a nonintrusive stochastic multiscale solver

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2011; 88:862–879Published online 18 April 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3201

A nonintrusive stochastic multiscale solver

Jacob Fish∗,† and Wei Wu

Multiscale Science and Engineering Center, Rensselaer Polytechnic Institute, NY, U.S.A.

SUMMARY

In this paper, we describe a practical nonintrusive multiscale solver that permits consideration of uncertain-ties in heterogeneous materials without exhausting the available computational resources. The computa-tional complexity of analyzing heterogeneous material systems is governed by the physical and probabilityspaces at multiple scales. To deal with these large spaces, we employ reduced order homogenizationapproach in combination with the Karhunen–Loeve expansion and stochastic collocation method based onsparse grid. The resulting nonintrusive multiscale solver, which is aimed at providing practical solutionsfor complex multiscale stochastic problems, has been verified against the Latin Hypercube Monte–Carlomethod. Copyright � 2011 John Wiley & Sons, Ltd.

Received 3 August 2010; Revised 28 December 2010; Accepted 27 February 2011

KEY WORDS: stochastic collocation; model reduction; multiscale homogenization; compositeuncertainties

1. INTRODUCTION

The premise of multiscale computations is that at a finer scale constitutive equations are betterunderstood than at the coarse scale and consequently requiring fewer parameters and/or experimentsto characterize material behavior. The enormous gains that can be accrued by this approach havebeen reported in numerous articles (see for instance [1–6]). Multiscale computations have beenidentified (see page 14 in [7]) as one of the areas critical for future nanotechnology advances. TheFY2004 $3.7-billion-dollar National Nanotechnology Bill (page 14 in [7]) states that: ‘approachesthat integrate more than one such technique (. . .molecular simulations, continuum-based models,etc.) will play an important role in this effort.’ Yet, despite of rapidly increasing computer power,engineering designs that resolve spatial and temporal scales are very rare primarily due to theirlack of predictability. There are two major reasons for this state of affairs:

(1) inability to resolve multiple spatial and temporal scales and(2) increased uncertainty at the finer scales.

The first barrier is purely computational. Historically, this challenge was addressed by the useof semi-analytical approximation methods, such as rule of mixtures, effective medium and self-consistent models [8–12]. In the past decade, computational homogenization methods based onthe mathematical homogenization theory [13–18] became increasingly popular but so far had littleor no impact on practitioners due to enormous computational complexity involved. More recentlymesomechanical or reduced order models, which introduce an intermediate mesomechanical scaleschematically depicted in Figure 1, emerged as leading candidates to deal with the so-called tyranny

∗Correspondence to: Jacob Fish, Multiscale Science and Engineering Center, Rensselaer Polytechnic Institute,NY, U.S.A.

†E-mail: [email protected]

Copyright � 2011 John Wiley & Sons, Ltd.

A NONINTRUSIVE STOCHASTIC MULTISCALE SOLVER 863

Figure 1. Linking micromechnical and macromechnical problems throughmesomechnical (reduced order) model.

Figure 2. Reduced precision due to increase in uncertainty and/or complexity.

of scales (or curse of scale) due to their ability to resolve coarse-scale material behavior withoutexhausting computational resources.

Methods belonging to this category are the Voronoi cell method [19], the spectral method [20],the network approximation method [21], the Fast Fourier transforms [22, 23], the transformationfield analysis (TFA) [24, 25] and the reduced order homogenization (ROH) method [26–29].

The second major barrier for utilization of multiscale approaches in practice is increasinguncertainty/complexity introduced by finer scales, which makes deterministic predictions oftenmeaningless (see Figure 2). Therefore, the use of multiscale methods has to be carefully weightedon the case-by-case basis. For example, in case of metal matrix composites with almost periodicarrangement of fibers, introducing finer scales might be advantageous since the bulk material typi-cally does not follow normality rules and developing a phenomenological coarse scale constitutivemodel might be challenging at best. The behavior of each phase is well understood and obtainingthe overall response of the material from its fine scale constituents can be obtained using homog-enization. On the other hand, in brittle ceramic composites (CMC), the microcracks are oftenrandomly distributed and characterization of their interface properties is difficult. In this case, theuse of multiscale approach may not be desirable.

