Giordano Tierra Chica

Transcription

1 NUMERICAL METHODS FOR SOLVING THE CAHN-HILLIARD EQUATION AND ITS APPLICABILITY TO MIXTURES OF NEMATIC-ISOTROPIC FLOWS WITH ANCHORING EFFECTS Giordano Tierra Chica Department of Mathematics Temple University In collaboration with: Francisco Guillén González María Ángeles Rodríguez Bellido Universidad de Sevilla, Seville, Spain Special Thanks to : Christian Zillinger 1 / 68

2 1 Motivation 2 Second order schemes and time-step adaptivity for the Cahn-Hilliard equation Cahn-Hilliard Model Second Order Schemes Approximations of potential term f k (φ n+1, φ n ) Time step adaptivity Numerical Simulations 3 Linear unconditional energy-stable splitting schemes for nematic-isotropic flows with anchoring effects Nematic Liquid Crystals Mixtures of Nematic Liquid Crystals with Newtonian fluid The model Numerical schemes Numerical simulations 4 References 1 / 68

3 Phase field or Diffuse interface models Sharp-interface models PDE for each phase + coupled interface conditions Very difficult numerically (interface tracking) Diffuse interface Phase-field models Phase function with distinct values (for instance +1 and -1) in each phase, with a smooth change in the interface (of width ε). Surface motion depending on the physical energy dissipation. When interface width ε tends to zero, recover a sharp interface model. 2 / 68

4 Motivation Design numerical schemes for diffuse-interface phase-field problems: 1 Efficient in time (Linear schemes, adaptive time-step). 2 Suitable to use (standard) Finite Elements (mesh adaptation) 3 Mimic properties of the continuous problem: Dissipative Energy law, maximum principle, mass conservation,... 4 Good finite and large time accuracy (infinite equilibrium states) Numerical analysis: 1 Large time Energy Stability 2 Unique Solvability of the schemes 3 Convergence of iterative algorithms approximating nonlinear schemes 3 / 68

5 Allen-Cahn and Cahn-Hilliard models The Allen-Cahn and the Cahn-Hilliard models are gradient flows for the same Free Energy (Liapunov functional): ( ) 1 E(φ) = E philic (φ) + E phobic (φ) := 2 φ 2 + F (φ) dx where F (φ) is a double-well potential taking two minimum (stable) values: F (φ) = 1 4ε 2 (φ2 1) 2 at φ = ±1 (polynomial potential: Ginzburg-Landau) Allen-Cahn : φ t + γ δe δφ = 0 ( Cahn-Hilliard: φ t M(φ) δe δφ Ω Maximum Principle ) = 0 Mass Conservation where δe δφ = φ + f (φ) with f (φ) = F (φ) = 1 ε 2 (φ3 φ). In both cases: d t E(φ(t)) 0. 4 / 68

6 Cahn-Hilliard Model Weak formulation: Find (φ, w) such that φ L ((0, T ); H 1 (Ω)) and w L 2 ((0, T ); H 1 (Ω)) satisfying ( ) φ t, w + γ w, w = 0 w H 1 (Ω) ( ) ( ) ( ) φ, φ + f (φ), φ w, φ = 0 φ H 1 (Ω). Energy Law: d dt E(φ(t)) + γ w 2 dx = 0. Ω Mathematical Analysis: Abels, Garcke, Grasselli, Miranville, Schimperna,... Numerical Analysis: Boyer, Elliot, Feng, Gómez, Hughes, Prohl,... 5 / 68

7 Second Order Schemes for the Cahn-Hilliard model Generic Second order Finite Difference schemes (Crank-Nicolson for linear terms) ( δt φ n+1, w ) ) + γ( w n+ 1 2, w = 0 w H 1 (Ω) ( ( φ n+1 + φ n 2 ), φ ) ( + f k (φ n+1, φ n ), φ ) ( ) w n+ 1 2, φ = 0 φ H 1 (Ω), where δ t φ n+1 = (φ n+1 φ n )/k (discrete time derivative). Discrete Energy Law: Testing by ( w, φ) = (w n+ 1 2, δ t φ n+1 ) δ t E(φ n+1 ) + γ w n L + 2 ND philic (φ n+1, φ n ) + ND phobic (φ n+1, φ n ) = 0, where ( ( φ ND philic (φ n+1, φ n n+1 + φ n ) := ), δ t φ n+1) ( ) 1 δ t 2 2 φn+1 2 = 0 and ND phobic (φ n+1, φ n ) := (f k (φ n+1, φ n ), δ t φ n+1) ( δ t Ω Ω ) F (φ n+1 ) 6 / 68

