Mechanics of Materials 38 (2006) 859–872
www.elsevier.com/locate/mechmat
Multi-scale goal-oriented adaptive modeling of
random heterogeneous materials
Albert Romkes
a
a,*
, J. Tinsley Oden b, Kumar Vemaganti
c
Department of Mechanical Engineering, University of Kansas, 1530 W. 15th St., Lawrence, KS 66045, USA
b
Institute for Computational Engineering and Sciences, The University of Texas at Austin,
1 University Station C0200, Austin, TX 78712, USA
c
Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati,
P.O. Box 210072, Cincinnati, OH 45221-0072, USA
Received 10 December 2004; received in revised form 3 March 2005
Abstract
This paper addresses the general problem of modeling local features of the response of highly heterogeneous elastic
materials with random distributions of the material constituents. The theory and methodologies of goal-oriented adaptive
modeling of heterogeneous materials [Oden, J.T., Vemaganti, K., 2000a. Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part I: Error estimates and adaptive algorithms. J. Comp. Phys. 164,
22–47; Oden, J.T., Vemaganti, K., 2000b. Adaptive modeling of composite structures: modeling error estimation. Int. J.
Comput. Civil Str. Eng. 1, 1–16; Vemaganti, K., Oden, J.T., 2001. Estimation of local modeling error and goal-oriented
adaptive modeling of heterogeneous materials. Part II: A computational environment for adaptive modeling of heterogeneous elastic solids. Comput. Methods Appl. Mech. Eng. 190, 6089–6124; Romkes, A., Vemaganti, K., Oden, J.T., 2004.
The extension of the GOALS algorithm to the analysis of elastostatics problems of random heterogeneous materials. ICES
Report 04-45, The University of Texas at Austin] are extended to incorporate uncertainty in the material data by using
classical Monte Carlo methods for calculating local quantities of interest. Techniques for estimating modeling error are
extended to cases in which the material data are random variables. Several numerical examples involving two-phase composites with random material properties are given.
2005 Elsevier Ltd. All rights reserved.
Keywords: Random heterogeneous materials; Goal-oriented adaptive modeling; Error estimation
1. Introduction
We address here a fundamental problem in the
mechanics of random heterogeneous media: the
*
Corresponding author. Tel.: +1 785 864 4397; fax: +1 785 864
5254.
E-mail address: romkes@ku.edu (A. Romkes).
analysis of local features (interfacial stresses, displacements, etc.) of the response of heterogeneous
solid bodies subjected to applied loads and displacements. The class of problems we have in mind consists of the micro-scale mechanics of multiphase
composites, constructed of materials with properties
characterized by random variables. Conventional
approaches to the analysis of heterogeneous media
0167-6636/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mechmat.2005.06.028
860
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
include the use of methods of homogenization, generally based on the assumption that the microstructure of the media is periodic, with known period e
(e.g. Bensoussan et al., 1978; Sanchez-Palencia,
1980; Jikov et al., 1994) or on techniques for determining effective properties of representative volume
elements (RVEs), statistically representative of the
microscale make up of the media (e.g. Christensen,
1979). A comprehensive account of techniques for
determining effective properties of random heterogeneous materials is given in the treatise of Torquato (2002).
Obviously, the standard averaging approaches
that give effective properties of heterogeneous materials cannot model local micromechanical effects
that are often critical factors in determining the
behavior and service life of structural components.
Moreover, the assumption of periodicity in the
microstructure is rarely valid for most materials.
Nevertheless, the use of effective properties to study
macromechanical effects has, until very recently,
been the principal approach employed to analyze
heterogeneous media, and has been a subject that
dominated the literature in the mechanics of materials for many decades. Zohdi and Wriggers (1999)
have proposed an alternative approach in which
the full micro-mechanical problem is solved on
a set of decoupled, computationally tractable,
subdomains.
A systematic approach toward analyzing local
effects in multiphase elastic heterogeneous materials
by using a multi-scale modeling method is embodied
in the GOALS algorithm, presented in Oden and
Vemaganti (2000a), Vemaganti and Oden (2001),
Romkes and Oden (2004). The idea is to identify
specific quantities of interest Q(Æ) related to the
material response, the goals of the analysis, to perform an initial analysis using effective properties,
and to sequentially improve the model of the
so-called material body by adding only enough complexity to control the estimated error in the target
quantities of interest. Since these quantities Q(Æ)
can characterize local features, such as interfacial
stresses or strains, the GOALS approach can yield
accurate results on micromechanical details that
are impossible to quantify or even detect by classical
methods. Mathematically, the quantities Q(Æ) are
defined by values of functionals on the solutions
of models of physical systems. Extensions of the theory on modeling error estimates, the foundation of
GOALS-type algorithms, to nonlinear problems
are presented in Oden and Prudhomme (2002).
In the present paper, the theory and methodology described in Oden and Vemaganti (2000a),
Vemaganti and Oden (2001), Oden and Prudhomme
(2002) is extended to the class of problems in elastostatics of heterogeneous materials, where the material properties are random and given as functions
of random variables with known probability distribution density functions. The model problem, notations, and variational formulation are introduced in
Section 2. A reinterpretation of the goal of the analysis within a stochastic setting is given in Section 3.
