Asymptotic Expansion for Multiscale Problems on Non-periodic Stochastic Geometries
AASYMPTOTIC EXPANSION FOR MULTISCALE PROBLEMS ONNON-PERIODIC STOCHASTIC GEOMETRIES
MARTIN HEIDA
Abstract.
The asymptotic expansion method is generalized from the periodic set-ting to stationary ergodic stochastic geometries. This will demonstrate that resultsfrom periodic asymptotic expansion also apply to non-periodic structures of a certainclass. In particular, the article adresses non-mathematicians who are familiar withasymptotic expansion and aims at introducing them to stochastic homogenization ina simple way. The basic ideas of the generalization can be formulated in simple terms,which is basically due to recent advances in mathematical stochastic homogenization.After a short and formal introduction of stochastic geometry, calculations in the sto-chastic case will be formulated in a way that they will not look different from theperiodic setting. To demonstrate that, the method will be applied to diffusion withand without microscopic nonlinear boundary conditions and to porous media flow.Some examples of stochastic geometries will be given. Introduction
Homogenization has become an important modeling tool for multiscale problems, i.e.phenomena that have causes and effects on multiple spatial scales. The applica-tion ranges from Physics (porous media flow, freezing processes in porous media,erosion[2, 7, 22]), engineering (composite materials, reaction diffusion equations incatalysts[19, 11]) to biology (processes in tissue[13], membranes[20]). Also the numer-ical investigation of such problems is of interest (see [16] and references therein).Homogenization considers physical, chemical or biological processes on large domainswith a periodic microstructure of period ε (cid:28) . On this periodic structure, a setof equations describing the several processes are set up and the limit behavior of thesolutions of these equations is investigated as ε → . Note that ε → is a mathematicalequivalent of the assumption ε (cid:28) , which means basically very small microstructurescompared to the domain of interest. For example, one may think of a solid structureperforated by a periodic structure of channels, which are filled by a Newtonian fluid, i.e.a fluid whose motion is described by the Navier-Stokes equations. Since one is usuallyinterested in a scale of meters and the pores have sizes of mm , it follows ε ≈ − . Inthe limit ε → , the velocity field would follow Darcy’s law [10] (see sections 2.2 and4.2).There are two ways to obtain the limit equations: Via strong mathematical calculations(proofs of convergence of solutions) or via rather formal calculations. One of the formalways to obtain averaged equations is the so called asymptotic expansion. Its ansatzis to expand the solution in a series of functions multiplied with increasing powers of ε , where the functions in this expansion depend on a global variable and a periodicvariable. We will demonstrate the basics of this approach below in section 2.Strong mathematical investigations where originally based on strong and weak conver-gence methods in Sobolev spaces, which is nowadays still a method of choice. However, Mathematics Subject Classification.
Key words and phrases. stochastic homogenization, asymptotic expansion, two-scale. a r X i v : . [ m a t h - ph ] J a n MARTIN HEIDA lots of other convergence methods entered homogenization theory such as Γ - and G -convergence, see [26] for an overview.In the beginning of the 1990’s, Allaire and Nguetseng [21, 1] developed the so calledtwo-scale convergence which was later extended by Neuss-Radu, Zhikov and Lukkassenand Wall [18, 25, 12]. As an interesting feature, two-scale convergence is related toan asymptotic expansion in the weak sense and therefore closely connected to theformal method. This similarity becomes even more striking by the method of periodicunfolding, developed by Cioranescu, Damlamian and Griso [5, 4], and which can beconsidered as a consequent generalization of two-scale convergence. In comparing themethods, one may get the idea that every result from formal asymptotic expansioncould be proved rigorously. However, for many homogenization problems there are lotsof technical difficulties like regularity proofs or a lack of Poincaré-inequalities whichare sometimes to hard to be overcome.There is, however, the justified criticism, that the methods above are based on theassumption of a periodic structure and that the resulting equations therefore maynot be valid for a non-periodic medium. Nevertheless, we expect that many of thederived equations may also hold for non-periodic microscopic geometries: Darcy’s law isobserved to hold in most porous media may it be sand, silt or loam. In a polycrystal or acomposite material (industrial ceramics) with their complex microstructure, we assumethat heat transport still follows Fourier’s law with averaged coefficients. Reactiondiffusion equations derived for a catalyzer should also hold if the microscopic structureis not perfectly periodic.So, it is an important question whether or not it is possible to apply homogenizedresults to non-periodic geometries and under what circumstances. Mathematicianswere aware of this problem and tried to apply homogenization techniques to non-periodic structures. An overview over many results before 1994 can be found in [26].In 2006, Zhikov and Piatnitsky [27] where able to introduce a two-scale convergencemethod on stationary, ergodic stochastic geometries on a compact probability space.Their results where generalized to arbitrary probability spaces by the author in [8]. Aformer attempt by Bourgeat, Mikelić and Wright [3] in the non-ergodic setting usedan averaged form of two-scale convergence and was not applicable to problems onmanifolds or complex boundary conditions. Also, due to the averaging, it was lessclose to the periodic setting than [27].The results in [8] provide a geometrical interpretation of subsets of the probabilityspace and demonstrates that all periodic quantities find their precise analogue in thestochastic case and vice versa. Moreover, if a stationary ergodic structure is periodic,the probability space has to be essentially the unit cell. The theory has been appliedsuccessfully to heat transfer in polycrystals in [9].Due to the connection between two-scale convergence and asymptotic expansion inthe periodic setting, these mathematical results give the basic idea to extend the as-ymptotic expansion to the stochastic case. Note that it is not the intention of thisarticle, to give a rigorous mathematical introduction to stochastic geometry or sto-chastic homogenization. Rather it is the aim of this paper to demonstrate that it canbe mathematically justified to apply results from asymptotic expansion to stochasticsettings and that there is formally no difference in the calculations. The most importanttheoretical results are cited in the appendix. For proofs of the fundamental theoremswhich are cited in this article, the reader is referred to [27, 8] and the references therein.This article is organized as follows: In the next section, the basic idea of periodicasymptotic expansion is explained and the resulting equations for two sample problems SYMPTOTIC EXPANSION ON STOCHASTIC GEOMETRIES 3 (diffusion and porous media flow) are given. In section 3, the concepts of stochasticgeometry will be introduced in a formal and mathematical non rigorous way. Theconnections to periodic geometries are explained. In section 4 the asymptotic expansionon stochastic geometries will be introduced and applied to thermal diffusion, porousmedia flow and diffusion processes with reactions on the microscopic boundary. Insection 5 some simple examples of stochastic geometries will be given.2.
