _nJb If- a" " "pu

= µ.fJa" {3

with

Similarly, for the contravariant basis vectors gU. if we write

g" = A;a{3

(17)

(I8)

( 19)

(20)

5=1.2

J. T. Odell. J.R. Cho I Complll. Methods Appl. Mech. t;ngr};. I36 (/996) 317-3-15

then. from thc relation of g,,' gf3 =5=.A; = 5; + OJh; + (O.1fh~h~ +.... 'Y = 1. 2

At any point of a body. the intlnitcsimal volume dv is givcn by

dv = dr3' (drl x dr2)

I ' l= g.1 . (gl x g2) dO dtr dO'

= ygdO

wherc dO = dO ' dO 2 dO.1. Morcovcr. yg = Vlgijl is related to Va as follows

yg=g.1·(gl Xg2)_ (" 13)- g.1' µ'I a" X a/3µ'2

= Iµ.Ivawherc Iµ.I = µ.: µ.; - µ.i µ.~. The infinitesimal surface ds on a surfacc of 03 = constant is given by

dv = 11 . (dr, X dr2)

= ygdw

whcrc dw = dO 1 dO2 .

The sccond kind of Christoffel symbols r;k are dcfincd by

r' i I rijk = g . gl.k = g . gk.j = kj

and. using the relations (18) and (20).

I,"" A" Ii 1'-" + A" Iif3y = a . all.y + oµ'(3.y = fly 1iµ'(3.y •

1'.1 - 3 ( y ) - y b,,(3 - g . µ'"ay .(3 - µ." yfl

r~f3= -bf3yaY' aliA: = -A~b~

whcrc t;y arc thc quantities dcfined on thc refcrcnce surface. Notc that 1';" = r~.1 = r~.1 = o.

3. Preliminaries

32]

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(22)

(23)

(24)

(25)

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Let V denote a displacement vector of a material particle located originally at r. Then. a ncw locationR is given by

R(O 1 .02. e 3) = r(O J .02,0.1) + V(O 1 .02. (3)

With defining Gj = iJR / iJOi and Gij = Gi' Gj•

, ~ . .dR- - dr- = (Gij - gij) dO' dOl

= 2£. dO; dOj'J

Thc strain measure £ij is exprcssed in the form of

2£ij = G,j - giJ

= gi . V.j + gj . V.i + V.I' V.j

(27)

(28)

(29)

We confine our attention to the linear theories of kinematics in which the assumption of infinitesimaldeformation leads to drop the non-linear terms V.I' V.j in thc strain-displacement relations. Thegradient of vector function V.I is expressed in the following form of

322 l.T. ()dell. l.R. ClIO I Complll. All'lllOds Appl. Medl. Ellgrg. /36 (1996) 317-345

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Thcn, (29) becomes

(31 )

We shall refer to the structure n as plate-like whenevcr b,,µ = 0 for all a. {3 = 1. 2 and shell-like whenb"fJ =;f 0 for any a. {3 = 1.2. We choose the orthogonal curvilinear lines 01

• f:J 2 such that they coincidewith lines of principal curvature. The Sobolcv norm Ilvlll.a in this curvilinear coordinate system is givenby

IIvll~.f1= In (IThI~+ Ivl:)yg dO

f {fdl2 };1 jk ij 3= OJ -d12 (Ig g v;ljvkIIl + Ig vjvil)Iµ.1 dlr vli d(r) (32)

In this paper. we focus on plate-likc structures and cylindrical shell-like structures for the applicationsof our mcthods described later. Wc list prcdefined quantities for the surface of a right-circular cylindcr(01 = O. (j: = z. 03 = r)

c=Og _ 2

11 - r .

b:=-I/R.µ.:=r/R.r~1= -r.

g~2 = g33 = I . others = 02 2 Ib: = bl = b2 = 0

: 2 Iµ.~= 1 . µ'I = µ'2 = ()

r:3=nl=l/r. others =0

(33)

where R is the radius of middle surfacc. Then, the components of the strain tensor are

ell = vl.1 + rv3·

21012 = (vI.: + v2.l) .

and the norm Ilvlll.fl is

10 =v33 3.3

(34)

(35)

(36)

It is useful to introduce the notion of membrane- and bending-dominated behavior, for thin elasticbodies with symmctric thickncss with respect to the middlc surface. For constant curvatures, thc evenmodes in the strain field, which arc cven-order polynomials in thc thickness coordinate, are

(11,,'µ)even = (11".# + lIyt:.# + 113baP)ev"n - 03b;byiu3)Odd

(1I"13)even = (11".3 + lIyt:.3t"en

= (U".3 )even - {li ;(uy ) even + f:J 3b;(uy )odd + }b~

(II 31" )even = (113.,.)",,,n - {li ;(Uy )e,'en + (J 3b;(lIy )odd + }b~

and vice vcrsa for the odd modes. These two modes becomc deeoupled in thc energy inner product asthe thickness tends to zero since yg can be approximated by Va. We then define the two portions ofstrain encrgies of the cven and odd strain modes.

J. T. Odell. J.R. Cho I Compw. Methods Appl. Alec/I. £Ilgrg. 136 (1996) 317-345

U(u)lc\'cn = U(Ejjlc"en)

U(U)l.ll1d = U(Ejjlodd)

323

(37)

(39)

We shall refer to a shell-like problem as membrane-dominated whenever thc even strain modes dclivcrdominating part of the total strain cnergy, and bending-dominated when vice versa. For plate-likebodies. this is discussed more fully in [21].

Referring to Fig. 4. we consider the structure with varying symmetric thickness. and suppose v iswritten as v(O)=V(Ot.02)Q(03). (OI.02)Ew. -d(OI.02)~2(J3~d(OI.{l2). Here. VE(H (w»3 andQEH1(-d/2.d/2). Then, we define a function Q(03)EH'(-dm/2.d",/2) with d",=max,ud(OI.02)such that

Q(03)={Q(03). -d(Ol ..O:)~203~d(O'.O:) (38)o . otherwtse

Note that, for plate-like bodies. (Ht(W»3 x H1(-dm/2. d".I2) is denoted as a convex hull of(HI(w»3 x H'(-d2. d/2).

The classical equations governing equilibrium of a material body occupying a region n E 1R3are.