The initial surrogate model and corresponding error
estimates in the stochastic GOALS-algorithm are
defined in Sections 4 and 5, respectively. An adaptive modeling process for locally enhancing the surrogate model is described in Section 6. In Section 7,
a brief overview of the extended version of the
GOALS-algorithm is given and numerical examples
involving applications to two-dimensional elastostatics problems are presented in Section 8. Lastly,
concluding remarks are collected in Section 9.
2. Model problem and notations
A model problem in the class of problems of
interest here involves a material body, occupying
an open and bounded domain D Rd ; d ¼ 1; 2; 3
(see Fig. 1), with boundary oD ¼ Cu [ Ct ,
Cu \ Ct = ;, meas(Ct) > 0, meas(Cu) > 0, Cu and Ct
being portions of oD on which displacements and
tractions are to be specified, respectively. The body
is in static equilibrium under the action of deterministic applied forces f 2 L2(D)d, a surface traction
t 2 L2(Ct)d1, and a prescribed displacement
U 2 L2(Cu)d on Cu. The body is assumed to be composed of a multi-phase, composite, elastic material
with highly oscillatory material properties. The
geometrical features and material properties of its
constituents are functions of a set of N random variables {xi 2 Xi}, collected in the random vector
x = {x1, x2, . . . , xN}. The collection of all possible
realizations of x is denoted X, i.e.
X ¼ X1 X2 XN .
By defining F as the r-algebra of all possible
subsets of X, the following probability measure
P : F ! ½0; 1 is introduced:
Z
Z
P ðAÞ ¼
dP ¼
pðxÞ dx; A X;
A
A
where p : X ! ½0; 1 represents the probability distribution density function. Let u = u(x, x) denote
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
861
Fig. 1. The model problem.
the random displacement vector field defined on
D · X and let $xu denote the spatial gradient. Then
the Cauchy stress tensor r(x, x) satisfies r(x, x) =
E(x, x)$xu(x, x), where E(x, x) is the fourth order
tensor of elasticities with components Eijkl(x, x) 2
L1(D, L2(X))d · d. We assume that the standard
symmetry and ellipticity conditions hold for a.e.
x 2 D and a.s. x 2 X:
The equivalent variational formulation of the model
problem (1) is then governed by
Eijkl ðx; xÞ ¼ Ejikl ðx; xÞ ¼ Eijlk ðx; xÞ ¼ Eklij ðx; xÞ.
a0 nij nij 6 Eijkl ðx; xÞnij nkl 6 a1 nij nij ; a1 P a0 > 0.
L : V ! R; B : V V ! R;
Z Z
Z Z
LðvÞ ¼
f : v dx dP þ
t : v ds dP ;
Find uðx; xÞ 2 fUg þ V such that :
Bðu; vÞ ¼ LðvÞ
For every x 2 X; find uðx; xÞ such that :
rx ðEðx; xÞrx uðx; xÞÞ ¼ fðxÞ; in X;
Erx u nðs; xÞ ¼ tðsÞ; on Ct ;
uðs; xÞ ¼ UðsÞ; on Cu ;
ð1Þ
ð3Þ
where the bilinear form Bð; Þ and linear form LðÞ
are defined as follows:
X
With these notations and conventions in force, the
linear elastostatics problem is formulated in terms
of the following system of stochastic PDEs:
8v 2 V ;
Bðw; vÞ ¼
X
D
Z Z
X
Ct
ð4Þ
Eðx; xÞrx w : rx v dx dP .
D
It is easily confirmed that the bilinear form Bð; Þ
defines an inner product on the space V · V and
consequently induces the following norm on V:
2
kvkV ¼ Bðv; vÞ.
ð5Þ
where and n and s, respectively denote the unit normal and position vector on oD. The assumptions
that the data on the right-hand-side of (1) are deterministic is made only for simplicity. The approach
we develop is easily generalized to handle random
data of this kind.
Lemma 1. There exists a unique solution u(x, x) 2
U + V to the stochastic boundary value problem (3).
The space of test functions V is given by
V ¼ v 2 H 1 ðD; L2 ðXÞÞ : vðx; xÞ ¼ 0 8x 2 Cu ; x 2 X ;
3. Goals of the analysis or quantities of interest
ð2Þ
where the Hilbert space H1(D,L2(X)) is defined as
follows:
H 1 ðD; L2 ðXÞÞ
Z Z
½rx v : rx v þ v : vdxdP < 1 .
¼ vðx; xÞ :
X
D
The proof of this lemma is straightforward and
can be found in Romkes et al. (2004).
A main attribute of goal-oriented adaptive
modeling, is the specification of quantities of interest
representing the goals of the analysis (Oden and
Vemaganti, 2000a,b; Vemaganti and Oden, 2001).
These quantities are characterized by functionals,
Q : V ! R, of the material response function
u(x, x). Let q(u), q : V ! L2 ðXÞ, denote a fine-scale
feature of the solution defined over a subdomain
of D. Examples of QðuÞ then include:
862
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
• The expected value or mean of a feature q(u):
Z
QðuÞ ¼ E½qðuÞ ¼
qðuÞ dP ;
X
• Variance of a feature q(u):
Z
QðuÞ ¼ Var½qðuÞ ¼
ðqðuÞ E½qðuÞÞ2 dP ;
X
• Covariance between features q1(u) and q2(u):
QðuÞ ¼ Cov½q1 ðuÞ; q2 ðuÞ
Z
¼
ðq1 ðuÞ E½q1 ðuÞÞðq2 ðuÞ E½q2 ðuÞÞ dP ;
ing the mean E[E(x, x)] of the elasticity tensor and
applying averaging techniques such as classical
homogenization (e.g. Sanchez-Palencia, 1980), in
the case the material has periodic microstructure,
or averaged Hashin–Shtrikman bounds (see Hashin,
1983). The averaging process yields a tensor E0 with
constant deterministic coefficients in D. By replacing
E(x, x) with E0, the surrogate problem can be
posed,
B0 ðu0 ; qÞ ¼ LðqÞ
B0 ðv; w0 Þ ¼ Q0 ðu0 ; vÞ
8q 2 V ;
8v 2 V ;
primal problem
dual problem
X
ð8Þ
• ith Order moment of a feature q(u):
Z
i
qðuÞi dP .
