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, ,,~.. ',' ' ~ Reprinted from Computer methods in applied mechanics and . - engineering Hierarchical nl0deling of heterogeneolls bodies Tarek 1. Zohdi*·I . .I. Tinsley Oden 2 Gregory .I. Rodin:; 7,',lIl,\ /tIS/iI/ill' for COII//,III.lliOI/(// tlllt! Applied "'(//11/'11/(/1/(5, The Ul/i""l.\iIY o.f f.',w" til AIIIIII/. A willI. TX 7S7/1. USA Received 15 May 1<)% ELSEVIER /..J-?;; , -,".)
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Page 1: Reprinted from Computer methods in applied mechanics … · Reprinted from Computer methods in applied mechanics and engineering.-Hierarchical nl0deling of heterogeneolls bodies Tarek

, ,,~.. ',' ' ~

Reprinted from

Computer methodsin applied

mechanics and. -engineering

Hierarchical nl0deling of heterogeneolls bodiesTarek 1. Zohdi*·I . .I. Tinsley Oden2

• Gregory .I. Rodin:;7,',lIl,\ /tIS/iI/ill' for COII//,III.lliOI/(// tlllt! Applied "'(//11/'11/(/1/(5, The Ul/i""l.\iIY o.f f.',w" til AIIIIII/. A willI. TX 7S7/1. USA

Received 15 May 1<)%

ELSEVIER

/..J-?;;, -,".)

Page 2: Reprinted from Computer methods in applied mechanics … · Reprinted from Computer methods in applied mechanics and engineering.-Hierarchical nl0deling of heterogeneolls bodies Tarek

COl\IPUTER METHODS IN AI'PLlEn MECHANICS ANn ENGINEERINGEDITORS: J,H. ARGYRIS, STUITGART and LONDON

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E1-'iEVIER Compu!. Methods Appl. Mech. Engrg. 138 (1996) 273-298

Computer methodsin applied

mechanics andengineering

. , . , 1: ·Hier'atchical ,-modeling of heterogeneous bodiesTarek I. Zohdi*'\, J. Tinsley Oden2

, Gregory J. Rodin3

Texas Inslilll/e JOT Compllla/iona/ and Applied Mathemalics.TheUniversilyofTexasIlIAuslin.A/lslin. TX 78712. USA

Received IS May 1996

Abstract

In this paper. a methodology is introduced for the development of adaptive methods for hierarehical modeling of elastil:heterogeneous hodies. The approach is based on the idea of computing an estimate of the modeling error introduccd by replacingthe actual line-scale material tensor with that of a homogenized material. and to adaptively reline the material description until aprespecilicd error tolerance is mel. This process generates a family of coarse,scale solutions in which the solution correspondingto the fine-scale model of the body. which emhudies the exact microstructure. loading allli boundary conditions. represents thehighest level of sophistication in :l family of continuum models. The adaptive strategy developed can lead to a new non-uniformdescription of material properties which reflects the loading ;lIld houndary conditions. A pOst·processing technique is alsointroducc:u whieh endows the coarse-scale solutions with fine-scale information, through a local solution process. Convergence ofthe adaptive algorithm is proven and modeling error estimates as a function of scale of nHIlc:rial description arc: presented.Preliminary results of several numerical experiments arc: given to cClnlirmestimates and to illustrate the promise of the approachill practical applications.

I. Introduction

In structural materials. the presence of fine-scale features, whether introduced by design or occurringnaturally. can greatly affect properties of the material. such as stiffness, yield strength. ultimate strengthand fracture toughness. In particulate and fiber-reinforced composi1es, macroscopic features of failureare commonly thought to be linked to microscopic damage features which preferentially nucleate atparticle/matrix and liber/matrix interfaces. Unfortunately. if all of the detailed microscale interactionsare taken into account in an analysis of the response of a structure. a problem of enormous size andcomplexity is typically cncountered which far exceeds the capacity of the largest supercomputersavailable or expected to be available for many generations.

Because of this fact the use of homogenized material properties are commonplace in classicalengineering analysis and design. Methods of homogenization are, of course, the foundation of theth~ory of composite materials and they have been the object of much study and discussion for manyyears. An account of the essentials of the theory, and its variants. can be found in [8]. Althoughhomogenization techniques have been proven effective in the determination of overall properties. it isobvious that fine-scale features are completely missed when using homogenized material properties instress analyses. Consequently. it is unreasonable to expect that the usc of uniform material propertiescan accurately capture local behavior necessary for realistic estimates of a structure's useful life. Forvery sensitive applications, numerical simulation, incorporating fine-scale features have been used to

• Corresponding author. E-mail: [email protected] Rese;lrch Assistant and CAM Fellow.2 Direclor of TICAM. Cockrell Family Regents Chair # 2 in Engineering,J Associate Professor. Aerospace Engineering and Engineering Mechanics.

004S-7!l2S/96/SI5.00 © 1996 Elsevier Science S.A. All rights reservedPH SO\l4S·7825(96)OI106-1

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

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274 T.I. Zohdi el ,,/. / Complll. Melhods App/. Mech. Engrg. 138 (1996) 273-298

resolve stress and strain fields throughout a body, There have been early attempts to incorporatefine-scale features in a global analysis. Noteworthy are the representative works of Fish and Belsky[1,2], based on multigrid methods, and Ghosh and Moorthy [3,4] whose techniques are based onVoronoi-cell finite elements.

In the present work, a method for obtaining solutions to problems in elastostatics involving bodies"composed of heterogeneous materials is developed. The appr~ach,is based on the idea of computing an

" . ~estimate'df the:mode/ing,£;ror:introduced by replacing theactualfine':scale material tensor with that of", '.;. ~. a homo'genized material;,and,.tqsu~cessively refiqe the materiarproperties until a preset error tolerance

has been met. The method is a: "technique of hierarchical modeling, in which the fine-scale model of astructural component, which embodies the exact microstructure, loading and boundary conditionsrepresents the highest level of sophistication in a family of methods.

Specifically, the approach consists of three steps, which are presented in the following order:(1) We first derive an explicit expression for an upper bound on the difference in a solution

generated by solving a linear elastostatics problem with a homogeneous elasticity tensor (thecoarse-scale solution) and solution generated by solving the same boundary value problem withthe actual elasticity tensor (the fine-scale solution). It is shown that this modeling error estimateprovides a global upper bound. Also. preliminary numerical experiments suggest that theestimate gives an adequate resolution of the local error. This bound is shown to hold,independently of the loading and boundary conditions, for any choice of an approximateelasticity tensor (not necessarily constant), provided that it satisfies standard ellipticity con-ditions.

(2) A technique is then developed to generate intermediate scales of the material description, ahierarchical family. which produces a sequence of boundary value problems whose solutionsconvergence to the fine-scale solution. in an energy norm. Scale-dependent modeling errorestimates arc presented.

(3) Finally. a relatively inexpensive post-processing techniquc is developed, which is designed to takeadvantage of the wide availability ot' parallel processing platforms. The method supplies thecoarse-scale solutions produced by the hierarchical process with fine-scalc information through alocal solution process. where the exact material properties are used. This process can be thoughtof as either a local perturbation I1lclhod. or. as a non-overlapping domain decompositiontechnique [9].

The results of several preliminary numerical experiments are given to confirm estimates and toillustrate the possible effectiveness of the presented approaches in practical applications. A fundamentalpoint of the presented work is that the error in solutions produced by classical homogenization methodscan be estimated and that the quality of the response can be improved by systematically accounting foreffects of fine-scale features of the material.

2. Preliminaries

2,], Notations and conventions

Throughout this work we use the L 2(n )-based Sobolev spaces fl"'(n) consisting of functions withgeneralized partial derivatives of order less than or equal to m in L ~(n) defined on an open domainn E IRN; N = I, 2 or 3. These spaces are equipped with the usual norms and seminorms

f "

~"~~

....:;. ;';.;:..... ,' ... :.: ••'~.:'>. '\. ' •• :"; ".',',' \ '.

(1)

We also lise the spaces flS(n) for non-integer s E IR.As is standard, C~n(n) is the subspace of Cm(n) consisting of functions with compact support in n.

