heterogeneous media: case studies in non-periodic ......heterogeneous media: case studies in...

Microstructure models of heterogeneous media Analysis of the micro-macro model Numerical illustration Justification of the formal homogenization Open issues

Heterogeneous media: Case studies in non-periodicaveraging and perspectives in designing packaging

materials

Adrian Muntean

Department of Mathematics and Computer Science, Karlstad University, Sweden

Karlstad, January 2015

Adrian Muntean Mathematics & Karlstad University

Microstructures


Outline of the Talk

Microstructure models of heterogeneous mediaDistributed microstructures“Structured” transportGeneric microscopic modelDerivation via formal homogenization of a micro-macro model

Analysis of the micro-macro modelWeak formulation. Basic estimatesGlobal existence and uniqueness of weak solutions

Numerical illustration

Justification of the formal homogenizationOutline of the proof for the corrector estimate

Open issues


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Heterogeneous media: Diffusion in a chessboard domain

Heterogeneous media and their homogeneous representation

Is an equivalent formulation posed in an homogeneous domain possible?


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Asymptotic homogenization for chessboard-type microstructures

∂tuε − div(Aε(x)∇uε) = fε in Ωε

uε = gε at black/white inner boundaries

−n · Aε∇uε = 0 at the outer boundary + i.c.

Here Aε(x) = A(x , xε

).Idea 1: Use

uε(t , x) = u0(t , x) + εu1(t , x ,xε

) + ε2u2(t , x ,xε

) +O(ε3)

to get PDEs for u0.Idea 2: Along the same line: show the corrector estimate

||uε − u0|| ≤ c(u1, . . . )εα, α > 0.


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Macroscopic equation for chessboard diffusion. Range of validity?

∂t (φu0)− div(φD(x ,w(x))∇u0) = φf0 in Ω

−n · φD(x ,w(x))∇u0 = 0 at the outer boundary

I φ is the chessboard porosity, i.e. φ = 116

I D(·,w(·)) is an effective transport coefficientI w is a parameter depending on the chessboard’s microstructure (cell

functions)

Rate of convergence (correctors):I ||uε − u0||L2 ≤ c1ε

I ||∂tuε − ∂tu0||L2 ≤ c2ε

I ||∇uε −∇u0 − oscillations||L2 ≤ c3√ε


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Averaging techniques in action

Towards understanding mixed transport fluxes:I Estimating speeds of macroscopic corrosion fronts in concrete

[large-time behavior for free boundary problems]I Ionic transport in porous media [homogenization of the

Stokes-Nernst-Planck-Poisson system]I Multiscale combustion [smoldering thin sheets, fast-reaction limits]I Cross-diffusion effects [averaging the Becker-Döring system for hot

colloids]I Thermo-diffusion effects [estimating permeability tensors for mass and

heat transport through porous membranes]


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Bridging length scales in heterogeneous media

I Averaging techniques (periodic homogenization, renormalization, ...)

I PDE models with distributed microstructure

1. two-scale models – A. Friedman, A. Tzavaras, P. Knabner2. distributed-microstructure models – R. E. Showalter and co-workers

(Walkington, Cook, Clark, Visarraga, ...) + M. Böhm, S. Meier, D. Treutler, J.Esher

3. dual- or double-porosity models – U. Hornung, W. Jäger, T. Arbogast, ...4. two-scale models with freely evolving micro-interfaces – C. Eck., H.

Emmerich, P. Knabner, A. Muntean (2 scale phase-field models), S. Meier,A. Muntean (2 scale fast-reaction asymptotics)

5. coupling multi-physics/discrete-to-continuum etc.


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Double-porosity structure of materials

Barenblatt, Zheltov, Kochina, PMM, 24(1960), 5, pp. 852–864


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Locally-periodic distributions of perforations

!


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Generic micro-model

uεt = ∇ · (Dh∇uε − qεuε)qε = −κ∇pε

∇ · qε = 0in Ωεh,

vεt = ε2∇ · (Dl∇vε) in Ωεl ,νε · (Dh∇uε) = ε2νε · (Dl∇vε)uε = vε

qε = 0on Γε,

uε(x , t) = ub(x , t)qε(x , t) = qb(x , t)

on Γ,uε(x , 0) = uεI (x) in Ωεh,

vε(x , 0) = vεI (x) in Ωεl ,


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Basic ideas of the (formal) two-scale homogenisation asymptotics

uε(x , t) = u0(x , x/ε, t) + εu1(x , x/ε, t) + ε2u2(x , x/ε, t) + ...

vε(x , t) = v0(x , x/ε, t) + εv1(x , x/ε, t) + ε2v2(x , x/ε, t) + ...

qε(x , t) = q0(x , x/ε, t) + εq1(x , x/ε, t) + ε2q2(x , x/ε, t) + ...