QðuÞ ¼ E½qðuÞ ¼
where
B0 ðw; vÞ ¼
Z Z
X
X
These quantities can, in turn, be used in estimating
the probability of occurrence events, e.g.
E0 rx w : rx v dx dP .
D
The proof of the following assertion can be found in
Romkes et al. (2004).
Lemma 2. The surrogate solution u0 is deterministic
and the unique solution to:
A ¼ fx 2 X : qðuðxÞÞ P a; a > 0g X;
1
i
P ðAÞ 6 i E½qðuÞ .
a
Find u0 2 U þ W :
b0 ðu0 ; qÞ ¼ lðqÞ 8q 2 W ;
Solving (3) with the quantity of interest QðuÞ is
equivalent to solving the following constrained minimization problem (Oden and Prudhomme, 2002):
Find u 2 V :
QðuÞ ¼ inf QðvÞ;
v2M
where
ð9Þ
where
W ¼ fv 2 H 1 ðDÞ : vCu ¼ 0g;
Z
Z
lðvÞ ¼
f : v dx þ
t : v ds;
D
Ct
Z
b0 ðz; vÞ ¼
E0 rx z : rx v dx.
D
M ¼ fv 2 V : Bðv; qÞ ¼ LðqÞ 8v 2 V g.
The solution of this saddle point problem is governed by the following pair of VBVPs:
Bðu; qÞ ¼ LðqÞ
Bðv; wÞ ¼ Q0 ðu; vÞ
8q 2 V ;
8v 2 V ;
primal problem
dual problem
ð6Þ
where the Gâteaux derivative of QðuÞ is defined as
follows:
1
0
Q ðu; vÞ ¼ lim h ½Qðu þ hvÞ QðuÞ;
h!0
u; v 2 V .
ð7Þ
Remark 3. If the quantity of interest QðÞ involves
statistical properties of u or its spatial derivatives
(i.e. displacements and strains), then the surrogate
dual solution w0 is also deterministic (see Romkes
et al., 2004).
5. Estimation of the modeling error
The errors incurred by using the surrogate model
are the random fields,
e0 ðx; xÞ ¼ uðx; xÞ u0 ðxÞ;
e0 ðx; xÞ ¼ wðx; xÞ w0 ðx; xÞ.
ð10Þ
4. Surrogate model
The residual functionals characterizing the accuracy
of the surrogate solutions are:
In our Random GOALS algorithm, the analysis
of (6) starts by introducing an approximate or surrogate deterministic model, established by employ-
Rðu0 ; vÞ ¼ LðvÞ Bðu0 ; vÞ;
0
v2V;
Rðu0 ; w0 ; zÞ ¼ Q ðu0 ; zÞ Bðz; w0 Þ;
z2V.
ð11Þ
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
By substituting (8) into (11), the following explicit
expressions for the residual functionals can be
obtained:
R R
Rðu0 ; vÞ ¼ X D EI0 rx u0 rx v dx dP ;
R R
Rðu0 ; w0 ; zÞ ¼ X D EI0 rx z rx w0 dx dP .
ð12Þ
Here, I0 ðx; xÞ is the deviation tensor:
I0 ðx; xÞ ¼ I E1 ðx; xÞE0 .
8z 2 V .
We can now call upon a theorem, proved in Oden
and Prudhomme (2002), that relates the error in
the quantity of interest to the residual functionals.
Theorem 4. The error in the quantity of interest QðÞ
when evaluated at the solution of the surrogate
problem is:
1
QðuÞ Qðu0 Þ ¼ Rðu0 ; w0 Þ þ ½Rðu0 ; e0 Þ
2
þRðu0 ; w0 ; e0 Þ þ rðe0 ; e0 Þ;
ð15Þ
where the remainder r(e0, e0) involves cubic terms in
the errors e0 and e0.
Substitution of (14) into (15) and neglecting the
higher order terms, yields:
ð16Þ
computable
The first term in the RHS of this equation is computable or can be approximated by employing
numerical approximation techniques such as Monte
Carlo discretizations. The error functions in the
remaining term are unknown and, therefore, have
to be estimated. Following Oden and Vemaganti
(2000a,b), Vemaganti and Oden (2001), we recall
the Parallelogram Law:
1
QðuÞ Qðu0 Þ ¼ Rðu0 ; w0 Þ þ kse0 þ s1 e0 k2V
4
1
2
kse0 s1 e0 kV ;
ð17Þ
4
where s 2 R n f0g.
