Lf56

Computer methodsin applied

mechanics andengineering

Compu!. Methods Appl. Mech. Engrg. 172 (]999) 3-25

Hierarchical modeling of heterogeneous solids1. Tinsley Oden*, Kumar Vemaganti, Nicolas Moes

T"xas JlISlitu/" for COII/plllllliollal wul Applied IHalhelllllt;cs. The Uni!.'ersily of Texa.\' at AII.\·till. Allstin. TX 78712. USA

Received 2 April 1998

Abstract

The modeling of microscale erfects required to describe physical phenomena such as the deformation of highly heterogeneous materialsmakes the use of standard simulation techniques prohibitively expensive. Most homogenizatioll techniques that have been proposed tocircumvent this problem lose small-scale information and as a result tend to produce acceptable results only for narrow classes of problems.

The concept of hierarchicaill/odt'/illg has been advanced as an approach to overcome the difljcultics of lIlultiscale modeling. Hierarchicalmodeling can be described as the methodology underlying the adaptive selection of mathematical models from a well-detined class ofmodels so as to deliver results of a prescl level of accuracy. Thus, il provides a framework for the autol1latic .lIld adaptive selection of themost essential scales involved in a simulation.

In the present paper. we review the Homogenized Dirichlet Projection Method (I-IDPM) lJ.T. Odcn and T.I. Zohdi. Compu!. MethodsAppl. Mech. Engrg. 148 (1997) 367-391: T.I. Zohdi. J.T. Oden and GJ. Rodin. Compu!. ~1ethods Appl. Mech. Engrg. 138 (1996)273-2981 and present several extcnsions of its underlying theory. Wc present global energy-norm and L 2 estimates of the modeling errorresulting from homogcnization. In addition. new theorems and methods for estimating error in local quantities of interest,such asmollifications of local stresses arc prcsented. These a posteriori estimates form the basis of thc HDPM. Finally, we extend the HDPM tomodels of local failure and damagc of two-phase composite materials. The results of scvcral llulllcrical cxperiments and applications aregiven. © 1999 Elsevier Science S.A. All rights reserved.

1. Introduction

The ability to analyze and accurately model heterogeneous and composite materials has assumed greaterimportance as the need to account for micromechanical effects in predicting the service life of machine parts andstructures is more broadly accepted. One of the main features of these materials is that their response to loadsand forces is often a complex multiscale. multiphysics phenomenon. Despite advances in computationaltechniques and computing power. direct simulation of heterogeneous materials is still not a viable option. Finiteelement models that can capture micromechanical effects generally must employ mesh sizes of the order of thesize of the microstructure and can result in an algebraic system with many millions of unknowns. On the otherhand. homogenization and averaging techniques for analyzing heterogeneous materials. while possibly leadingto manageable problem sizes. do not provide information about the microscopic fields needed, for example, topredict failure. Thus there is a need for accurate and computationally eflicient techniques that take into accountthe most important scales involved in the goal of the simulation while permitting the analyst to choose the levelof accuracy and detail of description desired.

Towards this end, the concept of hierarchical modeling was introduced [13,91 as a methodology that providesa multilevel description of the physical phenomenon of interest based, when possible. on a rigorousmathematical foundation. A hierarchy of descriptions of the physics of the problem is first set up. ranging from

* Corresponding author.

0045-7K25/99/s - see front mallcr © 1999 Elsevier Science S.A. All righls rescrved.PII: S0045-7825(9K)()()224-2

4 J.T. Odell et al. / COIIII'III. Methuds ApI'/. Mech. Elllirg. J72 (/999) ]-25

the coarsest possible description to the most detailed description contained in the class of models. Rather than toheuristically choose a level of description from the hierarchy, a posteriori estimates of the modeling errorassociated with a particular description are evaluated to enable the adaptive selection of a suitable characteriza-tion. Also, the level of description is allowed to vary spatially so that finer descriptions may be used in 'critical'regions.

Based on this concept of hierarchical modeling, the Homogenized Dirichlet Projection Method (HDPM) wasdeveloped in [13,9]. In this method, at the coarsest level in the hierarchy of models is a mathematical modelcharacterized by homogenized material properties. This is referred to as the homogenized problem and the lackof heterogeneity generally makes this problem computationally inexpensive compared to models of finer scale.The adequacy of the solution to this homogenized problem. compared to the fine-scale solution, is thenestimated using a posteriori modeling error estimates. In regions where the modeling error exceeds a presettolerance. a finer-scale model is used and a correction to the homogenized solution is computed. This process iscontinued until a simulation is obtained which is sufficiently accurate to satisfy preset error tolerances. Fullerdetails of this procedure are given later in this investigation.

While the use of hierarchical modeling permits the adaptive reduction of modeling error. it is equallyimportant to control the numerical error associated with the approximation of each model used in the analysis. Inour computations in this work, we employ an adaptive 3-d hp finite element method to control and minimize theeffect of numerical error on our results. The use of hp finite elements signitlcantly enhances the quality of thefinal solution obtained with HDPM while reducing the number of DOF required to solve both local and globalproblems.

The use of hierarchical modeling presents some intriguing observations on modeling of structures. The finalcomputational model achieved by, say. the HDPM to meet a given tolerance, may exhibit highly nonuniformmaterial characterizations that depend upon the data in the simulations (the geometry. loads, boundaryconditions, etc.) and the norms used to control the error. Thus, different tolerances and different norms lead todifferent material characterizations. We present here new error estimation results for control of L 2 error and errorin local quantities of interest.