pε(x , t) = p0(x , x/ε, t) + εp1(x , x/ε, t) + ε2p2(x , x/ε, t) + ...

|∇Sε| =1ε|∇y S|+ O(ε0)

νε = ν0 + εν1 + O(ε2),

Ωε` = S(x , x/ε) < 0 : x ∈ Ω,Ωεh = S(x , x/ε) > 0 : x ∈ Ω


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Micro-macro model

∂tv0(x , y , t) = Dl ∆y v0(x , y , t) |y | < r(x), x ∈ Ω,

∂t

(θ(x)u0 +

∫|y|<r(x)

v0 dy)

=

divx (DhA(x)∇x u0 − qu0) for x ∈ Ω,

q = −K(x)∇x p0 for x ∈ Ω,

∇x · q = 0 for x ∈ Ω,v0(x , y , t) = u0(x , t) for |y | = r(x),

u0(x , t) = ub(x , t) for x ∈ Γ,

q(x , t) = qb(x , t) for x ∈ Γ,u0(x , 0) = uI(x) for x ∈ Ω,

v0(x , y , 0) = vI(x , y) for |y | < r(x), x ∈ Ω.


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The porosity θ(x) of the medium is given by

θ(x) := 1− πr 2(x),

while the effective diffusivity D(x) := (aij (x))i,j and the effective permeabilityK(x) := (kij (x))i,j are defined by

aij (x) := Dh

∫y∈U | |y|>r(x)

δij + ∂yi Uj (x , y , t) dy ,

and

kij (x) :=

∫y∈U | |y|>r(x)

Vji (x , y , t) dy .


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x-dependent cell problems

∆y Uj (x , y) = 0 for all x ∈ Ω, y ∈ Y , |y | > r(x),

ν0 · ∇y Uj (x , y) = −ν0 · ej for all x ∈ Ω, |y | = r(x),

Uj (x , y) y -periodic,

andVj (x , y) = ∇yπj (x , y) + ej for all x ∈ Ω, y ∈ Y , |y | > r(x),

∇y · Vj (x , y) = 0 for all x ∈ Ω, y ∈ Y , |y | > r(x),

Vj = 0 for all x ∈ Ω, |y | = r(x),

Vj (x , y) and πj (x , y) y -periodic,

for j = 1, 2.


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Reduced micro – macro model

θ(x)∂tu −∇x · (D(x)∇x u − qu) = −∫∂B(x)

νy · (Dl∇y v) dσ in Ω,

∂tv − Dl ∆y v = 0 in Ω× B(x),

u(x , t) = v(x , y , t) at (x , y) ∈ Ω× ∂B(x),

u(x , t) = ub(x , t) at x ∈ ∂Ω,

u(x , 0) = uI(x) in Ω,

v(x , y , 0) = vI(x , y) at (x , y) ∈ Ω× B(x)


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Memory effects?


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Assumptions on data and parameters

(A1) S0 : Ω× U → R, which defines B(x) and also the 1-dimensionalboundary Ω× ∂B(x) of Ω× B(x) as

(x , y) ∈ Ω× ∂B(x) if and only if S0(x , y) = 0,

is an element of C2(Ω× U). Additionally, the Clarke gradient ∂y S0(x , y) isregular for all choices of (x , y) ∈ Ω× U.(A2)

θ, D ∈ L∞+ (Ω),

q ∈ L∞(Ω;Rd ) with ∇ · q = 0,ub ∈ L∞+ (Ω× S) ∩ H1(S; L2(Ω)),

∂tub ≤ 0 a.e. (x , t) ∈ Ω× S,uI ∈ L∞+ (Ω) ∩ H1,

vI(x , ·) ∈ L∞+ (B(x)) ∩ H2 for a.e. x ∈ Ω.


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Functional setting

V1 := H10 (Ω),

V2 := L2(Ω; H2(B(x))),

H1 := L2θ(Ω),

H2 := L2(Ω; L2(B(x))).

If 0 < |B(x)|, |∂B(x)| <∞, then the direct Hilbert integrals

L2(Ω; H1(B(x))) := u ∈ L2(Ω; L2(B(x))) : ∇y u ∈ L2(Ω; L2(B(x)))

L2(Ω; H1(∂B(x))) := u : Ω× ∂B(x)→ R meas. s. t.∫

Ω

||u(x)||2L2(∂B(x)) <∞

are separable Hilbert spaces with distributed trace:

γ : L2(Ω; H1(B(x)))→ L2(Ω, L2(∂B(x)))

given by

γu(x , s) := (γx U(x))(s), x ∈ Ω, s ∈ ∂B(x), u ∈ L2(Ω; H1(B(x)))


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Definition (Weak formulation)Assume (A1) and (A2). The pair (u, v), with u = U + ub and where(U, v) ∈ V, is a weak solution if the following identities hold∫

Ω

θ∂t (U + ub)φ dx +

∫Ω

(D∇x (U + ub)− q(U + ub)) · ∇xφ dx =

−∫

Ω

∫∂B(x)

νy · (Dl∇y v)φ dσdx ,∫Ω

∫B(x)

∂tvψ dydx +

∫Ω

∫B(x)

Dl∇y · ∇yψ dydx =

∫Ω

∫∂B(x)

νy · (Dl∇y v)φ dσdx ,

for all (φ, ψ) ∈ V and t ∈ S.