8 Energy Stability Definition Numerical schemes are energy-stable if δ t E(φ n+1 ) + γ w n , n. In particular, the discrete energy decreases, E(φ n+1 ) E(φ n ), n. Ω 7 / 68

9 Eyre s decomposition [Eyre] Splitting the potential term F (φ) = F c (φ)+f e (φ) with F c 0 (convex) and F e 0 (concave) Taking implicitly the convex term and explicitly the non-convex one, i.e. Properties: f k (φ n+1, φ n ) = f c (φ n+1 ) + f e (φ n ) = 1 ε 2 ((φn+1 ) 3 φ n ), First order accurate Nonlinear scheme Unconditionally unique solvable Unconditionally energy-stable 8 / 68

10 Midpoint (MP) Midpoint approximation of the potential term [Elliot], [Du], [Lin],... f k (φ n+1, φ n ) = F(φn+1 ) F(φ n ) φ n+1 φ n Then ND phobic (φ n+1, φ n ) = 0 δ t E(φ n+1 ) + γ w n L 2 = 0 Properties: Second order accurate Nonlinear scheme Conditionally unique solvable (k < ε 4 /γ) Unconditionally energy-stable 9 / 68

11 Midpoint (MP). Newton Scheme Theorem Solvability hypothesis Convergence hypothesis k < 4ε4 γ k 1/2 k ε 4 < C and l«ım (k,h) 0 h 2 = / 68

12 (US2) [Wang et al.] Splitting the potential term F(φ) = F c (φ) + F e (φ) with F c 0 (convex) and F e 0 (concave), Taking MP for the convex term and BDF2 for the non-convex: f k (φ n+1, φ n, φ n 1 )= F c(φ n+1 ) F c (φ n ) φ n+1 φ n + 1 ( ) 3f e (φ n ) f e (φ n 1 ). 2 Properties: Second order accurate Nonlinear scheme Unconditionally unique solvable Unconditionally energy-stable for a perturbed energy Ẽ(φ n+1 ) = E(φ n+1 ) + k 2 1 4ε 2 δ tφ n+1 2 dx, Ω 11 / 68

13 (US2). Newton Scheme Theorem Unconditionally unique solvable Convergence hypothesis (Idem MP) k 1/2 k ε 4 < C and l«ım (k,h) 0 h 2 = / 68

14 Optimal Dissipation Scheme (OD2) Aim: Design f k (φ n+1, φ n ), linear, second order accurate and ND phobic (φ n+1, φ n ) = O(k 2 ) Idea: Using a Hermite quadrature formula, F(φ n+1 ) F(φ n ) φ n+1 φ n = 1 φ n+1 φ n φ n+1 φ n f (φ)dφ = f (φ n ) + f (φ n ) 2 (φn+1 φ n ) + C f (φ n+ζ ) (φ n+1 φ n ) 2 We define f k (φ n+1, φ n ) := f (φ n ) (φn+1 φ n )f (φ n ) Properties: Second order Linear scheme Conditionally solvable (k < 8ε 4 /γ) Remark: We can not control the sign of ND phobic (φ n+1, φ n ) 13 / 68

15 (OD2-BDF2) Splitting the potential term F (φ) = F c (φ) + F e (φ) with F c 0 (convex) and F e 0 (concave), OD2 approximation of the convex term and BDF2 the non-convex one, f k (φ n+1, φ n, φ n 1 )= f c (φ n ) (φn+1 φ n )f c(φ n ) Properties: Second order Linear scheme Unconditionally solvable Remark: We can not control the sign of ND phobic (φ n+1, φ n ) ( 3fe (φ n ) f e (φ n 1 ) ). 14 / 68

16 Time step adaptivity. We have developed a new adaptive-in-time algorithm by using a criterion related to the residual energy law. Generic Algorithm: Given φ n, φ n 1,dt n 1, dt n, resmax and resmin: 1 Compute φ n+1 and RE n+1 := E(φn+1 ) E(φ n ) dt n + w n+1/2 2 L 2. 2 If RE n+1 > resmax, take dt n = dt n /θ and go to 1). (θ > 1) 3 If RE n+1 < resmin, take dt n+1 = θdt n. 4 Take t n+1 = t n + dt n and go to next time step. 15 / 68

17 Numerical Simulations. Spinodal decomposition. Comparative: OD2, MP, US2 and OD2-BDF2 P 1 -cont. FE for φ h, w h. Ω = [0, 1] 2, h = 1/90, γ = 10 4, ε = 10 2, resmax = 10 and resmin = 1. In Newton s method, a tolerance parameter tol = The time-step is reduced in the case that the method does not converge in 10 iterations. Random initial data (the same for all the schemes). 16 / 68