Owing to the above reasons, incorporation of stochastic processes into multiscale modelingwith accurate assessment of uncertainty propagation across scales is highly challenging. Thus,the primary objective of the present manuscript is to explore the feasibility of applying the bestcombination to practical problems of interest rather than to develop a new multiscale or stochasticapproach. Our choice of the ‘best of the two worlds’ is affected by the ability of integratingthe existing stochastic and multiscale capabilities. For the multiscale simulation engine we selectthe ROH method [26–29] due to its computational efficiency. For uncertainty quantification, we

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2011; 88:862–879DOI: 10.1002/nme

864 J. FISH AND W. WU

choose a non-intrusive approach based on stochastic collocation originally proposed by Mathelinand Hussaini [30], which when combined with the sparse grid approach [31, 32] and the Karhunen–Loeve (KL) expansion has been shown [33, 34] to outperform other non-intrusive methods, such asnon-intrusive polynomial chaos [35, 36], Monte–Carlo simulation and its improved version basedon the Latin hypercube sampling [37]. However, if software framework considerations were notan issue, various derivatives of spectral stochastic finite element method [38] may have offeredcomputational advantages.

In this paper we focus on uncertainty in the fine-scale constitutive model denoted by L(x,�),where x∈� and �⊂Rd is physical problem domain; �∈� where � indicates the set of possibleconstitutive law parameters, such as material elastic modulus or ultimate strength. The study willfocus on the following two cases:

(1) Low-dimensional probability space, i.e. the stochastic problem depends only on a relativelysmall number of random variables with a given joint probability distribution. This is thecase when constitutive law parameters are assumed as random variables that are invariant inphysical space.

(2) High-dimensional probability space, i.e. the input data may vary randomly in the physicaldomain � in which case the uncertainty is described in terms of random fields with a givencovariance structure. Although such random fields are properly described only by means ofan infinite number of random variables, the KL expansion [38] will be employed to describethe random fields in terms of small number of uncorrelated random variables.

Specifically, we will focus on dominant uncertainties affecting the coarse-scale quantities ofinterest (QoI), such as critical stresses and strains and maximum deformation. To quantify QoIit is necessary to quantify the dominant input uncertainties and to propagate them. This posesstochastic forward and inverse problems, of which the latter is not addressed here. The forwardproblem involves the study of how the coarse-scale QoI respond to variations in the input data.The stochastic collocation method (SCM) [39–41] in combination with the ROH approach willbe employed to solve the two-scale stochastic partial differential equation. The proposed SCM-ROH is nonintrusive multiscale method that leads to uncoupled deterministic problems and takesadvantage of precomputed database of influence functions. The latter is a consequence of the factthat elastic material properties can be often assumed to be deterministic.

This paper is organized as follows. In Section 2, a two-scale stochastic problem with randomconstitutive properties in the fine scale is defined and the ROH method is briefly outlined. Thenonintrusive stochastic framework is described in Section 3. Both the low-dimensional random-ness (random variables) and high-dimensional randomness (random fields) are considered. Thenumerical examples are given in Section 4. Finally, we conclude this paper with few remarks inSection 5.

2. ROH UNDER STOCHASTICS

2.1. Problem definition

We assume that various response fields such as displacements, strains and stresses denoted asg(x,y,g) dependent on coarse- and fine-scale spatial coordinates, x and y=x/�, respectively,related by 0<��1, as well as on a set of input-independent random variables g={�i }d

i=1 span-ning the d-dimensional support space �. In case of random fields, i.e. fine-scale constitutivelaws are assumed to be spatially varying, a stochastic model reduction scheme-based KL expan-sion is used to convert the infinite-dimensional space describing the class of spatially varyingconstitutive laws into finite-dimensional space. The finite-dimensional model expressed in termsof handful random variables represents the class of allowable constitutive laws in the fine scalethat satisfy experimental correlations. The stochastic model reduction technique is discussed inSection 3.



Since random variables may have different probability density functions f i (�i ) the joint prob-ability density function can be obtained by their product as

f (g)=d∏

i=1f i (�i ). (1)

A two-scale asymptotic expansion is employed to approximate the displacement field

ui (x,y,g)=u0i (x,g)+�u1

i (x,y,g)+�2u2i (x,y,g)+·· · . (2)

The resulting two-scale strong form of the boundary value problem is given by:

(a) Fine Scale

�ij,y j (x,y,g) = 0, x∈�, y∈�, g∈�, (3)

�ij(x,y,g) = Lijkl(x,y,g)

[εkl(x,y,g)−

I∑I

�kl(x,y,g)

], x∈�, y∈�, g∈�, (4)

εij(x,y,g) = u1(i,y j ) +u0

(i,x j )

= 1

2

[(�u1

i

�y j+

�u1j

�yi

)+(

�u0i

�x j+

�u0j

�xi

)], x∈�, y∈�, g∈�, (5)

u1i − periodic on ��. (6)

(b) Coarse Scale

�ij,x j (x,g)+ bi (x) = 0, x∈�, g∈�, (7)

�ij(x,g) ≡ 1

|�|∫

��ij(x,y,g)d�, x∈�, g∈�, (8)

u0i (x,g) = ui (x), x∈�u, (9)