ð18Þ
g
upp
¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
Z Z
EI0 rx su0 s1 w0 Þ : I0 rx su0 s1 w0 ÞdxdP ;
X
D
jRðsu0 s1 w0 ; u0 þ h w0 Þj
;
ku0 þ h w0 kV
Bðu0 ; w0 ÞRðu0 ; su0 s1 w0 Þ Bðu0 ; u0 ÞRðw0 ; su0 s1 w0 Þ
h ¼
.
Bðu0 ; w0 ÞRðw0 ; su0 s1 w0 Þ Bðw0 ; w0 ÞRðu0 ; su0 s1 w0 Þ
g
low ¼
ð19Þ
ð14Þ
QðuÞ Qðu0 Þ Rðu0 ; w0 Þ þBðe0 ; e0 Þ.
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
1
g
low 6 kse0 s e0 kV 6 gupp ;
where
Bðe0 ; vÞ ¼ Rðu0 ; vÞ 8v 2 V ;
Rðu0 ; w0 ; zÞ
R1
þ 0 Q00 ðu0 þ ne0 ; e0 ; zÞ dn
Lemma 5. Let u0(x) 2 W V be the solution to (9),
w0(x, x) 2 V be the solution to (8)2, and let the error
functions {e0, e0} be as defined in (10). Then the global
norms of the error functions in (17) are bounded as
follows:
ð13Þ
Conversely, by substituting the exact formulation
(6) into (11), the following set of variational problems governing the error functions e0 and e0 is
obtained:
Bðz; e0 Þ ¼
863
Proof. We begin with the equality
jBðse0 s1 e0 ; vÞj
kse0 s1 e0 kV ¼ sup
.
kvkV
v2V nf0g
By employing the linearity and symmetry properties
of Bð; Þ, substituting (14), and noting that
Rðu0 ; w0 ; vÞ ¼ Rðw0 ; vÞ (see (12)), the above expression can be rewritten as follows:
kse0 s1 e0 kV ¼ sup
v2V nf0g
jRðsu0 s1 w0 ; vÞj
.
kvkV
Applying the Schwarz inequality to the numerator
of this equation, leads to the expressions for the
upper bounds g
upp .
The lower bounds are derived by choosing
v = u0 + h±w0, h 2 R,
kse0 s1 e0 kV P
jRðsu0 s1 w0 ;u0 þ h w0 Þj
¼ jhðh Þj.
ku0 þ h w0 kV
An optimal choice for h±, determined as to establish
lower bounds with high accuracy, is obtained by minimizing the above fraction with respect to h±. h
Having upper and lower bounds to the terms
involving e0 and e0 in (17), it is straightforward to
establish lower and upper bounds on the error in
the quantity of interest.
Theorem 6. Let u0 and w0 be the solutions to
respectively (9) and (8)2. Then the error in the
quantity of interest is bounded by
glow 6 QðuÞ Qðu0 Þ 6 gupp ;
ð20Þ
864
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
where
Extensive experiments show that the upper bounds
g
upp in Lemma 5 are accurate estimates of the norms
kse0 ± s1e0kV. This suggests that the quantity
gest,upp defined by
1
1 2
2
glow ¼ Rðu0 ; w0 Þ þ ðgþ
low Þ ðgupp Þ ;
4
4
1 þ 2 1 2
gupp ¼ Rðu0 ; w0 Þ þ ðgupp Þ ðglow Þ
4
4
and where g
and
g
are
defined
in (19). h
low
upp
2
1
gest;upp ¼ Rðu0 ; w0 Þ þ 14 ðgþ
upp Þ 4 ðgupp Þ
The bounds hold for any nonzero value of the
parameter s, but its optimal value is given by
sffiffiffiffiffiffiffiffiffiffiffiffi
ke0 kV
.
ð21Þ
s¼
ke0 kV
Since ke0kV and ke0kV are unknown, the following
lemma is introduced.
Lemma 7. Let u0(x) 2 W V be the solution to (9),
w0(x, x) 2 V be the solution to (8)2, and the error
functions {e0, e0} be defined as in (10). Then the global
norms of the error functions are bounded as follows:
flow 6 ke0 kV 6 fupp ;
flow 6 ke0 kV 6 fupp ;
where
fupp ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z Z
EI0 rx u0 : I0 rx u0 dx dP ;
X
fupp ¼
ð22Þ
D
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
Z Z
D
flow ¼
jRðu0 ; u0 Þj
;
ku0 kV
flow ¼
jRðw0 ; w0 Þj
.
kw0 kV
v2V nf0g
is a convenient estimate of the homogenization
error in the quantity of interest.
6. Locally enhanced stochastic models
The homogenized surrogate model is generally
incapable of modeling small scale features that are
often of interest (e.g. crack initiation is one of many
phenomena that depend significantly on micro-scale
geometric features of the material). The Random
GOALS algorithm provides an iterative adaptive
modeling process in which the fine-scale problem
is solved on a small subdomain of the material by
applying the homogenized surrogate solution as
Dirichlet boundary data on the boundary of the
subdomain. The adaptive modeling process continues by increasing the size of the local problem until
the estimated error meets certain user-preset tolerances. A detailed description of this process of local
enhancement is given in the following two sections.