Recapitulation of the Periodic Case
In this section we will shortly recapitulate the asymptotic expansion technique forperiodic structures. It is not the intention of this section to go into the details ofcalculations but to rather explain what are the mathematical problems, the ansatzthat people use to solve these problems and what the results look like. For bothexamples, precise calculations will follow in section 4 below. We will first consider thestandard homogenization problem of diffusion (following [10]) and then discuss Navier-Stokes flow through porous media. In all calculations below and throughout the paper, n denotes the dimension of space (in a physical setting n = 2 or n = 3 ).2.1. The standard problem.
Given an open and bounded set Q ⊂ R n investigatediffusion processes (for example heat diffusion) with a source term and rapidly oscillat-ing coefficients which are due to a periodic microstructure. In particular, assume thatfor Y := [0 , n we have Y = A ∪ B ∪ Γ with A , B open in Y such that A ∩ B = ∅ and Γ = ∂A = ∂B in Y . We expand Y periodically to R n and identify A , B and Γ withtheir periodic continuation. We furthermore denote A ε = εA , B ε := εB and Γ ε := ε Γ .The diffusion constants in A and B are thought to be different.The mathematical problem reads ∂ t u ε − div ( D ε ∇ u ε ) = f on ]0 , T ] × Q u ε = 0 on ]0 , T ] × ∂ Q u ε (0 , · ) = a ( · ) on Q for t = 0 where D ε is given by D ε := D ( t, x, xε ) and D is a Y -periodic symmetric tensor withbounded entries, while f is a function on Q which may also depend on time. Asan example, we could have D ε ≡ on A ε and D ε ≡ on the interior of B ε with D ( t, x, y ) = 1 + 9 χ B ( y ) .For small enough microstructures, the oscillation in D may have a minor effect on u ε and the system may be described by some averaged diffusion coefficient D hom . Itmay therefore be sufficient to solve an approximating system with this smoothed D hom without the strong oscillations of D ε . Therefore, we investigate u ε as ε → by anansatz(2.1) u ε ( t, x ) = (cid:88) i ≥ ε i u i ( t, x, xε ) which is called the asymptotic expansion of u ε and u i : R ≥ × Q × Y → R which are Y -periodic functions in the third coordinate. The gradient operator turnsinto ∇ = ∇ x + ε ∇ y while the divergence reads div = div x + ε div y . Altogether, the MARTIN HEIDA expansions above inserted into the diffusion equation yield ∂ t u − ε div y ( D ε ∇ y u ) − ε div x ( D ε ∇ y u ) − ε div y ( D ε ∇ x u ) − ε div y ( D ε ∇ y u ) − div x ( D ε ∇ x u ) − div x ( D ε ∇ y u ) − div y ( D ε ∇ x u ) − div y ( D ε ∇ y u ) + O ( ε ) = f Assuming that the terms of order ε − and ε − vanish, it will be shown below in section4.1 that this system simplifies to a single equation for u ∂ t u − div x (cid:0) D hom ∇ x u (cid:1) = (cid:90) Y f with D hom defined in an appropriate way.2.2. Porous media flow.
Suppose that A ε is connected and consider the Navier-Stokes equations on Q ∩ A ε : ∂ t υ ε + ( υ ε · ∇ ) υ ε − div ( ν ∇ υ ε ) + ∇ p ε = f on ]0 , T ] × ( Q ∩ A ε ) div υ ε = 0 on ]0 , T ] × ( Q ∩ A ε ) υ ε = 0 on ]0 , T ] × ( ∂ Q ∪ Γ ε ) υ ε = 0 on ]0 , T ] × B ε υ ε (0 , · ) = a ( · , · ε ) on Q for t = 0 where f is an external force and a is Y periodic in the second argument. For an ansatz υ i : Ω × Y → R p i : Ω × Y → R ( x, y ) (cid:55)→ υ i ( x, y ) ( x, y ) (cid:55)→ p i ( x, y ) such that the solution υ ε and p ε can be described by υ ε = ∞ (cid:88) i =0 ε i υ i ( x, xε ) p ε = ∞ (cid:88) i =0 ε i p i ( x, xε ) , the resulting set of equations is − div y ( ν ∇ y υ ) + ∇ x p + ∇ y p = g div y υ = 0 (2.2) υ ( x, · ) = 0 on ∂Y We will see in section 4.2 that for a suitable choice of a matrix K , υ and p fulfill thefollowing equation: (cid:90) y υ = K ( g − ∇ x p ) which is Darcy’s law. 3. Stochastic Geometry
In order to develop the asymptotic expansion method for the stochastic (non-periodic)case, it is necessary to develop a suitable mathematical formalism. This will be the aimof the first part of this section. As a necessary condition, we expect that the periodiccase is but a specialization of the stochastic one, which will be shown in the secondpart. The basic idea of the theory described below is to replace Y by a probabilityspace Ω . It is then necessary to identify equivalents of A , B and Γ on Ω similar to theperiodic case. After that, the relations to periodic structures are pointed out, beforegoing on the asymptotic expansion and examples for stochastic geometries. SYMPTOTIC EXPANSION ON STOCHASTIC GEOMETRIES 5
A phenomenological introduction to stochastic geometry.