(}'.jj(u)L+f=O inn}u'=O on/;).l/j(u)nj = / on l~

Wherc ro and /~ are the Dirichlet and Newmann boundary portions of the boundary an; an =To urN' [~ n r.'1 = 0. The boundary an is composed of thc lateral boundary ant and the faces an~ ofthe body. as shown in Fig. 2:

ant = {o EIR31(OI.02)Eilw.lo31 <d/2}

an" = {O E 1R31 (0 1, 0 2) E w. 0 3 = ±d / 2} (40 )

an =afll uan:

For convenicnce of our study, we define thc projected boundaries awo and aWN by

awo = {(0 I . 02• 0) E ~)} = tV n ro

I' I 1 > / 'dWN={(/J .O·.O)Ew (0 ,O-.±d 2)E/N}

The strain-displacement relations and the constitutive equations arc

2Ejj(U) = (uilj + /iii;)" "kl

(}'I/(U) = E'I Ekt. 1 ~ k. I ~ 3

In (39). covariant derivatives (}'jjlk are dcfined byij _ jj Ij j kl;

(}' Ik - (}'.k + u r kl + u Tjl

Fig. 4. A shell-like body with varying symmetric thickness.

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(42)

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324 l. T. Odell. l.R. Cho I Complll. Melhods Appl. Mech. 1·:lIgrg. 136 (1996) 317-.1-15

and aii(u) are the contravariant components of Cauchy stress tensor

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Furthermorc. 1/i reprcsents thc components of a unit extcrior normal at a point II E an. t reprcscntsthe body force componcnts. and I' represents applied traction components on the boundary J~ of an.In (42). Eijkl is the fourth-order elastic moduli tensor of elasticities satisfying thc cllipticity condition(a> 0)

~ii = ~ii (45)

and Eiikl = Eiikl = E'ilk = Eklli. Throughout this work. we assume that !1 is sufficiently smooth(Lipschitizia1/ or smoother) and that the data rand ti are smooth: e.g. tEL \[1). ( E L2(J~).

We define yen) = {v = v(O) E (H '(n»3: v = 0 on To} so that the displacement fields have finitestrain energy U(v) < +X. Then. the variational fonnulation of the boundary valuc problem (39) is

Find II E V(!l) such that V v EVen)

a(lI. v) = t(v) (46)

where the bilincar function a: V(n) x V(n)-IR and the linear functional (': V(1l)-1R are in thefollowing forms:

(47)

Note that 2U(v) = a(v. v).Existence and uniqucncss of the solution of the problem (46) are proved using the continuity and the

V-ellipticity of the bilincar functional a(-. .) in the space V(n): there exist two positivc constants c l' C 2

such that for allY II. v E V (assuming mcas(rD) > 0)

(1) la(lI. v)1 ~ clllulll'lIvll,·(2) a(v. v) ~ c~lIvll~·

(48)

where eland c 2 arc positive constants and l'o is assumed to be (TD) >O. If r = 0 or meas(r;) = 0. thenthe solution to (46) is unique up to an arbitrary infinitesimal rigid motion.

4. Formulations of hierarchical models

In order to formulatc hierarchical models. we define the subspaccs Vt'(11) C V(n) with q =(q" Q2' (/3) as follows

{

V'/ = 2:.'~'~.Vi (0 t. 02 )P(O -') I vi (0' . (J 2) E H 1(lrJ) .}V'/(f1) = " I~() " I n

V~ =()on dWf)' O~j~qa' Cl' = 1.2.3

(49)

J. T. Odell. IR. Clio I Complll. Methods Appl. Mecll. ElIgrg. 13b (1996) 317-345 325

where Pi are polynomials of order j. The traction 1+ acts on the top surface iJfl+ and 1- acts on thebottom surface alL of plate- and shell-likc structurcs. and thc Dirichlct boundary rD is restricted to thelateral boundary ilJ2t. Then, the bilinear functional (/: V<J(f1) x vq(J2)-~ and thc linear functional( : V'I(Jl)- ~ can be cxprcsscd by the following dimcnsionally-reduccd forms. respectively.

f {Jd'2 }'l q I}kl 'I 'l'''' q q" Ja(u .v)= _ ,E (Uk./-II,) k/)(ui.j-u"r'I)!µ.!d8 vadww d'_ (50)

{(v'l)=f {Jdl: fU;lygd83}dw+ r {11;(OI.(J2.+dI2)l-Y;;: +u;(O'.o:.-dI2)ti-yg}dw(r) - (1/2 J ifWN

where £ijkl are clastic moduli of the thrce-dimensional elasticity and g + . g - represent the values of g onthe top and the bottom surfaces. respectively.

As a result. the variational formulation (46) becomes a dimensionally-reduced approximate formdefincd on the space V(!l).

Find u<JE V'l(fl) such that V vq E V'}(il)

a(u·l• v<J)= ((vq) (51)

A solution uq = u~g' of the problem (51). is defined as a projection of the solution of the three-dimensional elasticity problcm (46) on to the subspace V"(fl) with respect to a(·.·)

(52)

With specifying q = (ql' If:. (13) as a modcl order. we shall define a solution uq = (lIit.II!'. uj3) as the(q\. q:. q3)-hierarchical model. an approximation theory of the solution of fully thrce-dimensionalelasticity problem (46).

For the hierarchical models with lf3 ~ I of bending-dominated thin bodies. the modification of Eijkl isrequircd so that such models havc a correct limit. This modification is done by the rcplacement ofA = vEl {( 1+ 1/)(1 - 2v)} with vEI(l- II): [8.271. We shall denotc the (lfl' Q2' Q3)-hierarchical modclswith thc modified material moduli by thc (q\. q2' (13)* hierarchical models.

We ncxt summarize finite element approximations of (51); complete details are givcn in [10]. Thereferencc surface W is divided into a collection of .N = .N('lP) meshes. then domain il is totally coveredby finitc subdomains fll( over which piecewise continuous polynomial approximations uq

·h of the

q-hierarchical model are approximated.x

w = U w/( . W/( n WI. = 0, K 'I: LI(~I

."ti = U tiK • fl/( n ill. = 0. K 'I: L

I(=t

tiK n til. = eithcr empty or a common edge or a common smooth surfacc I~L

hI( = dia(wl()

PK = sup{ dia(B I() IB I( = ball in wK}

uq·IIII1K E 'lPPK(WI() X ~</K( -d 12, d 12)

uq·I'EVq·"(il)CVq(fl)+ {w: wEH1(f1)lrow=u on rl)}

wherc diamcter is measurcd on the unfoldcd flat clement for the curvilincar element. Thc restrictions ofthe finitc element approximation uq

·h to an clcmcnt fll( bclong to the product of two spaces. 'lPPK(WK) of

polynomials of dcgrec PI( over WI( (in which the order Pll of each variable 0" ~ Pl(' a = 1. 2). andg},</K(-dI2, d12) of polynomials of degrce ql( = (II .....(12K, lf3,..) in the thickness direction 1101. The mcshsize hK and the approximation order PK may vary over the mesh. but satisfy the quasiuniformity inmeshes (hl(lpK~(T=positive constant. and quasiuniform PK) so that (asymptotic) approximationproperties hold.