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Basic estimates

LemmaLet (A1) and (A2) be satisfied. Then any weak solution (u, v) of problem (P)has the following properties:

(i) u ≥ 0 for a.e. x ∈ Ω and for all t ∈ S;

(ii) v ≥ 0 for a.e. (x , y) ∈ Ω× B(x) and for all t ∈ S;

(iii) u ≤ M1 for a.e. x ∈ Ω and for all t ∈ S;

(iv) v ≤ M2 for a.e. (x , y) ∈ Ω× B(x) and for all t ∈ S;

(v) The following energy inequality holds:

‖u‖2L2(S;V1)∩L∞(S;H1) + ‖v‖2

L2(S;L2(Ω,V2))∩L∞(S;H2)

+ ‖∇x u‖2L2(S;H1) + ‖∇y v‖2

L2(S×Ω×B(x)) ≤ c1.

N.B. Assume (A1), (A2). Then uniqueness of weak solutions holds.


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Global existence

TheoremThere exists at least a weak solution of the micro-macro model.Proof. (Sketch)Schauder fixed-point argument in L2(S; L2(Ω)) framework

X1 := L2(S; L2(Ω)),

X2 := L2(S; H10 (Ω)) ∩ H1(S; L2(Ω)),

X3 := L2(S; V2) ∩ H1(S; L2(Ω; L2(B(x)))).

T : X1 → X1 with T := T3 T2 T1.


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T1 maps a f ∈ X1 to the solution w ∈ X2 of∫Ω

θ∂t (U + ub)φ dx +

∫Ω

(D∇x (U + ub)− q(U + ub)) · ∇xφ dx = −∫

Ω

fφ dx ,

for all φ ∈ H10 (Ω).

T2 maps a w ∈ X2 to a solution v ∈ X3 of∫Ω

∫B(x)

∂t (V + w)ψ dydx +

∫Ω

∫B(x)

Dl∇y (V + w) · ∇yψ dydx =∫Ω

∫∂B(x)

νy · (Dl∇y (V + w))ψ dσdx ,

for all ψ ∈ V2 and t ∈ S.T3 maps a v ∈ X3 to f ∈ X1 by

f =

∫∂B(x)

νy · ∇y v dσ.

LemmaT is well defined.


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Compactness stepLemmaThe operator T is compact.Step 1. Use Ψ : Ω× B(0)→ Ω× B(x).We call Ψ a regular C2-motion if Ψ ∈ C2(Ω× B(0)) with the property that foreach x ∈ Ω

Ψ(x , ·) : B(0)→ B(x) := Ψ(x ,B(0))

is bijective, and if there exist constants c,C > 0 such that

c ≤ det∇y Ψ(x , y) ≤ C,

for all (x , y) ∈ Ω× B(0). The existence of such a mapping is ensured by thefact that S0 ∈ C2(Ω× U), by (A1). If Ψ is a regular C2-motion, then

F := ∇y Ψ and J := det F

are continuous functions of x and y .

∇y v = F−T∇y v , ∂tv = ∂t v ,∫∂B(x)

νy · j dσ =

∫Γ0

JF−T νy · j dσ.


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The transformed equation can be now written as:Let w ∈ X2 be given.Find V ∈ L2(S; L2(Ω; H1

0 (B(0)))) ∩ H1(S; L2(Ω; L2(B(0)))) such that

∫Ω

∫B(0)

∂t (V + w)ψJ dydx +

∫Ω

∫B(0)

JF−1DlF−T∇y (V + w) · ∇yψ dydx =∫Ω

∫Γ0

νy · (JF−1DlF−T∇y (V + w))ψ dσdx ,

for all ψ ∈ L2(Ω; H10 (B(0))) and t ∈ S.

Denote by Γ0 the boundary of B(0).


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Step 2. Interior and boundary regularity (in y )Assume (A1) and (A2). Then Γ0 is C2 and

V ∈ L2(S; L2(Ω; H2(B(0)) ∩ H10 (B(0)))).

Step 3. Additional two-scale regularity (in y )Assume (A1) and (A2). Then

V ∈ L2(S; H1(Ω; H2(B(0)) ∩ H10 (B(0)))).

