On the solution of Bingham fluids and a Preconditioned Douglas-Rachford splitting method for Viscoplastic fluids Sofía López joint work with Sergio González LAWOC 2018 EPN Quito, September 6th 2018 1 / 42
Motivation: Bingham fluids 2 / 42
Motivation: Bingham fluids ◮ Exhibit no deformation up to a certain level of stress (yield stress). ◮ Behaves like a rigid solid when the shear stress is less than the yield stress. ◮ Above the yield stress, the material flows like a fluid. 2 / 42
Stationary Bingham-flow in Geometries of R d Flow equations for a Bingham fluid in Ω ⊂ R d : div u = 0 , √ 2 g E ( u ) q = 2 µ E ( u ) + E ( u ) � = 0 , if �E ( u ) � � q � ≤ g E ( u ) = 0 , if Div σ − ∇ p + f = 0 in Ω , + Boundary Conditions . where ◮ u : velocity field of the fluid, ◮ q : stress tensor, ◮ p : pressure ◮ E : strain velocity tensor 3 / 42
Stationary Bingham-flow in a pipe Simplified model: fluid flows under pressure drop, strain velocity tensor reduces to ∇ u . − ∆ u − div q = f , in Ω , q = g ∇ u |∇ u | , si ∇ u � = 0 | q | ≤ g si ∇ u = 0 u = 0 , on Γ . 4 / 42
Stationary Bingham-flow in a pipe Simplified model: fluid flows under pressure drop, strain velocity tensor reduces to ∇ u . − ∆ u − div q = f , in Ω , q = g ∇ u |∇ u | , si ∇ u � = 0 | q | ≤ g si ∇ u = 0 u = 0 , on Γ . Optimality system of the energy minimization problem (Mosolov-Miasnikov, 1965). 4 / 42
Energy minimization problem Energy minimization problem: nondifferentiable convex functional � � � J ( u ) := 1 |∇ u | 2 dx + g |∇ u | dx − fu dx . min 2 u ∈ H 1 0 (Ω) Ω Ω Ω 5 / 42
Energy minimization problem Energy minimization problem: nondifferentiable convex functional � � � J ( u ) := 1 |∇ u | 2 dx + g |∇ u | dx − fu dx . min 2 u ∈ H 1 0 (Ω) Ω Ω Ω Necessary and sufficient optimality condition (Variational inequality): find u ∈ H 1 0 (Ω) such that � � � � ( ∇ u , ∇ ( v − u )) dx + g |∇ v | dx − g |∇ u | dx ≥ f ( v − u ) dx , Ω Ω Ω Ω ∀ v ∈ H 1 0 (Ω) . 5 / 42
Duality Primal Problem � � � J ( u ) := 1 |∇ u | 2 dx + g min |∇ u | dx − fu dx . 2 u ∈ H 1 0 (Ω) Ω Ω Ω Dual Problem � − 1 |∇ u | 2 dx sup 2 | q ( x ) |≤ g Ω subject to: � ( ∇ u , ∇ v ) dx − ( f , v ) + ( q , ∇ v ) = 0 , for all v ∈ H 1 0 (Ω) Ω 6 / 42
Duality Primal approach: ◮ The location of the interface between the yielded and unyielded zones is crucial in solving flow problems. 7 / 42
Duality Primal approach: ◮ The location of the interface between the yielded and unyielded zones is crucial in solving flow problems. ◮ Previous contributions: replace the fluid properties with a bi-viscosity model. 7 / 42
Duality Primal approach: ◮ The location of the interface between the yielded and unyielded zones is crucial in solving flow problems. ◮ Previous contributions: replace the fluid properties with a bi-viscosity model. ◮ Regularized methods: replace the non-differentiable term � Ω |∇ u | dx by a regularization. g 7 / 42
Duality Primal approach: ◮ The location of the interface between the yielded and unyielded zones is crucial in solving flow problems. ◮ Previous contributions: replace the fluid properties with a bi-viscosity model. ◮ Regularized methods: replace the non-differentiable term � Ω |∇ u | dx by a regularization. g Difficulty: Important discrepancies arise with respect to the original problem. Primal approach: direct global regularization Khatib, Wilson (2001), Glowinski-Lions-Tremolieres (1976), Glowinski (1984), Frigaard-Nouar (2005), Dean-Glowinski-Guidoboni (2007),... 7 / 42
Duality Dual approach � 1 Ω |∇ u | 2 min 2 | q ( x ) |≤ g subject to: � Ω |∇ u | 2 + ( q , ∇ v ) = ( f , v ) , for all v ∈ H 1 0 (Ω) 8 / 42
Duality Dual approach � 1 Ω |∇ u | 2 min 2 | q ( x ) |≤ g subject to: � Ω |∇ u | 2 + ( q , ∇ v ) = ( f , v ) , for all v ∈ H 1 0 (Ω) No unique solution! 8 / 42
Duality Penalized Dual approach � Ω |∇ u | 2 + 1 1 2 γ � q � 2 min L 2 2 | q ( x ) |≤ g subject to: � Ω |∇ u | 2 + ( q , ∇ v ) = ( f , v ) , for all v ∈ H 1 0 (Ω) 8 / 42