326 l. T. Odell. l.R. Cho I Complll. Methods Appl. /Iledl. ElIgrg. 136 (1996) 317-345

The finite elcment approximation of (51) obtained in the space yq.h is characterized by the followingdiscrete problem

Find /1'1.1. E y'l.l'(n) such that

a(uq·iJ• v'I.I') = t(vq·h). V V</·h E yq.l'(il)

5. Error estimates

5. 1. A priori error estimates

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We review major results presented in [101 in which a priori error estimates of the hierarchical~odel~nr error eq E V(il). eq = /I -uq an~ the .finite elemen~ discreti.zation error <.h E y"(fl). eq

•1I =

ul - U I, are developed together WIth a dtscusslon of numencal lockmg and numencal results.The total error e measured with respect to the solution u of three-dimensional elasticity is given by

e E Y(il). e = eq + e'I.I' = u - uq·1I

These two errors arc orthogonal in the cncrgy norm IIvllf:({l) = Va(v, v).

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LEMMA 5.1. Let e'l E [H'(fl)]·l nYen) be rhe modeling error of a q-hiertlrchical model, and e,,·iJE[H'(fl)f n V'I(fl) be the finite element approximation error. Then. Ihe two errors are orthogonal in theenergy norm. i.e. a(eq

• e,,·h) = 0 and, consequenlly,

lIell~(tl) = lIell~(n) + lIeq·hll~({l)PROOF. Sce 110] 0

The following thcorem provides the global modeling error estimate.

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THEOREM 5.1. Let u be a solwion of problem (46) and uq be a solution of the dimemionally-reducedq-hierarchical model (51) with a model order q=(ql,q'.Cfl)' and let the applied loading be inIi 2/\/ (w) = {J E IJ2.11(w) : 'Y2;-1 flawr; = o. j = o. ].2, .... M}.- T/;en. there exists (/ constant Cq (dependenton q and M. but independent of d) such thm

lIu - u'lllnn) ~ Cq(d/2)("+ 112) (56)

where s = mine qm - 1. 2M) for Ihe membrane-dominated problems while s = mine qm' 2M) for thebending-dominated problems. qm = min(ql = (ho q3) + 0 with 0 = maxlqt - q31 ~ I.

PROOF. For details. see Cho and Odcn [91 which is an extcnsion of the approach of Vogelius andBabuska [28] to plane elasticity problems such as beam-. arch-. plate- and shell-like structures withuniform symmctric thickness with respect to the reference surface w. We just recall here the main stepsof the proof (for simplicity. we consider a uniform thickness beam-like body (0 I = x. 02 = y».

• Step 1: Pick the test function va in (49) to be of the form va(x. y) = Fa(x)wa(2y/d). F., E H~(w).waEHI[-d/2.d/2]nl-t;,. a=1.2. Thc spaces Wa define the membrane and bending modeaccording to the symmetric or anti-symmetric propertics of the displacement components. We writeu~ = I:j::11 U~(x).,,~(y).

• Step 2: With this definition, the variational form (51) is split into two parts with respect to wa'Then. wc obtain couplcd systcms of simultaneous equat ions defining the sequcnces {Ifin ;~nforeach mode and corresponding relations determining the coefficients {U:.} ;~Uin terms of loading.

• Step 3: We construct an approximate model uq using the obtained sequence {.,,~} '!~o andcoefficicnts {U~ }f=o for each mode. Wc then introducc u'l into the variational form (51) and

J. T. Odell. J.R. ClIO I Complll. Melhods Appl. Mech. Eflgrg. }3(, (1996) 317-345 327

evaluate the residual e'l = UX

- uq in the energy norm: then the estimation of residuals provides themodeling error.

Evcn though the 'proof is done for the structures with uniform symmetric thickness. this can beextcnded to those of varying symmctric thickness without difficulty. In such cases. we take d = dm =

I( tN-I 0muxw ( (J ., •.• 0 ).

This error cstimate is not a local estimatc. and Cq is not aceessiblc cxplicitly and can vary over thedomain. In thc next section. wc shall assume this is a local estimatc and that Cq is uniform.

With the prcvious assumption of the regularity of meshcs (e.g. quasiuniformity of h /\ and PK)' theapproximation error e'i.l' is [10]

(57)" ,

Ileq.hl12 ~ C ~ L -(dK' hK) IIuql12I.fl L.. 2"K ,.fl K

/\= I P

where C is a constant independent of elemcnt size h/\. approximation ordcr p/\ or ll'i, and Iluqllr.flK isdefined by

(58)

L(dK• hd rcflects the locking effect in the h-convergence which occurs only for bcnding-dominated thinstructures. so

L(dK• h/\) = h~K for bending-dominated thin cases

= h~K for the other cascs(59)

where IJ/\=(r-l). IL/\=min(PK.r-I). and µ'K=min(p/\-l.r-l) for plate-like bodies or ILK=mine P K - 2. r - 1) for shell-like bodies.

In view of the regularity of the solution uq in terms of the loadings [17.18]. (57) is assumed to bewritten in the following form

lIeq.lIIl~.fl~C £ L2(d~.h/\) 2/\=1 p·'K ,1/\ (60)

with

(61)

(62)

The unfolded curvilinear boundary aw of the referencc surface consists of a set of 'inc segmentsmeeting at a number Mr of vertices A with intcrior angle o'A• 0 ~ O'A < 21T. We define a by

21Ta = ()' V O'A E Mrmax O'A

(63)µ./\ = min(p/\ - 1. r - 1. a) or min(p/\ - 2. r - 1. a)IL/\ = min(p/\. r - I.a) .

When the corner singularity owing to the prcsence of corners. thc global rate of the p-vcrsion of finiteelement method in H'(W) is O(p-min

(r-1.2n) [7J. WC will therefore take

11/\ = miner - 1. 20')

So. we have sclective rates of h-convergence ILK and µ./\ so as to reflect thc locking phenomenon. Thestrength of the locking depends on severa' factors such as the thickness ratio with respect to thecharacteristic dimension of the body. the finite element mesh size. the boundary conditions. etc.However. the p-convergencc rates do not deteriorate due to the locking effect.

