on the asymptotically stochastic computational modeling of...

Future Generation Computer Systems 20 (2004) 409–424

On the asymptotically stochastic computationalmodeling of microstructures

Dennis D. Coxa, Petr Kloucekb,∗, Daniel R. Reynoldsba Department of Statistics, Rice University, 6100 Main Street, Houston, TX 77005, USA

b Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, Houston, TX 77005, USA

Abstract

We consider a class of alloys and ceramics with equilibria described by non-attainable infima of non-quasiconvex variationalintegrals. Such situations frequently arise when atomic lattice structure plays an important role at the mesoscopic continuumlevel.

We prove that standard variational approaches associated with gradient based relaxation of non-quasiconvex integrals inBanach or Hilbert spaces are not capable of generating relaxing sequences for problems with non-attainable structure.

We introduce a variational principle suitable for the computational purposes of approaching non-attainable infima of varia-tional integrals. We demonstrate that this principle is suitable for direct calculations of the Young Measures on a computationalexample in one dimension.

The new variational principle provides the possibility to approximate crystalline microstructures using a Fokker–Planckequation at the meso-scale. We provide an example of such a construction.© 2003 Elsevier B.V. All rights reserved.

Keywords:Microstructures; Crystalline materials; Weak white noise; Steepest descent algorithm; Weak convergence; Calculus of variations

1. Introduction

We address in this paper the necessity of us-ing a stochastic variational principle(SVP) lead-ing to the Fokker–Planck equation describing themeso-scale properties of various materials undergo-ing a martensitic transformation. The motivation forthis approach stems from the theory presented herewhich indicates that descent relaxation methods basedon a pseudo-gradient will not relax non-attainablenon-quasiconvex problems.

∗ Corresponding author. Tel.:+1-713-348-5724;fax: +1-713-248-5318.E-mail addresses:[email protected] (D.D. Cox), [email protected](P. Kloucek), [email protected] (D.R. Reynolds).

The successful relaxation of strain energies of con-strained crystalline materials allows us to determinewhich particular atomic structures produce meso-scopic strain. Such information can be used, e.g., todecide whether or not the stress-induced martensitictransformation can occur at the tip of a propagatingcrack. If the structure of the material does allow forthe martensitic transformation then the crack propa-gation may be stopped by a toughening accompany-ing the transformation from a less stable and moreductile state, say a tetragonal lattice structure, to amore stable and less ductile, say, monoclinic latticearrangement.

The model we consider here does not account fora surface energy, hence it lacks any spatial scale.Phenomenologically such models can be used onlyto investigate which variants, from the set of all the

0167-739X/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.future.2003.07.006

410 D.D. Cox et al. / Future Generation Computer Systems 20 (2004) 409–424

possible ones, will participate in relaxation of thestrain energy.

The paper is organized as follows. We describe arelation between microstructures and transformationtoughening inSection 2. We recall how non-attainablestates can be represented using Radon measures inSection 3. We describe a particular construction of apseudo-gradient inSection 4. We prove that gradientbased relaxation fails to achieve stochastic states inSection 5. We introduce the constrained finite elementmethod inSection 6. The next two sections introduce anotion of weak white noise and a corresponding varia-tional principle.Section 9demonstrates our variationalapproach by direct calculations of a Young Measureand its underlying probabilities.

2. Transformation toughening in zirconia

The purpose of this section is to discuss how certainmaterial properties can be drastically altered by chang-ing just the symmetry of their lattice structure, i.e.,by promoting certain specific arrangements of theiratomic lattices. We rely in this section on the overviewarticle [13].

Transformation toughening is the increase in frac-ture toughness of a material that is the direct result of aphase transformation occurring at the tip of an advanc-ing crack. The discovery of transformation tougheningin zirconia ceramics[12] indicates that traditionallybrittle ceramics can reach fracture toughness four ormore times higher when it undergoes themartensitictransformation. This is a lattice-distortive, virtuallydiffusionless structural change having a dominantdeviatoric component and associated shape changesuch that strain energy dominates the kinetics andmorphology during the transformation[9]. Hence, zir-conia ceramics are a good candidate for applicationswhere toughness is required and where an advantageof wear resistance, low density and of high meltingpoint characterizing ceramics can be taken advan-tage of.

The martensitic transformation is triggered bythe nucleation strain at the interface between a pro-gressing crack and the bulk zirconia. This ceramicundergoes the tetragonal to monoclinic transforma-tion. The mathematical challenge associated withmodeling this lattice transformation is to find the

Fig. 1. Schematic diagram illustrating the stages in thetransformation of a spherical tetragonal zirconia particle toself-accommodating monoclinic variants. The double arrows rep-resent the stresses in the surrounding untransformed material thatoppose continued growth of a particular variant and favor the nu-cleation of the self-accommodating variant with an opposing shearstrain. Nucleation of these self-accommodating variants is mostlikely to occur in the regions of stress concentration at the endsof transformed variants, i.e., in the regions marked X in (b). Thedrawing and caption are reproduced from[13].

microscopic distribution of martensitic variants par-ticipating in the mesoscopic strain. The distribution ofthese variants determines the structure of the atomiclattice. The lattice structure has a direct relation to thetransformation toughening: the more complex is themicrostructure, the more difficult it is for the crack topropagate.Fig. 1 illustrates a possible arrangement ofatoms resulting from original tetragonal zirconia. Theresulting microstructure, formed by monoclinic vari-ants, can be phenomenologically modeled as follows.We strive to find a continuous deformation conform-ing to the mesoscopic deformationg ∈ C0(Ω) in themicroregionΩ having the properties:

∇u ∈ F, a.e. in Ω ⊂ RN, N = 2,3,

u(x) = g(x), x ∈ ∂Ω. (1)

The setF represents crystallographically acceptableequilibrium variants of the zirconia particle. The

D.D. Cox et al. / Future Generation Computer Systems 20 (2004) 409–424 411

monoclinic variants (also low-temperature variants)are given by[2]:

Fdef=Fi|i = 1,2,

Fi = (I + η(−1)ie2 ⊗ e1)

N∑i=1

diei ⊗ ei,

di > 0, η > 0, i = 1,2. (2)

We refer to[13] for discussion of observations andmeasurements identifying these matrices during thetetragonal to monoclinic transformation in zirconia.Simple conditions for the existence of rank-one con-nection between martensitic wells are given in[11].These conditions are used to determine the structureof the setF in (2) constraining the deformation gra-dients. The rank-one connection is required by theHadamard jump condition to allow for a continuousdeformation with given gradients.