We also describe the extensions of the HDPM to a class of nonlinear problems involving local damage andpossible crack initiation. We describe an application of HDPM to a two-phase composite in which local failure isassumed to occur when the local stresses reach a prescribed limit value. When this limit is attained, the localstiffness in a neighborhood of the failure point vanishes and a local redistribution of stresses takes place. Cyclesof HDPM for the damaged structure are repeated until the damage is arrested or full failure occurs.

The outline of the presentation in this paper is as follows. We first present some preliminaries and notationand set up the model class of problems under consideration. In Section 3, we present energy and L 2 estimates ofthe error associated with the homogenized solution. In Section 4. we briefly describe the HDPM. Next, inSection 5. we develop an estimate of the homogenization error in quantities of interest described by linearfunctionals on the space of admissible displacements. This is followed by numerical examples in Section 6. Inthe numerical applications, we describe a struightforward extension of the HDPM to problems of local damageand of simulating accumulative damage in a two-phase composite. Finally, we offer some comments and discussfuture directions of research in this area.

2. Preliminaries

In this section we describe the notation and conventions to be adopted in our analysis. The model problemcharacterizing the exact or fine-scale problem and the homogenized problem are presented followed by ananalysis of the error introduced by homogenization.

2.1. Notation and the exact problem

We consider the familiar problem of linear elastostatics describing the deformation of a heterogeneous bodyin static equilibrium under the action of body forces f and boundary tractions t. The body occupies an openbounded domain II C [RN, N = 1,2, 3. The houndary all of the body is assumed to be Lipschitz and consists of

J.T. Oden et al. I Campl/t. Methods Appl. Mech. Ellgrg. /72 (/999) 3-25 5

a portion r" on which displacements are prescribed and a part I; on which tractions are prescribed andan = r" U r" J~nJ> 0.

Vector and tensor valued functions defined over n are denoted by boldface letters and repeated indicesindicate summation. As usual. H I(D) stands for the space of scalar valued functions with distributionalderivatives of order ~ I in L\a). We also define HI(n)~ (H I(D)t as the space of vector valued functions

. I·· 1 ddl IV I Iwhose components are all In H (n) and sImIlarly we denote L (D) = (L (D» . The spaces H (a) and H (n)are equipped with the norms

(I)

respectively. where Vi are the cartesian components of v.Values of functions v E H I(n) on r" are understood inthe sense of traces and denoted vir. It is also assumed that the loading is such that I E L \n) and tEL \1;).

lj .., 2

Next. the body is assumed to be characterized by an elasticity tensor E which is a bounded function in IRIV'xNand satisfies the following conditions of ellipticity and symmetry: 3 al• a" >0 such that 'r/ A E IRNXN• A = AT.

alA: A ~ A : E(x)A ~ a,.A :A. x E {}

EUkf(x) = Ejik/(X) = Eiilk(X) = Ekli/x) I~i, j, k, I ~ N •(2)

EijJx) being the cartesian components of the elasticity tensor E. The ellipticity condition states that the strainenergy of the body is positively finite for admissible non-zero strain fields; the symmetry condition restricts thenumber of independent components in the elasticity tensor.

The displacement boundary conditions on r" are specified as follows: 3 Ii E HI ({}) such that Ii I1;, = UU, whereI1Jj is the prescribed displacement data on r". Then the principle of virtual work governing the displacement fieldin the body leads to the following problem:

Find u E {Ii} + V(D) such that

[!lJ(u. v) = S;{v) V v E V(n) .

where the space of admissible functions V(a) is defined as

del' I I 'V(D) = {v : v E H (Il), v ru = O} .

The bilinear and linear forms are defined as

ddf f T2lJ(u. v) = Vv : EVu dx = tr[(Vv) EVu Idxn n

and

derf f9'(v) = I'vdx+ (·vds.n r,

(3)

(4)

(5)

(6)

If the solution to (3) and the data are sufficiently regular. which is rarely the case, then it satisfies the followingequations of classical elasticity.

-V· E(x)Vu(x) =I(x) x E nu(x) = OZL(x) x E r"n . (E(x)Vu(x» = t(x) x E r, .

(7)

6 IT. Odell et al. / Compll/. MetllOds App/. Mech. Ellgrg. /72 (/999) 3-25

(8)

2.2. 711ehOl/logellized problem

For the type of problems considered in this paper. E is a highly oscillatory function thus making the use ofconventional methods like tinite clements computationally expensive and in most cases. impossible. Thisproblem can however be made more amenable to computation through standard homogenization processeswhereby E is replaced by a function EO, often a constant, that is designed to characterize the macroscopicbehavior of the structure. The loading due to body force and tractions is assumed to remain unchanged. but thisassumption could also be relaxed without signiticantly complicating our analysis.

The homogenized elasticity tensor is also assumed to satisfy ellipticity and symmetry conditions similar to (2)with ellipticity constants {3, and {3", i.e. 3 {3,. {3" > 0 sllch that V A E [RNXN. A = AT,

Eo13r4 : A ~ A : (x)A ~ {3"A : A

The homogenized problem thus reads

•

Find 11° E {Ii} + Veil) such that.o/.j°(llo, v) = 8f(v) V v E V(fl)

with

II ° dCrJ 0 °q) (II ,v) = Vv : E VII dxf!

