multiscale models for flow in heterogeneous media with...

Multiscale models for flow in heterogeneous mediawith applications to fresh concrete

Borek Patzák1 Filip Kolarík1 Jan Zeman1,2

1Czech Technical University in Prague, Faculty of Civil Engineering2VŠB-Technical University of Ostrava, Centre of Excellence IT4Innovations

CST&ECT 2014 Naples, Italy, 2-5 September 2014

B. Patzák et al. (CTU . . . ) Homogenization for fresh concrete flow . . . CST&ECT 2014 1 / 34

Self-compacting concreteTarget applications

Courtesy of TailorCrete project

Complex structural shapes→ Conventional casting procedurescannot be usedHigh level of reinforcement→ FlowabilityNeed for homogenous concrete→ Problems with segregation


Computational modeling of concrete flow

Fresh concrete considered as a suspension of particles in a matrix.Depending on observation scale one can distinguish


Dicrete particle models

Work on very low scale, taking into account interactions betweenparticlesAllow to take into account mix compositionCan predict particle interlocking, segregationExamples: DEM (O. Petersson, 2003), DPD (N.Martys, 2005)Limited number of particles that can be dealtMany parameters could not be measured, requires fitting.


Suspension flow models

Simulate flow of particles (coarse aggregates) suspended in thefluid

Viscoplastic suspension element method (VSEM, Mori, Tanigawa,2005)Finite Element with Lagrangian Integration Points (FEMLIP, Moersi,2005)Lagrange multiplier/fictitious domain method for particulate flows(Glowinski, 1999)

Allow to study trajectories of particles (predictions of segregation)At present, not suitable for casting simulations (too manyparticles).


Homogeneous flow models

Work on macro scale, where the fresh concrete can be regardedas homogeneous suspensionProblem is described by Navier-Stokes equations, typically underincompressible flow assumptionsCan simulate real casting problems.On the other hand, particle blocking and segregation can bedifficult to model.


Homogeneous simulationsComputational modeling (PATZÁK & BITTNAR, 2009)

Example: LBox test

Flow of two immiscible fluids: Bingham (concrete) + Newton (air)Interface capturing methods: Volume-of-fluid/Level set methodStabilized P1/P1 formulation: SUPG/VMSImplemented into OOFEM


Homogeneous simulationsComputational modeling (PATZÁK & BITTNAR, 2009)

Example: V-Funnel test

Flow of two immiscible fluids: Bingham (concrete) + Newton (air)Interface capturing methods: Volume-of-fluid/Level set methodStabilized P1/P1 formulation: SUPG/VMSImplemented into OOFEM


Homogeneous simulationsFiber orientation modeling (F. KOLARÍK B. PATZÁK, L.N. THRANE, 2013)

Probabilistic approachThe orientation state described by the probability distributionfunction Ψ of the angle φ of the fiberThe evolution of the probability distribution is described byFokker-Planck equation (Folgar, Tucker, 1984)

Example: Plate formwork casting


Self-compacting concreteMain focus of the work

What is the effect of reinforcement ?


Effect of reinforcementPorous medium approaches

In practical simulations the explicit modeling of reinforcement isextremely demanding (fine meshes, level of observation one ortwo scales lower than macroscale)In the context of fluid mechanics the problem can be regarded asflow in porous medium:

continuum approach - porous medium treated as continuum,characterized by macroscopic properties, such as permeability,porosity, etc. Typical example: Darcy law -originally empiricalrelation, later derived using homogenization (Allaire, 1991,Sanchez-Palencia, 1986)bundle of tubes - porous medium treated as system of tubes.Neglect pore space morphology (Sochi, 2010)pore scale network model - network of flow channels with idealizedgeometry (Lopez et al., 2003)detailed simulations - detailed simulations on fully resolved porestructure


Effect of reinforcementMultiscale approach

Non-Newtonian porous mediaNonlinear behaviour at pore scale results in rather complexbehavior on the macroscaleHierarchy of models obtained using theory of mixtures withdifferent assumptions (Darcy, Brinkman, ...) [Rajagopal, 2007].Bingham plastic flow in porous medium studied by Lions andSanchez-Palencia, 1981, convergence proved by Mikelic, 1993.Recent advances in modeling lead to the development ofmultiscale computational approaches.