Using the classical variational approach, we can for-mulate the problem(1) as follows. Let us assume thatthere exists a strain densityW , having its local min-ima distributed along the orbits:

QFi|i = 1,2, where Q ∈ SO(N). (3)

The density must inherit the symmetry of the tetrago-nal phase, i.e.:

W(QiFQTi ) = W(F) for any Qi ∈ G, (4)

whereG is the symmetry group of the tetragonal phase.The densityW must satisfy the principle of frameindifference, i.e.:

W(RF) = W(F) for any rotationR ∈ SO(N). (5)

We also assume the following growth and coercivenessconditions for the densityW :

C1(|F |p + 1) ≥ W(F) ≥ C2|F |p − C3 (6)

for some suitablep > N andCi > 0, i = 1,2,3. It isalso reasonable to assume that, in the extreme case:

W(F) → +∞ if detF → 0+. (7)

We then define

E(u)def=∫Ω

W(∇u(x))dx (8)

and we investigate

inf E(u)|u ∈ W1,p(Ω), u = g on∂Ωfor some suitablep > N. (9)

The difficulty with finding the infimum or minimumin (9) is that we have to deal with a profoundlynon-quasiconvex problem which often has unattain-able structure. The properties of the minimizer de-pend strongly on the choice of the Dirichlet boundaryconditiong.

The complexity of microstructures corresponding to(9) can be considerable, and it may be accompaniedby a massive lack of uniqueness. Demonstration ofthese difficulties can be found in[15, Section 4.7].

3. Representation of non-attainable states

A typical relaxing sequence of(9) will convergeweakly, likely weakly-∗, to its expected value, or av-erage, in an appropriate Sobolev space. Since we donot expect the variational integral in(8) to be weaklylower semicontinuous, we assume that

lim infn→∞

∫Ω

W(Dun(x))dx <∫Ω

W(Du(x))dx. (10)

Here,unn∈N ⊂ W1,p(Ω) is a relaxing sequence andu ∈ W1,p(Ω) is its weak limit. Translation of(10) intothe framework of material science would imply thatthe functionu, representing the averaged deformationof a zirconia ceramic, does not carry any pointwiseinformation. Yet, a precise description of the spatialcomposition of deformation gradients is needed to de-scribe the transformation toughening as explained inSection 2. The lattice structure is also important andnecessary as an input for thermodynamic models[16](Fig. 2).

In order to overcome this difficulty we may supposethat for any densityW ∈ C0(Ω) which has at leasta quadratic growth at infinity there exists a functionW ∈ L1(Ω) such that

lim infn→∞

∫ω

W(Dun(x))dx =∫ω

W(x)dx (11)

for anyω which is a compact subset ofΩ. The func-tion W does not remember anything about the equi-librium structure of the material. This is encoded into


Fig. 2. The picture on the left shows twinned microstructure produced by F. Appel, 3GKSS Forschungszentrum, Institute for MaterialsResearch, Geesthacht, Germany. Deformation twins are generated during high-temperature deformation of Ti–48Al–2Cr (at.%),T = 1100 K,e = 8.9% [10]. The picture on the right shows how the twinned structures are created on the atomic level. The lattice is transformed intothe face centered tetragonal (FCT) (light gray lines) structure from the body centered cubic (BCC) arrangement (black lines). Picture byJ. Riordan.

the densityW . Though more importantly, the functionW does tell us about the mesoscopic distribution ofthe elastic energy densityW .

The functionW can be obtained via integral rep-resentation using a Radon probability measureµx.Namely:

W(x) =∫Rm×n

W(y)dµx(y). (12)

The Radon measureµx has a very natural structure inthe case of alloys undergoing simple symmetry phasechange. In the case of bi-stable alloys, it is possible toshow that[15]:

µx = λ(x)δF1 + (1 − λ(x))δF2, (13)

whereF1, F2 are the deformation gradients partici-pating in the equilibrium deformation. The functionλ = λ(x) represents the probability that at a pointx

the material is deformed with the deformation gradi-entF1. Usually, this ratio is called thevolume fraction.It follows directly from its definition that

λ(x) = limr→0+

limR→0+

limn→∞

measy ∈ BR(x)|‖Dun(x) − F1‖ ≤ rmeas(BR(x))

, (14)

where Dun are the gradients of the relaxing sequenceof the strain energy. For(14) to be valid, the minimiz-ing sequence must become stochastic in its derivative.This represents a major challenge for the optimization,numerical and computational approaches. We believethat the desirable methods for direct calculations ofYoung Measuresµx in (13)are those which generate astochastic state from an initial deterministic one. Thisproblem has not been solved. It seems that it is notwell understood what methods can achieve this goal.The crux of this problem can be represented by analmost a 70-year-old question posed by Heisenberg:what physical processes would serve as the source ofan origin of stochastic behavior?

It is possible to construct such sequences (pro-cesses) using a self-similarperiodic construction,e.g. [5]. It is however impossible to reconstructsuch sequences computationally using “off-the-shelf”tools. We address this issue inSection 5. We striveto compute the volume fraction using a StochasticVariational Principle (SVP) that will guarantee thatany minimizing sequence becomes asymptotically


a weak white noise, c.f., Section 8. The assumption isthat we must trigger the relaxation by a perturbationof a sharp initial state. This seems to be the key selec-tion principle. Such sequences can be reconstructedcomputationally despite all the limitations implied byany kind of finite-dimensional approximation, whichprohibit the convergence otherwise.

4. A pseudo-gradient

The relaxation of(9) in Banach spaces is compli-cated since it is very expensive to obtain thesteepestdescent direction. We instead use a pseudo-gradientwhich is in the presented form relatively inexpensiveto compute. The pseudo-gradient is often used in crit-ical point theory[8].