The closure of C~'(n) with respect to the H"'(n) norm in H"'(n) is denoted H~J(n) and the duals ofthe spaces H'(;(il) are the negative Sobolev spaces H-m(n). All spaces appearing in boldface are

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T.l. Zohdi el al. I CompU/. Me/hoils Appl. Mech. El/grg. 138 (J996) 273-298 275

vector-valued extensions of the preceding ideas, For example, H1n(fl) = [H"'(fl)]N is the space ofvector-valued functions whose components have partial derivatives of order less than or equal to m inL 2(fl) = [L 2(fl)]N. Accordingly.

N

IIvll;fmcn) = L IIv;lI~m(n) and;= 1

N

Ivl~m(fl) = L Iv;l~m(fl) .;=1

(2)

Fo/,' the- iI;lner pn;>duct of tensors'\7w, Vv E [L 2(fl )txN, U,_V E H 1(fl), we use the following notation.

.. T ~ iJu; iJvi 1Vv : 'Vu = tr[(Vv) 'Vu] = i.7:.1 aX

jaX

jE L (fl) . (3)

where auJ axj, avJ aXj are generalized partial derivatives of Ui and Vi' 1 ~ i. j ~ N. and where ui' V; areCartesian components of u and v. Throughout this paper we shall use the symbol 'vlan' for boundaryvalues of v E HflI(n), where boundary values are interpreted in the sense of traces.

2,2, Linear elasticity

We consider a material body composed of a linearly-elastic material and in static equilibrium underthe action of body forces f and surface tractions t. The body occupies an open bounded domain inn E IRN and its boundary is denoted an, For present purposes. it suffices to consider cases of which fl isregular: a simply-connected domain with Lipschitz boundary, The boundary an consists of a portion I:where the displacements are prescribed and a part r, where tractions are prescribed.

r"ur,=o. (4)

The data are assullled to be such that f E L 2(n), tEL "(r;).The space of admissible displacements, V(n). consists of those displacemcnt fields in the space III (fl)

which satisfy homogeneous (displacement) boundary conditions on r".

Thc displacements on I: are prescribed as follows: 3/1 E 1II([l).

ul/~ = Il\r. = '1., .

(5)

(6)

where UU is specified displacement data on r;,. Thus. the actual displacements of the body arc in thetranslation {II} -j- V(fl).

We consider the c.:lassical principle of virtual work characterized by thc following variationalboundary-value problem of elastostatics:

Find u E {u} + V(n) such that(7)

Here. g(J : H I(fl) X Ill(n) ~ IR is the bilinear form characterizing the virtual work and [ji(,) is a linearfunctional characterizing the work done by the cxternal forces.

9ll(u.v) = (Vv:E'VudxJa .'j;(v) = If' v dx + It. v d~.n 1',

(8)

where the mechanical (stress-strain) properties of the material are characterized by the elasticity tensorE which is assumcd to be a given function in [L "'(fl)t2xN2

.

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27(-, Fl. 20(,,1; e/al. I Complll. Me/hods Appl. Mee/'. ElIgrg. 138 (1996) 273-298

If the data in (7) are smooth and if (7) possesses a solution II that is sufficiently regular, then u is thesolution of the classical linear elastostatics problem,

-V, (E(x)Vu(x)) = f(x) x E nu(x) = ~?1.(x) x E r"

In . (EVll(X)) = t(x) x E r, ,I

(9)

2.2.1, The elasticity tensorThe elasticity tensor in (8) characterizes the mechanical properties of the material and, for materials

with highly heterogeneous microstructures, can be highly oscillatory. We require that the elasticities besubject to the following ellipticity and symmetry conditions: 3af• au> 0 such that V A E IRNXN

, A = AT(a.e. xE.Q),

(10)

Ei;k/(x) being the Cartesian components of E at point x.

2.2.2. HomogenizationThe problem described by (7) is usually far too complex to be solved by conventional computatinnal

methods due to the complex internal geometry of the body. characterized by a highly variable E. Toperform an analysis of the response of the structure it is customary to replace E by a uniform 'effective'elasticity t\:nsor. denotes E". This leads to a new more tractable problem.

Find ull E {II} + Yen) such that(J I)

Here,;?}J": H1(n) XH'(fl)~1R is the bilinear form characterizing the virtual work and 3'(-) is definedas before.

@U(llll.V)= r Vv:EIlVu°dx E1'(v)= r f'vdx+ { t·vds. (12)In In J/~In usual engineering calculations, Ell is a constant function in [L X(fl)]N!XN', and EO satisfies conditionsof symmetry and ellipticity: 3a~. a~>O such that VAEIRNXN

, A =AT (a.e. xEfl),

(13)

where E:~kl are Cartesian components of Ell. The boundary conditions are identical to those of the

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T.!. Zol1di e/ al. I Complll. Me/hods IIppl. Mecl1. El/grg. 138 (/996) 273-298 277

heterogeneous problem. Under the adopted conditions both (7) and (11) possess unique solutions u.1/ E {Ii} + V(fl), respectively,

The fine-scale energy norm of functions v E HI (ll) is defined by

IIvll~(!1) = @(v, v) = ( Vv: EVv dx. (14)Juwhere the:actual elasticity tensor E is used,

2.3. Componenls of the error

In this paper, we shall make the natural choice. 1111 -1I°llF.(!})' as the measure of the error betweenthe finest and coarsest scales, In general. even the exact solution ull cannot be obtained analytically anda finite dimensional approximation to it must be used; for example. a finite element approximation UO.

h.

Directly by the triangle inequality.

1111-uo.hllF.(u) ~ Ilu -1/1I£(!1) + Ilull-lI

o.hIlE(n) . (15)Modeling error Numerical error

Therefore. two sources of error occur in approximation of the heterogeneous problem. a modelingerror. due to the selection of homogenized properties and a discretization error inherent in the finitedimensional approximation of the homogenized problem. Discussion of the numerical error characteri-zation. which is not the subject of this paper. can be found in standard texts (see [10]). The remainderof the paper is concerned with the modeling error and its use in characterizing the material response.

3. Modelling error: The relationship hetween scales

3, J. Explicit (/ posteriori hOI/lids 011 tfte modelillg error

Key to Ihe ability to assess the quality of the homogenized solution generated by using EO, is thedevelopment of an accurate error estima1e which is independent of the loading and boundaryconditions. In this section we develop an explicit expression for the difference of the fine·scale andcoarse-scale solutions in the fine-scal,e e!lergy norm. lIu - ,/lIf:(fl)' We first record some properties ofthe elasticity operator E E ILX(ll)IN

'XN

' viewed as an L 2(ll).map. LeI

L:~m(fn= {,4E[L\fl)]NXN:/\T =A}.

and denote the L 2·inner product and norm on L:ym(fl) by

( 16)

«/L B» = fA: B dxu

IIIAIW = «A, A» . ( 17)

The operator E: L;ym(ll)~L;y,"(fl) is a self-adjoint. positive definite operator:

«EA. ~\»> 0 V A E L~y,"(n). A,.6 0 .,

«EA. B» = «A, EB» VA. B EL~y,"(n).

(18)

( 19)

it is well known that for such operators. the 'square-root' is well defined. generally in terms of theeigenvalue of E, and

Using this notation, it is readily verified that

@(U'V)=f Vu:EVvdx=«EI/~Vu.E"2VV» VU,vEV(n)./1

(20)

(21 )

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278 T./, Zohdi el al. I CompU/. Me/hods Appl. Mech, Engrg. 138 (1996) 273-298

so that

Ilvll~(n) = ~(v, v) = «EII2%,EI12%)).

With the previous definitions in place. we establish the following error bound.

(22)

THEOREM 3.1.1. Le,t E be a ~~nsor-vabiedfl4nctiQn. (1l.[L"'(!.l)]~;!x~2 satisfying (10) and let EO be a• ... J l ' '. 0 ~.... \t~nsP! .$ptisfyingil3»L,et u and u. be the corresponding soluti~ns to boundary value problems (7) and(11). respectively, The" ..

lIu-uO\\E(n)~III.foVuoIIlE(fl) .fo=I-E-1Eo, (23)

where explicitly

111.10 Vuolll~(fl) = In (I - E-1Eo)Vuo : E{l- £-IEo)%o dx.

PROOF. Denote by Il~(', '). the difference, for arbitrary vEV(il),

Il~ o(u", v) = ~(uo, v) - ~ o(uo. v) = In % : (E - EO)Vllo dx .

