DISK methods for yield fluids Karol Cascavita Directed by: - PowerPoint PPT Presentation

Yield Fluids Discontinuous Skeletal Methods Results Conclusions DISK methods for yield fluids Karol Cascavita Directed by: Alexandre Ern Supervised by: Xavier Château, Jeremy Bleyer University Paris-Est, CERMICS, NAVIER (ENPC) CERMICS Young Researchers Seminar

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Outline Yield Fluids 1

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Outline Yield Fluids 1 Discontinuous Skeletal Methods 2

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Outline Yield Fluids 1 Discontinuous Skeletal Methods 2 Results 3

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Outline Yield Fluids 1 Discontinuous Skeletal Methods 2 Results 3 Conclusions 4

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Yield Fluids: Motivation Growing interest due to a wide range of applications: Flow of viscoplastic(yield) fluids : civil engineering, materials processing, petroleum drilling operations, food and cosmetics industry. Bubbles in viscoplastic flows: Aerated building materials, mousse. Figure: Examples of applications

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Yield Fluids: Challenges Viscoplastic materials are non-Newtonian fluids that require a finite yield stress to flow (solid or fluid-like behavior) Yield stress fluids are governed by a non-regular and non-linear constitutive equation Solid/liquid boundary not known a priori Viscoplastic materials constitute a challenging problem theoretically and experimentally Scarce analysis, experimental and numerical data in the literature Classical FEM simulations methods are costly for iterative solvers: Not naturally fitted for parallelization. Discontinuous skeletal methods are a promising tool to replace FEM.

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Yield Fluids: Model problem I Let Ω ⑨ R d , d ➙ 1, denote a d-dimensional open bounded and connected domain For a source term f P L 2 ♣ Ω q d Momentum and mass conservation for incompressible flows: div σ t � f ✏ 0 in Ω , ✏ 0 Ω , div u in ✏ 0 ❇ Ω , u on with σ t total stress tensor and u the unknown velocity field. Spheric and deviatoric parts: σ t ✏ σ D ✁ 1 3 tr ♣ σ t q I (1)

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Yield Fluids: Model problem II Viscoplastic fluid model: Bingham model ✩ ❜ ❄ ∇ s u ✫ 1 σ D ✏ 2 µ ∇ s u � 2 τ 0 2 ⑤ σ D ⑤ → τ 0 when ⑤ ∇ s u ⑤ ✪ ∇ s u ✏ 0 otherwise with τ 0 ➙ 0 and µ → 0 denoting the viscosity and the yield stress respectively. ∇ s u the symmetric gradient. We use the Frobenius norm ⑤ τ ⑤ ✏ ❄ τ : τ Figure: Clasification of fluids

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Yield Fluids: Model problem II Glowinski [2] showed the PDEs can be recast as a minimization problem: ➺ ➺ ✏ D ♣ v q d Ω ✁ f ☎ v u arg min v P K ♣ 0 q Ω Ω where K ♣ u D q is the kernel of the divergence operator and defined as ✥ ✭ v P L 2 ♣ Ω q d ⑤ div v ✏ 0 P Ω , u ✏ u D P ❇ Ω K ♣ u D q ✏ , The dissipation energy ❄ D ♣ u q ✏ µ ⑤ ∇ s u ⑤ 2 � 2 τ 0 ⑤ ∇ s u ⑤ ,

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Yield Fluids:Augmented Lagrangian Algorithm Solve the saddle problem ♣ u , γ, σ q ✏ min max τ P L 2 L ♣ v , δ, τ q v P H 1 0 ,δ P L 2 New constraint γ ✏ ∇ s u Augmented Lagrangian: ➺ ➺ ➺ µ ⑤ γ ⑤ 2 d Ω � τ 0 L ♣ u , γ, σ q ✏ ⑤ γ ⑤ d Ω ✁ f ☎ u d Ω 2 Ω Ω Ω ➺ ➺ σ : ♣ ∇ s u ✁ γ q d Ω � α ⑤ ∇ s u ✁ γ ⑤ 2 d Ω � 2 Ω Ω with α → 0 is the augmentation parameter.

Yield Fluids Discontinuous Skeletal Methods Results Conclusions ALG: Uzawa-like algorithm, n th -iteration Weak formulation: Assume σ n and γ n , then find u n P V , ❅ v P H 1 0 ♣ Ω q d such that ♣ f , v q ✁ ♣ σ n ✁ 2 α γ n , ∇ s v q , 2 α ♣ ∇ s u n � 1 , ∇ s v q ✏ Solve point-wise ✂ ✟✡ X n � 1 ♣ x q � 1 γ n � 1 ♣ x q ✏ max ⑤ X n � 1 ♣ x q⑤ ✁ τ 0 0 , ⑤ X n � 1 ♣ x q⑤ ♣ 2 α � µ q where X n � 1 ✏ σ n � α ∇ s u n � 1 . Update the stress σ n � 1 ✏ σ n � α ♣ ∇ s u n � 1 ✁ γ n � 1 q .

