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

Compressible viscoplastic models for granular flows Duc Nguyen 1 1 LAMA, Université de Marne-la-Vallée CEMRACS 2019 Marseille, August 14, 2019. Duc Nguyen CEMRACS 2019 1 / 13

Mathematical model stress tensor σ : matrix symmetric n × n strain tensor Du := ∇ u + ∇ u T 2 ∂ρ ∂ t + div( ρ u ) = 0 ∂ ( ρ u ) + div( ρ u ⊗ u ) − div σ + ∇ p = f ∂ t σ ∈ ∂ F ( Du ) + Initial condition, Boundary condition Duc Nguyen CEMRACS 2019 2 / 13

Subgradient - Subdifferential Subgradient Duc Nguyen CEMRACS 2019 3 / 13

Newtonian and Non-Newtonian fluids Newtonian fluids F ( Du ) = η 2 | Du | 2 σ = η Du ⇒ Stress tensor is linearly dependent on Strain rate Example of Non-Newtonian fluid  Du Du � 0 fluid   | Du | F ( Du ) = | Du | σ = ⇒   | σ | ≤ 1 Du = 0 solid   Duc Nguyen CEMRACS 2019 4 / 13

Splitting scheme Using Finite Volume Method with Suliciu’s solver for: � ρ � � � � � ρ u 0 + div = ∂ t ρ u ρ u ⊗ u + pI N 0 Using Finite Element Method for: � ρ � � � 0 = ∂ t ρ u div σ + f Duc Nguyen CEMRACS 2019 5 / 13

Algorithm for viscoplastic models α u − div σ = f σ ∈ ∂ F ( Du ) Regularization method α u ε − div σ ε = 0 σ ε = F ′ ε ( Du ε ) In inviscid Bingham case:  Du Du � 0 σ 0 Du ε   | Du | σ = otherwise ⇒ σ ε = σ 0  � | Du ε | 2 + ε 2  | σ | ≤ σ 0   Necessarity of finding the optimal ε Advantages: Regularization method is natural, fast. Disadvantages (for inviscid Bingham): Cannot solve exactly plug zones Du = 0 Duc Nguyen CEMRACS 2019 6 / 13

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. Duc Nguyen CEMRACS 2019 7 / 13

Main results Proposition (Projection formulation) For any r > 0 : σ ∈ ∂ F ( Du ) ⇔ P r ( σ + rDu ) = σ where P r ( A ) = ( Id + r ∂ F ∗ ) − 1 ( A ) In inviscid Bingham case:   Du σ + rDu Du � 0 | σ + rDu | ≥ 1     | Du | | σ + rDu | σ = otherwise ⇔ σ =   σ + rDu | σ + rDu | < 1  | σ | ≤ 1      Duc Nguyen CEMRACS 2019 8 / 13

Algorithm I  α u − div σ = f   (1)  σ + rD ˆ  σ = P r (ˆ ˆ u )   Convergence [F .Bouchut, D.N.] Suppose: | Du | < L | u | . Condition: L 2 τ r < 1.  α u k + 1 − div σ k + 1 + u k + 1 − u k = f   τ        σ k + 1 = P r ( σ k + r ( 2 Du k − Du k − 1 ))    Duc Nguyen CEMRACS 2019 9 / 13

Algorithm II Convergence [F .Bouchut, D.N.] Suppose: | Du | < L | u | . Condition: L 2 τ 0 r 0 < 1. 1  θ k = √   1 + 2 ατ k − 1    τ k = θ k τ k − 1       r k = r k − 1  θ k − 1     σ k + 1 = P r k ( σ k + r k ( Du k + θ k ( Du k − Du k − 1 ))      α u k + 1 − div σ k + 1 + u k + 1 − u k  = f   τ k Duc Nguyen CEMRACS 2019 10 / 13

Numerical results 1D 1 1.8 "u.d" u 1:2 "rho.d" u 1:2 "uexact.d" u 1:2 "rhoexact.d" u 1:2 0.9 1.6 0.8 0.7 1.4 0.6 t(time) t(time) 0.5 1.2 0.4 1 0.3 0.2 0.8 0.1 0 0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 x x ε ≈ C dx 2 nx = 300 , dt Duc Nguyen CEMRACS 2019 11 / 13

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. Duc Nguyen CEMRACS 2019 12 / 13

THANK YOU FOR YOUR ATTENTION ! Duc Nguyen CEMRACS 2019 13 / 13