Substituting (25) into (11) gives

OO(ull, v) - ~9lJo(uo, v) = g?;(v) = 9lJ(u, v) ,

which directly leads to

@J(II - uo. v) = -Il@o(,,o. v) = -«'Yv. (E - EO)V/lo)) V v E V(il) .

S ' 0 .cttlllg V = u - /I • gives

1111 - //'II~ln) = -«V(u - 11°), (E - EO)Vuo))= - «V(II - /10), EE -I (E - EO)Vllo))

= _(£112V(1I _ uo), EI/2E-1(£ - £t1)'Y,,o))

= -(EI12V(u - uo). EI12$OVUO))

~ [«E 1I2V(1I - ,/'), E 1I2V(U - ,,0))) J"2[ «£ 1/2.111V/I 0 , E 1/2$0 v,,o))] 112

= Ilu - ,,011 F.(mll l..9IoV,/11 I EUJ) •

from which the assertion follows. 0

(24)

(25)

(26)

(27)

(28)

3.1.1, ObservationsIn Theorem 3,1, the proof makes use of the fact that the coarse-scale solution satisfies a global

variational statement, with the ,>ame boundary conditions as the fine-scale formulation. If we replace EOb EM d d' lOb !of h EM. '1 h' d' IY an. correspon mg y. u y u , were' IS not necessan y constant, t en we Imine late yobtain the following:

lIu-'/'IIEtm~III.f!ofVuMIIIE(/l) $M=I-E-1EM, (29)

This result proves to be useful in later analyses. An algorithm for construction of sequences of materialtensors EM is given in Section 5,

3.2. Connection with classical bOl/nds

Now we establish a connection between the presented error bound and classical bounds for theoverall elasticity tensor (see [61 for details). This tensor is defined by the relation

(u) =E*(E) (T = EE (30)

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T.I. Zohdi et at. I Compll/. Methods App/. ,\-tech. Ellgrg. 138 (1996) 273-298 271)

where (-) = lIlal In 'dx. In general, E* is not a material property, i.e. it depends on the data, If (u)and (E) are such that

(31 )

and/= 0, then there exist universal bounds for the eigenvalues of E*, This restriction, often referred toi,n,the literature as Hill's conditiqn"can be realized in several way-s [5,7]. The bounds for E* are givenby

(32)

(34)

This inequality means that the eigenvalues of the tensors E* - (E -I) -t and (E) - E* are non-negative.A special class of fields that fall under Hill's condition (31) are those produced in bodies with

specified boundary data of the following form: (1) pure displacements in the form II = g.x or (2) puretractions in the form t = fl· n; where the tensors g and ff are constant strain and stress tensors,respectively, It can be easily shown that under uniform conditions with no body forces. that for case(1): (E) = g, and for case (2): (u) = fl. These boundary conditions are referred to as uniform forobvious reasons. If we restrict our attention to uniform boundary conditions. the general error boundstake on special forms, We have the following (with EO constant).

1111 - uOIl ~(/l) ~ In (I - E -I EOWI/' : E(I - E -IEu)vuo dx

f (""II E-IEuM 0) (E" U EOo o)dx= vII -. • vII : ~vII - ~ ,"II

f ('" II Et"IO ,...,II EO" ° J.-IJ'Oo ° 1'0 o+E-IJ'oo °.1•00 0) (Ix= v'II:' vII - vII : J v II - ~ ~ VII : ~ vII ~ vII . l~ V II{J

=Vllu: (/:,')vIIUlnl-VI/I: EUYllolnl-Eovllo: v//lnl + (E-1)EOyuo: Eovl/llal·(33)

OBSERVA nON 1. For those cases having either type of uniform boundary conditions, wilh no boLlyforces, it is straightforward to show that if the classical upper bound is chosen, EO= (E). then

YUu: (E) v,,u\a\- Vllu : EIIVuU\al- EUvllo: vuulal + (E -I )EIlVIIO: Eovullial

= 0 - EOVllll: vlIlJlnl + (E-I )EIlVuu: EovI/lul

=EuVllu: {-I+ (E-I)(E)}Vllulnl

= (E)vl,o: {-(E)-I + (E-I)}(E)VuUla\

= uO: {-(E)-I + (E-1)}uolfll.

where UO(=EuEu) is the statc of strcss that exists in the uniform body. Therefore

lIu - uOlli(/l) ~ {- (E) -I + (E-I) }uo : uOlal . (35)

If the lower bound is chosen: EO = (E -1) -1, then for either type of uniform boundary condition

Vull: (E)VIIOlal- VuO:EOv,,ulal- EOVuo : vlIOlnl + (£-1 )£OVIIO: E °V,/' In I= Vuo: (E)VllOla\- VUO:(E-I

) -IVIIO\al + 0

= {(E) - (E-I)-I}VIIO: vl/'ini. (36)

Therefore.

1111 - uOIl~(Il) ~ {(E) - (E-I) -I}f/': EOlnl . (37)

where EO(=En-1uo) is thc state of strain that exists in the homogenized body. We observe that in both

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280 1'./. Zohdi 1'1 al. / Comp/ll. Melhods Appl, Mecll. Engrg. 138 (1996) 273-298

(38)

(40)

of these cases the difference in the classical bounds in (32) appear in the expressions for the modelingerror estimates.

OBSERVA TlON 2. It is straightforward to show that for the uniform displacement case, u = Y, x.where Y is arbitrary, that

II" . °11'" f r ~ ° II °1 I f IlU - Ii £(n) = VI.i :,EVudx+ Vu : E Vu £l - 2 . Vu : EVu dxn .I'.' II

=(e): (u)I£lI+Vull:EoVuol£lI-2eo: (u)lnl= {(E) - E*}Y : Ylnl:s;; {(E) - (E-I) -I}y: Ylnl .

where E* -I (u) = (e) = ell = Y. The last term on the right in (38) is a universal bound, i.e, it isindependent of the microstructure, Also, it is easy to show 1hat for the uniform traction case, t = fJ' n,where fJ is arbitrary that

1\ Il - 1/'11~({l) = f Vu: EVu dx + Vull : EOvuolnI - 2 f Vllo: EVil dxJ11 In= (e) : (cT)lnl + VI/': Eovi/'Inl- Zell: (cT)lnl= E*-I (u) : (u) Inl + Vuo: Eovulllnl- 2eo: (u)lnl= E*-l.~: fJlnl + VIIll: EOVI/'Inl- 2Eo: .1"lnl. (39)

It is easy to show with EO = (E). that E* -I fJ : fJln I + VIIIl: EllVI,uln I - 2e Il : fJln I IS a 11111111llUI11.

Therefore

Ilu - 1/'II~(fl) = E*-Ig : glnl + Vllo: Eovulllal- 2eo: glnl= £*-1 g :glnl + VI/': EOvulllnl- 2eo: .0/" InI= {(E)E*-I(E) - (E)}eo: elllni:s;;{(£)(E-1

) (E) - (E) }e" : EIlIDI.

where E* (E) = (u) = ull = g = E" E U The last term on the right in (40) is a minimum universal upperbound, The two universal bounding materials in the previous cases have the following ordering:

(41 )

In the next section we conduct numerical experiments on the global and local estimation of themodeling error with body forces and non-uniform boundary conditions in place.

4. Numerical experiments on error estimation

4. 1. Example 1: A heterogeneous bar

To eliminate any effect of numerical error on the study of the error bound, we first consider anexample with an analytical solution. To this end. we consider a linear elastic rod, of unit length, nxedon both ends, and subject to a constant body force (see the top three diagrams in Fig, 7). The nne-scaleand homogenized problems are

and

d ( dU(X»)dx E(x) C'i:r = -1 .