Step 4. Apply Lions-Aubin Lemma


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Concentration profiles of the two-scale model

Solution profiles of the two-scale model at different times: Up: R = 0.1; Bottom:R = 0.9.


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Numerical approximation of RD systems on two scales

I Convergent two-scale Galerkin approximationsA. Muntean, M. Neuss-Radu: A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in

complex media. JMAA 371 (2010), (2), 705-718.

I Convergence rates (a priori)A. Muntean, O. Lakkis (2010). Rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion system

with nonlinear transmission condition. RIMS Kokyuroku (Kyoto), 1693, pp. 85-98.

I Numerical example in 1D× 1D


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Corrector estimates – How goed is the averaging method?

I (uε, vε) solution vector for the micro problemI (u0, v0) solution vector for the macro problemI uε0, v

ε0 , u

ε1 macroscopic reconstructions

uε0(x , t) := u0(x , t)

vε0 (x , t) := v0(x , x/ε, t)

uε1(x , t) := uε0(x , t) + εU(t , x , x/ε)∇uε0(x , t)


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Justification of the formal asymptotics

TheoremAssume (A1) and (A2). Then the following convergence rate holds

||uε − uε0||L∞(S,L2(Ωε)) + ||vε − vε0 ||L∞(S,L2(Ω−Ωε)) +

||uε − uε1||L∞(S,H1(Ωε)) + ε||vε − vε0 ||L∞(S,H1(Ω−Ωε)) ≤ c√ε


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Outline of the proof for the corrector estimate

Step 1.Write weak formulations for both micro and macro pbs. (the later in terms ofmacro reconstructions)Step 2.Subtract the 2 weak formulations and choose suitable test functions

ϕ := uε − uε0(x , t) + εU(t , x , x/ε)∇uε0(x , t)

ψ := vε − vε0

Step 3. A technical lemma:Prove that

|ε∫

Sε

εD`∇vε · νεψdσ − 1|Y − B(x)|

∫∂B(x)

νy · D`∇y vε0ψdσ| ≤ cε||ψ||H1(Ωε).

Step 4. Bookkeeping of ε


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Open issues

1. Computability in 2D (including a priori / a posteriori error estimates)

2. Free micro interfaces?

3. Two-scale coupling between macro PDEs and micro phase transitions[deterministic/stochastic ODEs]. Analysis + computability issues +PDEs in measures(?)

4. Applications to harvesting geothermal energy – NWO-MPE project(ongoing)

5. Applications to design of smart packaging – Preparation phase


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Intermezzo: Multiscale smoldering combustion of thin sheets

I O. Zik, Z. Olami, and E. Moses (1998) Fingering instability in combustion, Phys. Rev. Lett. 81, 3868-3871.

I E. Ijioma, T. Ogawa, A. Muntean (2013). Pattern formation in reverse smoldering combustion : a homogenization approach.Combustion Theory and Modelling, 17(2), 185-223

I Ijioma, E.R., Muntean, A. Ogawa, T. (2015). Effect of material anisotropy on the fingering instability in reverse smoulderingcombustion. International Journal of Heat and Mass Transfer, 81, 924-938.

I Fatima, T., Ijioma, E.R., Ogawa, T. Muntean, A. (2014). Homogenization and dimension reduction of filtration combustion inheterogeneous thin layers.. Networks and Heterogeneous Media, 9(4), 709-737.


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Background of this work:

I A. Muntean, T. van Noorden: Corrector estimates for the homogenization of a locally-periodic medium with areas of low and highdiffusivity. Eur. J. Appl. Math. 24 (2013), (5), 657-677.

I T. van Noorden, A. Muntean: Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math.22 (2011), (5), 493-516.

I A. Muntean, M. Neuss-Radu: A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems incomplex media. JMAA 371 (2010), (2), 705-718.

I T. Fatima, N. Arab, E. Zemskov, A. Muntean: Homogenization of a reaction-diffusion system modeling sulfate corrosion inlocally-periodic perforated domains. J. Engng. Math. 69(2011), (2), 261-276.

I V. Chalupecky, T. Fatima, A. Muntean: Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro masstransfer limit. J. Math-for-Industry, 2(2010) (B-7), 171-181.

I A. Muntean, O. Lakkis (2010). Rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion systemwith nonlinear transmission condition. RIMS Kokyuroku (Kyoto), 1693, pp. 85-98.

I S. Meier, M. Peter, A. Muntean, M. Böhm, J. Kropp: A two-scale approach to concrete carbonation in Proc. Int. RILEM Workshopon Integral Service Life Modeling of Concrete Structures, Guimares, Portugal, 2007, 3–10.


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heterogeneous media: case studies in non-periodic ......heterogeneous media: case studies in...

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