(9)

(10)

and with the right-hand side as defined earlier. Again, if the solution to the homogenized problem represented by(9) is sufficiently regular. then it also satisties the following homogenized equations.

-v·Eo(x)Vut\x) = f(x) x E flI/(X) = ilJ1(x) x E r"

II ..., II i'II . (E (x)vu (x» = t(x) x E "

3. Analysis of the homogenization error

(11 )

The homogenized solution UO is obviously in error because material information is lost due to the process ofhomogenization. The homogenizatio!l or modeling error is defined as the difference between the exact solution

f) del 0and the homogenized solution, e = II - II .

To be able to develop adaptive methods of simulation, it is important to evaluate the quality of thehomogenized solution. In this section. we present various measures of the homogenization error ell as well as ameasure of the error in the stresses between the exact and homogenized solutions.

3.1. An ellerg)' estimate of the homogenization error

The following theorem is proveu in [13 J.

THEOREM 3.1. Let U lind Ull be the solutions to problems (3) alld (9), respectively. Then, the following holds:

( 12)

where

1.1' Odell 1'I al. I COIIIl'llf. Method.\' ApI'/. ,wee". ElIgrg. /72 (/999) 3-25 7

( 13)

oThus, if the solution to (9) is known. the homogenization error is bounded hy a quantity that can expressed interms of known quantities.

3.2. An L 2 estimate of the homogenization error

Another global estimate of the homogenization error is presented below.

THEOREM 3.2. Let /I and I/o be as ahove. Theil, the following estimate holds

(14 )

where at is defined in (2) and C(il) is a positive constant depending on the domain n.PROOF. It can be easily verified that the homogenization error eO is the solution to the following problem:

Find ell E V(!}) such that

9J(eo. v) = 9?0(v) V v E V(il)

where the right-hand side is defined as

9l0(V) = -J" Vv: E5>o'Vl/o dx .!l

Setting v = eO, we have

(}])oo f'l"'o""o f""o,.1,...,o£:JJ(e ,e ) = ve: Eve dx = - v'e: EofO vII dxn uThe left-hand side or (17) (,;an be bounded below,

f ""0. 'I'" l) ::>, f "",l) . "" l) dx - II"" °112ve . Eve dr ~ at H'. 't·e - a, ve I. 'I!!)U !!and the right-hand side can be bounded above using the Cauchy-Schwartz inequality.

f ,(). , 'r.' ° ,,:::II'""' Ilil II t1 '""' 0- Ve .E.!fl)'tll dx~ 'te /,",IJ, EofOHI 1.'(/11U

thus leading to

Now. we use the Poincare inequality

with (20) and the assertion follows. D

3.3. An estimate of the error in the stresses

(15)

(16)

(17)

( 1X)

(19)

(20)

(2 I)

The (inal estimate presented in this section provides an upper bound on the error in the stresses correspondingto the exact and the homogenized solutions.

THEOREM 3.3. Let II and 11° he as ab(JI'e. Define tire stress states associated with these solutions as follows:

8

Then,

u = EVu,

iT Oden et al. I Compllt. MetllOds Appl. Mech. Engrg. /72 (1999) 3-25

(22)

where a" is defined ill (2).

PROOF. First. we decompose the difference between u and uo'

... 0... 0 ... ° ... °u - Uo = E vII - E vu + E vU - E vU= EVeo + E.1>oVr/' .

Using the triangle inequality,

Ilu - uoIIL2(JJ) ..,,; liEVeoIIL2(fl) + liE.1>oVuoIIL2(Il} •

(23)

(24)

(25)

The first term on the right-hand side of (25) can be bounded above, since under the assumptions in place E has awell-defined square root.

liEVeoll~2(m2 = f EVeo: EVeo dx!J

J fD 0 - °= n (vEVe ): E(,JEVe ) dx..,,;a" f. (JEVeo) : (JEVeo) dx

!J

..,,;a f Veo: EVeo dxII J1

..,,;aJeoll;,w)2~allC

thus leading to

IIEVe(lIIL2(iJ) ..,,;Va;,?The second term on the right-hand side of (25) can also be bounded similarly:

IIE.1>oVuoll~2(m = In E.1>oVuo: E.1>oVuodx..,,;a" J .1>0Vuo :E.1>oVuo dx

fJ

"";a r2,,!>

(26)

(27)

Finally, combining (25), (27) and (28), the assertion follows. 0

We present other a posteriori estimates of modeling error in Section 5.

4. The Homogenized Dirichlet Projection Method (I-IDPM)

The HDPM is discussed in detail in L9,12,13]. The role of numerical error in HDPM is discussed in 18J. Inthis section, we summarize the major aspects of this method. We begin by noting that the homogenized solutionU

O can in general be a poor approximation to the fine-scale solution u. The error in the homogenized solution

J.T. Oden et al. / COlllpllt. Methods Appl. Mech. Engrg, J72 (/999) 3-25 9

depends mainly on the homogenized material properties. Typically, most homogenization techniques providesatisfaclOry solutions when the microstrucnlre is smail with respect to the size of the structure and is periodic. Inmost practical applications, however, the microstructure is random. Also. the homogenized stress state"0 = EI)Vuo can be quite inaccurate und hence cannot be Llsed for damage prediction. It is therefore natural toask if the homogenized solution can be improved without having to solve the original problem (3). The HDPMprovides a systematic way of enhancing the homogenized solution by solving relatively small local problems inareas of high modeling error. These local problems use the exact microstructure information and wherenecessary. the homogenized solution is used as Dirichlet data.