~ (EO dl/O(x») = -I,dx dl"

11(0) = 0 ,

1/'(0) = 0,

1/( I) = 0 (42)

(43)

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T.I. Zohdi e! til. I Complll. Methods App/. Mec/1. £/lgrg. 138 (/996) 273-298 281

(44)

Here. our interest is to determine performance of the error bound (23) in predicting global and localmodeling error throughout the body. To this end. for local resolution of the error we let 22 denote anarbitrary partition of n with a total number of subdomains, N = N(22), such that

N(3)

ii = U e" e", eL E 22 e" n eL = 0 if K =1= L 1:S;; K. L:s;; N(21.) ."=0

We define local error indicators. loc,al meas!Jres of the error, for eK C n by

(,,=III.1ovllOlltE(eK) I:s;;K:S;;N(22), (45)

For the global error estimator we simply take the global error bound (= III.1ovuoIlIE(n)' ClearlyN(!1.)

lIu - uOIl~(n):S;; 2: «(K)2. (46)"=1

The quality of the global and local error indicators is measured by effectivity indices 11 and 11K'respectively:

N(:!J.), ~ 2

7f= ~ 11K'K=t

(47)

In our example, the unit interval is divided into 10 000 equal intervals, and for each interval thematerial property is chosen at random to be either E = I or E = T. where T is the mismatch ratio. Equalamounts of hard and soft material are used. A coarser partition is overlaid representing the subdomainpartition. The following tests arc entirely representative of many realizations of the interval andsubdomain partitioning for this simple one-dimensional case. All of the following calculations arc doneanalytically. For convenience. and to illustrate the sensitivity of the results to the choice of EO. wechoose the classical upper and lower bounds in (32). Our objective is 10 illustrall' the dependence of theeffectivity indices on the domain partitioning and the choice of Ell.

01.J, J. ResultsFrom Tables I and 2. it is seen that an increase in the number of subdomains in the partition reduces

the quality of the local effectivity indices. This stems from the fact that the local estimates arc derivedfrom a global calculation that. due to its integral nature, is insensitive to local pointwise information.The local effectivity indices are closely clustered around unity for the harmonic average. independent ofthe mismatch. while this is not the case for the arithmetic average, Locally. the error indicators arc notguaranteed to be upper bounds on the error. and, as Tables 1 and 2 show. they underestimate the errorin some parts of the domain. The choice of homogenization affects the quali1y of the local estimates,

Table 1The effectivity indices for E" = (E) for 100 subdomains alld to subdomains. versus mismatch ratio

T lIu - uOllc\ll/liullclOi '1'/ max TfK N= 100. 10 mlllTfK N = 100. 10

10 0.804457 1.000186 1.38345 1.06~4R 0.529439 O.~334850 0.~56960 1.000205 1.6632~ 109153 0.494427 O,9188~

100 0.978203 1.000208 1.71799 1.09488 0.489773 0.91681

Table 2The effectivity indices for E" = (E -I) -, for 100 subdomains and 10 subdomains, versus mismatch ratio

T llu - uOllWl)/lIullcllll '1'/ maX'I'/K N= 100,10 minTfK N= 100.10

10 1.347399 1.000066 1.006312 1.000608 0.858558 0.99687050 3,276713 1.0000]8 1.001734 1.000158 0.955632 0.999186

100 4.680409 1.000009 1.000906 1.000082 0,976185 0.999578

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21\2 T.I. Zohdi e/ al. I Campll/. Me/hods Appl. Mech. Engrg, 138 (1996) 273-298

although this effect is barely perceivable globally, Interestingly, the error prediction for the harmonicaverage case becomes better with increasing mismatch ratio-tltis is strictly an anomaly associated withtltis specific one-dimensional problem, and, in general, is not necessarily tlte case. Note that even for thissimple one-dimensional problem, the error for the arithmetic average ranges from 80-90% of the actualsolution, and for the harmonic average, 134-468%. It easy to show that, with respect to the energynorm, the arithmetic average is within a fIaction of one perc~nt of being the optimal homogenizedmaterial£hoicetominiTllizethe:mo~elii1g ~rror'in energy for this specific problem.

4.2, Example 2: A three-dimensional analysis of a cube in shear

We consider a unit cube of material, with a heterogeneous two phase isotropic random 'checker-board' microstructure (Fig. 1). We choose EO to be isotropic, where the specific values of thehomogenized Lame parameters are simply the volumetric average of the Lame parameters of theinternal constituents, The unit volume is divided into 64 sub-cubes of equal volume, each withdimensions 1/4 x 1/4 x 1/4, Each sub-cube is randomly assigned, either a set of soft or hard materialparameters (Fig. I)

A=A'7 (48)

where 7 ~ 1 is a constant parameter which represents the mismatch ratio and where AS and µ: are theset of soft Lame material parameters whose relative ratio, for convenience, is taken to be that ofstandard grade steel. Equal amounts of hard and soft material are used,

The loading scenario for the heterogeneous cube is shown in Fig. 1, The virtual work formulation forthis problem is

Find ,/'EV(fl) such that

','" ....

J vv: E "VII (l dx = ft. v ds11 /'

"VvEV(fl) (49)

~

'J

"-'".1

Fig, 1. A test problem for three·dimensional dispersed cuboid microstructure.

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where

T.I. Zohdi e/ ai. I CampUl. Me/hOi/s Appl. Alech. EI/grg. 138 (/996) 273-298 283

(50 )

where• n corresponds to the interior of the 1 x I x I cube depicted in Fig. I: 0 < x I < 1, 0 < x2 < 1,

0<x3<1.~ ru-'corresponds to the plane x I = O.• r" corresponds to the plane x, = t'. where we specify t= {O. 0, l}T (uniform shear loading).• /=0,• all other faces of the cube are free surfaces.For the purposes of numerical experiment. the finite element method is employed to generate

approximations to u and UO denoted by ,/' and u°.!'. respectively. For a measure of the quality of theerror prediction locally, we use instead of (47). the local discrete effectivity indices.

(51 )

In all of the following numerical experiments. the finite element approximation consists of hexahedralelements, with trilinear polynomial interpolation. To illustrate the use of local error indicators, weadopt a simple unidirectional partitioning of the cube into four 'slabs': fJ\ : 0 <x I < 1/4. ('12 : 1/4 < X I <1/2, °3 : 1/2 < x I < 3/4 and 63 : 3/4 < x I < I. It is clear that the quality of the discrete error indicatorsis mesh dependent and. consequently. a series of tests are performed to illustrate mesh dependency forfixed mismatch ratios. These tests arc conducted in the following manner: keeping the microstructure ofthe cube fixed. for each sub-cube in the cube, we steadily increase the number of finite clements persub-cube, In this manner. we isolate the effects of the fineness of the tlnite clement mesh and themismatch ratio on the discrete effectivity indices.

./.2.1. Resultsl'ahlcs 3 and 4 contain resllllS which illllsirate the behavior of the global and local effectivity indices

with increasing tlnite clement mesh refinement for fixed, successively larger mismatches of materialproperties of 10. 50 and 100. The aCf/lal modelillg error varied betweell a few hLllldred percent (0 overJO()()%. depelldillg Oil the mismatch ratio alld ihe fillite element mesh. The effectivity indices. however,

Tahlc 3The global effeclivily indiccs for mismatch 7' = 10, 50. 100 and for incrcasingly tincr finitc clcment meshes

DOF Elcm/sub-cube .,.,". T = 10, 50, 100

375 I x I x I 1.21 1.44 1.5221R7 2x2x2 1.34 1.98 2.43b591 3x3x3 1.37 2,(5 2.71

14739 4x4x4 1.37 2.15 2.7227783 5x5x5 1.37 2.14 2.7146X75 6x6x6 1.37 2.14 2.72

Tahlc 4Thc local effectivity indices (for cach slab) for mismatch T = 10. 50, 100 and for increasingly finer finite clcmcnt meshes

DOF Elem/sub-cubc Slab I:.,.,~ Siah 2: .,.,~ Slab 3: 1)~ Siah 4: 1)~

375 1 x 1 x I 1.22 I Ali 1.55 0.93 1.05 1.12 1.44 1.67 1.74 \.32 I.IiIi 1.802187 2x2x2 1.311 1.94 2.41 1.55 2.44 3,OR 1.32 I.R3 2045 1.21 1.80 20456591 3x3x3 1.57 2.67 3.48 1.29 1.94 2.40 1.29 1.96 2,59 1.35 2.18 2.78

14739 4x4x4 1.43 2.32 2.99 1.27 1.92 2.40 1.30 1.96 2.45 1.51 2.52 3.2827783 5x5x5 1.31 2.03 2.58 1.39 2.20 2,79 1.41 2.17 2,79 1.38 2.18 2.7646875 6x6x6 \.36 2.09 2.63 1.37 2.17 2.77 1.35 2.116 2,77 1.40 2.25 2.91

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284 T.I. ZoJu/i e/ (/1. J Compll/. Me/hOlts Appl. Merh. Engrg. 138 (1996) 273-298

2.8

2,6

2.4

~ 2.2.~

:w 2~51Cl

1.8

1.6

·t.,. n •••• '., ••• "" T hn •• ,..', n,. ••• ".n , no.

mismalclllOO -mismalch 50 ,. n

mismalclll0 ...