Instead of theperiodic cell Y , consider a probability space (Ω , σ, µ ) with the probability set Ω , thesigma algebra σ and the probability measure µ . Assume that Ω is a metric space andthere is a family ( τ x ) x ∈ R n of measurable bijective mappings τ x : Ω (cid:55)→ Ω which satisfy • τ x ◦ τ y = τ x + y • µ ( τ − x B ) = µ ( B ) ∀ x ∈ R n , B ∈ B (Ω) • A : R n × Ω → Ω ( x, ω ) (cid:55)→ τ x ω is continuousThe family τ x is then called a Dynamical System. Additionally, we claim ergodicity of τ • , which is one of the following two equivalent conditions [ f ( ω ) = f ( τ x ω ) ∀ x ∈ R n , a.e. ω ∈ Ω] ⇒ [ f ( ω ) = const for µ − a.e. ω ∈ Ω][ P (( τ x ( B ) ∪ B ) \ ( τ x ( B ) ∩ B )) = 0] ⇒ [ P ( B ) ∈ { , } ] . (3.1)This condition seems to be only technical, but in fact, it is crucial for mathematicalhomogenization in [8, 27] and replaces periodicity as well as the requirement that Y issimply connected.In order to introduce stochastic geometries from a phenomenological point of view,assume that there is a measurable set A ⊂ Ω such that for the characteristic function χ A : Ω → { , } holds χ A ( τ • ω ) : R n → { , } is the characteristic function of a closedset A ( ω ) ⊂ R n µ − almost surely in ω . A ( ω ) is then called a random closed set . Notethat it possesses the property χ A ( ω ) ( x + y ) = χ A ( τ x ω ) ( y ) which is called stationarity.For some of the examples below, we assume at the same time that there is B ⊂ Ω suchthat B ( ω ) = R n \ A ( ω ) is the closure of the complement of A ( ω ) . The set Γ := A ∩ B is then also measurable and has the property that Γ( ω ) = ∂A ( ω ) . Indeed, as shown inthe appendix, Γ can be considered as an abstract manifold, since it can be assigned aHausdorff measure and a normal field.Since the space Ω is metric, there is a set of continuous and bounded functions C b (Ω) .At the same time, since A and B are measurable subsets, it is also possible to considercontinuous functions C b ( A ) and C b ( B ) on A and B . With help of the dynamical system τ • , it is possible to also define derivatives according to(3.2) ∂ ω,i f ( ω ) := lim h ∈ R → f ( τ he i ω ) − f ( ω ) h where ( e i ) ≤ i ≤ n is the canonical Basis on R n and f ∈ C b (Ω) . The space of continuouslydifferentiable functions is then C (Ω) := { f ∈ C (Ω) | limit (3.2) is defined ∀ ω ∈ Ω and ∂ ω,i f ∈ C (Ω) ∀ ≤ i ≤ n } . Similar spaces may also be defined on A and B . We may also define the gradient ∇ ω and the divergence div ω accordingly. For any set X , any function f : Ω → X andany ω ∈ Ω , the function f ( τ • ω ) : R n → X is called a realization ( ω -realization) of f and for any continuous or differentiable function the realization is also continuousor differentiable, respectively. This is due to the continuity of τ . In particular, for f ∈ C b ( A ) holds f ( τ • ω ) ∈ C b ( A ( ω )) and similarly for B and Γ . which means there is a distance function d : Ω × Ω → R ≥ such that d ( ω , ω ) = 0 ⇔ ω = ω , d ( ω , ω ) = d ( ω , ω ) and d ( ω , ω ) ≤ d ( ω , ω ) + d ( ω , ω ) for any ω , ω , ω ∈ Ω . But this is onlytechnical to be able to use the notion continuity. Mathematically, as shown in the appendix, a random closed set is defined via a mapping A ( ω ) intothe set of closed sets. The existence of sets A, B, Γ ⊂ Ω such that the above properties are fulfilled isshown afterward. For formal calculations, it seems more appropriate to start the other way round. MARTIN HEIDA
Finally, for a scaled random set A ε ( ω ) := εA ( ω ) holds χ A ε ( ω ) ( x ) = χ A ( ω ) ( xε ) = χ A ( τ xε ω ) , and for a function f ∈ C b ( A ) holds f ( τ • ε ω ) ∈ C b ( A ε ( ω )) for all the realizations ω .3.2. The periodic case.
It is helpful to compare the results above to the periodiccase. First, note that Y = [0 , n together with the Borel sigma algebra σ B and thestandard Lebesgue measure L can be considered as a probability space ( Y , σ B , L ) . Forany x ∈ R n let [ x ] ∈ Z n denote the vector of integers such that x ∈ [ x ] + Y . Definethe following family of mappings τ x : Y → Y y (cid:55)→ y + x − [ y + x ] which has all the properties we claimed for a dynamical system to hold. For any closedset A ⊂ Y , A ( y ) is automatically closed for all y ∈ Y and A (0) is the periodic contin-uation of A . The continuous functions on Y coincide with the Y -periodic continuousfunctions on R n and the derivatives ∂ y,i coincide with the classical derivatives ∂ i . B isthe closed complement of A in Y and Γ is the boundary of A in Y .It is therefore clear, that any method which is based on the formalism introduced aboveis at least applicable to the periodic setting. Section 5 will demonstrate that a muchbroader class of geometries is covered by this approach.4. Asymptotic expansion on stochastic geometries
We are now able to introduce asymptotic expansion on stochastic geometries. Themethod will be introduced via a sample calculation for the standard homogenizationproblem, which is diffusion with rapidly oscillating coefficients. This will be done usingan expansion (4.1) of the unknown similar to (2.1) in section 2.1. Afterward, themethod will be applied to porous media flow and diffusion with nonlinear microscopicboundary conditions. The formal calculations are well known for the periodic case andmost of them can be found in [10] among others. Here, they will be presented in thestochastic framework to demonstrate the similarity with the periodic case.4.1.
