Compressible viscoplastic models for granular flows Duc Nguyen 1 1 - - PowerPoint PPT Presentation

Compressible viscoplastic models for granular flows

Duc Nguyen 1

1LAMA, Université de Marne-la-Vallée

CEMRACS 2019 Marseille, August 14, 2019.

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Mathematical model

stress tensor σ : matrix symmetric n × n strain tensor Du := ∇u+∇uT

2

∂ρ ∂t + div(ρu) = 0 ∂(ρu) ∂t + div(ρu ⊗ u)− div σ + ∇p = f σ ∈ ∂F(Du) + Initial condition, Boundary condition

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Subgradient - Subdifferential

Subgradient

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Newtonian and Non-Newtonian fluids

Newtonian fluids

F(Du) = η 2|Du|2 ⇒ σ = ηDu Stress tensor is linearly dependent on Strain rate

Example of Non-Newtonian fluid

F(Du) = |Du| ⇒ σ =       

Du |Du|

Du 0 fluid |σ| ≤ 1 Du = 0 solid

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Splitting scheme

Using Finite Volume Method with Suliciu’s solver for: ∂t ρ ρu

• + div
• ρu

ρu ⊗ u + pIN

• =
• Using Finite Element Method for:

∂t ρ ρu

• =
• div σ + f
• Duc Nguyen

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Algorithm for viscoplastic models

αu − div σ = f σ ∈ ∂F(Du)

Regularization method

αuε − div σε = 0 σε = F′

ε(Duε)

In inviscid Bingham case: σ =        σ0

Du |Du|

Du 0 |σ| ≤ σ0

• therwise ⇒ σε = σ0

Duε

• |Duε|2 + ε2

Necessarity of finding the optimal ε Advantages: Regularization method is natural, fast. Disadvantages (for inviscid Bingham): Cannot solve exactly plug zones Du = 0

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Alternative approach

Augmented Lagrange Method Bermudez-Moreno Method [Bresch and al 2014] Bi-projection method [Laurent Chupin, Thierry Dubois 2015] Duality method [Chambolle, A. and Pock, T. (2011)] ...

Goal

Solving for the general viscoplastic model σ ∈ ∂F(Du) Proving the convergence in space for the scheme. Comparing with other methods.

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Main results

Proposition (Projection formulation)

For any r > 0: σ ∈ ∂F(Du) ⇔ Pr(σ + rDu) = σ where Pr(A) = (Id + r∂F∗)−1(A) In inviscid Bingham case: σ =       

Du |Du|

Du 0 |σ| ≤ 1

• therwise ⇔ σ =

      

σ+rDu |σ+rDu|

|σ + rDu| ≥ 1 σ + rDu |σ + rDu| < 1

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Algorithm I

       αu − div σ = f ˆ σ = Pr(ˆ σ + rDˆ u) (1)

Convergence [F .Bouchut, D.N.]

Suppose: |Du| < L|u|. Condition: L2τr < 1.              αuk+1 − div σk+1 + uk+1−uk

τ

= f σk+1 = Pr(σk + r(2Duk − Duk−1))

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Algorithm II

Convergence [F .Bouchut, D.N.]

Suppose: |Du| < L|u|. Condition: L2τ0r0 < 1.                          θk =

1 √ 1+2ατk−1

τk = θkτk−1 rk = rk−1

θk−1

σk+1 = Prk (σk + rk(Duk + θk(Duk − Duk−1)) αuk+1 − div σk+1 + uk+1−uk

τk

= f

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Numerical results 1D

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 t(time) x "u.d" u 1:2 "uexact.d" u 1:2 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2 2.5 3 3.5 4 t(time) x "rho.d" u 1:2 "rhoexact.d" u 1:2

nx = 300, ε ≈ C dx2 dt

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Numerical results 2D - Viscoplastic model

The proposed numerical scheme works for unstructured mesh. The second algorithm is not faster than the first one. Both scheme are faster than Lagrange Augmented and Bermudez-Moreno Method, but slower than Regularization method.

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THANK YOU FOR YOUR ATTENTION !

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