1.4

1,2o 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000

Degrees 01 Freedom

Fig, 2. Behavior of the effectivity index with changes in the number of degrees of freedom, for the cube in shcar with cubicalmicrostructure.

are remarkably accuratc. which mcans that we may predict the error well, even though it is quite large.Our global measure of the effectiveness of the error estimate, 1)". is relatively constant after a certainmesh fineness threshold (Fig, 2), which is approximately 27 trilinear hexahedra per sub-cube, and whilethe local effectivity indices also stabilize. they stabilize at a much slower rate. However, the localeffectivity indices arc clustered around the global values. "uggesting that we may obtain accurate localestimates of the error, We note thaI the choice of EO is not restricted to the classical relation of averagedquantities of stress and strain. and that any el:lsticity that slltisties the ellipticity conditions is admissible,While in pn1ctice the exact solution is usually never obtainable, the experiments lend some confidenceto the reliability of the estimator. Other loading cases. such as a cantilevered cube in uniaxial tensionand a cantilevered cube under the influence of constant body forces. were tested. with qualitativelysimilar results (not reported here) to the presented shear case,

5. Hierarchical scale construction

In general, the quality of the solution 11° may be poor. even with a good choice of a uniformhomogenized description. In order to improve solutions, it is clear that one must develop a methodwhich (efficiently) constructs solution scales between the coarsest-scale. corresponding to a perfectlyuniform material description, and the finest-scale, To obtain reasonable accuracy with numericalmethods based on discretization. the computational cell size must be less than. or equal to theinhomogeneity sizes encountered in the body, This makes the full fine-scale problem inaccessible. fortwo main reasons. First. if the full. coupled, fine-scale problem is discretized for an approximatenumerical method, it would not fit. due to memory limitations, on existing computers. Set:ond. even ifthe discrete problem were to fit into a hypothetical 'super-memory' machine, it would take on the orderof C,,2 operations to solve the resulting system. where 11 is the number of unknowns. and C is aconstant, greater than unity and is dependent on the condition number of the resulting discrete stiffnessmatrix, The value of 11 is so large (e.g. ,,- 0'(107» that such computations are not feasible,

To address this difficulty, a series of relatively inexpensive coarse-scale problems are generated.characterized by bodies comprised of subdomains, Each subdomain may contain different constantmaterial properties. The sizes of these subdomains are orders of magnitude larger than the in-homogeneity sizes encountered on the fine-scale (see Fig. 4). These coarser material descriptions form a

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T.I. Zohdi ('I al. I Campll/. Me/hods Appl. Mech. 1:·lIgrg. 138 (1996) 273-298 285

A B c

ABIIUA. The lubdoauin comspondin~ to 511IHlo, ACDFA

ABGIA, BCQGS.GQDEG.IGEFI=iub-$t:b-bom (OIt!Sp«tdin~ to sub-bo, ACDFA

ABGIA. BIlGS. GIIEG, IGEf1=tt1Is

F

12341. The boundill8 box

ACDfA.A sub-box

Q

D

Fig. 3, The nomenclature for the construction of a partition.

rtriml1 Uniform

Oneri",lon

•••'"- ~~"'~'~~ ~~~J.'~~

Fig. 4. Hierarchical elasticities produced by ALR for a single Cartesian subdornain.

hierarcilical family of scales of material description. The modeling error throughout the body isestimated at each scale. with the error estimator introduced previously. and the material description isrefined in those select regions where the modeling error is high. The global coarse-scale problem is thenresolved with the new material properties. and the error re-estimated. until a desired prespecifiedtolerance is met. However, in general, these hierarchical solutions do not capture much of the fine-scaleinformation. and. therefore, a post-processing method is applied which adds a local fine-scaleperturbation to the hierarchical solution, This process endows coarse-scale solutions with the necessaryinformation to be reasonably accurate for a local analysis. Therefore, the overall method is comprisedof two main stages: (1) generation of hierarchical scales and (2) local solution processes.

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286 '1'.1, Zolltli el "/. I ComplI/. jHelhods Appl. Mech. EJ/grg. 138 (/996) 273-298

5. 1. Construction of (J par/ilion

Although in the preceding theory the partition for the model error estimation has been based on anarbitrary partition of the domain. in practice. there must be some constraints, For definiteness, wederive a systematic construction which is amenable to numerical computations, We consider a generalcurvilinear, non-convex domain. and we introduce the idea of a bounding box (Fig. 3). In general,neither· th~slibdomain divisions' nor the partitions follow' the' material boundaries, We define thebounding box, 0, as simply an open rectangular parallelpiped such that

diaD· = inf dia{o : Ii CO} .

Furthermore, D· is subjected to a fixed partitionN - - 0U OK =0· OK n 01. = 0 K ~ L 1~ K. L ~ N(21 ).

kaJ

where each OK is of equal size, We define the subdumain e~en as

(52)

(53)

'. ". '.. " .~::',

N(~D)_

U e~=1i"nl

e~n e~= 0 K ~ L , (54)

We also define partitions within each subdomain e~, by first partitioning the corresponding 'sub-boxes'into equal pieces, In one dimension this corresponds to bisecting subdomains into equal line segments,while in two dimensions this corresponds to quadrasecting sub-boxes into equal rectangles. and in threedimensions it corresponds to octasecting sub-boxes into equal hexahedra with rectangular cross-scction,We denote these sub-sub-boxes by 0".1.' where

(55)

where N" is the number of sub-sub-boxes in sub-box K. Furthermore. we define

n n 0".1. = t)~.L (56)

and clearlyo DO

f)".LCO"-0 -0U (-)".1. = e"

,.}D1;,/.

(57)

where e~.L arc denoted as 'cells' (Fig, 3). It should be clear that sub-box 0" and e~ arc identical forCartesian domains. It is emphasized that the partitions for (1) the error indicators. (2) the hierarchicalscale generation and (3) the local pos1-processing are independent and in practice should be keptindt:pendent to obtain maximum performance from the method, However. for clarity of exposition, weuse the same partition as for the local error indicators throughout the presentation.

5.2, Stage I: An algorithm for generating hierarchical scales

Overview: A hierarchical family is generated by first starting with a homogeneous material.estimating where the corresponding homogenized solution is in large error. and locally relining of thematerial description where the error is high. This is an iterative procedure. For clarity of exposition, weillustrate the procedure with isotropic materials and with volumetric averaging for the homogenizationprocess, but neither of these conditions is necessary,

5.2.1, An adaptive algorilhm based 011 Adaptive Local Reaveragillg (ALR)• Step 0: Choose an error tolerance: 1111- u011 ~ 0 = error tolerance.• Step 1: Solve the initial coarse scale problem, with uniform material properties, For example.

suppose EO is a constant isotropic tensor with material constants,

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T./. Zohdi e/ al. I Complll. Melhocls Appl. Mech. Engrg. /38 (/996) 273-298 287

KO _ 1 r-1.01 In K dx (58)

(59)

(60)

where. as usual. K = A + t denotes the bulk modulus.• S/ep 2: Distribute the error tolerance in the following fashion.

.' le~1 ( °... 5K = 5 lflf V K = 1.2, .. , . N 2l )

• Step 3: Calculate the following in each subdomain e~.V K = L 2, . , .. N(fl 0).

'K = III.9'oV'uoIIIE(B]l) .9'0 = 1- £-1£0 ,

and check

'K~{)K VK=I,2, .. "N(210), (61)

• Step 4: If 'K > 5K, repartition the corresponding sub-box. OK' into sub-sub-boxes. as given in(55), and generate cells e~,LC e~. Calculate the following for each cell

I 1 f II 1 iK [] = - K dx [] =- dx 62leK.L leO I 0 J.L RK.1• leO I 0 J.L ( )K.L 8K.L K.L I/K.L

where le~.I,1 = meas(e~.L)' and where

EI(X)II'IX1.L=E~.1. XEe~.L· (63)

The new material generated by performing one scquence of Stcps 1-4 is dcnoted with superscript 1as opposed to O. which was used for the initial data and solution. For example. after completingthe first pass of Steps 1-4. the material propcrties are denoted as K I. µ.I and the correspondingglobal solution as /II.