Let us assume thatE is aC1-functional on a BanachspaceV . The substitution for a gradient in spaces lack-ing a scalar product is the following cone of locallyLipschitz continuous vector fieldsg(u) : V\0 → V

such that[18]:

‖g(u)‖V < 2 min‖dE(u, ·)‖V ∗ ,1, (15)

dE(u, g(u)) > min‖dE(u, ·)‖V ∗ ,1‖dE(u, ·)‖V ∗ ,

(16)

where dE(u, ·) denotes the Gateaux derivative ofEwhich is a linear functional onV . A remarkable re-sult [19, Lemma 3.2]guarantees that anyE ∈ C1(V)

admits a pseudo-gradient vector field. This resultguarantees that a gradient flow is well defined andasymptotically leads to descent to a critical point. Inparticular, this means that any vector field satisfyingthe conditions(15) and (16)can be used to define agradient descent method for minimization of varia-tional integrals such as(8). Assuming that the energyE, (8), is defined onV = W1,p, p > 2, we define aparticular pseudo-gradient below.

First, let us denote the distributional gradient of theenergyE by G, i.e., for anyu ∈ W1,p(Ω), the func-tionalG ∈ W−1,q(Ω,RN), whereW−1,q(Ω,RN) de-notes the dual space ofW1,p

0 (Ω,RN),1/p+1/q = 1,is given by the variational relation:

〈G(u), ϕ〉W−1,q(Ω,RN),W

1,p0 (Ω,RN)

def= d

dtE(u + tϕ)|t=0 =

∫Ω

DW(∇u(x)) : ∇ϕ dx (17)

for all ϕ ∈ W1,p0 (Ω), whereA : B=defTr(ATB),A,B ∈

RN×N . We have the following Lemma.

Lemma 4.1 (pseudo-gradient).Let us assume thatΩ is bounded with smooth boundary. Let g(u) ∈W

1,p0 (Ω,RN) be defined by

∆g(u)(x) = div(∇g(u)(x)‖∇g(u)(x)‖−1+q/p),

x ∈ Ω, g(u)(x) = 0 on ∂Ω, (18)

where‖ ·‖ is a matrix norm andg(u) ∈ W1,q0 (Ω,RN)

is given by

g(u) = −∆−1G(u). (19)

Here −∆−1 : W−1,q(Ω,RN) → W1,q0 (Ω,RN). Fi-

nally we define

g(u) = g(u)‖g(u)‖p−2

W1,p0 (Ω,RN)

. (20)

Theng(u) is a pseudo-gradient of the strain energyE in the sense of the conditions(15) and (16) withV = W1,p(Ω,RN).

Remark 4.2. We have

g(u) ∈ W1,q0 (Ω,RN)

in view of the growth conditions(6).

Remark 4.3. We note that the−∆−1 mapping isisometric fromW−1,q(Ω,RN) to W

1,q0 (Ω,RN) [1].

Hence,‖g(u)‖W

1,q0 (Ω,RN)

= ‖G(u)‖W−1,q(Ω,RN).

Remark 4.4. The pre-gradientg(u) is unique for agivenG. Namely, assuming that∂Ω is piecewiseC∞,and takingF ∈ W−1,q(Ω) represented by

〈F, ϕ〉def=∫Ω

N∑i=1

fi(x)∂ϕ(x)

∂xi+ f0(x)ϕ(x)dx,

ϕ ∈ W1,p0 (Ω),

‖F‖W−1,q(Ω,RN)def=(

N∑i=0

‖fi‖qLq(Ω)

)1/q

, (21)

we have[14, Theorem 3.8]. Letp ∈ [2,∞), then thereexists a unique solutiong(u) ∈ W1,q(Ω,RN), u(x) =


u0(x) on ∂Ω, u0 ∈ W1,q(Ω), of∫Ω

∇g(u)(x)∇ϕ(x)dx = 〈F, ϕ〉

for all ϕ ∈ W1,p0 (Ω). (22)

Moreover, there exists a constantC(q,Ω) such that

‖g(u)‖W1,q(Ω) ≤ C(q,Ω)(‖u0‖W1,q(Ω)

+‖F‖W−1,q(Ω)). (23)

Proof of Lemma 4.1. We observe that(18) yields∫Ω

∇g(x)∇g(x) dx =∫Ω

|∇g(x)|q dx. (24)

The Hölder inequality and(24) yields∫Ω

|∇g(x)|p dx ≥∫Ω

|∇g(x)|q dx. (25)

It follows from Remark 4.4and(18) that there existsa constantC, depending onp andΩ, such that∫Ω

|∇g(x)|p dx ≤ C(p,Ω)p∫Ω

|∇g(x)|q dx. (26)

Hence(25) and (26)yield∫Ω

|∇g(x)|p dx = C(p,Ω)p∫Ω

|∇g(x)|q dx. (27)

Since∫Ω

|∇g(x)|p dx

= C(p,Ω)p∫Ω

|∇g(x)|q dx

= C(p,Ω)p∫Ω

∇g(x) : ∇g(x) dx

= C(p,Ω)p∫Ω

DW(∇u(x)) : ∇g(x) dx, (28)

we have either with respect to the semi-normW1,p0 (Ω)

or from the Poincaré inequality (assuming that theconstant appearing in it is bounded by 1):

‖g(u)‖pW

1,p0 (Ω,RN)

= C(p,Ω)p dE(u, g(u))

≤C(p,Ω)p‖dE(u, ·)‖W−1,q(Ω,RN)‖g(u)‖W1,p0 (Ω,RN)

.

(29)

Thus(20) and (29)yield

‖g(u)‖W

1,p0 (Ω,RN)

= ‖g(u)‖p−1

W1,p0 (Ω,RN)

≤ C(p,Ω)p‖dE(u, ·)‖W−1,q(Ω,RN). (30)

Thus we assume

C(p,Ω)p < 1 (31)

in order to accommodate both requirements(15) and(16) simultaneously. For(16) see below.