ð23Þ
Proof. In the following, only the proof for the
bounds flow and fupp are presented. The proof for
flow and fupp is done analogously. Using (14)1, we
note that:
ke0 kV ¼ sup
ð24Þ
6.1. Goal-oriented error indicators
EI0 rx w0 : I0 rx w0 dx dP ;
X
2
jBðe0 ; vÞj
jRðu0 ; vÞj
¼ sup
.
kvkV
kvkV
v2V nf0g
Applying the Schwarz inequality, leads to the
expression for the upper bound fupp. Conversely,
the lower bound follows by applying the definition
of the supremum and taking v = u0. h
The approximation s* of the optimal value of s is
now introduced as
sffiffiffiffiffiffiffiffi
fupp
.
s ¼
fupp
We begin by partitioning the domain D into M
subdomains {Hk},
!
M
[
D ¼ int
Hk ;
ð25Þ
Hi \ Hj ¼ ; 8i 6¼ j.
k¼1
Recalling (16) and using the continuity of Bð; Þ, we
obtain the following (approximate) upper bound to
the error in the quantity of interest:
QðuÞ Qðu0 Þ 6 Rðu0 ; w0 Þ þ ke0 kV ke0 kV .
Applying the upper bounds derived in Lemma 7 and
the Schwarz inequality, yields:
QðuÞ Qðu0 Þ 6 fupp kw0 kV þ fupp fupp ;
where fupp and fupp are given in (23). This expression
suggests that the restrictions of each of the terms in
the RHS to a subdomain Hk represent the contributions of this subdomain to the total error in the
quantity of interest. Thus, the following stochastic
error indicators are defined:
def
bk ¼ fk kw0 kV ;k þ fk fk ;
k ¼ 1; 2; . . . ; M;
ð26Þ
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
where
Z Z
2
fk ¼
2
fk ¼
Hk
X
Hk
Z Z
2
kw0 kV ;k ¼
Thus, the actual fine-scale material properties are
taken into account inside DL, whereas the homogenized solution u0 is applied on the boundary of the
critical domain that has no prescribed tractions. By
solving (28) and applying the extension operator:
EI0 rx u0 : I0 rx u0 dx dP ;
X
EI0 rx w0 : I0 rx w0 dx dP ;
Z Z
X
865
vL : V ðX; DL Þ ! V ;
vL ; in X DL ;
vL ðvL Þ ¼
0; in ðX DÞ n ðX DL Þ;
Erx w0 : rx w0 dx dP .
Hk
6.2. The locally defined VBVP’s
Having computed the error indicators {bk}, the
algorithm searches for the subdomain DL D, or
domain of influence, that has the highest contribution to the error in the quantity of interest. If the
quantity of interest is defined over a subdomain
DQ D, an initial guess for DL is made by taking
the union of all subdomains Hk that intersect with
DQ:
!
M
[
DL ¼ int
Hk \ DQ .
ð27Þ
ð29Þ
one arrives at the following locally enhanced stochastic solution,
def
~u ¼ u0 þ vL ð~uL u0L Þ.
ð30Þ
Remark 8. To derive an estimator for the error in
the target quantity, we note that:
QðuÞ Qð~uÞ ¼ ½QðuÞ Qðu0 Þ ½Qð~uÞ Qðu0 Þ
gest;upp ½Qð~uÞ Qðu0 Þ .
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
computable
k¼1
Next, the algorithm searches for neighbors of DL
that have a high contribution to the error in the
quantity of interest. If the error indicator for
X · DL is denoted as bL (see (26)), then DL is
updated by adding those neighboring Hk to DL for
which
bk > dTOL bL ;
where dTOL is a user-defined tolerance. A local function space is defined on X · DL,
n
o
V ðX; DL Þ ¼ vL 2 H 1 ðDL ; L2 ðXÞÞ : c0 ðvL ÞjCLu ¼ 0 ;
where c0 : H 1 ðDL ; L2 ðXÞÞ ! H 1=2 ðoDL ; L2 ðXÞÞ denotes the zeroth order trace operator on DL and
CLu ¼ oDL n ðoDL \ Ct Þ. By defining the restriction
of the homogenized solution u0jXDL as u0L, we
now seek the local solution ~
uL 2 V ðX; DL Þ to the
following local boundary value problem:
Find ~
uL 2 fu0L g þ V ðX; DL Þ such that :
BL ð~
uL ; vL Þ ¼ LL ðvL Þ
8vL 2 V ðX; DL Þ;
X
def
LL ðvL Þ ¼
X
DL
DL
f : vL dx dP þ
Z Z
t : vL ds dP .
X
Ct \oDL
def
gest;upp ¼ gest;upp ½Qð~uÞ Qðu0 Þ.
ð31Þ
The accuracy of this estimator obviously depends
on the accuracy of the estimate of the homogenization error gest,upp. Numerical experiments reveal
that gest,upp exhibits high accuracy when the quantity of interest is the mean of a local response
feature. Hence, for this quantity of interest, gest;upp
is an accurate estimator. A number of improvements of gest;upp is under study to estimate the error
of the enhanced solutions in other quantities of
interest and will be reported in a forthcoming paper.
7. The random GOALS algorithm
The GOALS analysis can now be summarized as
follows (see Fig. 2):
ð28Þ
where the localized bilinear and linear forms are:
Z Z
def
BL ðwL ; vL Þ ¼
Erx wL : rx vL dx dP ;
Z Z
The alternative estimator is therefore defined as
follows:
Step 1: The domain D is partitioned into subdomains {Hk} according (25). Also, the error
tolerance for the quantities of interest,
aTOL, and the error indicators, dTOL, are
specified.