The standard homogenization problem.
The standard problem in the sto-chastic setting reads ∂ t u ε − div ( D εω ∇ u ε ) = f on ]0 , T ] × Q u ε = 0 on ]0 , T ] × ∂ Q u ε (0 , · ) = a ( · ) on Q for t = 0 where now D εω = D ( t, x, τ xε ω ) . An example would be D ( · , · , ω ) = 1 for ω ∈ A and D ( · , · , ω ) = 2 for ω ∈ B .While the original idea is to expand the functions u ε asymptotically by (2.1) withfunctions u i : R ≥ × Q × Y → R , it will now be based on a stochastic ansatz. Inparticular, for a given choice of the microscopic geometry ω with the micro structures A ε ( ω ) , B ε ( ω ) and Γ ε ( ω ) , we may expand u ε by(4.1) u ε ( t, x ) = (cid:88) i ε i u i ( t, x, τ xε ω ) . SYMPTOTIC EXPANSION ON STOCHASTIC GEOMETRIES 7
The gradient and the divergence have the expansions(4.2) ∇ = ∇ x + 1 ε ∇ ω , div = div x + 1 ε div ω Using an expansion (4.1) for u ε with (4.2) will lead to ∂ t u − ε div ω ( D ε ∇ ω u ) − ε div x ( D ε ∇ ω u ) − ε div ω ( D ε ∇ x u ) − ε div ω ( D ε ∇ ω u ) − div x ( D ε ∇ x u ) − div x ( D ε ∇ ω u ) − div ω ( D ε ∇ x u ) − div ω ( D ε ∇ ω u ) + O ( ε ) = f which means we split up the equation in terms ε − ( . . . ) + ε − ( . . . ) + ε ( . . . ) + O ( ε ) = 0 . Since the latter equation should hold for all ε , it followsdiv ω ( D ( t, x, ω ) ∇ ω u ) = 0 . Note that in the periodic case, the latter equation implies u ( y ) = const , while in thestochastic case, this is not clear. However, due to Gauss’ theorem 6 for dynamicalsystems it follows by testing the equation with u : (cid:90) Ω D ( t, x, ω ) |∇ ω u ( t, x, ω ) | dµ ( ω ) = 0 which is ∇ ω u = 0 . Definition (3.1) of ergodicity yields u ( ω ) = const . The terms oforder ε − yield div ω ( D ε ∇ x u ) = − div ω ( D ε ∇ ω u ) . Now, if φ j is a solution to the cell problemdiv ω ( D ( t, x, ω ) ∇ ω φ j ) = − div ω ( D ( t, x, ω ) e j ) for j = 1 , . . . , n , the function u can be expressed by u ( t, x, ω ) = n (cid:88) j =1 φ j ( t, x, ω ) ∂ j u ( t, x ) . Existence of φ j with (cid:82) Ω φ j = 0 can be shown with help of the Poincaré inequalitiesfrom the appendix and the Lax-Milgram theorem.Finally, the zero order terms add up to(4.3) ∂ t u − div ω ( D ∇ x u − D ∇ ω u ) − div x ( D ∇ x u ) − div x ( D ∇ ω u ) = f . By integrating the latter equation over Ω with help of Gauss’ theorem 6, one obtains ∂ t u − div x (cid:18)(cid:90) Ω D dµ ∇ x u (cid:19) − (cid:90) Ω div x ( D ∇ ω u ) = (cid:90) Ω f dµ . The third term on the lefthand side can be reformulated to (cid:90) Ω div x ( D ∇ ω u ) = div x (cid:32)(cid:90) Ω D (cid:88) j ∇ ω φ j ∂ j u (cid:33) which finally yields ∂ t u − div x (cid:0) D hom ∇ x u (cid:1) = (cid:90) Ω f dµ . Here, D hom = (cid:0) D homi,j (cid:1) = (cid:90) Ω ( e i D ∇ ω φ j + D i,j ) dµ is a symmetric positive definite matrix. The proof is the same as in [10]. MARTIN HEIDA
Remark.
The obtained limit problem is independent on the particular choice of therealization ω which we used for homogenization. This is a feature of the ergodicityand stationarity and reflects our expectation that the averaged equations should notdepend on the specific geometry but on “the type” of geometry. We will see that theother examples share this property of the limit equations.4.2. Porous media flow.