With respect to the second condition(16) we havein view of (18), (27) and (28):

sup‖ϕ‖

W1,p0 (Ω,RN)

≤1|dE(u, ϕ)|

= sup‖ϕ‖

W1,p0 (Ω,RN)

≤1

∫Ω

∇g(x) : ∇ϕ(x)dx

≤ ‖g‖W

1,q0 (Ω,RN)

= C(p,Ω)−p/q‖g‖p/qW

1,p0 (Ω,RN)

= C(p,Ω)−p/qC(p,Ω)p/q dE(u, g)1/q. (32)

Thus

sup‖ϕ‖

W1,p0 (Ω,RN)

≤1|dE(u, ϕ)| ≤ dE(u, g(u))1/q. (33)

Hence forq ∈ (1,2), i.e., forp > 2:

‖dE(u, ·)‖qW−1,q(Ω,RN)

≤ dE(u, g(u)). (34)

Assuming‖dE(u, ·)‖W−1,q(Ω,RN) < 1, taking someε ∈ (0,1) so thatq+ ε < 2, and using(33), we obtain

‖dE(u, ·)‖2W−1,q(Ω,RN)

≤ ‖dE(u, ·)‖q+ε

W−1,q(Ω,RN)

≤ ‖dE(u, ·)‖εW−1,q(Ω,RN)

dE(u, g(u))

= ‖dE(u, ·)‖εW−1,q(Ω,RN)

‖g(u)‖2−p

W1,p0 (Ω,RN)

×dE(u, g(u)). (35)

It remains to show that

‖g(u)‖2−p

W1,p0 (Ω,RN)

‖dE(u, ·)‖εW−1,q(Ω,RN)

< 1. (36)

It follows from (29) and (31)that

‖g(u)‖1−1/p

W1,p0 (Ω,RN)

< ‖dE(u, ·)‖1/pW−1,q(Ω,RN)

. (37)


Thus

‖g(u)‖2−p

W1,p0 (Ω,RN)

‖dE(u, ·)‖εW−1,q(Ω,RN)

< ‖dE(u, ·)‖(1/p)(2−p)(1−1/p)−1+ε

W−1,q(Ω,RN). (38)

Taking ε sufficiently close to 1 so thatp < (2 − ε)/

(1 − ε) we have

1

p(2 − p)

(1 − 1

p

)−1

+ ε > 0

and(36) follows from the assumption‖dE(u, ·)‖W−1,q(Ω,RN) < 1.

Definition 4.5 (descent algorithm). Letu0 ∈W1,p(Ω,RN), p > N, such thatE(u0) < 1, be given.Let us assume that the pseudo-gradient is given by(18)–(20). Let αn ∈ (0,Λ−1) be a result of the linesearch

minα∈[0,1/Λ)

E(un − αg(un)), (39)

whereΛ is such that‖D2W(Q)‖N2×N2 ≤ Λ‖Q‖p−2N×N ,

Q ∈ MN×N . We choose the new iteration to be

un+1 = un − αng(un) in Ω. (40)

We haveun ∈ C0(Ω,RN), n ∈ N, in view of thecontinuity of the embedding ofW1,p(Ω,RN) into con-tinuous functions,C(Ω), for p > N.

5. Failure of pseudo-gradient based relaxation

The purpose of this section is to show that for anyu0 ∈ W1,p(Ω,RN), p > N, the sequenceunn∈N ⊂W1,p(Ω,RN), p > N, generated during a gradientnavigated relaxation of(8) with the Dirichlet boundarydata such that(9) does not admit minimizer convergesstrongly to a function inW1,p(Ω,RN). The minimizerrepresents a local minimum of the stored energyE.The derivatives of the minimizing sequence thus do notconverge to a measure-valued distribution even thoughthe infimum of the stored energy is not attainable inany Sobolev space. Namely:

un→u strongly inW1,p(Ω,RN), and consequently,

limn→∞ E(un) = E(u) > 0. (41)

This result indicates that we need a different struc-ture of the strain energy in order to obtain themeasure-valued distributions. We address this modi-fication inSection 8.

We assume that the densityW has the followingstructural properties:

E(u)≥λp/2∫Ω

‖∇u(x) − (Π∇u)(x)‖p dx, λ > 0,

(42a)

‖D2W(Q)‖N2×N2 ≤ Λ‖Q‖p−2N×N, Q ∈ MN×N,

(42b)

1 ≤ ‖un‖W1,p(Ω,RN) ≤ Λ0, (42c)

√λ − Λp−2

λ(1/2)(p−1)> 0, (42d)

2C1/p1√λ

≤ 1, (42e)

whereΠ is a projection on the set of equilibria ofEgiven by (3).

We prove the following two lemmas that will beused in the proof ofTheorem 5.3.

Lemma 5.1. Let the sequenceun∞n=0 ⊂ W1,p

(Ω,RN), p > N, be generated by the descent algo-rithm, Definition 4.5. Let us assume that(42) holdstrue. Let

lim infn→∞ E(un) = J. (43)

Then for anyt ∈ [0,1] we have

C(p,Ω, λ,Λ)(E(un+1 + tg(un)) − J)1−1/p

≤ ‖g(un+1 + tg(un))‖W1,p0 (Ω,RN)

, (44)

where

C(p,Ω, λ,Λ)def=C(p,Ω)p−1

(√λ − Λp−2

λ(1/2)(p−1)

).

The constantC(p,Ω) originates in the estimate(23).

Proof. We abbreviate

un(t)def=un+1 + tg(un)


and we assume without loss of generality thatJ = 0.It follows from the Hadamard jump condition[2] thatthere exists awn ∈ C0(Ω) such that

∇wn ∈ SO(N)F1 ∪ SO(N)F2,

‖un − wn‖Lp(Ω,RN) ≤ C3‖∇un − ∇wn‖Lp(Ω,RN×N)

≤ C4‖∇un − Π∇un‖Lp(Ω,RN×N). (45)

whereΠ is a projection on the set of equilibria ofE.ConsequentlyE(wn) = 0.