Self-compacting concreteHomogenization-based approach

Γ

ΩFS

∂ΩFS

ΩS

∂ΩS

Fluid domain ΩS

Perforated domain ΩFS

Interface Γ

Working assumptions: No free surface, steady-state


Self-compacting concreteHomogenization-based approach

Γ

ΩDΩS

∂ΩS RVE Ω

Homogenization of Stokes flow in ΩFS → Generalized Darcy flow in

porous media ΩD

Porous media response computed on subscale problem (RVE)


Formulation of the problemStokes flow in perforated domain

Strong form of Stokes flow

−∇ · τ + ∇p = ρb in ΩFS

∇ · u = 0 in ΩFS

u = 0 in ∂ΩFS

(τ − pI) · n = −pn in ΓP

u = unn in ΓV

Γ

ΩFS

∂ΩFS

Γ = ΓV ∪ ΓP


Formulation of the problemStokes flow in perforated domain

Weak form of Stokes flow: Find (u, p) ∈ S ×Q

∫ΩF

S

∇δw : τ (u) dx−∫

ΩFS

(∇ · δw)pdx+

∫ΩF

S

δq(∇ · u) dx =∫ΩF

S

δw · ρb dx−∫

ΓP

δw · pnds

for all (δw, δq) ∈ V ×Q


Formulation of the problemHomogenization process - based on approach introduced in (SANDSTRÖM &LARSSON, 2013)

By introducing p = pM + pS , δq = δqM + δqS , integration by partsand separation of the scales, we get

for all δw :

∫ΩF

S

∇δw : τ (u) dx+

∫ΩF

S

δw ·∇pM dx

−∫

ΩFS

(∇ · δw)pS dx =

∫ΩF

S

δw · ρb dx

for all δqS :

∫ΩF

S

δqS(∇ · u) dx = 0

for all δqM :

∫ΩF

S

∇δqM · u dx =

∫ΓV

δqMun ds



Decomposition of the domain ΩD =∑

i Ω,i (RVEs)Volume averaging as 〈f〉 = 1

|ΩF|∫

ΩFf dx

It can be shown that following relation holds:∫ΩF

S

f dx =

∫ΩD

φ 〈f〉 dx

ΩDΩFS

⇔



Macro-scale problem (Darcy)

for all δqM :

∫ΩD

∇δqM · φ 〈u〉︸︷︷︸w(∇p−ρb)

dx =

∫ΓV

φ⟨δqMun

⟩ds

Sub-scale problem (RVE)

for all δw :

∫ΩF

∇δw : τ (u) dx−∫

ΩF

(∇ · δw)pS dx

= −∫

ΩF

δw · (∇pM − ρb) dx

for all δqS :

∫ΩF

δqS(∇ · u) dx = 0


Formulation of the problemStokes-Darcy system

Γ

ΩDΩS

∂ΩS

n

−∇ · τ + ∇p = ρb in ΩS

∇ · u = 0 in ΩS

u+ w(∇p− ρb) = 0 in ΩD

∇ · u = 0 in ΩD

(uS − uD) · n = 0 on Γ

β (uS − uD) · t = n · τ · t on Γ

pS − pD = n · τ · n on Γ



Γ

ΩDΩS

∂ΩS

n

−∇ · τ + ∇p = ρb in ΩS

∇ · u = 0 in ΩS

u+ w(∇p− ρb) = 0 in ΩD

∇ · u = 0 in ΩD

(uS − uD) · n = 0 on Γ

β(uS −*0

uD ) · t = n · τ · t on Γ

pS − pD = n · τ · n on Γ



Stokes domain:

∫ΩS

∇δw : τ (u) dx−∫

ΩS

(∇ · δw)p dx−∫∂ΩS

δw · (τ − δp) · nds

−∫

Γ(δw · t)β(uS · t) ds+

∫ΩS

δq(∇ · u) dx =

∫ΩS

δw · ρb dx,

Darcy domain:

∫ΩD

δw · udx+

∫ΩD

δw · w(∇p− ρb) dx+

∫Γ(δw · n)pD ds

+

∫ΩD

δq(∇ · u) dx.