321'\ i. T. Odell. i.l? ClIO I CompUl. Melhocls Appl. Alt'cll. Engrg. 136 (1996) 317-345

5.2. An a posteriori error estimate

We summarize the theory presented in [151 on the devclopment of an a postcriori error estimate. Letthe lincar functionals /I: V(fl)-~ and 1: V(I1)-~ be dellned. rcspectively. by

217(v) = a(v. v) - 2f(v)

J(v) = I/(v) - Il(u'l·h)

It is not difficult to verify that we have

1- -2 11e11~(I1)= 1(u) = I/(u) - Il(u</·h) = inf fl(v) - 1/(U'lI')

t'EV(n)

We define the local solution spacc V and V(g» in a subdomain fl ....by

V....= {v E [HI(fl ....)J3: v = 0 on an n I~}.I'WI

V(9P)= n V....""~1

(64)

(65)

(66)

(68)

(67)

so that V(fJ) C V(g». The elcmcnts v E V(g» arc discontinuous across thc intcrclement boundaryscgments I~J .. Also. we define the local solution space V'~(fJ) and V'I(g» in a subdomain n....as

'I _{v: =~ V~'(.' .• ')I'i(/I')IV~,(", .• ')E.HI«(r)",,), }V ....(n) - J=U

V~=Oona(r) ....na(r)D' O~j~qa' 0'=1.2.3.1'(9' )

V'I(g» = n V1""=1

so that V'I(Il)CV'I(g». We dcfine the jump of vas

{v ....- vL jf K> L

ITvD= .vL - v.... If L> K

We introduce the averaging function aKL (5) on the intcrclcment boundary I~L as

aKJ. : 1......1. _ ~3

a~L(s)+a~ .....(s)=l sEI~L' k=I.2.3

Techniques for computing the function a~l_ so as to equilibrate the residual fluxcs are discussed in[2.3,15]. For any aKJ.' we can define the averaged k-component of traction acting on element IlK alongrKL as

(69)

We extend 1I to thc space V(g» and V'I(g». respectively. and consider the difference of functionals1'3':V(g»-~.

1~(v) = l/(v) -ll(U'I.h). v E V(g» (70).~~){ }= L 1/ ....(V)-l1 ....(U'l·h)-j VT(r(lIK)u(u'l·h)I_"df....= 1 ilSlKlilll

+ L r. ~vTll(f'(n ....)(T(U".lI) I-a d\' (71)K>LJIKL

where Il ....(u'l·h) is defined as the restriction of ll(u'll') to the elemcnt fJ.....and r(n) is the matrixcontaining components of thc outward unit normal vector fl. We insure that the interelement jumps ITvD

l.T. Udel/. l.R. Cho I COli/pili. Melhods Appl. Mech. Engrg. 136 (1996) 317-J.J5 329

arc constrained to bc zero by using a Lagrange multiplier µ. Thc clements of .f{ are formally fluxes(tractions) in 11/\=1 (H-I12(iJfl,J)3 and perform work through their actions on the jumps [vB; for v E V;the elements of .N. are identified with normal traccs of stresses in H( div. n); sec 141. This leads to theintroduction of a Lagrangc functional Y:. Y: : V(gz» x .tt -IR defined hy

I£(v. µ) = J~(v) - µ([vD)

where the .Ai denotes the space of Lagrange multipliers. Using (64).

1--2 lIell~(JJ) = inf J(v) = in£. sup Y:(v. µ)

"EI'(/l) ,'EI'(~) µE.II

~sup inf I£(v. µ)µE.II·'EI/(,,,)

(72)

(73)

With the choice of a Lagrange multiplier it in [15]. an upper bound of the error is given as follows:

S(:l')

11e11~,(1I):s:-2 L inf {nK(V)-I1/\(II"·J.)- r VT(f'(fl/\)U(II<1'''»h,dS} (74)K~ I "EI/.: Jafl/.:\,)!)

Hence. in order to obtain an upper bound of the approximation error. we need to solve the followinglocal element-wise problems;

find UK E V/( such that V v E VK

aK(UK• v) = e/(v) + r VT (r(II/JU(II"''') 1_" dl'J afl/.:\afl(75)

where aK(-.·) and t'K(') are restrictions of a(-.·) and t(·). rcspectively. to the element fl/(. Obviously.equilibrated local problems are dcfincd on the three-dimensional broken space V(11/(». With this UK'

-2 inf {nK(v) - llK(U'/''') - r vT (r(flK)U(U"'''» 1-" d5} = a,;(liK. Ii,;) + 2llK(u".I,) (76)vEV/.: Ja/lk\')!}

and (74) leads to

.qil')

_" ( . ".h· ".11)- L.. aK II,;-U .UK-UK=1

Let us denote

2 _ (' ".J. - q."TJK-aK UK-U .UK-U )

THEOREM 5.2. Let TJ/\ be as above. Then

(77)

(78)

S(iI'l

11e1l~(fI):S: L TJ~ (79)K= I

PROOF. This follows immediately from (77). 0

The quantities TJK thus serve as error indicators and provide a measure of the error bound contributedto the global error by the error in element flK•

330 J. T. Odell. J.H. ClIO I Compll/. Melhods Appl. Mt·ch. Ellgrg. 136 (ll)l)6) 317-345

(80)

6. An hpq-adaptive strategy

Now. we introducc an hpq-adaptive strategy based on a variation and extension of a three-stephp-adaptive strategy proposed by Oden ct al. 117.18]. Thc basic idea of this stratcgy is to split the totalerror into the portions of the modeling error and the finite clement approximation error, to use the apriori estimates for the modeling error and the approximation error. and to use the a posteriori errorestimator introduccd in the preceding section. The procedure allows us to determine approximately themodel level. thc mesh size and the approximation ordcr for each element. To achieve these ends. twomajor difficulties discussed in the prcvious section are cncountercd and should bc rcsolved: (I)non-local a priori modeling error estimate and unmeasurable portions of two types of errors in theestimated total error. and (2) the locking phenomenon. In our four-step algorithm. wc consider asequence of two hierarchical models: qo and qf' and a sequcnce of three mcshes: Ilo· IlJ and Ilf. Here.we define new notations for the developmcnt of our algorithm.

lIell £(11) .

TI= II II . crror IlldcxII E(I1)

OK = local error indicator estimatcd by a posteriori error estimator (=lleIiEIOK»

(

,\' ) 1/20= L ()~ global crror indicator ("'" IleIIE(l/l)

K-t

(

.\' ) II!

i\ = L A~ -'\K is a constant defincd hy (61)K~l

where }( represents the number of elements.As mentioncd earlicr. the estimated error lie II H(II) is composed of the modcling error lIe'lll E(II) and

the finite element approximation error Ile'l·h II £(11); unfortunately. the proportions of the two types oferrors are difficult to measure. Thc ratio of the two portions depends on the model level and themeshing as wcll as the type of problem; furthermore. it may vary element by element. We denote by a.(0 <a < I) thc ratio of the modcling error to the total crror. We shall assume a is uniform over thedomain:

(81)

Thcn. the portion of the finite c1emcnt approximation error is

• Setting. Define a target error index

( lIeIlJ:'(/n)TIT == TlTargcl = 111111 E(n)

then. from Lemma 5.1.