4.1. Construction of local problems

We tirst consider a non-overlapping partition gp of the domain il into N subdomains (;,)k' k = 1.2, ...• N( gp)(Fig. I) such that

N(.'i')

(29)

The boundary af)k of each subdomain Bk consists of a portion I; on which tractions are prescribed and aportion T." on which displacements are prescribed '

(30)

Local function spaces are defined as

dcfV(fo)k) = {v: v E Veil), vlme = o. vl~ = O}.

k 1z.(31 )

For each subdomain. we define an operator ~k that extends functions from the local space V( Bk) to V(Il) asfollows:

del'~k : V(6D:3 Vk ~ v E V(n). vl(-) = Vk• VIfJ\A = O. (32), ,

The restriction of the homogenized solution to each subdomain is defined as II ~~ u °1 f) . We denote by Ii ~ theI

solution to the following boundary value problem

Find ii ~E {u~} + V( (ojk) such that

P1Jk(li~. vk) = ~(Vk) V vk E V((~k)'

for I ~k~N(gp) with

Fig. I. A non-ovcrlapping partition of the domain.

(33)

10

and

J.T. Odell el al. I Compll/. Melhods Appl. Me ...h. EIIgrg. J 72 (1999) 3-25

(34)

(35)derI I~(Vk)= f·vkdx+. (·vkds.l~)1.: Il..

r

The displacements on I~ are prescribed as II ~I(' = u71 r. ' i,e. the homogenized solution is used as Dirichlet dataH k" kIt

on the l~ portion of each subdomain's boundary. In particular. this data is used on the interior parL of eachsubdomai~)'s boundary given by afoJ\a.f2. As a result, the local problems are uncoupled. Finally. a global solutionis constructed from the local solutions in the following manner

ii 0 E {Ii} + V(fl) , (36)

which will be referred to as the HDPM solution. Clearly. this new solution is continuous across subdomains.

4.2. Characterization of the HDPM soliltion

We begin by detining the potential energy of functions wE {II} + V(f2) asdd I

Jew) = 2' 98(w. w) - ~(w) .If II is the exact solution to (3). then it is well known that

,j{u) ~ }i(w) V w E {Ii} + v(n) .The following result guarantees that the HDPM solution is indeed an improved solution.

rMEOREM 4.1. With the previous definitions in force.

,j(1/0) ~,I(I/).and hence

PROOF. See 19J. 0

(37)

(38)

(39)

(40)

This result implies that the HDPM solution is always closer to the exact solution than the homogenized solutionregardless of the homogenized material properties. Proof of two corollaries that follow immediately from thistheorem are in [9J;

COROLLARY 4.1. Let II be the exact solution to (3). Additionally, aSSIIlJlethat V· (EVli7). f E H -1(Bk) andEn-o) H-1/2 r.. 1'/( ~ ...11k . 1/ E (k) lu'n.

II -°11' . rlel -" n •II - U ;,'([1) ~ IV = 2(.!-(1I ) - ,j(1I )) + (' . (4] )This result provides an estimate of the enor in the correction Ii" of the homogenized solution. It can be seen thatthe term Cf(lill) - $(uo») is negative so that 1/1 ~? always. The next result is a very useful sensitivity property:

COROLLARY 4.2. Define ?k by

(42)

-0 _odd_O 0and ek by ek = Ilk -Ilk' Then, for] ~k~N('?P)

IT. Odell et al. I COII/pllt. Methods API". Mech. E/lgrg. J 72 (/999) 3-25 II

(43)

The above corollary predicts the improvement that can be obtained by solving a local problem. If ?k is small in asubdomain, then there is little gained by solving a local problem. On the other hand, if the above estimate issharp, and ?k is high in a subdomain. the homogenized solution can be significantly improved by solving thelocal problem (33). We discuss this issue in more detail in the section on numerical examples.

Finally, we present an L 2 estimate of the difference between the HDPM solution and the homogenizedsolution.

71-1EOREM 4.2. Let al he as defined in (2). Then, for I ~ k ~ N( P/')

(44)

where C(0k) is constalll that only depends on the subdomain ek•PROOF. The proof is essentially the same as the proof of Theorem 2. First. we note that e~ is the solution tothe following boundary value problem on subdomain f),:

Find e~E V( ('1k) such that(45)

with the right-hand side

.¥~(Vk) = - f VVk: E.1>oVu~ dx . (46)(.,),

The rest of the steps carryover directly. 0

4.3. The overall algorithm

The overall adaptive algorithm is as follows:

Step I. Given the initial data fl, r". 1,'. E. f, {i and t. construct a partition of the domain r;p = {(-U~'~I' Choosea homogenized material tensor EO. Specify sensitivity and error tolerances al and (\'2 so that

(47)dd II °11I/llol = a2 1I £(!ll·dcf ° I f), I(?k)"d = a11lu 11£,111X Inl 'Step 2. Solve the homogenized problem (9) to obtain 1/.Step 3. Compute the local sensitivities ?k using (42) for k = I, .... NUl) and form a set ,j of subdomains

which are above the prescribed sensitivity tolerance

(4R)

Step 4. For the subdomains that fail to satisfy the sensitivity tolerance. k E,j, solve the local problems (33)to obtain ii ~.