Hence

E(un(t)) = E(un(t)) − E(wn(t))

=∫Ω

∫ 1

0

d

dτW(τ∇un(t)(x)

+(1 − τ)∇wn(t)(x))dτ dx

=∫Ω

∫ 1

0DW(τ∇un(t)(x)

+(1 − τ)∇wn(t)(x))dτ : ∇(un(t)(x)

−wn(t)(x))dx. (46)

Writing

DW(τ∇un(t) + (1 − τ)∇wn(t))

= DW((1 − τ)(∇wn(t) − ∇un(t)) + ∇un(t)) (47)

we have

DW((1 − τ)(∇wn(t) − ∇un(t)) + ∇un(t))

−DW(∇un(t))

=∫ 1

0

d

dωDW(ω(1 − τ)(∇wn(t) − ∇un(t))

+∇un(t))dω(1 − τ)

∫ 1

0D2W(ω(1 − τ)(∇wn(t)

−∇un(t)) + ∇un(t))dω(∇wn(t) − ∇un(t)).

(48)

Thus it follows from(46) and (48)that

E(un(t)) =∫Ω

DW(∇un(t)(x)) : ∇(un(t)(x)

−wn(t)(x))dx

−∫Ω

∫ 1

0(1 − τ)

∫ 1

0Y(·)dω dτ(∇wn(t)

−∇un(t),∇wn(t) − ∇un(t))dx, (49)

where the tensorY is given by

Y(·) = D2W(ω(1 − τ)(∇wn(t) − ∇un(t)) + ∇un(t)).

Since∫Ω

DW(∇un(t)(x)) : ∇(un(t)(x) − wn(t)(x))dx

=∫Ω

∇g(un(t)) : ∇(un(t)(x) − wn(t)(x))dx (50)

we have in view of the definition of the pseudo-gradient,(18) and(42):∣∣∣∣∫Ω

DW(∇un(t)(x)) : ∇(un(t)(x) − wn(t)(x))dx

∣∣∣∣≤ C(p,Ω)−p/q‖g(un(t))‖W1,p

0 (Ω,RN)

1√λE(un)

1/p.

(51)

Using the assumption(42b), the last integral in(49)can be estimated by∫Ω

∫ 1

0(1 − τ)

∫ 1

0‖∇zn(x)‖p−2 dω dτ

×‖∇un(x) − ∇wn(x)‖2 dx, (52)

where∇zn=defω(1− τ)(∇wn(t)− ∇un(t))+ ∇un(t).Denoting

f(x) =∫ 1

0(1 − τ)

∫ 1

0‖∇zn(x)‖p−2 dω dτ,

using(42a)and the Hölder inequality, the integral in(52) can be estimated by(∫

Ω

f(x)p/(p−2) dx

)(p−2)/p

‖un − wn‖pW

1,p0 (Ω,RN)

≤(∫

Ω

f(x)p/(p−2) dx

)(p−2)/p 1

λp/2E(un). (53)

The triangle inequality, the assumptions(6) and (42a)yield

‖∇zn‖Lp(Ω,RN×N)

≤ ‖∇un‖Lp(Ω,RN×N)

×(

1+ω(1−τ)‖∇wn(t) − ∇un(t)‖Lp(Ω,RN×N)

‖∇un(t)‖Lp(Ω,RN×N)

)

≤ Λ(1 + ω(1 − τ)). (54)


The last inequality is obtained as follows. We have

C1(‖∇un‖pLp(Ω,RN×N)+ 1) ≥ E(un)

≥ λp/2‖∇(wn(t) − un(t))‖pLp(Ω,RN×N). (55)

Thus, in view of(42b) and (42e)

1 ≥ 2C1/p1√λ

≥ ‖∇(wn(t) − un(t))‖Lp(Ω,RN×N)

‖∇un‖Lp(Ω,RN×N)

. (56)

Consequently, it follows from(54) that(∫Ω

f(x)p/(p−2) dx

)(p−2)/p

≤ Λp−2. (57)

Thus(49), (51) and (57)yield

E(un(t)) ≤ C(p,Ω)−p/q‖g(un(t))‖W1,p0 (Ω,RN)

× 1√λE(un)

1/p + Λp−2

λp/2E(un). (58)

Hence(1 − Λp−2

λp/2

)E(un(t))

≤ C(p,Ω)−p/q 1√λ

‖g(un(t))‖W1,p0 (Ω,RN)

×E(un(t))1/p, (59)

which yields(44).

Lemma 5.2. Let g(un)∞n=0 ⊂ W1,p(Ω,RN), p >

N, be the sequence of pseudo-gradients, defined by(18), (19) and (20), corresponding to the sequenceun∞n=0 ⊂ W1,p(Ω,RN), p > N, generated by thedescent algorithm, Definition 4.5. Then there ex-ists a positive finite constantβ, independent ofn,such that∫Ω

∇g(un+1 + tg(un))(x)∇g(un)(x)dx

≥ β‖g(un+1 + tg(un))‖W1,q0 (Ω,RN)

×‖g(un)‖W1,p0 (Ω,RN)

, t ∈ [0, αn]. (60)

Proof. We prove the lemma for the casep = q =2. The casep > 2 can be handled similarly usingYoung’s inequality. We have in this caseg = g. Wedenote

gn(t)(x)def= g(un+1 + tg(un))(x), gn(x)

def=g(un)(x).

We will not indicate the dependence onx in the courseof the proof to make the formulas shorter and easierto read. We have

∫Ω

|∇gn(t)|2 dx +∫Ω

|∇gn|2 dx

= 2∫Ω

∇gn∇gn(t)dx +∫Ω

|∇(gn − gn(t))|2 dx.

(61)

We show below that∫Ω

|∇(gn − gn(t))|2 dx ≤ (αnΛ)2‖gn‖2

W1,20 (Ω)

. (62)

Hence, it follows from(61) and (62)that

∫Ω

|∇gn(t)|2 dx + (1 − α2n)

∫Ω

|∇gn|2 dx

≤ 2∫Ω

∇gn∇gn(t)dx. (63)

Definition 4.5requiresαn ∈ [0,Λ−1). Thus

0 < (1 − α2n) < 1 for any n ∈ N. (64)

Consequently, we obtain the inequality(60) with

βdef=max

n∈N(1 − (αnΛ)

2)−1

from (63)and the inequalitya2+b2 ≥ 2ab. It remainsto verify (62).