Suppose that A ε is connected and consider the Navier-Stokes equations on Q ∩ A ε : ∂ t υ ε + ( υ ε · ∇ ) υ ε − div ( ν ∇ υ ε ) + ∇ p ε = f on ]0 , T ] × ( Q ∩ A ε ( ω )) div υ ε = 0 on ]0 , T ] × ( Q ∩ A ε ( ω )) υ ε = 0 on ]0 , T ] × ( ∂ Q ∪ Γ ε ( ω )) υ ε = 0 on ]0 , T ] × B ε ( ω ) υ ε (0 , · ) = a ( · , · ε ) on Q for t = 0 where f is an external force and a is Y periodic in the second argument. From physicalinvestigations, we know that the flow through a porous medium obeys Darcy’s law andit is the aim of the following calculations to demonstrate that it is possible to obtainit as a limit problem for vanishing porescale in the stochastic setting.For an ansatz υ i : R ≥ × Q × Ω → R p i : R ≥ × Q × Ω → R ( t, x, ω ) (cid:55)→ υ i ( t, x, ω ) ( t, x, ω ) (cid:55)→ p i ( t, x, ω ) such that the solution υ ε and p ε can be described by υ ε = ∞ (cid:88) i =0 ε i υ i ( t, x, τ xε ω ) p ε = ∞ (cid:88) i =0 ε i p i ( t, x, τ xε ω ) , (4.4)the resulting set of expanded equations is up to order :(4.5a) ε − ( − div ω ( ν ∇ ω υ )) + ε − (( υ · ∇ ω ) υ − div ω ( ν ∇ ω υ ) − div x ( ν ∇ ω υ ) + ∇ ω p )+ ε (( υ · ∇ x ) υ + ( υ · ∇ ω ) υ + ( υ · ∇ ω ) υ )+ ε ( − div ω ( ν ∇ ω υ ) − div x ( ν ∇ x υ ) + ∇ x p + ∇ ω p − g ) + O ( ε ) = 0 (4.5b) ε − div ω υ + div x υ + div ω υ + ε ( div x υ + div ω υ ) + O ( ε ) = 0 (4.5c) (cid:88) i ε i υ i ( x, · ) = 0 on ∂Y , (cid:88) i ε i υ i ( · , ω ) = 0 on ∂ Ω For each power of ε , a set of equations is obtained, which has to hold independently onall the other equations such that the whole group of equations is valid for all choicesof ε .The order of − in (4.5a) together with the order − in (4.5b), and (4.5c) yields for υ ε − : − div ω ( ν ∇ ω υ ) = 0 on A div ω υ = 0 on A (4.6) υ ( x, · ) = 0 on Γ . SYMPTOTIC EXPANSION ON STOCHASTIC GEOMETRIES 9
Again, in the periodic case, it would immediately follow υ ≡ . However, Gauss’theorem 6 again provides the necessary framework since (cid:90) A ν |∇ ω υ | dµ = 0 implies ∇ ω υ = 0 which is together with ergodicity (3.1) υ ( ω ) = const and with (4.6) finally υ = 0 .The latter result together with the order − in (4.5a), order in (4.5b) and (4.5c)yields for υ − div ω ( ν ∇ ω υ ) + ∇ ω p = 0 div ω υ = 0 (4.7) υ ( x, · ) = 0 on ∂Y which is again υ ≡ due to (cid:82) A υ · ∇ ω p = 0 and (4.7) .Using these results in the zero-order term in (4.5a), order 1 in (4.5b) and order in(4.5c), we get − div ω ( ν ∇ ω υ ) + ∇ x p + ∇ ω p = g div ω υ = 0 υ ( x, · ) = 0 on Γ (4.8)Assuming that there are solutions to the problems − div ω ( ν ∇ ω u i ) + ∇ ω Π i = e i div ω u i = 0 u i ( x, · ) = 0 on Γ where e i is the i-th coordinate vector of R , it is easy to see that there is p and υ := (cid:80) ( g − ∇ x p ) i u i such that ( υ , p , p ) is a solution to (4.8). Existence of u i maybe again shown similarly to the previous example. Defining a matrix K by K i,j := (cid:90) A ∇ ω u i · ∇ ω u j = (cid:90) A u i · e j one may check that we obtain indeed Darcy’s law which was expected: (cid:90) A υ = K ( g − ∇ x p ) Diffusion with nonlinear boundary conditions.
Based on the calculationsin section 4.1, consider the following problem ∂ t u ε − div ( D εω ∇ u ε ) = f on ]0 , T ] × ( Q ∩ A ε ( ω )) u ε = 0 on ]0 , T ] × ∂ Q ( D εω ∇ u ε ) · ν Γ ε ( ω ) = εg ( u ε , U ε ) on ]0 , T ] × (Γ ε ( ω ) ∩ Q ) u ε (0 , · ) = a ( · , τ • ε ω ) on Q ∩ A ε ( ω ) for t = 0 combined with the nonlinear boundary problem ∂ t U ε + g ( u ε , U ε ) = 0 on ]0 , T ] × (Γ ε ( ω ) ∩ Q ) U ε (0 , · ) = ˜ U ( · , τ • ε ω ) on Γ ε ( ω ) ∩ Q for t = 0 . This system describes diffusion with a production or uptake due to reactions on themicroscopic boundaries. The ε -factor in the boundary condition takes into accountthat the microscopic surfaces increase with a factor ε − . From physical perspective, we expect that the reactions on the walls will lead to a macroscopic production or uptaketerm in the limit problem.Using an expansion (4.1) for u ε and a similar expansion for U ε , one would immediatelyconclude for w ∂ t w + g ( u , U ) = 0 on ]0 , T ] × Q × Γ U (0 , · ) = ˜ U ( · , ω ) on Q × Γ for t = 0 . For the expansion of u ε one again obtains ∂ t u − ε div ω ( D ε ∇ ω u ) − ε div x ( D ε ∇ ω u ) − ε div ω ( D ε ∇ x u ) − ε div ω ( D ε ∇ ω u ) − div x ( D ε ∇ x u ) − div x ( D ε ∇ ω u ) − div ω ( D ε ∇ x u ) − div ω ( D ε ∇ ω u ) + O ( ε ) = f together with the following boundary conditions on Γ :(4.9) ε ( D ∇ ω u ) · ν Γ + 1 ε D ( ∇ x u + ∇ ω u ) · ν Γ + D ( ∇ x u + ∇ ω u ) · ν Γ − g ( u , U ) = 0 . It is again possible to obtain with slightly modified argumentation in the partial inte-grations u ( ω ) = const as well as u ( t, x, ω ) = n (cid:88) j =1 φ j ( t, x, ω ) ∂ j u ( t, x ) , where now the φ j are solutions todiv ω ( D ( t, x, ω ) ∇ ω φ j ) = − div ω ( D ( t, x, ω ) e j ) on A ( D ∇ ω φ j + De j ) · ν Γ = 0 on Γ . The zero order terms in (4.3) are now integrated over A which yields together with thezero order boundary condition in (4.9): ∂ t u − div x (cid:0) D hom ∇ x u (cid:1) − (cid:90) Γ g ( u , U ) dµ = (cid:90) Ω f dµ . where now D hom = (cid:0) D homi,j (cid:1) = (cid:90) A ( e i D ∇ ω φ j + D i,j ) dµ . We therefore find the nice property that the macroscopic unknown u is produced orconsumed by microscopic reactions on the surfaces, which was expected.5. Non-periodic models captured by stochastic geometries
We will now describe some basic modeling tools for stochastic geometries and give somesimple but interesting models for different applications in microstructures. Note thatfor many real world applications, it is up to now not possible to give a model for thenatural geometries. For example natural soil with roots, wormholes and other obstaclesis currently out of reach. Therefore, the models given here can only be considered asvery rough models for porous media and other applications. Also, we will restrict onthe most simple models. For more models in stochastic geometry, refer to Stoyan,Kendall and Mecke [23].