Step 2: The homogenized and deterministic tensor
E0 is derived by applying an averaging
technique to the mean of the tensor E (see
866
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
Fig. 2. The GOALS-algorithm.
Section 4). We obtain the mean hEi using a
MC approximation of N random samples
of {xk}:
Z
N
1 X
E dP Eðxk Þpðxk Þ;
ð32Þ
hEi ¼
N k¼1
X
where p(Æ) is the probability distribution
density function.
Step 3: The surrogate primal problem (9), that
employs the elasticity tensor E0, is deterministic and can be solved for u0 with
relatively low computational cost. As mentioned in Remark 3, the surrogate dual
problem can be deterministic depending
on the characteristics of the quantity of
interest QðÞ. In the case it is not deterministic, w0(x, x) is computed by using a MC
discretization of X.
Step 4: The modeling error in the quantity of interest is estimated by computing the estimates,
given in (24). Since these estimates involve
integrations over X, MC approximations,
analogous to (32), are used.
Step 5: If the error estimates are less than aTOL Qðu0 Þ, then the analysis stops and provides
the analyst with Qðu0 Þ.
Step 6: The initial guess of the domain of influence
DL is determined by taking the union of all
subdomains that intersect with the subdomain DQ upon which the quantity of
interest is defined.
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
Step 7: The MC approximation of the error indicators {bk}, given in (26) and bL are
computed.
Step 8: DL is updated by adding those neighbors of
DL for which the following tolerance is
exceeded:
bk > dTOL bL ;
Step 9: The local problem (28) is solved for ~
uL ðx; xÞ
by using a MC discretization {xk} of X and
subsequently solving the integrands of (28)
for every xk, yielding a set of f~
uL ðx; xk Þg.
The MC approximation of the locally
enhanced solution f~
uðx; xk Þg is then constructed by using (30).
Step 10: The modeling error QðuÞ Qð~
uÞ is estimated
by computing the MC approximation of the
estimates given in (31). If the estimated
errors do not exceed aTOL Qð~
uÞ, then the
analysis stops. Else, the iteration process
continues by returning to Step 8.
Remark 9. Errors due to MC approximation of the
stochastic integrals are ignored here: it is assumed
that the number of MC samples is sufficiently large
that these errors are negligible.
8. Numerical experiments and examples
To demonstrate the methods developed earlier
and to test the effectiveness of the error estimation
867
and adaptation modeling techniques, we consider
here numerical examples in which the randomness
exists only in the Youngs moduli and Poisson ratio
of the material constituents. If the stochastic space
X is one-dimensional, standard Gaussian quadrature is used to discretize X. Otherwise, an overkill
Monte Carlo sampling of X is made. To solve the
deterministic surrogate problem or the locally
enhanced problems at samples in X, the finiteelement code ProPHLEX (Altair Engineering,
2000) is used.
8.1. One random variable
We consider an aluminum–boron composite
material in which the matrix material and cylindrical inclusions are both isotropic. The volume fraction of the boron inclusions is 0.3 and the
inclusions are randomly dispersed in the aluminum
matrix material as shown in Fig. 3. The material is
loaded by tractions t1 and t2 and has prescribed
homogeneous displacements at y = 0. The Youngs
modulus and Poisson ratio of the aluminum are
deterministic and given by Em = 69 GPa and
mm = 0.3. The material properties of the inclusions
are random and governed by one random variable
x 2 X = (0, 1), i.e.
Eincl ðxÞ ¼ Eincl;min þ ðEincl;max Eincl;min Þx;
mincl ðxÞ ¼ mincl;min þ ðmincl;max mincl;min Þx;
Fig. 3. Aluminum–boron structural component under applied stresses t1 and t2 and homogeneous prescribed displacements at y = 0.
868
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
where
Eincl;min ¼ 420 GPa;
Eincl;max ¼ 500 GPa;
mincl;min ¼ 0:17;
mincl;max ¼ 0:22.
We assume that the probability distribution density
function p : X ! ½0; 1 is given by the following
truncated Gaussian distribution:
2
xx
pffiffiffi0
A
pðxÞ ¼ pffiffiffiffiffiffi e r 2p ;
r 2p
ð33Þ
2
;
A¼
0ffi
pffiffiffi0ffi erf x
pffiffiffi
erf 1x
r 2p
r 2p
where x0 = 0.5 and r = 0.7. In Fig. 4, a subdomain
partition of the composite material into 34 subdomains {Hk} is shown. The goal of the analysis
Z Z
1
QðuÞ ¼
eyy ðx; xÞpðxÞ dx dx
jDQ j X DQ
Fig. 5. Strain field e0yy for the deterministic surrogate problem of
the aluminum–boron composite.
is then to determine the statistical average of the
average strain eyy in a small circle DQ in subdomain
H13, with a radius of 0.028 m (see Fig. 3). Thus,
since the microstructure is nonperiodic, the homogenized deterministic properties for the initial surrogate model are taken to be the average of the
Hashin–Shtrikman upper and lower bounds (see
Hashin, 1983). The corresponding surrogate solutions u0(x) and w0(x) are computed and the resulting
strain field e0yy ðxÞ is shown in Fig. 5.