We define an intcrmcdiate error lcvel

TIl = 'YTIT

and the parameter T

crf .T=-. T>O

all

(82)

(83)

(84)

(85)

(86)

whcre au. af arc two ratios of the modeling error of thc initial hicrarchical model qo and the finalhierarchical model qf' rcspectively. In view of our assumption on a. T is automatically uniform overthe domain. By decreasing T. we decrease the portion of the modeling error in the final total error.

l. T. Odt'll. l.R. Clio I COn/pili. Mel/iods Appl. Mech. ElIgrg. J3fl (1996) 317-345 331

• Step 1. Introduce an initial mesh fl(J of .N~ elements with a uniform hierarchical model qo withmodel orders q(J= (q I . q, . q3 ) and uniform approximation ordcr p = Po and mcsh size h = hl)

u -0 0small enough to produce an error which falls within the asymptotic convcrgence ratc of the errorproduced by pure h-refinement. In particular. in ordcr to avoid bad initial data owing to thclocking. Po is chosen so that Po ~ 2 for the bending-dominatcd thin plate- and shell-like bodies.Then. we solve the problem on the initial mesh flo and estimate the global and the local errorindices 00' OK using the a posteriori error estimator:

° ·,"ll

lIell~(n)= (J~ = L (J~ •",,'=1 (I

(J,. = lIellE(fl)K)"II

(87)

From the orthogonality of the error to the space V 'I.i1([l ).

and

(88)

(89)

• Step 2. For an optimal q-refincd hierarchical modcl. we nced to (i) rcducc thc global modelingerror. and (ii) equidistribute the modeling error. We compute thc tinal model level qK for each

felemcnt so as to achieve the target modeling error. Assuming Theorem 5.1 providcs a localestimate. we have for the initial hierarchical model qo

lIeqOIlE(!1Ko) = C"Ko(dKI2)(SKo+6K,) = £¥o1JKoIIUoIIE(/I)

And for the final (= targct) modeling error

So.

(90)

(91)

(92)

(93)

where 8K and 8K are dcfined in Thcorem 5.1. Then" ,£¥f1JT C"K/ (dK) t(SK,-SKo)·qDK/-D,,)1

£¥o~IIUoII1,(!I)1JKIl CqKu 2

To simplify matters. we take (ql . - q3. ) = (qt. - C{3. ). so that (8K - 8K ) = O. and furthermore."" Ku K, K, '0

we take

C"K, = (qKu)2C'IK

IIC{K,

Then. finally. we have the following cquation for calculating the final model level:

(94)

(95)

Then. we enrich the modellevcl for each element to obtain the final hicrarchical modcl qf' We thensolve the problem on the q-refined mcsh (e.g. flo with qf)' and estimate the global and local errorindices O{l" 0Ko.' From the a priori error estimatc (60) and [17.181, we compute parameters 11K

according to

332 l. T. Odell. l.R. Cho I Complll. Melhods Appl. Alech. ElIgrg. 136 (1996) 317-345

8I;wp;;/1I; =---,;r-

where v and µ.. rates of h- and p-convcrgcnce. respcctively. are given in (03).

(96)

• Step 3. For an optimal h-refined mesh. we need to (i) rcduce the global approximation crror. and(ii) equidistribute the error. We calculate the number of new elements 111; for each clement in theinitial mesh flo:

(97)

where l!uo.11wn is the encrgy norm estimated on the q-rcfined mesh. flo with (I,· Here. X.. thetotal elcments of an intermediate mcsh fll' will bc known only after the intermediate mesh isknown. We usc itcration method to solve Eg. (97) using thc following rclation

,.\'

L 111; = .N-IA'~ I

(98)

(99)

Having 11K' wc redistributc insidc each clcment using a weighting schemc bascd on errors in thcq-refined mesh. If 1/1; < 1 we can consider unrefinements. Next. we solve the problem on thcIlq-refined mesh and compute an a posteriori error estimatc 81 at the global level and 81; at thc

I

clement level.• Step 4. Based on the actual estimated intermcdiate error distribution 8J• we computc a distribution

of polynomial degrees PI; to construct the final mesh nr which dclivcrs thc final error index 7J,satisfying the target error' 7Jr The parameters AIK on thc Ill/-refined mesh arc

"0Po Iii/11.=-/µ

Ii 11K

and the target error has to satisfy

( toO)

(tol)

where IIVIIIE(!/) is computed on the Ill/-rcfincd mesh (1IVIII~'(lI) = 1111111;:(11) + 0;). Then. we havethe equation for calculating PI; :

f

h-I'µ/\'~ .11',11- K 1\

PK = 'I 'f 7J1- I VIII f:(l1I

Although the exponents. v and µ.. may change wjth the choice of PI; . we neglcct this to avoid thcf

non-linearity in handling (101). Wc thus enrich P 1;, on each clement for the final mcsh fl, and solvcthe problem on this final mesh. We cstimate the final error tI, and IIVfllw/)' (IIV,II;,-ft/) = 1I1I,1I~:(tn+H;) to compute the final error index 7J, = 0,1 IIVf il f:(!1)'

• Postloop calmlario1l. If 7J, ~7JT the computation is terminated: otherwise steps 2-4 may berepeated starting with D, as the initial mesh for the next loop.

The detailed derivation of the steps 3.4 can be found in [17.181.

REMARK 6.1. The choice of the parameter T is closely related to the function Cqli

IC"1i . Therefore.I •

controlling of the model level using this parameter leads to relaxation of the choice of the functionc.. Ie .~K qK, .

l. T. Odell. l.R. Cho / Compll/. Melhods Appl. Meck ElIgrg. 136 (1996) 317-345 333

REMARK 6.2. We estimate \IUIIE(f}J at every step because it is affccted by thc locking phcnomenonfor bending-dominatcd thin bodies. However. IIUnlln/l) can bc used for all steps if locking is notprescnt.

REMARK 6.3. According to thc thickness ratio, the choice of thc initial approximation ordcr Po andthe ratcs of h-convcrgence JL for bending-dominatcd thin bodics requires some consideration. From thcnumerical results presentcd in Illll. the intensity of the locking and rates of "-convergence arc differentfor different thicklH;ss ratio.

7. Numerical experiments

We eonsider four model probelms; a plate-like problcm. bending- and membrane-dominated shell-like problems with the uniform thickness, and a bending-dominatcd shell-like problem with linearlyvarying thickness.

As a first application. we consider the rectangular plate-like structure shown in Fig. 5. This structureis a thin elastic body with edges anI clamped (ulafl = 0) and subjectcd to a uniform tractiong:=4.0X(dla)3Ibs/inz on its LIpper surface iJ[}+ no~mal to the planc gz=4.0x(dla)3Ibs/in2

.

Material constants arc speciflcd in the figure and the thickness ratio aid is 20. For this problcm. wespccify a uniform initial hicrarchical model with q = (I. 1.0) and a uniform initial mcsh with h = 114.P = 2. as shown in Fig. 7.

a= 1.0 in

d =Thickness

E = Ynung's Modulus

ICY psiPoisson's Ratio = 0.3

g,= 4.0, (dla)' IWon2

Fig. 5. Clampcd squarc platc-like structure under action of a uniform normal traction 8, on Ihc uppcr surface.