SYMPTOTIC EXPANSION ON STOCHASTIC GEOMETRIES 11
Figure 5.1.
Left: Poisson-Voronoi tessellation with its underlying Pois-son point process; Right: The corresponding Delaunay tessellation. Thepoint processes and tessellations where constructed using the “StochasticGeometry 4.1” software, developed at the TU Bergakademie Freiberg,Institut für Stochastik.5.1.
Point processes.
A point process (PP) is a measurable mapping A : Ω → ( R n ) N .Let P ( N, Q, p ) denote the probability to find N points in the bounded and open set Q ⊂ R n for the point process p . For some open set V ⊂ R n with V (cid:51) , define theintensity λ p by λ p := lim t →∞ t n | V | E ( P ( · , tV, p )) , where E ( P ( . . . )) is the expectation value of N . According to [23] a PP is stationaryif its characteristic is invariant under translation, i.e. P ( N, Q, p ) = P ( N, Q + x, p ) forall x ∈ R n . From a stochastic point processes, one may construct higher dimensionalstructures in a deterministic way. Such structures would still be stationary randomclosed set. We will come to that point below.A prominent example of a PP is the so called stationary Poisson point process whichhas the property that P ( N, Q, p ) = λ Np | Q | N N ! exp ( − λ p | Q | ) An other useful class of PP are so called hard core PP: From a given random PP anypoint is erased if its nearest neighbor has a distance less than a certain value d .5.2. Voronoi-Tessellations and Delaunay-Diagram.
Starting from a PP, one mayconstruct the so called Voronoi-tessellation as the set of all points in R n , whose nearesttwo neighbors of the PP have the same distance. This can be used for example as amodel of polycrystals[8]: The point process represents the set of crystallization nucleiwho start to grow at the same time with the same speed and who stop growing themoment they hit each other.A generalization of this model which models crystal growth starting at different timeswith different speed is the so called Johnson-Mehl-tessellation.As a dual of the Voronoi-tessellation, one can consider the Delaunay-diagram whichconnects all points of the PP who share a common border in the Voronoi-tessellation.Connecting the points with cylinders (pipes) instead of lines, one would already get amodel for a porous medium. Examples of both are shown in figure 5.1 Figure 5.2.
A ball process based on a Mattern hard core point processof type 2 (generated with Stochastic Geometry 4.1)5.3.
Ball- and Grain-models.
Based on a hard core PP, assign to each point of thePP a ball with a fixed radius (Figure 5.2). This model could be used to model a porousmedium in 2D. However, in 3D we expect the grains of the porous medium matrix totouch some neighbors and that none of them is free without touching any neighbor.To achieve such a geometry, consider the following construction:From a Voronoi-tessellation consider for each point P c of the PP the set of points P b,i where the corresponding Delaunay-diagram hits the grain boundaries. We may theninterpolate these points by a smooth manifold to mark the surface of a grain. As anexample, define d ( P c , x ) := (cid:88) i ( x − P c ) ( P b,,i − P c ) (cid:89) j (cid:54) = i ( x − P b,,j ) ( P b,i − P b,,j ) , G ( P c ) := { x ∈ R n : d ( P c , x ) ≤ } The sets G ( · ) could then model the grains of sand and the complement of the G wouldbe the porous medium. 6. Conclusion
We saw that asymptotic expansion can also be applied to stochastic geometries if theyfulfill certain conditions, in particular stationarity and ergodicity. The method wasapplied to diffusion with and without nonlinear microscopic boundary conditions andto porous media flow. Some examples for stochastic geometries where given for solidand porous microstructures. Note that we did not treat homogenization of problems ∂ t u ε − div ( ε D ε ∇ u ε ) = f with ε -scaled diffusion. However, such problems can betreated even easier than the example calculation above with results of the form ∂ t u − div ω ( D ∇ ω u ) = f .The results of this article can be considered as one more step towards understandinghomogenization in non-periodic heterogeneous media. A big advantage of the presentedapproach is, that formal calculations require only few new theory. The only formaldifference in the calculation is, that Y has to be changed into Ω and y into ω . Althoughthe mathematical theory behind the rigorous calculations gets much more complex,the method’s user is not bothered by that. Indeed, besides some critical points in thecalculations which were pointed out, one may not even think of the fact that one isdoing stochastic calculations. The limit equations are independent on the particularrealization which was used for homogenization. This is a feature of ergodicity andstationarity of the geometries.A major issue for further investigations is the search for suitable models for geometriesof natural heterogeneous media. Also, from the mathematical point of view, it would SYMPTOTIC EXPANSION ON STOCHASTIC GEOMETRIES 13 be nice to develop a “stochastic unfolding” corresponding to the periodic unfolding as afurther theoretical foundation of the presented asymptotic expansion. Finally remark,that the intention of this paper was not to give a rigorous introduction to stochastichomogenization or the mathematics behind it but only to demonstrate that for non-mathematicians, switching from periodicity to stationary ergodic geometries can beeasily achieved and would not bother the calculations or the results.