To assess the accuracy of the error estimator
gest,upp (see (24)), the fine-scale solutions u(x, x)
and w(x, x) are computed and the statistical average
of the corresponding strain field eyy(x, x) is shown
in Fig. 6, where the area of interest has been magnified. In Table 1, the relative error in the quantity of
Fig. 6. Ensemble average of the strain field eyy for the fine-scale
problem of the aluminum–boron composite.
Fig. 4. Subdomain decomposition of the aluminum–boron
composite.
Table 1
Relative homogenization error and error estimates for the
aluminum–boron composite
gest;upp
fupp
QðuÞ Qðu0 Þ
QðuÞ
QðuÞ Qðu0 Þ
ke0 kV
0.557
0.962
1.240
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
interest and the effectivity indices of the estimates
gest,upp and fupp (see (23)) are given. The relative
error appears to be large, but since the effectivity
indices of the estimates are close to unity, both estimates succeed in accurately predicting the error in
the quantity of interest and in the global norm kÆkV
of the error.
The analysis subsequently proceeds by performing six steps of local enhancements, according the
Random GOALS algorithm given in Section 7.
For steps 1, 3, and 6, the statistical averages of the
corresponding strain fields ~eyy ¼ o~
uy ðx; xÞ=oy are
shown in Figs. 7–9, respectively. The boundaries
of the domains of influence oDL are highlighted
red. In Table 2, the relative errors and the effectivity
indices of the error estimator gest;upp (see (31)) are
shown for all the six steps. Initially, the error
decreases rapidly with a factor of about 0.5, but
then appears to reach stages where it stagnates.
The first stage consists of steps 2 to 3, and 4, and
the second stage of steps 5–6. The accuracy of the
error estimator is maintained with effectivity indices
between 80% and 85%.
869
Fig. 8. Strain field ~eyy for the locally enhanced surrogate problem
of the aluminum–boron composite—step 3.
8.2. Multiple random variables
We next consider a structure composed of a carbon–epoxy composite material, shown in Fig. 10,
with elastic isotropic constituents and circular
Fig. 9. Strain field ~eyy for the locally enhanced surrogate problem
of the aluminum–boron composite—step 6.
Table 2
Relative error and error estimates for the locally enhanced
solutions of the aluminum–boron composite
Step no.
Fig. 7. Strain field ~eyy for the locally enhanced surrogate problem
of the aluminum–boron composite—step 1.
1
2
3
4
5
6
gest;upp
QðuÞ Qð~uÞ
QðuÞ
QðuÞ Qð~uÞ
0.283
0.140
0.140
0.138
0.111
0.109
0.852
0.852
0.852
0.850
0.814
0.814
870
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
Fig. 10. Carbon–epoxy composite under applied stress t and
homogeneous prescribed displacements at y = 0.
carbon inclusions that are periodically distributed in
the epoxy matrix material. The material is loaded by
a traction t and has prescribed homogeneous displacements at y = 0. Both constituents have random
Youngs moduli and Poisson ratios, which are prescribed by four random variables {x1, x2, x3, x4} =
x. It is assumed that the random variables are independent and each variable xi 2 [0, 1]. Thus,
X ¼ ½0; 1 ½0; 1 ½0; 1 ½0; 1.
The material properties are defined in terms of the
random variables as follows:
Eincl ðxÞ ¼ Eincl;min þ ðEincl;max Eincl;min Þx1 ;
mincl ðxÞ ¼ mincl;min þ ðmincl;max mincl;min Þx2 ;
Em ðxÞ ¼ Em;min þ ðEm;max Em;min Þx3 ;
mm ðxÞ ¼ mm;min þ ðmm;max mm;min Þx4 ;
where
Eincl;min ¼ 110 GPa;
Eincl;max ¼ 130 GPa;
mincl;min ¼ 0:28;
mincl;max ¼ 0:32
and
Em;min ¼ 5 GPa;
mm;min ¼ 0:325;
Em;max ¼ 7 GPa;
mm;max ¼ 0:365.
Each variable xi has a truncated Gaussian distribution density function pi(xi), as defined in (33), with
x0 = r = 0.5. Since the random variables are assumed to be independent, the overall probability
distribution density function p : X ! ½0; 1 is obtained by applying:
Fig. 11. Subdomain decomposition of the carbon–epoxy
composite.
pðxÞ ¼
4
Y
pi ðxi Þ.
ð34Þ
i¼1
In Fig. 11, a subdomain partition of the composite
material into 16 subdomains {Hk} is shown. The
goal of the analysis is again to determine the statistical average of the average strain eyy in a small
circle DQ in subdomain H10, with a radius of
0.028 mm.
The homogenized deterministic properties for the
initial surrogate model are taken to be the homogenized properties (see Sanchez-Palencia, 1980) of the
ensemble average of the material properties on the
periodic unit cell (i.e. any Hk). The corresponding
surrogate solutions u0(x) and w0(x) are computed
and the resulting strain field e0yy ðxÞ is shown in
Fig. 12. As in Section 8.1, the results using the error
estimator gest,upp (see (24)) are considered. To assess
the accuracy of this error estimator, the fine-scale
solutions u(x, x) and w(x, x) are computed and the
statistical average of the corresponding strain field
eyy(x, x) is shown in Fig. 13. The relative error in
the quantity of interest and the effectivity index of
the estimator are shown in Table 3. Again, the relative error is large, but the effectivity index of the estimate is very close to unity.