• Step 5: Repeat Steps 1-4 until the error tolerance has been mel. After M stages (Steps 1-4). i.e.after ,\1 steps in which at least one sub-box has be repartiti~pe(,I, during a stage. we ,arrive at amaterial characterization defined by the tensor £'11 E (L To (£1) 1'" x '\ •• such that V A E \R" . N, A = AT

1E,I/(x)A = KM(X) '3 (tr A)l + µ.M(X) dev(/\) a.e. x E n

where dev(A) = A - H tr A)l. and where

(64)

,111 1 rK 8fj!.L = le~ I )Hq K dx

K.1. K.1.

5,3. Convergence of the ALR so/lltions

(65)

It can be easily shown (8] that the ellipticity constants appearing in (10) arc related to K and µ. in thefollowing manner

0< af = min{3K, 2µ.}(66)

x> au = max{3K. 2µ.} .

Furthermore. we observe that for any e~." C fl.

af~I~\~ 110 3K(X)dx=-I.~ 110 3A(X)dx+-I.~ 11 J 2µ.(x)dx~au·0K.1. 8K.L f)K,L RK.L f)K.L H~.1.

(67)

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288

and

T./. Zohdi e/ al. I Compll/. Me/hods Appl. Mech. Engrg. 138 (1996) 273-298

a, ~ I ~ 11 2J.L(x) dx ~ au·e K.L B~.L

Therefore. as one would expect.

(68)

(69)

In other words, the material properties of the intermediate scales are bounded by the original ellipticityconstants.

THEOREM 5.3,], LeI u be the solution to (7) and let 11M be the solution to the boundary value problem,

Find uM E {u} + V(fl) such that(70)

where g) M : III([1) x If I(fl) >-+ IR is the bilinear form characterizing the virtual work for (he Mth-scagematerial.

(71 )

with EM given by (64), and fI(') is defined by (8), Then

PROOF. Let

AOOM(u"', v) = OO(u.ll, v) - @'I!(UM• v) = In Vv: (£ - E,II)VuM dx,

Substituting (73) into the equation associated with (70)

OO(uM• v) - AOOM(u,ll. v) = fJ(v)

and subtracting the result from (7) yields

OO(u _UM. v) = -il!IJM(UM, v).

Allowing v = u - uM, and by Cauchy-Schwarz

OO(u -1/', u- UM) ~ IIV(u - uM)1I1.2(1l)II(E - £M)VuMII1.2({ll .

Because of the assumed ellipticity of the elasticity tensor (recall (10)),

lIu -IlMII~(n} L M II;;;. V(u - u ): V(u - u' ) dx .

at n

Therefore

(72)

(73)

(74)

(75)

(76)

(77)

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T.i. Zohdi e/ {II. I Comp,,/. Methods Appl. Alech. E/lgrg. 138 (1996) 273-298

~ ...~

Fig. 5. A hierarchy of scales. for the heterogeneous bar. illustrated for ten subdomains.

289

(78)

In general. estimates such as this are asymptotically exact in the sense that as gl/ approaches E in[L "(fl)IN~XN~. as M--H'J, we cxpect 11M-II in V(fl) by virtuc of (72),

5.4. Sfage II: Post-processing the coarse-scale soillfiol/S "/rollgh local fine-scale pertllrbtliion

Overview: While the solutions produced by ALR t:onverge to the fine-scale solution in the energynorm. they may not adequ:llcly capture the line-scale features. Hcr..:. we illustraW how the hierarchicalsolutions can be post-processed in an inexpensive local manner. which is proven to yield superiorsolutions in an energy sense. The process is as follows: thc solutillfl generated by the homogenizedmaterial description is used to wnstruct approximate local displaccment boundary conditions forinterior subdomain boundaries produced by a partition of the problem domain. The exterior boundarywnditions are not altered (Fig. 6). The decoupled local problems are then solved. The motivation hereis that the detailed solution to several decoupled sub-problems posed on smaller subrtomains, arc moretractable. and can be computed easily in parallel if desired, The final global solution is then 'recon-structed' by simply reassembling the local slJlutions. which arc conforming. We refer to this procedureas the Homogenized Dirichlet Projection Method. HDPM. The operation counts involved in this

/1\~/, :W='\: ~\' :-r.::r=. w~:W" 'W' :w:-,", ., J S :J.:.iJ-: • • • • II -.- ~! . .... i -

JEt if :Q:iiIllfltB: it• , • ., • • • \. • II

• _ -- wi I' _ 'f: • It .'. • ~ "I II. .. • II • ,'/ I' _ ,,.

Fig, 6. An example of ·post·processing· or 'decomposition of the domain' or 'local perturbation' using the coarse·scale solution.

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290 T./. Zohdi el /II. I Complll, Me/hods AppJ. Mech, Engrg. 138 (1996) 273-298

procedure are significantly less than those required to solve the hypothetical, coupled, tine-scaleproblem.

5.5. Construction of local subdomain problems

We define the boundary of an individual subdomain e~, as i)e~ consisting of a portion l-;"u where~ ---, the'displacell)ents are prescribed and a part rK1 where tractions are prescribed:

rKU = ae~\l'K' I~, = i)e~ n r, , (79)

We define the following space

V(e~) = {v EV(n), v = 0 on n\e~, vir". = O} , (80)

Let ~K' l~K~N(2lo), denote the operator from V(e~) into V(n) that identifies each vKEV(e~)with a function v in V(n) such that

vle~=vK vIO\e~=O' (81)

We define

M '\/1 EH1(r.:\o)IIK = II eQ ,':'J' K .i:.

(82)

We denote ii~ as the function that is zero outside of e~ and that is equal to the solution to thefollowing local variational boundary-value problem of elastostatics,

Find li~ E {II~} + V«(-)~) such that(R3)

where @K : H I(e~) x H I«(-)~) t-) ~ is the bilinear form characterizing the local virtual work and g;K( , )is a local linear functional characterizing the work dOile by the external forces:

@K(li~.VK)= r VVK:EVli:~dx ~K(VK)= !o!'VKdx+ r, t·vKds. (84)Je1! eJ( J/Kl

The displacements on I~u are prescribed as follows:

-.\11 '\/1IIK~,=IIKr.·....u Ku

(85)

On rK1the given external tractions. t, are prescribed, The global solution is constructed in the followingmanner:

(86)

where it is natural to think of the above parenthetical terms as local perturbations to the homogenizedsolution u'\', The question now becomes: if we were to solve the local problems and assemble the localsolutions together (according to (86)). will the overall solution be improved? Consider the fact that thelocal problems are solved with inexact internal boundary conditions. which may be grossly in error.Fortunately. as we now shall see. this local construction and solution process will guarantee a superiorsolution in the energy norm.

THEOREJ\;j 5,5.1. Let II be the exact solution to (7). Then with the previous definitions

1117,\1 - 1111 £(n) ~ 11,/' - u II "(ll) . (87)

PROOF. Let u· be an arbitrary element. such that u· E {Ii} + Veil). Then

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1'.1. Zohdi el aI, / CompUl. Me/hods Appl. Mech. £ngrg. /38 (1996) 273-298

lIu* - ull~l!1) = oo(u -u*, u - u*)

= a3(u, u) + :1J(u*. u*) - 2:1J(u, u*)

= :1J(u*, u*) - oo(u, u) - 2:1J(u. u*) + 2B?J(1I.II)

= oo(u*, 11*) - @(u. u) - 2B?J(1I, 11* - u)

= OO(u*, u*) - :1J(u. u) - 2£¥(u* -u)

= @(u*. u*) - 2~(u*) - (@(u, u) - 2£¥(u»

= 2fJ'(u*) - 2.'1(u) .

291

(88)

Since each ii~ is solution to the local boundary value problem governed by (83). then each il~minimizes the local potential.

1 .II 0fJ'K(W)=2'OOK(W,W)-£¥K(W) 'v'wE{uK}+V(BK) I~K~N(E2),

Clearly. from the construction of the global function in (86)

fl("') "'ar("') "'ar(-M) ar(-M)J U = L.J oJ K UK ~ L.J :::J K II K =.J U ,K K

Therefore

fi(ll,lI) - fJ'(/l)::::; fi(u"') - fJ'(II) .