Step 5. Construct the HDPM solution

(41) )

Step 6. Compute the estimated error in the HDPM solution

I/J ~r [2(,j(/i'0) _ ,j(uo» + (] 1/2 (50)Step 7. If If; ~ ifflu" STOP. Else. repeat Steps 2-7 with improved material properties. A general algorithm for

12 J.T. Odell et al. I Campl/t. Methods Appl. Mech. Erlgrg. 172 (1999) 3-25

choosing improved properties can be found in [9]. If the error tolerance is not satisfied with improved materialproperties, go to Step 8.

Step 8. Coarsen the partition and repeat Steps 2-7.Step 9, Relaxation. (This step is optional), At the conclusion of the adaptive process when all global and

subdomain errors meet the assigned tolerances, tractions are discontinuous across subdomain boundaries. Anumber of Schwarz-type iterations relaxing the boundary constraints on displacements can be performed toreduce the stress discontinuity,' remove spurious singularities in the displacement derivatives, and furtherimprove the accuracy of local features of the solution.

REMARK 4.1. While in principle, it is possible to use traction boundary conditions to solve the local problems,the need for flux-equilibration introduces certain difficulties. Moreover, the final solution may not be continuousacross subdomains. The use of Dirichlet boundary conditions obviates these complications and allows for easyparallel solution of the local problems [II].

REMARK 4.2. There is a considerable amount of literature on the use of global-local and multiscale analysesfor modeling heterogeneous materials. This includes the works of Fish and Belsky [3,4], Ghosh and Moorthy15], Ghosh and Mukhopadhyay [6] and more recently Moes et at. [7]. In the HDPM, the effects of multiplescales are automatically included in the analysis by allowing the a posteriori estimates to choose the mostappropriate scales. The coarsening of the partition in Step 8 of the overall adaptive algorithm above correspondsto an increase in the disparity between scales.

5. Homogenization error in quantities of interest

Recent work in error estimation in the context of finite element analysis has focused on obtaining bounds onthe numerical· erratin .quantities of interest other' than' the energy norm [I, 10]. In this section, we use', theapproach of Prudhomme and Oden 110] to obtain an upper bound on the homogenization error in otherquantities. We assume that we are interested in estimating L(eo) = L(u} - L(uo), where L is a continuous linearfunctional on V(fl), LEV'. For instance, L may represent something more localized than the global estimate(12) such as the error in u or Vu over a small region in fl. The main objective here is to relate L(eo} to the'source' of the homogenization error, the right-hand side of (16). So, we would like to find a linear functional"IJi' E V", if it exists, such that

...

(51 )

The functional "'If' is known as the influence function(al) since it indicates the influence of the residual on thequantity of interest. Since, V is reflexive. we have that 3 'w E V such that

and hence (51) becomes

L(el) = 0j>l)(w).

Using (16). we obtain

The influence function w can thus be obtained as a solution to the global dual problem

Find w E V(ll) such that

,93(v, w) = L(v) V v E V(n) .

(52)

(53)

(54)

(55)

It then follows that w exists and is unique. The dual problem (55). however, is as difficult to solve as theoriginal problem (3). A natural way to simplify this problem is to solve the homogenized dual problem

V v EV(n) ,

1.7'. Odt'll el al. / COll1plll. Methods Appl. Meeh. EI/[?rg. /72 (/999) 3-25

Find WO E V(fl) such that

.o/Jll(v. wo) = L(v) V v E V(Il).

It immediately follows that the modeling error in the influence function eO d~f w - W 0 satisties

I m(v. eO) = ,0/[ o(v)with

Q7JI)( ) - -f ~ o. E tl. 'Ii' dx;:1(. v - "W..7o ,..·v .Ii

We also note that ell satisfies the following bound (analogous to (12»:

Next. we note that

L(eo) = m(eo. w) = £XJ(eo,e) + 9J(eo, wll)

and hence

IL(e")1 ~ I27.J(eo. e)1 + 193(1'°, wO)1~ Ileollt:'IJl)llellllr:UJI + Ileoll."w)llwollw}J .

13

(56)

(57)

(58)

(59)

(60)

(61 )

Finally. using (12) and (59). we arrive at the following bound for the homogenization error in the quantity ofinterest.

I ° dol - II 011IL(e )1 ~.B = (( + (w Wll' (62)Thus. the estimation of the homogenization error in the quantity of interest requires the solution of a global dualproblem. In our analysis above. we assume that the homogenization parameters chosen for the dual problem arethe same as the ones chosen for the primal (original) problem. As a result. the two problems have the sameleft-hand side which is computationally convenient. This assumption. however. can be relaxed without majorchanges to our analysis. Indeed. this may not be an unattractive choice considering the facl that thehomogenized problem Iypically requires far fewer degrees of freedom than a local problem with microstructure(see Example 6.1). Later in this paper. we present some I-D examples to illustrate the performance of ourestimate.

6. Numerical examples

6./. £xalllple /

Consider a composite slab (dimensions 8 X I X 2) divided into 16 equal subdomains. The body is subjected toa uniform compressive load over one subdomain as shown in Fig. 2. We assume that the microstructure isprovided by 1024 spherical inclusions distributed uniformly in the matrix material so that each subdomain has a4 X 4 X 4 arrangement of inclusions (Fig. :I). While the matrix has the properties £ = 400.0 MPa. 1) = 0.2, theinclusions have the propeJ1ies £ = 4000.0 MPa. 1) = 0.2. The volume fraction of the inclusions is assumed to be0.2. To obtain the homogenized material properties, we use the arithmetic average of the Hashin-Shtrikmanbounds. Finally. we use (?k),,,1 = 0.511"[)1It:([J I X I('?kl/lfli.