One step of local enhancement is performed (see
Section 7) and the resulting strain field ~eyy ¼
o~uy ðx; xÞ=oy is shown in Fig. 14. The relative error
and the effectivity index of the error estimator
A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
871
Table 3
Relative homogenization error and error estimate for the carbon–
epoxy composite
gest;upp
QðuÞ Qðu0 Þ
QðuÞ
Qðu0 Þ
QðuÞ
0.746
0.993
Fig. 12. Strain field e0yy for the deterministic surrogate problem of
the carbon–epoxy composite.
Fig. 14. Strain field ~eyy for the locally enhanced surrogate
problem of the carbon–epoxy composite.
Table 4
Relative error and error estimate for the locally enhanced
solution of the carbon–epoxy composite
gest;upp
QðuÞ Qð~uÞ
QðuÞ
QðuÞ Qð~uÞ
0.472
0.968
9. Concluding remarks
Fig. 13. Ensemble average of the strain field eyy for the fine-scale
problem of the carbon–epoxy composite.
gest;upp (see (31)) are shown in Table 4. The error has
significantly decreased and the error estimate proves
to be very accurate.
The goal-oriented adaptive modeling technique
proposed here is an extension of the GOALS algorithm (Oden and Vemaganti, 2000a,b; Vemaganti
and Oden, 2001) for the analysis of deterministic
heterogeneous materials. In the Random GOALS
algorithm, the unsolvable fine-scale, or base, problem is replaced by an initial, solvable, surrogate
model that uses deterministic, averaged properties
(e.g. Bensoussan et al., 1978; Sanchez-Palencia,
1980; Hashin, 1983). The accuracy of the surrogate
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A. Romkes et al. / Mechanics of Materials 38 (2006) 859–872
solution is quantified in terms of a posteriori estimates of the modeling error in a user-specified
quantity of interest. Adaptive control of the modeling error is then realized by a systematic process of
local enhancement. In this process, the stochastic
fine-scale problem is solved on a subregion, or
domain of influence, and the initial surrogate solution is applied as a prescribed displacement field
on the boundary. The domain of influence is
increased stepwise by adding neighboring subregions with high error contributions, until the estimate of the error meets preset error tolerances.
The numerical examples presented in Section 8
indicate that the developed a posteriori error estimates are quite acceptable for these problems, the
effectivity indices being around 0.815–1.00. The
adaptive process does control the modeling error
and quite reliable results can be obtained for very
complex non-periodic problems.
The adaptive procedure employed here can likely
be improved. Since the error estimators driving the
selections of subdomains to be added to the surrogate models is based on a global representation of
the residuals, some pollution of the local errors by
remote residuals can occur. There are a number of
ways that the pollution could be reduced, but these
are not explored in the present work. In any case,
the fact that the local error estimates in quantities
of interest are of good precision, makes it possible
to always judge if the error inherent at any step of
the process is acceptable or if further adaptation is
needed.
Acknowledgement
The support of this work, under ONR Grant No.
N00014-99-1-0124, is gratefully acknowledged.
References
Altair Engineering, Inc., 2000. ProPHLEX User Manual Version
3.0. Austin, Texas.
Bensoussan, A., Lions, J.L., Papanicolaou, G., 1978. Asymptotic
analysis for periodic structuresStudies in Mathematics and its
Applications, Vol. 5. North-Holland, Amsterdam.
Christensen, R., 1979. An Introduction to the Mechanics of
Composite Materials. John Wiley and Sons, New York.
Hashin, Z., 1983. Analysis of composite materials, a survey.
J. Appl. Mech. 50, 481–505.
Jikov, V.V., Kozlov, S.M., Oleinik, O.A., 1994. Homogenization
of Differential Operators and Integral Functionals. SpringerVerlag, Berlin, Heidelberg.
Oden, J.T., Prudhomme, S., 2002. Estimation of modeling error
in computational mechanics. J. Comp. Phys. 182, 496–515.
Oden, J.T., Vemaganti, K., 2000a. Estimation of local modeling
error and goal-oriented adaptive modeling of heterogeneous
materials. Part I: Error estimates and adaptive algorithms.
J. Comp. Phys. 164, 22–47.
Oden, J.T., Vemaganti, K.S., 2000b. Adaptive modeling of
composite structures: modeling error estimation. Int. J. Comput. Civil Str. Eng. 1, 1–16.
Romkes, A., Oden, J.T., 2004. Adaptive modeling of wave
propagation in heterogeneous elastic solids. Comput. Methods Appl. Mech. Eng. 193, 539–559.
Romkes, A., Vemaganti, K., Oden, J.T., 2004. The extension of
the GOALS algorithm to the analysis of elastostatics problems of random heterogeneous materials. ICES Report 04-45,
The University of Texas at Austin.
Sanchez-Palencia, E., 1980. Nonhomogeneous media and vibration theoryLecture Notes in Physics, Vol. 127. SpringerVerlag, Berlin Heidelberg New York.
Torquato, S., 2002. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer-Verlag,
New York.
Vemaganti, K., Oden, J.T., 2001. Estimation of local modeling
error and goal-oriented adaptive modeling of heterogeneous
materials. Part II: A computational environment for adaptive
modeling of heterogeneous elastic solids. Comput. Methods
Appl. Mech. Eng. 190, 6089–6124.
Zohdi, T.I., Wriggers, P., 1999. A domain decomposition method
for bodies with heterogeneous microstructure based on
material regularization. Int. J. Solids Struct. 36, 2507–2525.