Since for anyφ ∈ W1,20 (Ω,RN):

∫Ω

∇g(un) : ∇φ dx =∫Ω

DW(∇un) : ∇φ dx,∫Ω

∇g(un+1 + tg(un)) : ∇φ dx

=∫Ω

DW(∇(un+1 + tg(un)) : ∇φ dx

=∫Ω

DW(∇(un + (t − αn)g(un))) : ∇φ dx, (65)

we have in view of the assumptions(42b) and (42c)for t ∈ [0, αn]

418 D.D. Cox et al. / Future Generation Computer Systems 20 (2004) 409–424∣∣∣∣∫Ω

∇(g(un) − g(un+1 + tg(un))) : ∇φ dx

∣∣∣∣= |t − αn|

∣∣∣∣∣∫Ω

∫ 1

0D2W(∇(un + τ(t − αn)g(un)))

×dτ(∇g(un),∇φ)dx

∣∣∣∣∣≤ αnΛ‖gn‖W1,2

0 (Ω,RN)‖φ‖

W1,20 (Ω,RN)

. (66)

The proof of (62) follows by takingφ = g(un) −g(un+1 + tg(un)).

We now prove the summability of theW1,p(Ω,RN)-norm ofg(un), which is sufficient to obtain the strongconvergence of the minimizing sequences generatedby the descent algorithm,Definition 4.5.

Theorem 5.3. Let the sequenceun∞n=0 ⊂ W1,p

(Ω,RN), p > N, be generated by the descent algo-rithm, Definition 4.5. Let us assume thatE(u0) < 1and let us assume that(42) holds true. Then thereexists a finite constantC such that∞∑n=0

αn‖g(un)‖W1,p0 (Ω,RN)

≤ C < +∞. (67)

Proof. We denote again

gn(t)(x)def=g(un+1 + tg(un))(x).

Lemmas 5.1 and 5.2and the definition of the descentalgorithm(39) yield for anyθ ∈ [0,1/2]:

d

dtE(un+1 + tg(un))

θ

= θE(un+1 + tg(un))θ−1

×∫Ω

DW(∇un+1 + t∇g(un))∇g(un)dx

Definition 4.5= θE(un+1 + tg(un))θ−1

×∫Ω

∇gn(t)∇g(un)dx

Lemma 5.2≥ βθE(un+1 + tg(un))θ−1‖gn(t)‖W1,q

0 (Ω,RN)

×‖g(un)‖W1,p0 (Ω,RN)

Lemma 5.1≥ CβθE(un+1 + tg(un))θ−1

×E(un+1 + tg(un))1−θ‖g(un)‖W1,p

0 (Ω,RN). (68)

Integrating(d/dt)E(un+1 + tg(un))θ over (0, αn) weobtain

E(un+1 + αng(un))θ − E(un+1)

θ

≥ Cβθαn‖g(un)‖W1,p0 (Ω,RN)

. (69)

SinceE(un+1 +αng(un)) = E(un), summing up(69)overn ∈ N we obtain

E(u0)θ − lim

n→∞ E(un)θ

≥ Cβθ

∞∑n=0

αn‖g(un)‖W1,p0 (Ω,RN)

, (70)

which proves(67).

Theorem 5.4 (strong convergence).Let the sequenceun∞n=0 ⊂ W1,p(Ω,RN), p > N, be generated bythe descent algorithm, Definition 4.5. Let us assumethatE(u0) < 1 and let us assume that(42)holds true.Then there exists a functionu ∈ W1,p(Ω,RN) suchthat

un → u, strongly in W1,p(Ω,RN). (71)

Consequently

limn→∞ E(un) = E(u)

> Jdef= inf E(v)|v ∈ W1,p(Ω,RN). (72)

Proof. Let us assume that the sequenceun∞n=0 ⊂W1,p(Ω,RN) generated by the descent algorithm,Definition 4.5, minimizes the energyE. Then thereexists its weak limitu ∈ W1,p(Ω,RN). Thus∫Ω

(Dun(x) − Du(x))ϕ(x)dx

=∞∑m=n

∫Ω

(Dum(x) − Dum+1(x))ϕ(x)dx

for all ϕ ∈ Lq(Ω,RN). (73)

Since∫Ω

(Dum(x) − Dum+1(x))ϕ(x)dx

=∫Ω

−αmDg(um)(x)ϕ(x)dx, (74)

we have, using the Poincaré–Friedrichs inequality:

D.D. Cox et al. / Future Generation Computer Systems 20 (2004) 409–424 419∣∣∣∣∫Ω

(Dun(x) − Du(x))ϕ(x)dx

∣∣∣∣=∣∣∣∣∣

∞∑m=n

∫Ω

−αmDg(um)(x)ϕ(x)dx

∣∣∣∣∣≤ C‖ϕ‖Lq(Ω,RN)

∞∑m=n

αn‖g(um)‖W1,p(Ω,RN). (75)

It follows from Theorem 5.3that∞∑m=n

αn‖g(um)‖W1,p(Ω,RN) → 0 as n → ∞. (76)

Hence

un → u strongly in W1,p(Ω,RN). (77)

6. Langevin equation and the Fokker–Planckdynamics

Based on the result ofSection 5, we assume thatthe elastic densityW = W(Du) describing equilibriaof crystalline materials with non-attainable structureshould be written in the form (Fig. 3):

W(s)def=Wmeso(s) + Wmicro(s), s ∈ R

N×N. (78)

The contributionWmesoencodes the information aboutthe equilibrium state of a given material and theWmicrocontribution guarantees that any minimizing sequencebecomesweak white noisein the sense ofDefinition7.1. In particular, we assume that

∂Du Wmicro(Du)(x, t) = σ(Du)(x, t)η(x, t), (79)

Fig. 3. The periodic or quasi-periodic structures often found in micrographs of metallic alloys or zirconia ceramics are invisible on theLangevin scale due to the stochastic nature of the description at the atomic scale. These structures are recovered by solutions of theFokker–Planck equations. The picture on the right is a photo of a complex martensitic microstructure by C. Chu and R.D. James[4].

whereη becomes asymptotically (ast → ∞) weakwhite noise. A possible construction yielding such amicroscaling term usingsubgrid projection methodisstudied both analytically and computationally in[6].