An approximation to ,,0 is generated using the adaptive lip tinite element program ProPHLEX 121 and isdenoted by ,,0.11. The lip mesh used to solve the homogenized problem (see Fig. 4) has 7407 degrees of freedomwith an estimated relative numerical error of 4.0% in the energy norm.

14 J.T. Oden et al. I Comput. Methods Appl. Mech. Engrg. 172 (1999) 3-25

SUB DOMAINBOUNDARIES

DIMENSIONS: 8xlx2Fig. 2. Schematic of the composite bar partitioned into 16 subdomains. Subdomain numbering is also shown.

Fig. 3. The microstructure in each subdomain.

..

On computing the Ck I ~ k ~ 16, it is found that 4 subdomains fail to satisfy the sensitivity criterion (see (48))so that ,j = {I. 2, 9. 1O}.In these subdomains, we find approximate solutions to the local problems (33). k E,jand denote these by ii~·h The hp mesh for subdomain I is shown in Fig. 5. The HDPM solution is constructedusing

-O.h.1f = O.lf +" CP (-O.h _ 0.1f)U U LJ Ok Uk Uk .

kEJ(63)

It is found that (If I lIu°11 E(fl) = 0,604 and l/Ih.H IlluoIIE(fl) = 0.065. Thus, the HDPM can dramatically improve thehomogenized solution.

1.T. od"l/ et al. / COlIIl'lIt. Methods Apl'l. Alech. EI/f(rg. 172 (/999) 3-25

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Fig. 4. hI' mcsh for thc homogenizcd problcm with 7407 degrees of freedom.

~y

Fig. 5. hI' mesh for subdomain I with 93261 degrecs of freedom.

15

16 i,T. Odell e/ al. I COlI/pUI. Me/hods Appl. Mech. Engrg. 172 (/999) 3-25

A quantity that is of interest in stress analysis is the Von-Mises stress. The maximum Von-Mises stress (seeFig. 6) predicted by the homogenized problem is 170.3 MPa at x = () whereas the maximum Von-Mises stresspredicted by the HDPM solution is 534.98 MPa in subdomain I at x = (0.177,0.072,0.072) (see Fig. 7). Thusthe use of the homogenized solution for making design decisions without further processing can be quitedangerous.

Finally, we note that the homogenized problem needs 7407 degrees of freedom and each local problem on anaverage requires about 90000 degrees of freedom. For this problem, it is estimated that a direct simulationrequires about I 800000 degrees of freedom.

6.2. Example 2

Now, we consider an example focusing on the difficulties posed by the highly oscillatory nature of E, themost significant of these being the integration of functions of E. One such quantity is the local sensitivityindicator (42).

We choose the problem of a beam clamped to a wall at one end and loaded by tractions on the other end. Thebeam has reinforcing bars as shown in Fig. 8. The mismatch ratio of the two materials is assumed to be 5.0. Wehomogenize the beam using the Hashin-Shtrikman bounds and then compute an approximation to thehomogenized solution 11°. The next step is to compute

(64)

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(65)

Obviously. the intcgrands above are highly oscillatory. We use two methods to evaluate the above expressions.

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I~~

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17

In the first method, we use a conforming mesh that respects the boundaries of the microstructure as shown inFig. 9(a) and in the second method, we use a uniform mesh as in Fig. 9(b). The two methods are compared forGaussian integration rules varying from the 1 x I x 1 integration rule to the lOX lOX 10 integration rule ineach element. In the first case, the material properties are known element-wise and in the second, it is necessaryto check if a given integration point lies in the matrix or an inclusion. From Fig. 10. we see thai the resultoblained using a non-conforming mesh is highly oscillatory both for ?k = I and ? This shows that the use ofconforming meshes, though expensive due to time spent in mesh generation, is necessary for integrating highlyoscillatory functions. On the other hand, non-conforming meshes are relatively easier to generate but Ihe numberof integration points per element required for accurate results make their use an unattractive choice.

6.3. Exalllple 3

We now present a preliminary study on the performance of the estimate (62) for the following I-D problem.Consider an clastic bar of unit length lixed at both ends and subjected to a constant body force. The primalfine-scale and homogenized problems arc

Itl .l.T. Odell er al. I COli/pilI. Merhods Appl. Mech. Ellgrg. 172 (1999) 3-25

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- - .... - - ....,,- ,,-I \ I \

1 , , I\ I \

,\ I \ I..... ,,- .... '"- -

- - .... - -,,- ,,- ....I \ I \

I \ , I\ I \

,\ I \ I

.... - ,,- .... ,,-- -(b)

Fig. 9. (a) Conforming and (b) non·conforming meshes for computing (, and (.

and

d ( dll)-- E(x)- = -1,dr ell:

d (,0 dll")- dr E pCx) dr = - I ,

1/(0) = 0 ,

1/"(0) = o.