18 Numerical Simulations. Dynamic Figura: Dynamic of the model for the random initial condition 17 / 68

19 Numerical Simulations. Mixing energy OD2 OD2BDF2 MPNW US2NW 2000 Mixing Energy Time Figura: Mixing energy in [0, 0.5] 18 / 68

20 Numerical Simulations. Mixing energy OD2 OD2BDF2 MPNW US2NW Mixing Energy Time Figura: Mixing energy in [0.5, 1] 19 / 68

21 Numerical Simulations. Mixing energy OD2 OD2BDF2 MPNW US2NW Mixing Energy Time Figura: Mixing energy in [1, 5] 20 / 68

22 Numerical Simulations. Mixing energy OD2 OD2BDF2 MPNW US2NW Mixing Energy Time Figura: Mixing energy in [5, 8.5] 21 / 68

23 Numerical Simulations. Equilibrium solution of MP Figura: Equilibrium solution of MP 22 / 68

24 Numerical Simulations. Time steps OD2 OD2BDF2 MPNW US2NW Time Step Time Figura: Time steps in [0, 0.5] 23 / 68

25 Numerical Simulations. Time step Time steps OD2 OD2BDF2 MPNW US2NW 0.03 Time Step Time Figura: Time steps in [0.5, 1] 24 / 68

26 Numerical Simulations. Time steps OD2 OD2BDF2 MPNW US2NW Time Step Time Figura: Time steps in [1, 5] 25 / 68

27 Numerical Simulations. Time steps OD2 OD2BDF2 MPNW US2NW Time Step Time Figura: Time steps in [5, 8.5] 26 / 68

28 Numerical Simulations. Time steps Mixing Energy OD2 OD2BDF2 MPNW tol=10 3 MPNW tol=10 8 MPNW tol=10 12 MPNW tol=10 13 US2NW tol= Time Figura: Mixing energy in [6, 8.5] 27 / 68

29 Numerical Simulations. Efficiency Computational cost: MP OD2 OD2-BDF2 US2 # Time steps # Linear systems solved (OD2 with constant time step k = iterations) Conclusions: Figura: Features of schemes 28 / 68

30 3D Numerical Simulations. OD2 time scheme. Finite element discretization in space, with φ h and w h in P 1 -cont. FE Ω = [0, 1] 3, h = 1/30, γ = 10 4, ε = 10 2, resmax = 10 and resmin = 1. Random initial data. 29 / 68

31 Applications LINEAR UNCONDITIONAL ENERGY-STABLE SPLITTING SCHEMES FOR A PHASE-FIELD MODEL FOR NEMATIC-ISOTROPIC FLOWS WITH ANCHORING EFFECTS 30 / 68

32 Types of Liquid Crystals Figura: Types of Liquid Crystals 31 / 68

33 Types of Liquid Crystals d d n n θ d Nematic Smectic-A Smectic-C Figura: Types of Liquid Crystals 32 / 68

34 Nematic Liquid Crystals Ginzburg-Landau formulation (penalized version of Ericksen-Leslie system): u t + u u + p σ vis σ nem = 0, u = 0, d t + (u )d + γ nemw = 0, w = δenem δd, where (δ /δd) denotes the variational derivative with respect to d, γ nem > 0 is the relaxation time coefficient, (1) and σ vis = 2νDu, σ nem = λ nem( d) t d, ( ) 1 E nem(d) = Ω 2 d 2 + G(d) dx with G(d) = 1 4η 2 ( d 2 1) 2. It is known that this system satisfies the following energy law, d dt [E kin(u) + λ neme nem(d)] + 2 ν Du 2 δe nem 2 dx + λ nem γ nem Ω Ω δd dx = / 68

35 Nematic-Isotropic. The variables of the problem The following variable will take part in the description of the model: the solenoidal velocity u(t, x), t (0, T ), x Ω R 3 the pressure of the fluid p(t, x), the director field d(t, x), that represents the average orientation of the liquid crystal molecules, the function c(t, x) localizing the two components along the domain Ω R d (d = 2 or 3) filled by the mixture, 1 in the Newtonian Fluid part, c(t, x) = ( 1, 1) in the interface part, 1 in the Nematic Liquid Crystal part. 34 / 68