�ij(x,g)n j (x) = ti (x), x∈�t , (10)

where Equation (4) describes the constitutive relation which assumes an additive decom-position of total strain εij into elastic and inelastic components, or more generally referredto as eigenstrains I �kl, where the left superscript I denotes various eigenstrain types,such as inelastic deformation, thermal change, moisture effects, etc. For simplicity, weassume that coarse-scale essential and natural boundary conditions as well as fine-scaleperiodic boundary conditions are deterministic. Furthermore, we assume the coarse- andfine-scale geometry to be deterministic. Only fine-scale material constitutive parametersare assumed to be random fields in coarse-scale domain with a given joint proba-bility distribution and covariance structure. The extreme case, where the correlationlength is equal or greater than the coarse-scale problem size, is of lowest computationalcomplexity.

2.2. ROH

In the nonintrusive framework, the solution in probability space � is decoupled into a set ofindependent deterministic problems on �×�. The two-scale deterministic problem is solved usingROH. By this approach we construct residual-free fine-scale displacement field u1

i (x,y,g) for eachrealization g∈� to ensure that the stress field in a unit cell satisfies equilibrium equations for



arbitrary eigenstrains I �0ij and eigenseparations �n

u1i (x,y,g) = uel

i (x,y,g)+∑I

I u�i (x,y,g)+u�

i (x,y,g)

= Hmicikl (y,g)εkl(x,g)+∑

I

∫�

I hmic_�ikl (y, y,g)I �0

kl(x, y,g)dy

+∫

Shmic_�

i n (y, y,g)�n(x, y,g)dy. (11)

The resulting fine-scale displacement gradient is given by

u1i,y j

(x,y,g) = Gmicijkl (y,g)εkl(x,g)+∑

I

∫�

I gmic_�ijkl (y, y,g)I �0

kl(x, y,g)dy

+∫

Sgmic_�

ijn (y, y,g)�n(x, y,g)dy, (12)

where Gmicijkl , I gmic_�

ijkl , and gmic_�ijn are influence functions [26–29] for coarse scale strain, eigenstrain,

and eigenseparation, respectively. Note that when elastic constitutive parameters are considered asrandom variables, these influence functions need to be recomputed at each realization by solvinga sequence of elastic boundary value problems.

The reduced order model is obtained by discretizing the eigenstrain and eigenseparation fields as

I �0ij(x,y,g) =

nI∑�=1

I N (�)(y)I �(�)ij (x,g),

�n(x, y,g) =m∑

�=1N (�)(y)�

(�)n (x,g),

(13)

where nI and m are the number of partitions of phases and interfaces, respectively; I �(�)ij and

�(�)n are the average eigenstrain and eigenseparation in phase partition � and interface partition �,

respectively. N (�)(y) is piecewise constant shape function defined as

N (�)(y)={

1, y∈�(�),

0, y /∈�(�),(14)

whereas N (�)(y) is a linear combination of piecewise linear finite element shape functions definedover the surface partition �. The resulting reduced order system of equations is given by

(i) Fine Scale

ε()ij (x,g)−

N∑I=1

nI∑�=1

I P (�)ijkl (y,g)I �(a)

kl (x,g)−m∑

�=1Q(�)

ijn (y,g)�(�)n (x,g) = A()

ijkl(y,g)εkl(x,g),

−N∑

I=1

nI∑�=1

I C (ϑ�)nkl (y,g)I �(�)

kl (x,g)+ t (ϑ)n (x,g)−

m∑�=1

D(ϑ�)nm (y,g)�

(�)m (x,g) = B(ϑ)

nkl (y,g)εkl(x,g),

(15)

(ii) Coarse Scale

�ij(x,g)= L ijkl(y,g)εkl(x,g)+N∑

I=1

nI∑�=1

I E (�)ijkl(y,g)I �(a)

kl (x,g)+m∑

�=1F (�)

i j n(y,g)�(�)n (x,g), (16)

where tn =G(�n) represents the traction along interface. The coefficient tensors, whichdepend on elastic properties only, are determined prior to non-linear coarse-scale analysis[26–29].



3. A NONINTRUSIVE STOCHASTIC SOLVER

In this section we describe a nonintrusive stochastic solver aimed at quantifying the influence ofvarious uncertainties in the fine scale on the coarse-scale quantities of interests. The microstructuraluncertainties include elastic constitutive parameters of phases (such as Young’s modulus, thePoisson ratio), inelastic constitutive parameters of phases and interfaces (such as damage lawparameters) and geometric parameters describing material microstructure (such as volume fractionof fiber). The above uncertainties can either vary from one unit cell to another (random fields) orbe constant throughout the coarse-scale domain (random variables) as shown in Figure 3.