III I) = 0 (66)

(67)

Here, E~ indicates the homogenization parameter for the primal problem. The tirst linear functional we consider(as indicated by the subscript I) is

(68)

J.T. Odefl et al. I Compl/t. Methods Appl. Mech. Eflgrg. /72 (/999) 3-25

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and

.I.T. Odfll et al. / CO/J/pat. Methods Apl'l. Mech. EII{iI-g. 172 (/999) 3-25

d (,(j dW~)- dx Eix) dx = !l(x - 0) - H(x - h). w~(I)=O. (73)

The unit interval is divided into 10000equal intervals, and for each interval the material properly is chosen atrandom to be cither E = I or E = T. where T is the mismatch ratio. Equal amounts of hard and soft material areused. All of the following calculations are performed analytically. Also, E~ and E~; are independently chosen tohe the arithmctic average (E) or tbe harmonic average (E-1) -I. Finally, we define the effectivity index by

dell U ITJ = L(e ) / f3.From Tablcs I and 2 it is secn tbat when thc harmonic avcrage is chosen to homogenize the primal prohlcm,

pointwise errors are very slllall but the estimate performs poorly. Tbe poor performance results from Ihe fact thatthe estimate (62) does not account for any cancellations. On thc other hand. when the arithmetic average ischosen to homogenize the primal problem. pointwise errors are high and the estimate performs relatively well.The effect of cancellations is nol very significant in this case. In either case, using the arithmetic average for thedual problem improves the estimate.

6.4. Example 4

In the final example of this paper. we demonstrate a simple scheme for the study of damage mechanics ofcomposites in the framework of hierarchical modeling. We consider a unit cube of material. with a two-phaseisotropic spherical microstructure (see Fig. II). The volume fraction of the inclusion material is 0.25 and thematerial properties are chosen to be those used in Example I. The cube is fixed on the face y = 0 and loaded byshear on the face v = I. We employ a unidirectional partitioning of the cube into four simple 'slabs':0-' (E) 0.0000067 0.0077 1150.210.0 (E) (£-') -, -0.00229 0.01 4.410.0 (£) (t.') -0.00229 O.IXl44 1.9

100.0 (E-') , (E-')-' 0.()OOOO74 0.1772 23966.4100.0 (E-')-' (E) 0.00IXl074 0.0209 2g25.3100.0 (E) (£-') , -O.IXlJOI 0.0350 11.6!IXl.O (E) (E) -0.00301 0.0041 1.37

J.T. Odell ('t al. I CO/llp/ll. Methods Appl. Mee". EII~:rR. /72 (/999) 3-75 21

(b)

Fig. II. Schemalic of (a) the compositc cubc and (bl thc micro,tlUclurc employcd in studying damagc mechanics.

generated using the arithmetic average of the Hashin-Shtrikman bounds and the homogenized problem is solvedhp adapti vely to' obtain II 0. increment the iteration count and OOTOStep 2. Else, STOP.

The process of 'disabling' elements leads to a redistribution of stresses and the new stress state mayor maynot satisfy the failure criterion. In case the criterion is not satisfied, the above algorithm can be carried out untila certain volume fraction of the subdomain has failed. Note that Step 3 in the algorithm above can be modifiedto include other failure modes such as debonding by considering different failure criteria. The essential idea isthat these analyses can be performed in the context of hierarchical modeling.

The above algorithm is applied to subdomain 4. We assume that the inclusion material fails at 490.0 MPa andthe matrix material fails at 400.0 MPa. Fig. 12 shows two views. A and B. of the Von-Mises stress distribution inthe subdomain at the zeroth iteration. i.e. before the initiation of damage. View B is obtained by rotating view Aby 180" about the ~ axis. It is found that the maximum Von-Mises stress at this iteration is 497.24 MPa. Fourinclusion elements and 8 matrix elements fail the tolerance test (Step 3 above). These elements are disabled andthe problem is resolved. The Von-Mises stress field for iteration I is shown in Fig. 13. Now the maximumVon-Mises stress in the subdomain increases to

22 .I.T. Odell et III. I COlIIl'lIt. lHethod,\' "1'1'1. "'tech. ElIgrg. /72 (/999) 3-25

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J.T. Od"1/ et al. I COlliI'll!. Methods Appl. Meek Engrg. /72 (/999) 3-25

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24 J.T. Odell et III. / COIIII'III. Methods Al'pl .• lv/(·cll. Ellgrg. 172 (/91)9) 3-25

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7. Final comments

IT. Odell er a/. I COlliI'll!. .~lethods Appl. Ml'CIJ. EII!!r/!,. /72 (1999) 3-25 25

The concept of hierarchical modeling, and its implementation through the Homogenized Dirichlet ProjectionMethod. provide a systematic family of approaches toward the resolution of mulliscale problems, pal1ieularlyproblems of analyzing heterogeneous materials. The approach bypasses or generalizes many of the traditionallimitations of homogenization theory and the theory of composite materials. To mention a few:

• no periodicity of microscale constituents is assumed• the approach (therefore) does not rely on the existence of RVEs (Representative Volume Elements)• homogenization methods are merely artifacts of the overall adaptive strategy and are not goals of the

modeling process in themselves (however. the choice of homogenization technique has significant impacton the performance of error estimators and the success of the method)

• extensions to multi scale modeling problems are possible• as shown in the present study, nonlinear behavior can be accommodated in the modeling process in a

straightforward manner. albeit at more computational expense.A number of extensions and generalizations of the hierarchical modeling concept represent challenging but

critical areas for future work and some of these are currently under study. These include the development ofadaptive modeling strategies more integrated with the homogenization steps to allow for systematic modeling ofmulliscale phenomena. These could be used to control multiple (internal) iterations of the HDPM strategy toaddress many different levels of scales present in many applications. More work is needed to refine ourapproach to local error estimation and to study the limitations of the approach for model elTor analysis describedhere. Finally. modeling of a wide range of nonlinear phenomena is within reach. including a more generalsimulation of progressive damage accumulation, crack initiation and propagation. life cycle prediction andmicromechanical effects. such as local diffusion phenomena. We plan to explore these issues more deeply infuture works.