We interpret the gradient flow inW−1,p(Ω)-topologyas a Langevin equation having the form

d

dtu(x, t) = div(∂Du Wmeso(Du(t, x))

+σ(Du)(x, t)η(x, t)). (80)

Writing (80) in its weak form and using a piecewiseaffine finite element approximation ofu, we obtaina system of ODEs for the coefficientsα=def(αi)

N(h)i=1

which now become functions of time. Namely:

d

dtα(t) = A

−1(b(α) +∑

(α)η(t)). (81)

The Fokker–Planck equation corresponding to(81)forthe probability densityp = p(t, α) > 0 reads[17]:

d

dtp(t, α) = −divα(A

−1b(α)p(t, α))

+1

2D2α(A

−1∑

(α)p(t, α)). (82)

Let E denote the expectation operator. Thus

E(αi)(t) =∫R

ypi(t, y)dy, where

pi(t, y)def=p(t, α1, . . . , αi−1, ·, αi+1, . . . , αN(h)).

(83)

The functionpi(t, y) represents the probability den-sity for αi. We note that for sharp values ofαi wehavepi(t, y) = δαi(y). Consequently, we define the


mesoscopic, or averaged, solution of(80) by

uMesoh (x, t)

def=E(uh)(x, t)

=N(h)∑i=1

E(αi)(t)ϕi(x), t > 0, x ∈ Ω.

(84)

This approach is adopted in[7].

7. Multi-dimensional weak white noise

We have shown inSection 5that the descent al-gorithm, Definition 4.5, applied to the double-wellproblem does not generate relaxing sequences. Inother words, there exists a deterministic limiting statecorresponding to a deterministic initial guess and nota limiting stochastic state describing the infimum ofthe energy. This means in terms of any minimizing se-quenceunn∈N that it converges strongly to a functionin the norm of some Sobolev space. In this case, thereconstruction procedure(14) for computing the vol-ume fraction cannot provide a reliable approximation.

In computational practice, the initial iteration fordescent algorithms is obtained by a superposition ofthe averaged state (the weak limit) with numeric noise.Typically this leads almost immediately to a conver-gence of the minimizing sequence to the nearest lo-cal minimum. The evaluation of the volume fractionbased on such sequences using(14)yields results withalmost 50% error. We demonstrate this phenomenonin Section 9.

We introduce a variational principle that imposesa forcing mechanism prohibiting the minimizing se-quences to adhere to any state where possible spatialcorrelation can occur. This mechanism prohibits theminimizing sequence to converge to any of the manylocal minima of the variational integral(8), and leadsto the representation(79).

The following definition establishes how we inter-pret the notion of weak white noise.

Definition 7.1. We say that a sequenceukk∈N ⊂W1,∞(Ω,Rm),Ω ⊂ R

n, n = 1,2,3,m = 1,2,3,that converges weakly-∗ in W1,∞(Ω,Rm) to a Lips-chitz continuous functiong, becomesasymptotically

weak white noiseif the following three conditions holdtrue. Let

zk(x)def=Duk(x) − Dg(x) ∈ R

m×n. (85)

The first condition is

Eω[zk] = 1

meas(ω)

∫ω

zk(x)dx → 0

for all open ω ⊂ Ω as k → ∞.

The second and the third conditions are that for almostall τ ∈ R the covariance operatorE has the followingtwo properties:

Eτ,ω[zk] = 1

meas(ω)

∫ω

zk(x) ⊗ zk(x + τ)dx σ−1(ω)

→I ∈ R

m2×n2if τ = 0,

0 ∈ Rm2×n2

if τ = 0

for all open ω ⊂ Ω as k → ∞,

(86)

whereσ represents “standard deviation” given by

σ(ω)def=E0,ω[zk]

1/2. (87)

We assume that the functions Dun are extended peri-odically ontoR

m×n for uk(x+ τ) to be defined wherex + τ /∈ Ω. We note thatE is non-negative.

8. An SVP in one dimension

For the one-dimensional analog of the SVP, we con-sider the case where the set of allowable gradientsA = ±1, and the bodyΩ = (0,1). In addition tothe coercitivity of the variational integral in(8), wealso assume, similar to(42a), that

Wmeso(u′) ≥ λdistu′, ±12, λ > 0. (88)

This is enough to guarantee that any minimizing se-quence will converge macroscopically to a functiong,i.e.:

un g weakly in, e.g., W1,2((0,1)). (89)

In order to obtain a computationally feasible problem,we insist that among all (uncountably many) possibleminimizing sequences having the property(89), theones which become asymptotically weak white noise


are computationally desirable. Our approach consistsin the introduction of aWmicro(x, s) term which canbe relaxed only by such sequences. We demonstrate apossible construction on a one-dimensional problemin the framework of finite element approximation.Let us assume thatuh is an element of some finiteelement space defined on a regular mesh with sizeh.Then we define

Wmicro(u′h)(x)

def= 1

Nh

Nh/2∑k=0

(zkh(x)zkh(x) − 1)2,

where

zkh = N1/2h

∫ 1

0

u′h(x) − g′

h(x)√1 − g2

h′(x)e−2πikx dx. (90)

Here gh represents the projection of the weak limitg in the given finite element space. We evaluate thecoefficientszkh using the fast Fourier transform. Letus seth = 1/n.

Theorem 8.1. There exist asymptotically weak whitenoises sequencesu′

1/nn∈N .