From the computational complexity point of view and in the context of ROH method, we identifysix levels of computational complexity (Category I being the highest computational complexity):

(I) Fine-scale geometry as random fieldSince fine-scale geometry defines residual-free fields in the unit cell the influence func-

tions will vary in the coarse-scale domain. Thus, the coefficient tensors in Equations (5)and (6) need to be evaluated on the fly during non-linear simulation at each quadraturepoint in the coarse-scale domain.

(II) Fine-scale geometry as random variableAlthough the influence functions are constant over the coarse-scale domain, they have

to be recomputed due to randomness of the unit cell geometry. This will require recreationof the unit cell CAD and FE models.

(III) Elastic constitutive parameters of each phase as random fieldsThe influence functions depend on elastic properties and therefore need to be computed

on the fly for each coarse-scale domain’s quadrature point during non-linear coarse-scalesimulation.

(IV) Inelastic constitutive parameters of each phase/interface as random fieldsIf elastic properties and fine-scale geometry are assumed to be deterministic, then the

influence functions need to be computed only once throughout the entire stochastic analysis.The inelastic constitutive laws of fine-scale phases and interfaces are allowed to vary inthe coarse-scale domain.

(V) Elastic constitutive parameters of each phase as random variablesThese parameters affect the influence functions; however, in this case the influence

functions are the same from one coarse-scale quadrature point to another.(VI) Inelastic constitutive parameters of each phase/interface as random variables

This level presents the simplest and the most inexpensive scenario.

The general framework of the nonintrusive stochastic multiscale solver (NSMS) that can addressall the six levels of complexity is shown in Figure 4.

The NSMS consists of the following building blocks:Random Field Classifier—the stochastic partial differential equation is classified into one of the

six categories described above.

Figure 3. Random variables and fields.



Postprocessing

Preprocessing

Random Variable Decoupler

Coarse Scale Statistics Abstractor

Stochastic Model Reducer /Decoupler

Deterministic MDSStochastic Solver

(MC, SC)

Random Field Classifier

Figure 4. Block diagram of nonintrusive stochastic multiscale solver.

Random Variable Decoupler—the original dependent random variables are transformed intoindependent random variables.

Stochastic Model Reducer/Decoupler—the infinite-dimensional probability space is reduced toa finite-dimensional space of random variables using the KL expansion.

Stochastic Solver—the Monte–Carlo (MC) or stochastic collocation (SC) methods are appliedto transform stochastic partial differential equations into deterministic equations in the physicalspace that can be solved by the deterministic multiscale solver.

Coarse-scale Statistics Abstractor—it computes statistical moments in the quantities of interestand failure probabilities.

In this study, we focus on three simplest cases: levels IV–VI. We assume inelastic constitutiveuncertainties as random fields, but neglect variation of elastic parameters in the coarse-scaledomain. Furthermore, without loss of generality, we assume that different types of random fields(or variables) to be mutually independent [34, 39–42]. In the following sections, stochastic modelreducer and solver using collocation methods is outlined.

3.1. Stochastic model reduction: KL expansion

Uncertainties that vary from one material point to another in coarse-scale domain belong to theinfinite-dimensional probability space. Typically, spectral methods are employed to expand theuncertainty into infinite series, which is then truncated based on some energy-like criterion. Onesuch method is KL. It is based on the spectral expansion of covariance function of random fields.Let h(x,h) be the random field function, x be the physical spatial coordinates in coarse-scaledomain � and h be the probability spatial coordinates. Let Rhh(x1,x2) be its covariance functionof two points x1, x2 in the physical domain �. By definition, the covariance function is real,symmetric and positive definite. All eigenfunctions are mutually orthogonal and form completeset spanning random space. The KL expansion can be written as

h(x,h)= h(x)+∞∑

i=1

√i�i (x)�i (h), (17)



where h(x) is the mean of random field, and {�i (h)}∞i=1 forms a set of spatially independent randomvariables with zero mean and unit variance; �i (x) and i are the eigenfunctions and eigenvaluesof the covariance function, respectively, which follow from∫

Rhh(x1,x2)�i (x2)dx2 =i�i (x1). (18)

Among various decompositions of random fields, the KL expansion is considered to be optimal, butrequires a priori knowledge of covariance function. To truncate the KL expansion, an eigenvalueenergy-like criterion is often used to find the minimum L such that

h(x,h)∼= h(x)+L∑

i=1

√i�i (x)�i (h), s.t.

∑Li=1 i∑∞i=1 i

��, (19)

where � is the so-called energy threshold, and usually defined as 90%.