36 Nematic-Isotropic. Energy The total energy of the system is given by E tot (u, d, c) = E kin (u) + λ mix E mix (c) + λ nem E nem (d, c) + λ anch E anch (d, c) with E kin (u) = 1 u 2 dx kinetic energy, 2 Ω ( ) 1 E mix (c) = Ω 2 c 2 + F(c) dx mixing energy, ( ) 1 E nem (d, c) = I(c) 2 d 2 + G(d) dx elastic energy, Ω where F(c) = 1 4ε 2 (c2 1) 2, G(d) = 1 4η 2 ( d 2 1) 2, and we represent their derivatives as f (c) := F (c) and g(d) := G (d). 35 / 68

37 Nematic-Isotropic. The anchoring effect At the interface between the nematic and newtonian fluids, liquid crystals prefer to orientate following a certain direction (called as easy direction). Three effects can be described: the parallel case, where the director vector is parallel to the interface, the homeotropic case, where the director vector is normal to the interface, no anchoring. NO ANCHORING PARALLEL ANCHORING HOMEOTROPIC ANCHORING 36 / 68

38 Nematic-Isotropic. The anchoring effect E anch (d, c) = 1 2 Ω ( δ 1 d 2 c 2 + δ 2 d c 2) dx where the anchoring energy will take different forms depending on the anchoring effect considered, that is, (0, 0) no anchoring, (δ 1, δ 2 ) = (0, 1) parallel anchoring, (2) (1, 1) homeotropic anchoring. 37 / 68

39 Nematic-Isotropic. The localizing functional I(c) x It represents the volume fraction of liquid crystal at each point x Ω and its derivative will be denoted by i(c) := I (c). It could take different forms but any admissible form must satisfy the following properties: I C 2 (R), I(c) = 0 if c 1, I(c) = 1 if c 1, I(c) (0, 1) if c ( 1, 1). We consider the following interpolation function I(c) := and its derivative is defined as i(c) := I (c) = 0 if c 1, 1 16 (c + 1)3 (3c 2 9c + 8) if c ( 1, 1), 1 if c 1, (c + 1)2 (c 1) 2 if c ( 1, 1), 0 otherwise / 68

40 Nematic-Isotropic. The model Combining the Least Action Principle (LAP) and the Maximum Dissipation Principle (MDP), we arrive to the following PDE system, fulfilled in the time space domain (0, T ) Ω: u t + u u + p σ tot = 0, u = 0, d t + (u )d + γ nem w = 0, w = δetot δd, c t + u c (γ mix µ) = 0, µ = δetot δc. (3) 39 / 68

41 Nematic-Isotropic. The total stress tensor The total tensor reads, being: σ tot = σ vis + σ mix + σ nem + σ anch, σ vis = 2νDu viscosity, σ mix = λ mix c c mixing tensor, σ nem = λ nem I(c)( d) t d nematic tensor, and the anchoring tensor σ anch has the form: ] (σ anch ) ij = λ anch [δ 1 d 2 c c + δ 2 (d c) ( c d) 40 / 68

42 Nematic-Isotropic. The expression for w and µ Taking into account that the total energy of the system is given by E tot(u, d, c) = E kin (u) + λ mix E mix (c) + λ neme nem(d, c) + λ anch E anch (d, c) then the variational derivatives of E tot are w = δe tot δd = λnem[ (I(c) d) + I(c) G (d)] + λ anch δe anch δd, and µ = δe tot δc ( ) 1 = λ mix [ c + F (c)] + λ nemi (c) 2 d 2 + G(d) + λ anch δe anch δc, where the anchoring terms will depend on each case: 0 No anchoring, δe anch = (d c) c Parallel anch., δd c 2 d (d c) c Homeotropic anch.. (4) and δe anch δc 0 No anchoring, = [(d c) d] Parallel anch., [ d 2 c (d c) d] Homeotropic anch. (5) 41 / 68

43 Nematic-Isotropic. The model The PDE system (3) is closed with the following initial and boundary conditions: u t=0 = u 0, d t=0 = d 0, c t=0 = c 0 in Ω, u Ω = ( I(c) d ) n Ω = 0 in (0, T ), ( c n = δe ) tot n Ω δc = 0 in (0, T ), Ω (6) 42 / 68

44 Nematic-Isotropic. Reformulation of the stress tensor Lemma The following relation holds: where σ mix σ nem σ anch = µ c ( d) t w + ϕ ϕ = λ nem I(c) ( ) ( ) d 2 + G(d) +λ mix 2 c 2 + F(c) + λ anch W (d, c), 2 with W (d, c) = ( δ 1 d 2 c 2 + δ 2 d c 2). 43 / 68