From (88) the desired result follows.

IlliM - 1I11£l!1l ~ 1111.11- "11':.'(/1)' 0

(89)

(90)

(91)

(92)

5.5,1. ObservmionsWe make three important observations.(1) For three-dimensional problems, the computational cell size is on the order of the inhomogeneity

siz{;, the cost of the HDPM procedure, in terms of operation counts. is N times ch{;aper thansolving a problem with a direct numerical discretization technique. where N is 1he number ofsubdomains. Since the subdomain problems are completely decoupled. parallel processingtechniques can be employed. As a consequence. one may attempt to also reduce the time tosolution furt!ler by a factor of P, where P is the number of (equal speed) processors available.

(2) With the HDPM construction of the local problems. we can bound the difference in thehomogenized solution and its local perturbation. the 'sensitivity' beforehand. In other words. wemay determine locally where the local solution process will produce a significant change in thesolution. With the construction of the local problems in Theorem 5.5 we have, directly fromTheorem 3.1.

(93)

(94)

Note that the quantity on the right-hand side of (93) has been bounded beforehand, and isidentical to the local error indicator. and therefore incurs no extra work, if the same partitionsare used.

(3) Central to the success of the method is the choice of EM to minimize the final solution error,lIu -11.\111£(11)' The final error can be characterized in a straightforward manner. For anyadmissible virtual displacement v, we have

r V(II - tiM): EVv dx = r f' v dx + r t, v ds - r Vii''': EVv dxIn In Jr, In=2': {f f'vdx+ r t'vds- r WiM

: EVVdx} .R~ e~ J'j;" Jf)~

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292 T.I. Zolrdi e/ al. I Complll. Me/hods Appl, Mech Engrg. 138 (1996) 273-298

Noting that V· (EVil' v) - V, (EVil)' V = EVil: 'ilv, and using the divergence theorem, we have

( I

J V(u - ;;.11) : EVv dx = L {J 0 (f + V, (EV;;M) , v dxn R~ 8"

,+ irK' (t - EV;;ol: '11K)' V ds - irK" EV;; 'IIK' v dS}= 0 ~ 0 - L {J. E'ili'iM

. 11K' v dS} ,8~ /K,,\ru

(95) I

Therefore

lIu - iIMII~(fl) = L J EViiM '11K' (11M

- u) ds . (96)e~ rKu\/:,

The right-hand side can be interpreted as the work done by the jumps in traction moving throughthe difference in the actual and homogenized displacement on interior subdomain boundaries.Critical to the success of the method is, therefore. the choice of EM to minimize the work doneby the traction jumps on the subdomain boundaries. The choice for gil is certainly problem-dependent.

5.5.2, Example I.' Applying H DPM direcrly ro rhe (zero-scale) homogenized SOlllliollIn this example, we illustrate the dependence of the HDPM solution on the choice of EM. To this

end, we return to one-dimensional example considered before. and first consider M = O. Ell. Thefine-scale and homogenized problems arc

d ( du(x»)dx E(x)~ = -I, 1/(0)=0 II( I ) = 0 (97)

and

~ ( ,I( , du·I/(x))_ 11.11(0) = 0 . uM(I)=O.d\.' E (,\) dt - - I . (98)

For each subdomain K = 1.... , N(f!l ). the local H D PM problem is

d(

d-'I/()1I K x) -.II ,1/dx E(x) dx =-1. xE(X .....XK+1) uK(X ....)=1I (X ....),

(99)

where XK, XK+I arc the endpoints of each subdomain (Figs, 7 and 8),In our example. the unit interval is divided into 10 000 equal intervals, and for each interval the

property is chosen at random either E = 1 or E = T, where T is the mismatch ratio, Equal amounts ofhard and soft material are used. A coarser partition is overlaid representing the subdomain partition.All of the following calculations are done unalyrica/ly. For convenience, and to illustrate the sensitivityof the results to the choice of EM. we choose the classical uPEer and lower bounds in (32),

From Tables 5 and 6 it is clear that the harmonic average (E f = (E -1) -I) produces far superior finalsolutions. This is due to the facI that the internal boundary conditions are of displacement type and thatthe harmonic average produces the superior overall displacement compared to that of the arithmeticaverage. In this one-dimensional case, the flux jumps play a minor role, and therefore the error isessentially governed by the quality of the displacement data, Of course, in general, we seek an EM thatproduces the exact displacement on the internal subdomain boundaries, Initially, the solution producedby the harmonic average is in gross error, but after HDPM it has a far superior solution. with respect tothe energy norm compared to that of the arithmetic average. Increasing the number of subdomains

"

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u-£a~.

oU _ UomoO"

T./. ZoluJi e/ al. I CompU/. ,He/hods Appl. Mech, ElIgrg. 138 (1996) 273-298 293

Fig. 7. The heterogenous bar, with depictions of the exact solution. the homogenized solution, and local perturbations to thehomogenized solution,

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

Displ.lC~,"I!'I't

Subdomain 1

Subdom.inBound.ry roint

-1 .. u--Subdomain 2

Etc .._...

l!omosenized Solution

Subdom.in N

Fig, 8. The heterogeneous bar, and construction of the local problems.

Table 5The homogenizcd solution error and HDPM error for 1-;" = (E) for 100 subdomains and 10 subdomains. versus mismatch ratio

T

1050

100

11/111£(111

0.1852840.1731240.171544

11/1 - 11°11£cn ,111/111 £(11)

0.8044570.9569600.978203

11/1 - 11"'11£(11/111111£(11) N = 100, 10

0.649229 0,6449980.916105 0.9115360.957011 0.952321

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294 T.I. Zohdi 1'1 al. / COli/pili. Melhods Appl. Mech. ElIgrg. 138 (1996) 273-298

Table 6The homogenized solution error and HDPM error for E'" = (E-1

) -, for 100 subdomains and 10 subdomains. versus mismatehratio

T

1050

100

1I"IIElu,0.18528401731240.171544

11/1- /I"IIE(u/liuIiElu,1.34739932767134.680409

11,,- u"lIwl)/Ii/iIlE1u, N= 100.10

0.100066 0.032339[),125849 0.0403900.129001 0.041567

0.6

0.4 ...... n111/111 Rill II H

0.2

c:.~ 0iii

·0.2

'0.4

Exact --HDPM .---Homo 0

.~,,"

-0.6 o 0.1 0.2 0.3 0.4 0.5 0.6LenQlh 01 Iho Specimen

0.7 0.6 0.9

Fig. 9. A misl11a:dl of 20: 1 with 150 intervals. cach wilh randomly assigned material properties. HI'DM wilh :' subdomains.using the harmonic average solution to construct local boundary cond:lions.

0.5

0.4

0.3

0.2

-0.1

-0.2

·0.3

-0.4

('"

Exact -HDPM ----,Homo 00

.. _-- .. ~- '":,-', ... ''';4: ...... -t.;,..:.::-•.:..:.-~_;..•-.~.

·0,5o 0.1 0,2 0.3 0.4 0.5 0.6

Lenglh 01 Ihe Specimen0.7 0,8 0.9

Fig. 10. A mismatch of 20: 1 with 150 intervals, each with randomly assigned materiai properties. HPDM with 5 subdomains,using the arithmetic average solution to construct local boundary conditions.

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T.I. Zohdi 1'/ al. / COn/put. Me/llOds Appl, Mech. £ngrg. 138 (1996) 273-298 295

produces a solution which is increasingly worse, since more inexact information is introduced on moresubdomain boundaries.

Strain fields for cases with less material variations, which are easier to visualize on the printed page.are included in Figs, 9 and 10, and it can be seen that for the harmonic averaging case the HDPMsolution seems to capture local fluctuations in the solution quite well.