3.1.1. Numerical solution of eigenvalue problem. For a given covariance function the eigenvalueproblem (18) is solved numerically by evaluating the covariance function at M quadrature pointsin the coarse-scale domain. Thus, covariance function Rhh(x1,x2) is reduced to finite size (M × M)covariance matrix Rhh. The criterion to finding the number of modes L�M in Equation (19) ismodified appropriately to reflect the reduced space

h(x,h)∼= h(x)+L∑

i=1

√i�i (x)�i (h), s.t.

∑Li=1 i∑Mi=1 i

��. (20)

In this study, inelastic fine-scale constitutive parameters are described as random fields whereaselastic properties are assumed to have no spatial variation. In constructing reduced order stochasticmultiscale solver it is noteworthy to mention that for any realization of {�i }L

i=1 (see Figure 5) allthe coefficient tensors in (15), (16) and eigenpairs

√i�i (x) remain fixed and therefore can be

precomputed in the preprocessing stage.The eigenpairs are spatially varying in the coarse-scale domain. A deterministic multiscale solver

that employs commercial finite element code as a coarse-scale (macro) solver has to interface theseeigenpairs throughout non-linear analysis. In most commercial finite element package materialsubroutine, such as UMAT in ABAQUS, invokes point identity during iterative solution of thecoarse-scale problem. Such an identity is usually represented by ‘i th integration point on theelement with id j’. Therefore, we can store precomputed eigenpairs of the discretized covariancematrix Rhh using a table-like data structure, such as hash-table. Each key of one entry is thepoint identity and corresponding value is a set of eigenpairs of that point. During non-linear

Figure 5. Illustration of KL expansion of fine-scale constitutive parameter as random field (with meanfield subtracted) in macrodomain.



simulation, each integration point of the coarse-scale model looks up the corresponding eigenpairsto update their local fine-scale constitutive parameters using Equation (20) for each realization of{�i }L

i=1. Consequently, a stochastic multiscale problem involving random fields has been reducedto stochastic multiscale problem with finite random variables.

3.2. Stochastic collocation

Having stated stochastic multiscale problem in terms of finite random variables, which are eitherfine-scale material constitutive parameters defined a priori or those obtained from the KL expan-sion (17), we now consider stochastic collocation method that approximates probability space byinterpolation from a set of collocation points (referred to as realizations). Let g(�1

j1,�2

j2, . . . ,�d

jd) be

a deterministic solution of the two-scale problem (15)–(16) at a collocation point (�1j1,�2

j2, . . . ,�d

jd)

in the d-dimensional probability space, where jk denotes the jk th node index in the k-direction (orkth random variable). Denote the total number of grid points in k-direction as mk . The interpolatedsolution in random space is given by

g(g)=m1∑

j1=1· · ·

md∑jd=1

g(�1j1, . . . ,�d

jd )N 1j1 (�1) · N 2

j2 (�2) · · · N djd (�d ), (21)

where N kjk

is an interpolation function in the k-direction, such as Lagrange polynomials, satisfyinginterpolation property

N kjk (�k

js )={

1 if js = jk,

0 otherwise.

The quantities of interest are typically the statistical moments of g(g). The pth statistical momentdenoted by Mp can be calculated using numerical integration that takes advantage of the functionevaluation at the collocation points

Mp =m1∑

j1=1· · ·

md∑jd=1

[g(�1j1, . . . ,�d

jd )]p f (�1j1, . . . ,�d

jd )(w1j1 ·w2

j2 · · ·wdjd ), (22)

where f (�1j1, . . . ,�d

jd) is a joint probability density function; (w1

j1·w2

j2· · ·wd

jd) are weights at quadra-

ture points (�1j1, . . . ,�d

jd).

3.2.1. Equal grid. One intuitive approach to selecting collocation points is uniform equal grid.As shown in Figure 6, each direction in random space is divided into equal intervals. For the mkpoints along the k direction in [−1,1]d hypercube, the grids k are given as

�kjk =−1+ 2

mk −1( jk −1), jk =1, . . . ,mk (23)

and a grid point in multidimensions is defined by a tensor product ( 1 ⊗·· ·⊗ d ).

3.2.2. Sparse grid. To realize the enormous computational complexity of full tensor product rulein multidimensions, consider a tensor product of two random variables with 10 nodes in �1 and �2

directions, which has a complete polynomial order of up to 9. From the Pascal triangle it followsthat there are 55 monomials forming complete polynomial expansion of up to order 9. Yet, thetensor product involves summation of 100 terms, i.e. 45% of terms are wasted. The percentageof wasted monomials grows exponentially with increase in the number of random variables.For instance, in 3d , 78% of terms are wasted, in 4d , it increases to 92%, and in 5d and 6d , thenumber of wasted terms reaches 97 and 99%, respectively. This is often referred to as a curse ofdimensionality.