Appendix A. Stochastic Geometries
This section follows [8] to introduce some basic mathematical concepts on randomgeometries. In particular, random closed sets and random measures will be definedrigorously. The connection to the periodic case was explained in detail in [8] and isbased on similar ideas as the connections between periodic and stochastic asymptoticexpansion. It will be shown that the assumptions on A , B and Γ in section 3.1 arereasonable and can be made w.l.o.g.. For more information on random closed sets,refer to Matheron [14] or Molchanov [17].Let F ( R n ) denote the set of all closed sets in R n . Then for K ⊂ R n compact and V ⊂ R n the following sets can be defined: F V := { F ∈ F ( R n ) | F ∩ V (cid:54) = ∅} V ⊂ R n open set(A.1) F K := { F ∈ F ( R n ) | F ∩ K = ∅} K ⊂ R n compact set(A.2)The Fell topology on F ( R n ) is created by the sets F V , F K for all open V and compact Kand according to [14], ( F ( R n ) , T F ) is compact, Hausdorff and separable. The Matheron- σ -field σ F is the Borel- σ -algebra created by the Fell-topology.For a probability space (Ω , σ, µ ) , a Random Closed Set (RACS) is a measurable map-ping ˜ A : (Ω , σ, µ ) −→ ( F , σ F ) . It is the aim of this short introduction to demonstrate that such RACS have the niceproperties which we assumed on A , B and Γ from section 3.1. Remark that up to now,a RACS is a mapping from a probability space into the set of closed subsets of R n .In what follows, M ( R n ) respectively M denotes the set of all locally finite Borelmeasures on R n . In Particular, L denotes the Lebesgue measure and H m the m -dimensional Hausdorff measure. The smallest topology such that M → R , ˜ µ (cid:55)−→ (cid:90) f d ˜ µ is continuous for all f ∈ C ( R n ) is in general called the Vague-topology T V on M ( R n ) .The Borel- σ -field of this topology is denoted by σ V respectively by B ( M ) . Theorem . [6] ( M , T V ) is a separable metric space. The σ -algebra σ V on M createdby the Vague topology T V is the smallest one, such that µ (cid:55)→ µ ( B ) is measurable forevery bounded and measurable set B ⊂ R n .A random measure on R n is a measurable mapping (Ω , σ, µ ) → ( M , σ V ) , ω (cid:55)→ µ ω ,where (Ω , σ, µ ) is a probability space. Random closed sets and the random measuresare related due to the following Lemma, which implies that a random closed set alwaysinduces a random measure. Lemma . ([24] Theorem 2.1.3 resp. Corollary 2.1.5) Let F m ⊂ F be the space of closed m-dimensional sub manifolds of R n such that thecorresponding Hausdorff measure is locally finite. Then, the σ -algebra σ F ∩ F m is thesmallest such that M mB : F m → R Γ (cid:55)→ H m (Γ ∩ B ) is measurable for every measurable and bounded B ⊂ R n .It was part of the argumentation in [8] that due to Lemma 2 for any initial randomclosed manifold A : (Ω , σ, µ ) → F m the assumptions Ω ⊂ M and σ = B ( M ) ∩ Ω canbe made w.l.o.g.. We introduced dynamical systems in section 3.1 and also definedergodicity. However, one normally assumes only measurability of τ but the continuityis a rather direct consequence of Ω ⊂ M [8].A random measure is said to be stationary if for every ω ∈ Ω : T ( x ) µ ω = µ τ x ω with T ( x ) µ ω ( B ) := µ ω ( B + x ) ∀ B ∈ B ( R n ) and a RACS A is called stationary if χ A ( τ y ω ) ( x ) = χ A ( ω ) ( x + y ) where χ A ( ω ) is thecharacteristic function of A ( ω ) . This is slightly different introduction of stationaritythan in section 3.1. However, we will see below that these definitions of stationarity andergodicity already guaranty that RACS have the properties which we claimed in section3.1. First, it is necessary to show that there is an equivalent of a Hausdorff-measureon Γ . This Hausdorff measure will be µ Γ , P , stated by the following Theorem . (Mecke [15, 6]: Existence of Palm measure)Let L be the Lebesgue-measure on R n with dx := d L ( x ) and (Ω , σ, µ ) as above. Thenthere exists a unique measure µ P on Ω such that(A.3) (cid:90) Ω (cid:90) R n f ( x, τ x ω ) dµ ω ( x ) dµ ( ω ) = (cid:90) R n (cid:90) Ω f ( x, ω ) dµ P ( ω ) dx for all B ( R n ) × B (Ω) -measurable non negative functions and all µ P × L - integrablefunctions. µ P as in Theorem 3 is called the Palm measure of µ ω . Note that µ is the Palm measureof L . For a random closed manifold Γ : (Ω , σ, µ ) → F m ω (cid:55)→ Γ( ω ) the randomHausdorff measure will be denoted by ω (cid:55)→ µ Γ( ω ) and the