In the one-dimensional example, among the two effective material choices of £'1( tested, the selcc1ionis straightforward. the harmonic average delivers a superior final solution. As we have noted. in higherdimensions this choice is unclear. '

5.5.3, Example 2: Applying HDPM to the ALR generated solutionsReturning to the heterogeneous bar. we now apply HDPM to the ALR solutions. The main points to

be observed in this example are that further error reduction. beyond applying HDPM directly to thezero-scale solution. uo, can be made by using hierarchical structures generated by ALR. We present twolevels of uniform refinement of the material description, Level 0 corresponds to a solution generated bya uniform elasticity throughout the body; level 1 corresponds to a solution generated by using anelasticity corresponding to the average in each subdomain: level 2 corresponds to solution generated byusing material properties obtained by bisecting each subdomain and averaging the material propertyover each subdornain half (Fig. 5). All calculations were done analytically. For convenience. and toillustrate the dependence of the final solution on the choice of the local reaveraging technique, welocally reaverage according to the classical bounds. (E) and (E-1) -1.

As before, even when ALR and HDPM are combined, Tables 7 and 8 illustrate that Ihe choice of the

Taole 7The level O. I and 2 solution errors for E" '" (E). 10 000 IlHllCrial vari<itions. 10 suodomains with inereasing mismatch ratio

T

III

511100

11"- Ij"IlEtll/lll1l1nlll

n.64499!in.91153fin.952321

1111 -lj'II1:""/IIIIII';(I"0.6439140.9112600.952175

11"- ,j:II/:(/I/II"I1F.(11O.fi43fi440.9111790.952134

0.5

0.4

0.3

0.2

0.1

c: 0.~

en ,0.1

-0.2

,0.3

-0.4

-0.5

,{),60 0.1 0.2 0.3 0.4 0.5 0.6

Length of Ihe Specimen0.7 0.8

Exact -HOPM .....

0.9

....

Fig, 11. Level I scale used for HDPM. A mismatch of 20 : 1 with 150intervals. each with randomly assigned material properties.HDPM with 5 subdomains. using the harmonic average solution to construct local houndary conditions,

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296 TI. Zolrdi e/ al. I Complll. ,\lte/hods Appl. Mech. ElIlIrg. 138 (/996) 273-298

Table 8The Icvel 0, 1 and 2 solution crrors for E" = (E-1

) -I. 10 000 material variations. 10 subdomains with increasing mismatch ratio

T

1050

IOU

III/ - ,;°11Eln/ III/II nil)

0.0323390.0403900.lM1567

III/ - ,; III £(n/ III/II f:(1l)0.0042030.0052440.005392

III/ - ,;°IlE(n/III1I1£(1l)0.0026290.0032830.003377

·..\jl....~_···_·~·····'~,..~·

0.5

0.4

0.3

0.2

0.1

<:'§ 0in

·0.1

-0.2

-0.3

,0.4

,0.50 0.1 0.2 0.3 0.4 0.5 0.6

Length 01 Ihe Specimen0.7 0.8

Exact -HDPM ----

0.9

Fig. 12. Lcvel I scale used for IIDI'M, t\ mismatch of 20: I with 150 intcrvals. each with r,lIluol11ly assigncd material propertics.liD!'1\.! with 5 subdomains, using the arithmctic average solution 10 construct lueal i)ounullry conuilions.

0.5

0.4

0.3

0.2

0.1

o

-0.1

-0.2

-0.3

-0.4

ExaCI -HDPM .n,.

~I

-0.5o 0.1 0.2 0.3 0.4 0.5 0.6

Length 01 the Speelmen0.7 0.8 0.9

Fig. 13, Level 2 scale used for HDPM. A mismatch of 2U ; 1 with 150 intervals. cach with randomly assigncd material properties.HDPM with 5 subdomains. using the harmonic averagc solution to construct local boundary conditions.

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7'.1. Zohdi el al. I Complll. Mc/hods IIppl. Alecll. Engrg, U8 (1996) 273-298 297

Exact -HDPM -- ...

-~ : ~ ~ -..--- :'::;;;:;;-:.:,;.;'

0.5

0.4

0.3

0,2

0.1 H'I .......I

-0.1

-0.2

-0.3

-0.4

·0.5o 0.1 0.2 0,3 0.4 0.5 0.6

Lenglh allhe Specimen0.7 0.8 0.9

Fig. 14. Level 2 scale used for HDPM. A mismatch of 20: 1 with 150 intervals, each with randomly assigncd matcrial properties.IlDPM with 5 suodomains. using the arithmetic avcrage solution \0 construct local boundary conditions.

(locC/l) homogenized properties is criticC/IIO the qllC/lit)' of the fil/al solwioll. It is possible. as illustrated inTable g to achieve error that is on the order of a fraction of a percent. for this one-dimensionalproblcm. Figs. 11-14 make a clear point that the methods produce cxtremely accurate resolution of thestrain field locally, As can be seen. for this example. in the harmonic averaging case. one may obtainmore than one order of magnitude reduction of the error after only two levels of refinement.

':X~~<':';":t~OC¥b:"":

OBSERVA nON. One can interpret the ALR solutions as pr,)viding better local boundary conditionsfor subdomains in HDPM, and in this light, the method as a whole can be thought of as a non-overlapping domain decomposition method [91.

6. Summary

In this paper a methodology that consists of the following three main ideas has been introduced:(1) A global explicil estimate of the solution error introduced in using a homogenized (coarse)

elastici1y tensor and using the actual fine-scale elasticity tensor is derived. This estimate onlyrequires the calculation of the coarse-scale solution, Numerical experiments suggest that theestimate gives reasonable estimation of the local error as well.

(2) A procedure for generating a hierarchical family of material descriptions and correspondingsolutions is developed, The solutions corresponding 10 the members of the hierarchy are shownto converge to the solution of the fine-scale problem in an energy norm. In this mC1hod. theline-scale model of a structural component. which embodies the exact material dc..;cription,represents the highest level of sophistication in a family of continuum models,

(3) A post-processing procedure which endows the coarse-scale solutions with fine-scale informationis developed. The process requires the solution of local. decoup\cd subproblems posed onsubdomains inside the body. This process is trivially parallelizable _ due to the decoupled natureof the method, It is proven that the solutions generated by this 'post-processing' procedure willalways yield superior solutions to the original coarse·scale solution. Error reductions of orders ofmagnitude over the classical homogenized solution realized in simple one-dimensional examplesand may also be obtainable in higher dimensions.

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298

Acknowledgment

T.J. Zohdi e/ 01. I Complll. Me/hods Appl. ,Hech, ElIgrg. 138 (1996) 273-298

The authors would like to thank Drs Leszek Demkowicz and Peter Shi for insightfui discussionsrelated to the presented subject matter. The authors gratefully acknowledge the support of this work byNational Science Foundation under grant ECS-9422707 and the Office of Naval Research under grant95-1-0401.

References

[1) J. Fish and V. Belsky, Multigrid method for periodic heterogeneous media Parll: Convergence studies for one dimensionalcase. Compu!. Methods Appl. Mech. Engrg. 126 (1995),

[2] J. Fish and V. Belsky. Multigrid method for periodic heterogenous media Part II: Multiscale modeling and quality control inmultidimensional case. Compu!. Methods App!. Mech, Engrg. 126 (1995).

13] S. Ghosh and S.N. Mukhopadhyay. A material based finite clement analysis of helerogeneous media involving Dirichletlessalations, Comput. Methods Appl. Mech, Engrg. 104 (1993) 211-247,

[41 S. Ghosh and S.N, Moorlhy. Elastic·plastic analysis of helerogeneous microstruclUres using the Voronoi-cclllinite elementmethod. Compul. Methods Appl. Mech. Engrg .. to appear.

[5] S. I-Iasanov and C. Huet. Order relationships for boundary conditions effeet in heterogeneous bodies smaller than therepresentalive volume. J. Mech. Phys. Solids 42 (1994) 1995-2011.

[6] R. Hill. The clastic behaviour of a crystalline aggregate. Proc. Phys. Soc, (Land.) A65 (1952) 349-354.[7] C, Huet. Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys, Solids 38 (1990)

813-841.[81 V.V. Jikov. S.M. Kozlov and O.A. Olenik, Homogenization of Differential Operators and Integral Funclionals (Springer-

Verlag. 1994).[91 (', Le Tallec. Domain decomposilion methods in computational mechanics. Compu!. Mech. Adv. 1(2) (1994).

1101 J,T. Oden and G.F, Carey. Finile Elements: Mathcmalical Aspects. Vol. IV (Prellli<:c-Hall, Englewood Cliffs. NJ. 1983).

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",til

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