Acknowledgments

We gratefully acknowledge the support of this work by the Office of Naval Research under grantNOOO14-95-1-040 1 and by NSF through NPACI. the National Partnership for Advanced ComputationalInfrastructure. grant 10152711.

References

[I] /. Babuska. 1. Strouhoulis. C.S. Upadhyay and S.K. Gangarnj, A postcriori cstimation and adaptative control of the pollution error inthc iI-version of thc tinite clement method. Int. J. Numer. Methods Engrg. 38 (1995) 4207-4235.

12] Computational Mechanics Company, Inc .. Austin, TX. ProPHLEX User Manual for Version 2.0. 1996.131 1. Fish and V. Bclsky. \olultigrid method rnr periodic heterogeneous media. Part I: Convcrgencc studies for onc-dlmensional casc.

Compul. Methods Appl. Mech. Engrg. 126 (1995) 1- t 6.141 J. Fish and V. Bclsky, Multigrid method for pcriodic hcterogencous media. Part 2: Multiscale modeling and quality control in

multidimensional case. Comput. Mcthods Appl. Mech. Engrg. 126 (1995) 17-38.151 S. Ghosh and S. :\'loorthy. Elastic-plastic ,malysis of mhitrary hcterogenellus matcrials with thc voronlli cell finite clement method.

COIIlPUt. Mcthods Appl. Mcch. Engrg. 121 (1995) 373-409.(6] S. Ghosh and S.K Mukhopadhyay. A llIaterial based finite elcment analysis of heterogeneous media involving Dirichlet tesselations.

Comput. Methods Appl. :\kch. Engrg. 104 (1993) 211-247.17] N. Moes. J.T. Oden and K. Vemaganti. A two-scale strategy and a postcriori error estimation for modeling hererogelleous structurcs, in:

P. Ladeveze and J.T. Oden. cds .. On New Advances in Adaptive Cllmpulational Mcthods in Mechanics (Elsevier publication, t997) toappear.

18] ~. Moes. J.T. Oden and 1.1. Zohdi. Investigation of the interactions hetweell the numerical ,mel the modeling e'Tors in the homogenizedDirichlet projcction methud. Comput. Methods Appl. Mech. Engrg .. 10 appear.

191 J.T. Oden and T./. Zohdi. Analysis and adaptivc modeling of highly helcrngeneous elastic stmcturcs. Comput. ~1cthods Appl. Mcch.Engrg. 148 (1997) 367-391.

1.10] S. Prudhomlllc and J.T. Odcn. Goal oricntcd adaptivity and local error estilllatilln , f "", l) . "" l) dx - II"" °112 f ,(). , 'r.' ° ,,::: II'""' Ilil II t1 '""' 0 ( 1 X)

page6titlesu = EVu, = EVeo + E.1>oVr/' . Ilu - uoIIL2(JJ) ..,,; liEVeoIIL2(fl) + liE.1>o VuoIIL2(Il} • (24) (25) liEVeoll~2(m2 = f EVeo: EVeo dx = n (vEVe ): E(,JEVe ) dx ..,,; a" f. (JEVeo) : (JEVeo) dx ..,,; a f Veo: EVeo dx ..,,; aJeoll;,w) 2 IIE.1>oVuoll~2(m = In E.1>oVuo: E.1>oVuo dx ..,,; a" J .1>0 Vuo :E.1>o Vuo dx (26)

page7titlesV(fo)k) = {v: v E Veil), vlme = o. vl~ = O}. ~k : V(6D:3 Vk ~ v E V(n). vl(-) = Vk• VIfJ\A = O. (32) , ,

page8titlesderI I ii 0 E {Ii} + V(fl) , Jew) = 2' 98(w. w) - ~(w) . (37) II -°11' . rlel -" n • (42)

page9titles(43) (44) dd II °11 dcf ° I f), I (?k)"d = a11lu 11£,111 X Inl '

page10titles5. Homogenization error in quantities of interest ...

page11titles.o/Jll(v. wo) = L(v) V v E V(Il). ;:1(. v - "W..7o ,..·v . IL(e")1 ~ I 27.J(eo. e)1 + 193(1'°, wO)1 ~ Ileollt:'IJl)llellllr:UJI + Ileoll."w)llwollw}J . IL(e )1 ~.B = (( + (w Wll' 6./. £xalllple /

page12titlesSUB DOMAIN DIMENSIONS: 8xlx2 ..

page13titles15

page14titles6.2. Example 2 (65)

page15titles.. I

page16titlesd ( dll) d (,0 dll") - dr E pCx) dr = - I , III I) = 0

tablestable1

page17titlesO"r '- d ( dU'l) (69) (70) f', LJu) = v(x) dr. " (a, b) C (0. I). (71 ) d ( dl-L) - -. E(x) ---=-. = H(x - ((') - H(x - b) dr dr . (72)

page18titlesd (,(j dW~) - dx Eix) dx = !l(x - 0) - H(x - h).

tablestable1table2

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page21titleso a

page22tablestable1

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