Palm measure by µ Γ , P . Theorem . (Ergodic Theorem [6])Let the dynamical System τ x be ergodic and assume that the stationary random mea-sure µ ω has finite intensity. Then(A.4) lim t →∞ t n L ( A ) (cid:90) tA g ( τ x ω ) dµ ω ( x ) = (cid:90) Ω g ( ω ) dµ P ( ω ) for µ almost surely in ω for all bounded Borel sets A ⊃ {| x | < } and all g ∈ L (Ω , µ P ) For µ ε Γ( ω ) ( B ) := ε n µ Γ( ω ) ( ε − B ) the latter theorem yields after a transformation ofvariables(A.5) lim ε → (cid:90) Q f ( x, τ xε ω ) dµ ε Γ( ω ) = (cid:90) Q (cid:90) Ω f ( x, ω ) dµ Γ , P dx . for f = φ ( x ) ψ ( ω ) with φ ∈ C ∞ ( Q ) and ψ ∈ C b (Ω) and therefore also for f ∈ C ∞ ( Q ; C b (Ω)) . This is a first indicator why we may consider µ Γ , P as a Hausdorffmeasure. It will be more striking by the following Lemma . [8] There is a measurable set, also denoted by Γ ⊂ Ω , with χ Γ( ω ) ( x ) = χ Γ ( τ x ω ) for L + µ Γ( ω ) -almost every x for µ -almost every ω . Furthermore µ (Γ) = 0 and µ Γ , P (Ω \ Γ) = 0 . SYMPTOTIC EXPANSION ON STOCHASTIC GEOMETRIES 15
The same proof also provides such a characteristic function for a n -dimensional randomclosed set ˜ A . ˜Γ := ∂ ˜ A is also a random closed set, which can be verified using F V from(A.1) with V = ˜ A \ ∂ ˜ A . In case ˜Γ is regular enough, there are subsets A, Γ ⊂ Ω such that χ ˜ A ( ω ) ( x ) = χ A ( τ x ω ) and χ ˜Γ( ω ) ( x ) = χ Γ ( τ x ω ) . This is equally possible for ˜ B := (cid:16) R n \ ˜ A (cid:17) ∪ ˜Γ . Thus, the assumptions made on A , B , and Γ made in section3.1 are now justified. Note that it is also possible to define ν Γ : Γ → R n such that ν Γ( ω ) ( x ) = ν Γ ( τ x ω ) and ν Γ thus can be considered as normal field on Γ .Note that in section 3.1, we started with A, B, Γ ⊂ Ω and defined A ( ω ) , B ( ω ) and Γ( ω ) to avoid these rather technical preliminaries. We conclude the mathematical partby some remarks on function spaces on Ω as well as on Gauss’ theorem and Poincaréinequalities.Continuity and differentiability of functions were introduced in section 3.1. It is ofcourse also possible to define arbitrarily high differentiability C kb (Ω) . Since Ω is aseparable metric space equipped with a measure µ , it is possible to define L p (Ω , σ, µ ) := (cid:26) f : Ω → R : f is σ − measurable , (cid:90) | f | p dµ < ∞ (cid:27) These L p -spaces have good properties, in particular they are separable and for anyfinite measure µ on (Ω , B (Ω)) exists a countable dense set of C ∞ b (Ω) -functions in L p (Ω , B (Ω) , µ ) ( ≤ p < ∞ , µ σ -finite )[8].It is of course possible to define various kinds of Sobolev spaces on Ω . The reader isreferred to [26, 27, 8].Finally, in order to justify some of the calculations in section 4, it is vital to proof aGauss-like theorem on stochastic spaces as well as some Poincaré inequalities. Lemma . [8] For all ψ ∈ C b (Ω) and φ ∈ C b (Ω) n holds:(A.6) (cid:90) Ω ψ ∇ φdµ = − (cid:90) Ω ( ∇ ψ ) φdµ Since the technique is very often used, we give the abbreviated proof from [8]:Define Q m := [ − m, m ] n . Then, since φ and ψ are bounded, the Ergodic Theorem 4leads to: (cid:90) Ω ψ ∇ φdµ = lim m →∞ m ) n (cid:90) Q m ψ ( τ x ω ) ∇ φ ( τ x ω ) dx = lim m →∞ m ) n (cid:18) − (cid:90) Q m ( ∇ ψ )( τ x ω ) φ ( τ x ω ) dx + (cid:90) ∂Q m ν Q m ( x )( φψ )( τ x ω ) d H n − ( x ) (cid:19) = = lim m →∞ m ) n (cid:18) − (cid:90) Q m ( ∇ ψ )( τ x ω ) φ ( τ x ω ) dx + O ( m n − ) (cid:19) = − (cid:90) Ω ( ∇ ψ ) φdµ where the integral over the boundary vanishes due to the fact that the surface of Q m grows with (2 m ) n − .QEDNote that (A.6) would also hold for an integral over A if ψ = 0 on Γ or ∇ ψ · ν Γ = 0 on Γ . From the corresponding Poincaré inequalities on R n , one may conclude in the sameway (cid:90) Ω (cid:0) ψ + |∇ ψ | (cid:1) ≤ C (cid:18)(cid:90) Ω ψ + (cid:90) Ω |∇ ψ | (cid:19) ∀ ψ ∈ C b (Ω) (cid:90) A (cid:0) ψ + |∇ ψ | (cid:1) ≤ C (cid:90) A |∇ ψ | ∀ ψ ∈ C b ( A ) with ψ = 0 on Γ where we have to assume that the constant C for the realizations is independent on ω : (cid:90) Q m ∩ A ( ω ) (cid:0) ψ ( τ x ω ) + |∇ ψ ( τ x ω ) | (cid:1) ≤ C (cid:90) Q m ∩ A ( ω ) |∇ ψ ( τ x ω ) | + O ( m n − ) Once such inequalities are established for C b -functions, it is easy to expand them tocorresponding Sobolev spaces on Ω . (See [8, 9] for more complicated examples) References [1] G. Allaire. Homogenization and two-scale convergence.
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