The basic idea of the sparse grid method, originally proposed by Smolyak [43], is to construct ahierarchal basis of one-dimensional interpolants and then to consider a tensor product of interpolantsthat only contribute to the completeness of the polynomial one wants to approximate.



Figure 6. Equal grid in 1-D case.

Starting from one-dimensional univariate condition, the hierarchical interpolation is given by

g(�i )=mi∑j=1

g(�ij )N i

j (�i ). (24)

Here i denotes the interpolation level. Similarly, for the (i −1)th interpolation level, one has

g(�i−1)=mi−1∑j=1

g(�i−1j )N i−1

j (�i−1), (25)

where mi−1 and mi denote the total number of basis nodes in levels i −1 and i , respectively. Inthe following we will consider a Clenshaw–Curtis grid that provides a nested structure of basisnodes, i.e. the basis nodes in level i −1 are a subset of those in level i . Since level i interpolantscan exactly represent g(�i−1) we have

g(�i−1)=mi∑j=1

N ij (�

i )

[mi−1∑k=1

g(�i−1k )N i−1

k (�i−1j )

]. (26)

The difference between the two subsequent levels is defined as

�i = g(�i )− g(�i−1)=mi∑j=1

g(�ij )N i

j (�i )−

mi∑j=1

N ij (�

i )

[mi−1∑k=1

g(�i−1k )N i−1

k (�i−1j )

]

=mi∑j=1

[g(�ij )− g(�i−1

j )]N ij (�

i ). (27)

Since g(�ij )= g(�i−1

j ) on mi−1 nodes of level i −1, Equation (27) can be rewritten as

�i = g(�i )− g(�i−1)=m�

i∑j=1

[g(�ij )− g(�i−1

j )]N ij (�

i ), (28)

where m�i =mi −mi−1 denotes the number of new nodes that are added to level i from level i −1.

The Smolyak algorithm constructs the sparse interpolation space in d-dimensions as follows.Let ik , k =1,2, . . . ,d be the interpolation level along the k-direction and i= (i1, i2, . . . , id ) themulti-index with |i|= i1 +·· ·+id . The Smolyak algorithm builds the interpolation function in



multi-dimensions by adding a combination of one-dimensional functions of order ik subject to theconstraint q −d +1�|i|�q , which yields

g(g)|q = g(g)|q−1 + ∑|i|=q

(�i1 ⊗·· ·⊗�id ), (29)

(�i1 ⊗·· ·⊗�id ) =m�

1∑j1=1

· · ·m�

d∑jd=1

g(�i1j1, . . . ,�id

jd)

−g(�i1j1, . . . ,�id

jd)|q−1]N i1

j1(�i1 ) · N i2

j2(�i2 ) · · · N id

jd(�id ), (30)

where q −d�0 denotes the sampling level in the sparse grid. Equation (29) states that given anapproximation in the previous level g(�)|q−1, new sampling points are selected so that monomialsthat satisfy |i|=q are added to the approximation in the new level q .

The Clenshaw–Curtis grid points are defined as

�ikjk

= −cos

(�( jk −1)

mk −1

), jk =1, . . . ,mk

�ik1 = 0 if mk =1.

m1 = 1 and mk =2ik−1 +1 for ik>1.

(31)

Note that sparse grids and equal grids are defined on a hypercube [−1,1]d . A mapping function� is employed to translate points in the hypercube into the random space spanned by randomvariables g.

4. NUMERICAL EXAMPLES

4.1. Fine-scale constitutive uncertainties as random variables

In the following we present an algorithm accounting for fine-scale constitutive uncertainties asrandom variables.

Algorithm 1: nonintrusive stochastic multiscale solver for random variables.1. Set the level of sparse grid or number of grid points of each direction of equal grid.2. Construct the grids in the hypercube [−1,1]d and apply mapping to transform it into the random

variable space [�1lb,�1

ub]⊗·· ·⊗[�dlb,�d

ub], where �klb and �k

ub are lower bound and upper bound of

random variable �k .3. Group the grid points with the same elastic constitutive parameters Gel

1⋃

Gel2⋃ · · ·⋃Gel

n , i.e.

(gel11 , . . . ,gel1

k1)︸︷︷︸

Gel1

⋃(gel2

1 , . . . ,gel2k2

)︸︷︷︸Gel

2

⋃· · ·⋃

(geln1 , . . . ,geln

kq)︸︷︷︸

Geln

,

where gelmj denotes the j th point in the group Gel

m .

4. For each group Gelm , 1�m�n

(1) compute coefficient tensors from Equations (1) and (16).

(2) for each grid point in the group, gelmj , 1� j�km

- update inelastic constitutive parameters and perform simulation,

- record coarse scale quantities of interests g(gelmj ),

- record joint probability f (gelmj ) and integration weight w(gelm

j ).