Duality Penalized Dual approach � Ω |∇ u | 2 + 1 1 2 γ � q � 2 min L 2 2 | q ( x ) |≤ g subject to: � Ω |∇ u | 2 + ( q , ∇ v ) = ( f , v ) , for all v ∈ H 1 0 (Ω) Theorem There exists a unique solution ( q γ , y γ ) ∈ L 2 (Ω) × H 1 0 (Ω) to the penalized dual problem. Multiplier approach: use of dual information Glowinski (1984), Glowinski-Le Tallec (1989), Sánchez (1998), Roquet-Saramito (2003,2008), Huilgol-You (2005), Dean et al. (2007), Muravleva-Muravleva (2009), Olshanskii (2009) 8 / 42
Regularized optimality system ◮ This penalization corresponds to a regularization of the primal problem. 9 / 42
Regularized optimality system ◮ This penalization corresponds to a regularization of the primal problem. ◮ It changes the functional structure locally. 9 / 42
Regularized optimality system ◮ This penalization corresponds to a regularization of the primal problem. ◮ It changes the functional structure locally. Let us introduce, for γ > 0 , the function ψ γ such that: � g | z | − g 2 if | z | > g 2 γ ψ γ : z → ψ γ ( z ) = γ if | z | ≤ g 2 | z | 2 γ γ . 1 γ =5 γ =10 0.9 γ =50 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 9 / 42
Regularized optimality system ◮ This penalization corresponds to a regularization of the primal problem. ◮ It changes the functional structure locally. Let us introduce, for γ > 0 , the function ψ γ such that: � g | z | − g 2 if | z | > g 2 γ ψ γ : z → ψ γ ( z ) = γ if | z | ≤ g 2 | z | 2 γ γ . � � � J ( u ) := 1 |∇ u | 2 dx + g ψ γ ( ∇ u ) dx − fu dx . min 2 u ∈ H 1 0 (Ω) Ω Ω Ω ◮ Primal problem turns into the minimization of a continuously differentiable function. ◮ Dual problem is a constrained minimization of a quadratic functional. 10 / 42
Optimality system Recall the extremality conditions: − ∆ u − div q = f , in Ω , q = g ∇ u |∇ u | , si ∇ u � = 0 | q | ≤ g si ∇ u = 0 u = 0 , on Γ . 11 / 42
Optimality system Recall the extremality conditions: − ∆ u − div q = f , in Ω , q = g ∇ u |∇ u | , si ∇ u � = 0 | q | ≤ g si ∇ u = 0 u = 0 , on Γ . Regularized optimality system − ∆ u − div q = f , in Ω , max ( g , γ |∇ u γ | ) q γ = g γ ∇ u γ , for γ > 0 . 11 / 42
Optimality system Recall the extremality conditions: − ∆ u − div q = f , in Ω , q = g ∇ u |∇ u | , si ∇ u � = 0 | q | ≤ g si ∇ u = 0 u = 0 , on Γ . Regularized optimality system − ∆ u − div q = f , in Ω , max ( g , γ |∇ u γ | ) q γ = g γ ∇ u γ , for γ > 0 . 11 / 42
Optimality system Discretization: finite differences q − � A h � u + Q h � f = 0 , e , γξ ( ∇ h � q − g γ ∇ h � max( g � u )) ⋆� q = 0 for γ > 0 . 12 / 42
Optimality system Discretization: finite differences q − � A h � u + Q h � f = 0 , e , γξ ( ∇ h � q − g γ ∇ h � max( g � u )) ⋆� q = 0 for γ > 0 . Difficulty for Newton type algorithm: max function is not differentiable! 12 / 42
Semismooth Newton method Definition (Newton differentiability) If there exists a neighborhood N ( x ∗ ) ⊂ S and a family of mappings G : N ( x ∗ ) → L ( X , Y ) such that �F ( x ∗ + h ) −F ( x ∗ ) − G ( x ∗ + h )( h ) � Y lim � h � X → 0 = 0 , � h � X then F is called Newton differentiable at x ∗ . 13 / 42
Semismooth Newton method Definition (Newton differentiability) If there exists a neighborhood N ( x ∗ ) ⊂ S and a family of mappings G : N ( x ∗ ) → L ( X , Y ) such that �F ( x ∗ + h ) −F ( x ∗ ) − G ( x ∗ + h )( h ) � Y lim � h � X → 0 = 0 , � h � X then F is called Newton differentiable at x ∗ . Differentiability of the max function The mapping y �→ max ( 0 , y ) from R n → R n with � 1 if y ≥ 0 g ( y ) = 0 if y < 0 as generalized derivative, is Newton differentiable. 13 / 42
Semismooth Newton method Definition (Newton differentiability) If there exists a neighborhood N ( x ∗ ) ⊂ S and a family of mappings G : N ( x ∗ ) → L ( X , Y ) such that �F ( x ∗ + h ) −F ( x ∗ ) − G ( x ∗ + h )( h ) � Y lim � h � X → 0 = 0 , � h � X then F is called Newton differentiable at x ∗ . Semi-smooth Newton step x k + 1 = x k − G ( x k ) − 1 F ( x k ) . De los Reyes, González (2009,2012) 14 / 42
Regularization + Parallelization How to solve this type of systems for a big amount of discretization points? 15 / 42
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