0.1

q-rcf(meshl) -h-rcf(q=l) -p-rcf(q=l) .~ ...

errJ~otaI) q-rcf(mesh2) -w-.-____ h-rcf (q=2) -..--..... -+----_ p-rel' (q=2)

e .-----------.. ~---.+~;~~:~~~_j::.::~~h--......~:.:..::":'.:~:~:......

e~~~~:~~ _

u_ ... _ .. _ .~-~.~-~.- ~;:~-~:~~.:: -~ -----.

100 1000Degree of freedom

10000

Fig. 6. Estimated modeling errors and finite element approximation errors for the plate-like problem.

334 J.T. Ot/ell. J.R. ClIO I Complli. Melhot/s App/. Mel'''. ElIgrg. 136 (1996) 317-345

Numerical rcsults arc shown in Fig. 6 which is concerned with Lemma 5.1. For this purpose twoinitial meshes. meshl and mesh2. are chosen: meshl being constructed with a uniform hierarchicalmodel with q = (I. 1.0) and a uniform mesh with h = l/3. P = 2 while mcsh2 is defined hy a uniformhierarchical modcl with q = (2. 2. 2) and a uniform mesh with" = 1/4. P = 3. Each curve is obtaincd byuniform lJ-. 11-. and p-rcflncments. respectivcly. starting from the initial mcsh. Thc global relative crrordefined by

lie II Em)Global relative error = I . III' (102)

2U(Il) -

is estimated by the a posteriori error estimator introduced in this paper.We notice two important aspects from the figurc: (1) thc crrors in the uniform q-relinement are

boundcd by the approximation error lahelled by err(approx) while that in the uniform 11- or p-refinement arc bounded by thc modelling error labelled by crr(model) and (2) the total error labelledhy err(total) is orthogonally dccomposed of the modeling error and thc approximation error. This resultindicates that the control or the total crror should be donc by two indepcndcnt refinements (q- or h-.p-refinements) so as to· guarantce thc target crror of any accuracy.

In order to sclect an appropriate value of the parameter 'T. wc simulate Eq. (95) with sctting specificdin Table]. wherc the global relative modeling error is computcd using

~ I/ ". IU(Il) - U(llq)1 -Glol'al rclative modeling crror = Illle

2U(1l )( 103)

where qT is a distribution of modcl orders computed using Eq. (95) according to the paramcter 'T. andU(uq

') ~s a strain en.crgy· of the qT -hierarchical model approximated with a uniform approximation ordcrp = 5 on the initial mesh. On the other hand. U(u) is approximated using the uniform hierarchicalmodel with, q = (7.7. 7) with p = 7 on thc initial mesh. from the results contained in Tahle I. we scethat as 'T decreases the inodel orders increase while thc global relative modeling error decreasesmonotonically: From the data. the parameter 'T should be smaller than at least lUll to achieve thc targeterror index TIT = 0.05.

Fig. 8 shows a distribution of local relative errors of the approximate solution obtaincd on the initialmodel and thc initial mesh. The parameters 'T and 'Yare ehoscn to be 0.001 and 2X With thoseparametcr values. adaptive hierarchical models shown in Fig. 9 are obtained. and the IIp-finitc clementmesh shown in Fig. 10 is produced. Relatively higher model orders appear at the boundary.

Aftcr four steps (Fig. II), wc obtain the linal crror index which is within the targct valuc TIT = 0.05.as shown in Fig. 12. where local relativc crrors are uniformly distributcd. Fig. 6 prescnts theconvergence rates of uniform hq-. pq-refinemcnts. and the four-step hpq-adaptive refinement. For Izq-and p.q-·rclincments. a uniform hierarchical model with q = (2, 2, 2) is assigncd for next three steps.The.se two refinements produce the total error hounded by the modeling crror. so taking the propermodel orders qT is essential to meet a givcn target error.

figs. 13- 16 are the altcrnative results whcn we start a differcnt initial model. a uniform model withq = (I. I. 2) instead of q = (1. 1. 0). with 'T = 0.112 and 'Y= 0.25. In particular. we notice that a differentadaptive hicrarehical model is obtained.

cct: DiOI Ml2h ..., R.c!Mi ... P.mw

~:--iClQ.,

~~:::::TOTs .497Il+OO\ .

~mo;_ .411E-OI 9MA.XL!SOI!+OO "DOP. 325

p-

nOF: 390,

A, .mul .P46B-01 .J4Z8400 .119HlOO.1-, ~

~

Fig. 7. Initial mesh for the plate-like problem wilh aid = 20 (initial (I. !. 0) model).Fig. K. Distribution of local relative errors 01 inillal mesh.

~=tcQ.,

'"

:::'"g.

".:;~

t:,;;...~~.....

.~w.....,I'.......<",

MnbProl«" .I'!.0lModelProICCl: viOl

'" 1

001'0 4632

.1-,p. 1

OOP_ 4632,

A,Fig. 9. Final model for the plate-like problem (1)T = O.IlS. T = 1l.001 and 'Y = 2.8).Fig. Ill. Final mesh for the plate-like problem (1)T = 0.05 ... = 0.01l] and 'Y = 2.8).