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Discontinuous Skeletal methods There are differente kinds of DiSk methods: MFD, HFV, HDG, HHO . Di scontinuous Sk eletal methods approximate solutions of BVPs by using discontinuous polynomials in the mesh skeleton attaching unknowns to mesh faces Salient features: Dimension-independent construction Supportgeneral meshes(conforming and non-conforming) Arbitrary polynomial order

Yield Fluids Discontinuous Skeletal Methods Results Conclusions DISK methods In this work we use the Hybrid High Order method, introduced recently by Di Pietro et Ern [1, 3] for linear elasticity. Attractive features: Designed from primal formulation Arbitrary order of polynomials. Suitable for hp-adaptivity. They can be applied to a fair range of PDE’s. Gradient reconstruction based on local Neumann problems.

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Degrees of freedom I We consider as model problem the laplace equation: ✁ ∆ u ✏ f P Ω Cell-Face based method: Hybrid method. DoFs are polynomials of order k ➙ 0 attached to the mesh cells and their faces. k ✏ 0 k ✏ 1 k ✏ 2 Figure: DOFs for k =0, 1, 2. We define for all T P T h the local space ★→ ✰ U k h ✏ P k P k d ♣ T q ✂ d ✁ 1 ♣ F q F P F h

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Gradient reconstruction I The local potential reconstruction operator: r k � 1 : U k T Ñ P k � 1 ♣ T q T d The local gradient reconstruction operator: ∇ r k � 1 : U k T Ñ ∇ P k � 1 ♣ T q T d T , then ∇ r k � 1 v ✏ ∇ s with s P P k � 1 Let v P U k ♣ T q T d ∇ s solves the local problem for all w P P k � 1 ♣ T q d ➳ ♣ ∇ s , ∇ w q T ✏ ♣ ∇ v T , ∇ w q T � ♣ v F ✁ v T , ∇ w ☎ n TF q F F P F T Reconstruction operator derives from integration by parts formula. ➺ ➺ r k � 1 v T then the reconstructed function is in P k v ✏ d ♣ T q and is Set T T T unique.

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Gradient reconstruction II Local interpolation operator I k T : H 1 ♣ T q Ñ U k T , that maps a given function v P H 1 ♣ T q into the broken space of local collection of velocities. I k T v ✏ ♣ π k T v , ♣ π k F v q F P F T q , Conmuting diagram property For all u P H 1 ♣ T q and all w P P k � 1 ♣ T q d ♣ ∇ r k � 1 I k T v , ∇ w q T ✏ ♣ ∇ u , ∇ w q T (2) T Thus, r k � 1 T is the elliptic operator on P k � 1 I k ♣ T q T d

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Reconstruction operator III Reconstruction operator r k � 1 v is used to build the following bilinear form on T U k T ✂ U k T : a ♣ 1 q T ♣ v , w q ✏ ♣ ∇ r k � 1 v , ∇ r k � 1 w q T T T Note how ♣ ∇ r k � 1 v , ∇ r k � 1 w q T mimics locally the l.h.s. of our original T T problem Find u P H 1 ❅ v P H 1 0 ♣ Ω q s.t. ♣ ∇ u , ∇ v q Ω ✏ ♣ f , v q Ω , 0 ♣ Ω q

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Stabilization operator I T , the reconstructed gradient ∇ r k � 1 For v P U k v is not stable: T ∇ r k � 1 v ✏ 0 does not imply that v T and v ❇ T are constant functions T taking the same value. We introduce a least-squares penalization of the difference between functions in the faces and function in the cell � ✟ S k T v : ✏ π k v ❇ T ✁ ♣ v T � r k � 1 v ✁ π k T r k � 1 v q⑤ ❇ T , ❇ T T T

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Stabilization operator II Using the stabilization operator just defined, we build a second bilinear form on U k T ✂ U k T : ➳ h ✁ 1 F ♣ S k T v , S k s T ♣ v , w q ✏ T w q F , F P F ❇ T where h F denotes the diameter of the face F . The stabilization as defined allows HHO to converge as k � 2 in L 2 norm The simpler stabilization considering the difference v ❇ T ✁ v T would limit convergence to k � 1

Yield Fluids Discontinuous Skeletal Methods Results Conclusions Global spaces Local discrete spaces U k T , for all T P T , are collected into a global discrete space U k h : ✏ U k T ✂ U k F , where U k T : ✏ P k d ♣ T q : ✏ t v T ✏ ♣ v T q T P T ⑤ v T P P k d ♣ T q , ❅ T P T ✉ , U k F : ✏ P k d ✁ 1 ♣ F q : ✏ t v F ✏ ♣ v F q F P F ⑤ v F P P k d ✁ 1 ♣ F q , ❅ F P F ✉ . For a pair v h : ✏ ♣ v T , v F q in the global discrete space u k h , we denote v, for all T P T , its restriction to the local discrete space U k T , where v ❇ T ✏ ♣ v F q F P F ❇ T Homogeneous Dirichlet BCs are enforced strongly by considering the subspace U k h , 0 : ✏ U k T ✂ U k F , 0 , where U k F , 0 : ✏ t v F P U k F ⑤ v F ✑ 0 , ❅ F P F b ✉ .