Linearized Darcy problem states as follows: Find (v, q) ∈ S ×Qsuch that

∫ΩD

δw · v dx︸︷︷︸M

+∫

ΩD

δq(∇ · v) dx︸︷︷︸GT

+∫

Γ

(δw · n)q ds︸︷︷︸GΓ

+∫

ΩD

δw ·

Dtan︷︸︸︷∂w(∇p− ρb)

∂∇p·∇q dx︸︷︷︸

dK

=

−∫

Γ

(δw · n)pD ds︸︷︷︸fΓ

−∫

ΩD

δq(∇ · u) dx︸︷︷︸fp

−∫

ΩD

δw · udx︸︷︷︸fu

−∫

ΩD

δw · w(∇p− ρb) dx︸︷︷︸fw

holds for all (δw, δq) ∈ V ×Q



After discretization using P1/P1 elements:

Darcy: (M dKD +GΓ

GT 03x3

)vq

=

fDext,u − f

Dint,u

fDext,p − f

Dint,p

Similarly for Stokes:(

dKS −MΓ G

GT 03x3

)vq

=

fSext,u − f

Sint,u

fSext,p − f

Sint,p



Stabilized VMS Darcy problem: Find (v, q) ∈ S ×Q such that for all(δw, δq) ∈ V ×Q

∫ΩD

δw · v dx︸︷︷︸M

+∫

ΩD

δq(∇ · v) dx︸︷︷︸GT

+∫

Γ

(δw · n)q ds︸︷︷︸GΓ

+∫

ΩD

δw ·

Dtan︷︸︸︷∂w(∇p− ρb)

∂∇p·∇q dx︸︷︷︸

dK

−∑e

[∫Ωe

(δw −∇δq)τ ·(v +

∂w

∂∇p·∇q + u+ w(∇p− ρb)

)dx

]=

−∫

Γ

(δw · n)pD ds︸︷︷︸fΓ

−∫

ΩD

δq(∇ · u) dx︸︷︷︸fp

−∫

ΩD

δw · udx︸︷︷︸fu

−∫

ΩD

δw · w(∇p− ρb) dx︸︷︷︸fw



Final stabilized matrix form yields:

Darcy: (M dKD +GΓ

GT +Cv Cq

)vq

=

fDext,u − f

Dint,u

fDext,p − f

Dint,p

Stokes: (

dKS −MΓ G

GT CSq

)vq

=

fSext,u − f

Sint,u

fSext,p − f

Sint,p


ExamplesPermeability

0.25 µ = 1

0 5000 10000 15000 20000 25000 30000

Number of nodes

0.28

0.29

0.3

0.31

0.32

Perm

eabili

ty

Due to symmetry: K = KI

Sensitive to mesh size


ExamplesUnidirectional flow

4×

1

µ = 1

β=

1

4× 14

vn = 0

vn = 0

v=

[1,0

]

p=

0

Fully resolved Stokes Homogenized Darcy–Stokes


ExamplesEffect of interface conditions

8×

0.5

variable β

8× 0.5

vn = 0

v=

[1,0

]

p=

0

22

3 3vn = 0

µ = 1I II

Parametric study on influence of β


ExamplesEffect of interface conditions

Fully resolved Stokes Homogenized Darcy–Stokes

optimized β


ExamplesEffect of interface conditions - Section I

0 2 4 6 8Position

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5T

angential velo

city

Homogenized solutionDirect simulation

β


ExamplesEffect of interface conditions - Section II

0 2 4 6 8Position

-0.09

0.41

0.91

1.41

1.91

2.41

No

rma

l ve

locity

Homogenized solutionDirect simulation

β


Conclusions and outlook

Multi-scale approach to fresh concrete flow by homogenizationhas been proposedLeads to the (generally non-linear) Stokes-Darcy systemWhat remains to be done

1 “Unsaturated” simulationsVolume-of-fluid/Level set approach for Stokes-DarcyFree surface problem

2 Extensive Validation&Verification

AcknowledgmentThis work was supported by the Czech Science Foundation throughproject No. 13-23584S.


multiscale models for flow in heterogeneous media with...

Documents