'....WVI

'"'"0-

~:--;

~~~

TOT- .430&01 I ?"MIN_ .IXZn.OZ QMAX-.l43Jl.OI :::>Dt)p. 46:12.

~.1"-m.oo ,I,9fI,tool A,~::~::-<:>a-:".

"<::

~-;......"g.~

ciQ~-w:>-.

---~'0,e=:'-lI

W....'""I

DOF= 4SfII)

J-,

Re..\.a1i"c Error maul..

Mesh

,47)s.o1

01

01

Pn>

p_ 1

10000

A,OOf= 4S69

Fig. II. Estimated convergence rates of hpq-finile clements for the plate-like problem.Fig. 12. Distribution of local relative errors ('7r = 0.05. T = U.IXlI and')' = 2.S).

"'_I

hq (uniform) -+-pq (uniform)hpq (4-step) ·0 ....

]000Degree of freedom

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

"""---':::::"':"'"•.••......••.....

"<J

0.1

0.01100

'-- I"'j,'-" plOI

.- I

g"">',=.,i!;a.0o6

Fig. 13. Final modeling for a uniform initial modcl with q = (I. J, 2) ('7T = 0.05, T = 0.112 and,. = 2.5).Fig. 14. Final mcsh for a uniform initial model with q = (I, 1,2) .('7'1' = 0.05. T = 0.] 12 and')' = 2.5).

Project: DlOI ReWI..,Enor cc:c pta. RoL&tlve Enor /IlQaI.

.Jcn&O\ ..... &01 .(I()Q\OI .1'2In.OO

lur- .315E<OOMIN= .2521!-0 IMAX ... II5E..ooDOP= 455

,..-1-.7

.J<J21I.<)\ .oo<'J.'" .1>00&0, .I:nn.oo

rur- .427E~1MIN= .15lli-02MAX=.127lHl1DOF= 4869

.1-7

:-~c:::~~:-?JQc

~

~Fig. 15. Distribution of local relative errors of initial mesh for a uniform initial model with q = (I, 1.2).

Fig. 1<i. Distribution of local relative errors of final mesh for a uniform initial model with q = (1. 1.2) (".' = 0.05. T = 0.112 and1'=2.5).

L=2.5 inR = 1.0 in

d = Thiclcnessg(9,x):4.0 x (l!Rl]lbslin

1

E=to'PSJ

Poisson'slUtio = 0.3

l'toloot: .WI

p- I

·Mcah

DOF= 390

.1-,

::::~ci}~~~'>-"g.£:1~~-....0-

""~~'-....'-lI

~'J.

Fig. 17. Quad-cylindrical shell-like structure witb normal tractions g(O. x).Fig. 18. ]nitial mesh for the bending-dominated shell-like problem with WI = 20 (initial (I. I. 1) Jl1odcl).

t..lt..l-J

.72$E-(U ,"'~lttOO .2118+00 ~(F..oo

TOT= .693E..ooMIN= .121E-01MAX=.276E..oo!.lOP= 390

,Ay

Pn>iecc :sbOl

q-

Model

LlOI'= 5337,

Ay

wwen

~:--ia""'~~?:lQC5

"§~

Fig. ]9. Distribution of local relative errors of initial mesh.Fig. 20. Final model for the bending-dominated shell-like problem (7)T = 0.05. T = 0.002 and 'Y = 2.5).

-;.....:::.:::-

~...~~-;.-"?-~

ciii~....wC\

.......:c'0~W-'lI

W....'-J,

10000

-+--

'-Duo,

hq (uniform)pq (uniform)hpq (4-step)

\\\\.\,

\.\\\_- -------._-::~~

'c

1000Degree of freedom

0.1

0.01100

c;t:Q)

Q)

>':;l

'"e<a.&>o{3

,Ay

DOF= 5337

MC'.h

p-

---1!0icct: JlbO 1

Fig. 2]. Final mcsh for the bending-dominatcd shell-like problcm (7)T = 0.05. T = 0.002 and 'Y = 2.5).Fig. 22. Estimatcd convcrgence ratcs of hpq-finitc c1cmcnts for thc bending-dominated shcll-likc problem.

l. T. Oden. l.R. ClIO I Complll. A-IelJwds Appl. Mel'''. Engrg. 136 (1996) 317-345 339

For a discussion of the effects of the choices of thc paramctcr y which dctcrmines the mcsh density.thc reader is referrcd to [17.18].

The next example is a quad-cylindrical shell-like structurc clamped at one side and subjcctcd to auniform traction g(O, x) = 4.0 x (t / R)~ lbs/ in2

• which is bending-dominatcd. Due to symmetry. wcconsidcr a half of the body. Hcrc. the thiekncss ratio R/t is 20 and material data are given in Fig. 17.Wc start a uniform initial model with q = (I, I. 1) and a uniform mesh with h = 1/4. P = 2. as shown inFig. 18. Estimated local relativc errors of the approximatc sl~lution obtained arc prcscnted in Fig. 19.

Referring to thc results in Tablc 2. wc can choose the parameter 'T by which the modeling error is lessthan our targct error (lh = 0.05). With the parametcrs T and y of 0.002 and 2.5. respectively. we obtainthe adaptive hierarchical model shown in Fig. 20 and the adaptive hp-finite elemcnt mesh shown in Fig.21. The reason for sclecting the smaller value T is to tighten the modeling error and to loosen theapproximation error for a computational purpose.

In Fig. 20. high model orders appear ncar the clamped boundary. In Fig. 21 line meshes occur nearthe boundary while high approximation orders are generatcd in the region distant from the boundary.The convergence rates of uniform hq-. pq-. and the four-step hpq-adaptive rcfincments are reprcscntcdin Fig. 22. Finally. we obtain the result with the global rclative error TIt = 0.041. and with a uniformdistribution of local errors shown in Fig. 23.

A third example involves a cylindrical can clamped at both ends and subjected to a uniform internalpressure Po = 4.0 x (t / R) Ibsl in2

• as shown in Fig. 24. We analyzc thc shaded octant part of the bodyshown. and take the thickness ratio to be R It = 20 (Fig. 25). This problcm is membranc-dominatcdexcept near the boundaries where thc displacement field is strongly bending-dominated. We take auniform initial hierarchical model with q = (1. 1. 1) and a uniform initial mesh constructed with h = 1/4.P = 2 as in thc cascs of the previous two model problems. Fig. 26 shows a distribution of computed localrelative errors of the approximate solution obtaincd on thc initial model and thc initial mesh.

Table 3 prescnts the model ordcrs and the corresponding global relative modcling error versus theparameter 'T. Compared with the previous two bending-dominated problems. this example has muchlower global modeling error. Based on the computcd values. parameters T of around 0.1 can be choscnto mect the targct error TIT = 0.05. Hcre. the paramctcrs T and yare chosen to be 0.3 and 0.2.respectively.

Figs. 27 and 28 show the q-adaptive hierarchical modcl and hp-adaptive finite elemcnt mcsh.respectively. Higher model orders and fine meshes appear at the clamped boundary. We obtain a totalrclative error TIl = 0.036 which is within the preset tolerancc, as in Fig. 30. In Fig. 29. convergencc ratesfor uniform hlJ-. pq-. and the four-stcp hpq-refinements are compared.

Thc final example is a blade-like body shown in Fig. 3 I which has linearly varying thickness. rangingfrom R It = 5 to 25. and subjcctcd to a uniform radial traction Pu = 4.0 lbs/in~ on the outer surface ofthe body. Figs. 32 and 33 show an initial mesh of the upper half of the body constructed by uniform 15quadratic elements employing the uniform initial (1. I. 1) hierarchical model and estimated localrelative error distribution. respectively.