5. Denote the total number of grid points as N , and compute statistical moment of QoI’s by Mp =∑Ni=1[g(gi )]p f (gi )w(gi ).



Figure 7. The fibrous unit cell (left) and the 0/90/90/0/90/90/0 composite laminate (right).

Figure 8. Isotropic damage model for fine-scale phases.

Table I. Fine-scale material properties.

E (MPa) v S (MPa) G

Matrix 2.E+4 0.2 13 0.04Fiber 4.E+4 0.2 160 0.1

We consider fibrous composite microstructure as a fine-scale model and seven-layer compositelaminate 0/90/90/0/90/90/0 subjected to tensile loading as a coarse-scale model shown inFigure 7. For the constitutive model of fine-scale phases we consider an isotropic damage modelas shown in Figure 8, where S denotes the elastic proportional limit stress and G is the strainenergy density. The material properties are considered in Table I.

A typical stress–strain curve is plotted in Figure 9. In this paper, we will investigate the effectof variations in the fine-scale material inelastic parameters on the coarse-scale ultimate stress. Thevariations of elastic fine-scale parameters have been studied in the previous paper in [44].

Problem 1: Random variables. In this example, we consider inelastic properties of the twophases as random variables and the coarse-scale ultimate stress as a quantity of interest. Statistical



Figure 9. Strain/stress curve of tensile test of composite laminate.

properties of input variables are listed in Table II. The two collocation methods are compared inFigure 10 with the reference solution of 20 000 Latin Hypercube Samplings. It is found that thesparse grid converges faster than the equal grid for the mean value of the ultimate stress. As forthe variance, both methods converge after around 2000 realizations.

4.2. Fine-scale constitutive uncertainties as random fields

The Algorithm 2 below is employed for ROH with inelastic fine-scale constitutive parameters asrandom fields.

We consider the same fibrous composite microstructure as in Section 4.1. The fine-scale materialproperties are given in Table I. We consider a coarse-scale model with tensile loading in the fiberdirection as shown in Figure 7. The specimen dimension is 13.9×4.13×1.27mm3. The following

Algorithm 2: nonintrusive stochastic multiscale solver for random fields.

Step I: Construct eigenpairs databaseGiven: coarse scale model with Nel elements, nipt integration points in each element totaling M = Nel ×nipt

integration points, and covariance function Rhh(x1,x2) for random field parameter h.1. Compute M × M discretized covariance matrix Rhh using Rhh(x1,x2).2. Compute eigenvalues and eigenvectors {i ,Vi } from RhhV =V , where i :1→ M .

3. Find dominant eigenpairs {√

i Vi }, where i :1→ L , such that∑L

i=1 i∑Mi=1 i

�90%.

4. Create a lookup table T, each entry has a key ‘p_q’ stating with the qth integration point of element pand its value as {

√i�i } for i :1→ L , where �i =Vi [p×nipt +q].

Step II: Perform the two-scale reduced order simulation1. Set level of sparse grid or number of grid points in each direction of equal grid.2. Read the eigenpairs table T and the number of dominant eigenpairs L .3. Construct the grids in a hypercube [−1,1]L and transform it into the random variable space [�1

lb,�1ub]⊗

·· ·⊗[�Llb,�L

ub], where �klb and �k

ub are lower bound and upper bound of random variable �k .4. For each grid point (�1

j1,�2

j2, . . . ,�L

jL)

(a) for qth integration point of element p in coarse scale model, p :1→ Nel, q :1→nipt look up eigenpairs

{√

i�i } in table T with a key ‘p_q’.(b) update random fine scale inelastic constitutive parameters with

h(x,g)= h(x)+L∑

i=1

√i�i (x)�i

ji

(c) perform deterministic simulations and record coarse scale quantities of interests g(g j ), joint probabilityf (g j ) and integration weight w(g j ).

5. Denote the total number of grid points as N and compute statistical moments of quantities of interest byMp =∑N

i=1 [g(gi )]p f (gi )w(gi ).



Table II. Inelastic random variables for Problem 1.

Standard DistributionMean deviation Cov (%) type

S of matrix 13 0.65 5 GaussianG of matrix 0.04 0.002 5 UniformS of fiber 160 8 5 GaussianG of fiber 0.1 0.005 5 Uniform

100

101

102

103

104

105

10-4

10-3

10-2

10-1

100

number of realizations

abso

lute

rel

ativ

e er

ror

Mean of Ultimate Stress

sparse gridequal grid

100

101

102

103

104

105

10-4

10-2

100

102

104


abso

lute

rel

ativ

e er

ror

Variance of Ultimate Stress

sparse gridequal grid

Figure 10. Absolute relative error in mean and variance of ultimate stress for Problem 1.

exponential covariance function has been considered:

Rhh(x1,x2)=�2h exp

(−|x1 −x2|lc

), (32)

where �h is the standard deviation and lc is the correlation length. Note that larger lc values givefaster decay rate of eigenvalues of Rhh, i.e. require smaller L in the KL expansion (20).