Proof. We give a sketch of the proof. We constructindependent random variablesY1n, Y2n, . . . , Ynn tak-ing value±1 with P [Ykn = 1] = (1 + g′(x))/2 andP [Ykn = −1] = (1 − g′(x))/2. Let

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

u'(x

)

u'(x): Steepest Descent, f=0, N=2048

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

u'(x

)

u'(x): Stochastic Variational Principle, f=0, N=2048

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

0

5x 10

u(x)

u(x): Steepest Descent, f=0, N=2048

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

0

5x 10

u(x)

u(x): Stochastic Variational Principle, f=0, N=2048

Fig. 4. The solutions (left) and their derivatives (right) using both the standard descent method (top) and the proposed SVP method(bottom). Note that the standard descent method produces considerably wilder behavior in both the solution and derivative.

u′1/n(x) =

n∑i=1

Yiφ[(i−1)/n,i/n)](x), (91)

where φA(x) denotes the characteristic function ofthe setA, i.e.,φA(x) = 1 if x ∈ A andφA(x) = 0 ifx /∈ A. Standard probabilistic arguments are used toshow that this is an asymptotically weak white noisewith probability one, hence there exist sequencessatisfying the definition.

Suppose now thatu′h is a statistical white noise,

meaning that

E

u′

h(x) − g′h(x)√

1 − g2h′(x)

u′h(y) − g′

h(y)√1 − g2

h′(y)

=

1 if x andy are in the sameAj,

0 otherwise.(92)

Here,Aj are partition intervals. Then

E(zkhzkh) =Nh

∫ 1

0

∫ 1

0E

u′

h(x)−g′h(x)√

1−g2h′(x)

u′h(y)−g′

h(y)√1−g2

h′(y)

e−2πik(x−y) dx dy. (93)

By the white noise property, the expectation inside theintegral is 0 unlessx andy belong to the same interval


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

λ(x)

λ(x): Steepest Descent, f=0, N=2048

Error: 0.093091

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

λ(x)

λ(x): Stochastic Variational Principle, f=0, N=2048

Error: 0.0054918

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1λ(

x)

1 λ(x): Steepest Descent, f=0, N=2048

Error: 0.089319

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1λ(

x)

1 λ(x): Stochastic Variational Principle, f=0, N=2048

Error: 0.0054918

Fig. 5. The computed volume fractions (see(14)), using derivative tolerancer = 15% and averaging ball sizeR = 30 h in each direction.The plots on the left show the volume fractions corresponding to the deformation gradientF = 1, while those on the right show thosecorresponding to deformation gradientF = −1. Again, the upper plots result from the standard descent algorithm, and the lower plotsresult from the SVP method. We note that the true volume fractions for each should giveλ = 1/2.

in the partition, in which case the expectation is 1.Thus

E(zkhzkh) = Nh

Nh∑j=1

∫Aj

∫Aj

exp[−2πik(x − y)] dx dy

= 1

2π2k2

Nh∑j=1

[1 − cos(2πk(bj − aj)]. (94)

0 50 100 15010

10

10

10

10

10

10

Iteration

Ene

rgy

Minimization histories for each energy term: SVP

total wmacrowmeso wmicro

0 1000 2000 3000 4000 5000 6000 7000 800010

10

10

10

10

10

10

10

10

Iteration

Ene

rgy

Minimization histories for each energy term: SD

total wmacrowmeso

Fig. 6. The minimization histories of each term in the objective function per iteration (solid line—total objective function; dottedline—solution’s deviation from the shape functiong(x); dashed line—solution’s deviation from the prescribed derivativesF1 and F2;dot-dashed line—extra term from SVP). Those on the left correspond to the standard descent algorithm, while those on the right correspondto the SVP method. In both we see that the overall shape term does little to aid in the minimization, but in the SVP method the additionalterm seems to drive the minimization of the other terms, quickly beating the standard descent method.

If we suppose that the partition intervals are uniform,i.e., Aj = [aj, bj) = [(j − 1)/N, j/N), then the lastexpression is

N2

2π2k2

[1 − cos

(2πk

N

)]

= N2

4π2k2

[2π2k2

N2+O

(1

N4

)]= 1 +O

(1

N2

)


asN → ∞. Hence, we see that the statistical whitenoise property implies that “on average” theWmicroterm will be minimized. In fact, one can show thatif the u′

h is a statistical white noise, then the randomvariableszkhz

kh will not converge to 1 with probability

1—the variance does not even go to 0. This is relatedto the fact that the periodogram is not a consistentestimator of the spectral density function. We may beable to achieve better results by using an analogueof the spectral density estimates which “smooth” theperiodogram. More discussion on these topics may befound in [3].

9. One-dimensional computational example

Using the SVP introduced inSection 8, we at-tempt to solve the simple one-dimensional problemwith constant target functiong(x) = 0, x ∈ [0,1].We use gradient-based relaxation. Using a finite el-ement discretization containing a regular mesh of2048 elements, the proposed variational principle in-deed performed much better than descent algorithm,Definition 4.5, applied to non-perturbated functionalin the generation of the proper microscopic structure.Figs. 4–6show the differences.

Acknowledgements

Cox was supported in part by the grant NSFDMS–9971797. The other two authors were sup-ported in part by the grant NSF DMS-0107539,by the Los Alamos National Laboratory ComputerScience Institute (LACSI) through LANL contractnumber 03891-99-23, as part of the prime contractW-7405-ENG-36 between the Department of Energyand the Regents of the University of California, bythe grant NASA SECTP NAG5–8136, and by thegrant from TRW Foundation.

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Dennis D. Cox received his PhD in Math-ematics from the University of Washing-ton in 1980. He has held positions inmathematics and statistics at the Univer-sities of Wisconsin and Illinois beforejoining the Department of Statistics atRice University in 1993. His primary re-search interests are in statistical functionestimation, spatial statistics, design andanalysis of computer experiments, MonteCarlo methods, and stochastic dynamicalsystems.

Petr Kloucek received his PhD in Math-ematics from the Charles University,Prague, Czech Republic, in 1991. He hasheld positions in ETH Zurich, EPF Lau-sanne and University of Minnesota be-fore he joined Department of Computa-tional and Applied Mathematics at RiceUniversity in 1996. His primary researchinterests are nonlinear partial differentialequations applied to problems in mate-

rials science, finite element method applied to approximation ofmeasure-valued solutions of non-quasiconvex problems and math-ematical modeling of electrolytic processes.

Daniel R. Reynolds received his PhDin May 2003 from the Rice Universityfor his work on a thermodynamic modelgoverning the physics of shape memoryalloy wires.

on the asymptotically stochastic computational modeling of...

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