To obtain the results within a target accuracy TIT = 0.05. we select the parameters T of 0.007 and y of2.0. respectively. As expected from Theorem 5.1. the nearer elements are located to the clamped

Table 2Hierarchical model levels (q) and estimated global relative modeling errors according to T for Ihc bending-dominated shellproblem (initial mesh: q = (1. 1. 0). P = 2.11 = 1/4 and initial global relative error Tlo = 0.69. target relative error TJ-r = 0.05)

T Elcment no. GlobalreI. error

2 3 4 5 6 7 8 9 10 11 12 ]3 14 15 16

1.0 1 I I I I I 1 I I I I I 1 I I I 0.29144ll.1 1 I I 2 I I 2 2 I I I 2 I I 2 2 0.02103llO] 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0.014820.00] 2 2 2 3 2 2 3 4 2 2 3 4 2 2 3 4 O.ll02420.0005 2 3 4 5 2 3 4 5 2 3 4 5 2 2 4 5 0.00075

LJ2

'Q:t.: .hOl

o _&01 ,t3lB+OO .I'IR.OO .2IUto!:tQ)

TOT= .41llE-01MIN •• 125E-m

. MAX=.U3E-O IDOI'= 53)7

~, LJ2

o____m_--@-~

L=2.5inR= 1.0 in

d = Thickness

Po=4.0 x (lIR) IbsliJ

E= 107psi

Poisson's Ratio = 0.3

'........a

:--;

~~:-?:l9~

~~

Fig. 23. Distribution of local relative errors (lIT = 0.05, T = tl.tlO2 and 'Y = 2.5) .Fig. 24. Cylindrical can c1ampcd and inflated by a uniform internal prcssure {lo·

:::"'...,==-

.....'II....

.t..

'""

~;;:~'.oJ0-

.-...'-C'-C

~

=:~~...'6"~

~,

~- .156£<001~= .399B-02

~1AX=.7311!-01 :OOF= 390

:n ..J.l.-o1.~cn.01-'"7Il-01.19'J8-01

z

A,

cd..: ab02

DOF= 390

Moh• 1002Pro'

p-

Fig. 25. Initial mcsh for thc membrane-dominatcd shcll-likc problcm with U {I = 20 (initial (1, I, I) modcl).Fig. 26. Distribution of local relative errors of initial mcsh.

Pro!ect: &h02 Model

DOF- 2484

J'roicct: a.h02 Maoh

q- 1 1 :t 4 , 6 7 I I I $I"

Fig. 27. Final modcl for the mcmbrane-dominated shcll-like problcm ('1, = 0.05. T = 0.3 and 'Y = 2.0).Fig. 28. Final mcsh for thc membrane-dominated shcll-likc problem ('IT = 0.05. T = 0.3 and 'Y = 2.0).

...0

~">.;>

0.]'"11<=3.D0(5

hq (uniform) -pq (uniform) .....--.hpq (4-step) .-a ....

.......:\.""" l?-,<>_ Tj,

"-----...

Proia::t: 1'h02

0.01100 1000

Degree of freedom]0000 .193E~1 .J.rnl·OI SSc&ol

Fig. 29. Estimatcd convcrgencc rates of hpq-finite elements for the mcmbranc-dominated shell-likc problem.Fig. 30. Distribution of local relativc errors ('I, = (l.OS. T = 0.3 and 'Y = 2.0).

Fig. 31. Blade-like body with linearly varying thickncss subjected to a uniform radial traction P" on the outer surface (Ril, = 5and RJI, = 25).

w...N

~~9;r~?='!'"-:C

001'= 372

%

I ~)-..,~--:::-c~:l.

"=~:::....3--~.,;;~-'..;0\

-:::10100-'-....,-.....I~....'"

IDOl'; 4386

~,

MCIh

Model

Proi",,: bIOI

Pmloct' NOI

p_ I

q-

TOT:a: .J6..1E ...OOMIN ... .449£-01MAX~.16m.oo001"" 372

.11tEt-OO

L = 1.0in.

R = 1.0 in.

//= 0.20i1l.

11 = 0.04 in.

Po = 4.0 /bsl m.1

.177u...oo

/1

Fig. 32. Initial mesh for an uppcr half of the blade-likc problcm (initial (1. 1, I) model).

"'<J:!-<>lA25n-ol

Fig. 33. Distribution of local relative errors of initial mesh.Fig. 34. Final model for the bladc-like problem (7)T = O.ll5. ,. = 0.007 and)' = Z.O).

J.T. Odm, J.U. Cho / Compll/. Melhods Appl. Mech. Engrg. 136 (1996) 317-345 343

Tablc 3Hierarchical model levels (q) and estimated global relative modeling crrors according to ... for the membrane-dominated shellproblem (initial mesh; q = (I.], ]), p = 2, h = ]/4 and initial global relative error 1/0 = 0.16. targct rclative error Tir = 0.05)

.. Elcmcnt no . Globalrcl. error

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1.0 1 1 1 1 1 1 1 1 1 I ] ] ] 1 ] 1 0.056680.1 2 2 2 2 2 2 3 2 I I 1 I ] I I 1 0.019890.01 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0.018420.001 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0.001970.0005 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0.00196

Proiect: blO 1 Mab

r- 1

DOF. 4386,

~J

Fig. 35. Final mcsh for the bladc-like problcm ('1, = 0.05 ... = 0.007 and y = 2.0!.

Proiccl: hlO I

1

RcLuh'c En... (Bqui.

.13U"HOO

Tar •. SOOR-OIMIN: .64Sli,...03MAX-.I69l!-<J1nop. 4386

Fig. 36. Distribution of loeal rclative errors (1/, = 0.05, .. = O,()()7 and y = 2.0).

344 l. T. Ode". l.R. ClIO I Complll. Melhods Appl. Mech. E"grll. 136 (1996) 317-345

boundary thc higher the ordcrs of thc modcls appear, and this is illustrated in the rcsults shown in Fig.34.

As can be observed in Fig. 35, thc algorithm produces a fine mesh ncar the boundary and thc bladetip while high-ordcr c1cments appear in thc middlc of the body. A linal accuracy index 1/1 = 0.05 isobtaincd which is within thc preset tolerance. and the local relative error distribution is shown in Fig.36.

8. Conclusions

In this study, we have developcd a four-step hpq-adaptive strategy based on an a priori hierarchicalmodcling error estimate [91, an a priori crror estimate of hp-finite clement methods [101, and an apostcriori error estimator [15]. The lip-adaptive part of this scheme is based on the three-stepalgorithm.

This lipq-adaptive scheme is able to control (in an cncrgy norm sense) both types of errors. themodeling crror and the approximation error. For this schcme. we introduced a new paramctcr T whichcontrols thc ratio of portions of the' modeling errors in the linal and the initial total crrors andovercomes the difficulties involved in decoupling two types of crrors.

According to the theoretical and numerical rcsults obtained herein. thc following main observationscan bc made.

• The two basic error components are orthogonally decomposcd in the total error. i.e. e = eq .L eq·" in

the cncrgy inner product. So. refinement in any single q- or h-, p-direction alone may fail to obtainresults satisfying predefined crror tolerance.

• The parameter T decouples thesc two a priori unknown error componcnts. and controls the modelordcrs for a reasonable hierarchical modeling strategy.

• Whilc thc basic adaptive sehemc involves four steps. the second step is generally quite incxpcnsivebccause it involves solving the problem on the initial mesh with only q-refined modcls.

Acknowledgments

The support of this work receivcd from the Office of Naval Research through Grants No. NOOOI4-92-5-1161 is gratefully acknowledgcd.

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