Problem 2: Random fields. We consider fiber strength S as the random field in the coarse-scaledomain. We investigate the mean and variance of the ultimate tensile stress. The random field isassumed to be Gaussian process, so that all the random variables �i in the KL expansion are ofnormal distributions with zero mean and unit variance. We first select lc to be 24 mm resulting intwo dominant modes covering 90% eigenvalue energy. The coefficient of variation (cov) is definedas the ratio between the standard deviation and mean. The mean fiber strength S is 160 MPa.Figure 11 compares the results for cov=1%. Here, we use 10 000 Latin Hypercube Monte–Carlo(LHMC) samples as a reference solution. Figure 11 also compares LHMC with 2000, 3000 and4000 samples with stochastic collocation methods based on the sparse grid and equal grid. It canbe seen that stochastic collocation methods converge faster than LHMC. The equal grid seems toperform better at small number of realizations, but sparse grid possess higher rate of convergenceas evidenced with increasing number of realizations.

We also study the performance of various collocation methods for cov=5 and 10% as shown inFigures 12 and 13, respectively, where LHMC with 10 000 samples is used as a reference solution.The relative performance of the collocation methods remains the same, but as expected, withincrease in cov, the rate of convergence becomes slower. Finally, we reduce lc to 16 mm and considerthe first four modes in the KL expansion. The higher dimension of random variables necessitates106 LHMC samples for the reference solution. Figure 14 shows the results for cov=10% withfour random variables. Our observations remain the same: equal grid performs better for smallnumber of realizations but for higher accuracy sparse grid is superior.



100 101 102 103 104 10510-5

10-4

10-3

10-2

10-1

100


abso

lute

rel

ativ

e er

ror


sparse grid

equal grid

LHMC

100 101 102 103 104

10-3

10-2

10-1

100

101

102

103

104


abso

lute

rel

ativ

e er

ror


sparse grid

equal grid

LHMC

Figure 11. Absolute relative error in mean and variance of ultimate stress forProblem 2 (lc =24mm, cov 1%).

100 101 102 103 10410-4

10-3

10-2

10-1

100

number of realizations100 101 102 103 104


abso

lute

rel

ativ

e er

ror


sparse grid

equal grid

10-3

10-2

10-1

100

101

102

103

104ab

solu

te r

elat

ive

erro

rVariance of Ultimate Stress

sparse grid

equal grid

Figure 12. Absolute relative error in mean and variance of ultimate stress forProblem 2 (lc =24mm, cov 5%).

100 101 102 103 10410-4

10-3

10-2

10-1

100

101


abso

lute

rel

ativ

e er

ror


sparse grid

equal grid

100 101 102 103 10410-4

10-3

10-2

10-1

100

101

102

103


abso

lute

rel

ativ

e er

ror


sparse grid

equal grid

Figure 13. Absolute relative error in mean and variance of ultimate stress forProblem 2 (lc =24mm, COV 10%).



100 101 102 103 104 105 10610-3

10-2

10-1

100

101

102


abso

lute

rel

ativ

e er

ror


sparse grid

equal grid

100 101 102 103 104 105 10610-2

100

102

104

106


abso

lute

rel

ativ

e er

ror


sparse grid

equal grid

Figure 14. Absolute relative error in mean and variance of ultimate stress forProblem 2 (lc =16mm, COV 10%).

5. CONCLUSION

This paper explores the potential utilization of commercial deterministic finite element solver toovercome the so-called combined curse of scale and dimensionality. The building blocks of theproposed stochastic multiscale solver have been carefully chosen to comply with the standarddeterministic finite element code architecture at the expense of suboptimal computational efficiencycompared with some of the intrusive schemes. By utilizing ROH [26–29] the computational costof a single realization has been greatly reduced in comparison with the direct homogenization andonly fractionally higher than of the phenomenological constitutive models. The KL expansion [38]in combination with sparse grid [30–34] and massively parallel computer architecture where oneor few realizations are assigned to a single processor have the potential of making the methodologyfeasible in practical applications.

Yet, despite the promise, a better understanding of proper balance between deterministic errorsintroduced by scale separation and spatial model reduction on one hand, and KL expansion incombination with sparse grid on the other hand, is needed (see [45] for some of the discussion).For instance, if deterministic errors are dominant, then perhaps perturbation-based methods [46]would suffice in some cases.

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a nonintrusive stochastic multiscale solver

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