45 Nematic-Isotropic. The variational formulation ( ) u t, ū + ((u )u, ū) + (ν(c)du, Dū) ( p, ū) ( d) t w, ū + (c µ, ū) = 0, ( u, p) = 0, d t, w + ((u )d, w) + γ nem(w, w) = 0, λ nem(i(c) d, d) + λ nem(i(c) g(d), d) δe anch + λ anch = (w, d), δd (c t, µ) (c u, µ) + γ mix ( µ, µ) = 0, [ ] ) d λ mix ( c, c) + λ mix (f (c), c) + λ nem (i(c) 2 δe anch + G(d), c + λ anch = (µ, c), 2 δc for each (ū, p, w, d, µ, c) H 1 0 (Ω) L2 0 (Ω) H1 (Ω) H 1 (Ω) H 1 (Ω) H 1 (Ω). 44 / 68

46 Nematic-Isotropic. Continuous energy law Using adequate test functions, we can prove that the previous system satisfies the following (dissipative) energy law: d Etot(u, d, c) + ν(c) Du 2 dx + γ nem w 2 dx + γ mix µ 2 dx = 0. dt Ω Ω Ω From the energy law, we deduce the following regularity for a (possible) solution: u L (0, T ; L 2 (Ω)) L 2 (0, T ; H 1 (Ω)), w L 2 (0, T ; L 2 (Ω)), c L (0, T ; L 2 (Ω)), µ L 2 (0, T ; L 2 (Ω)), Ω F (c)dx L (0, T ), ( ) Ω I(c) 1 2 d 2 + G(d) dx L (0, T ) (7) E anch (c, d) L (0, T ), c L (0, T ; H 1 (Ω)), Ω I(c) d 4 L (0, T ), d L (0, T ; L 2 (Ω)). 45 / 68

47 Nematic-Isotropic. Numerical schemes For simplicity, we describe our numerical scheme using an uniform partition of the time interval: t n = nk, where k > 0 denotes the (fixed) time step. Moreover, hereafter we denote δ t a n+1 := an+1 a n. k 46 / 68

48 Nematic-Isotropic. Numerical schemes Definition A numerical scheme is energy-stable if it satisfies δ t E tot (u n+1, d n+1, c n+1 ) + ν(c n+1 ) Du n+1 2 dx Ω +γ nem w n+1 2 dx + γ mix µ n+1 2 dx 0, n. Ω In particular, energy-stable schemes satisfy the energy decreasing in time property, i.e., Ω E tot (u n+1, d n+1, c n+1 ) E tot (u n, d n, c n ), n. 47 / 68

49 Nematic-Isotropic. Coupled Nonlinear Implicit Scheme Given (u n, p n, d n, w n, c n, µ n ), find (u n+1, p n+1, d n+1, w n+1, c n+1, µ n+1 ) such that, ( u n+1 u n ) ( ) k, ū + (u n+1 )u n+1, ū (p n+1, ū) + 2(νDu n+1, Dū) ( ) ( d n+1 ) t w n+1, ū + (c n+1 µ n+1, ū) = 0, ( u n+1, p) = 0, ( ) d n+1 d n (, w + (u n+1 )d n+1 ), w + γ nem(w n+1, w) = 0, k ( ) λ nem(i(c n+1 ) d n+1, d) + λ nem(i(c n+1 ) g(d n+1 ), d) δeanch + λ anch (c n+1, d n+1 ), d (w n+1, d) = 0, δd ( ) c n+1 c n, µ (c n+1 u n+1, µ) + γ mix ( µ n+1, µ) = 0, k [ ] ) λ mix ( c n+1, c) + λ mix (f (c n+1 ), c) + λ nem (i(c n+1 d n+1 2 ) + G(d n+1 ), c 2 ( ) δeanch +λ anch (c n+1, d n+1 ), c (µ n+1, c) = 0, δc (8) Disadvantages of this scheme: High computational cost (Coupled + Nonlinear system) it is not clear that any iterative method to approximate the nonlinear scheme will converge(several nonlinearities) Energy-stability? 48 / 68

50 Nematic-Isotropic. Splitting schemes We have designed two splitting first-order schemes, denoted by or (d n+1, w n+1 ) (c n+1, µ n+1 ) (u n+1, p n+1 ), (c n+1, µ n+1 ) (d n+1, w n+1 ) (u n+1, p n+1 ), decoupling computations for nematic part (d, w) from the phase-field part (c, µ) (or the contrary in the second case) and from the fluid part (u, p). 49 / 68

51 Nematic-Isotropic. Splitting schemes STEP 1: Find (d n+1, w n+1 ) D h W h s. t, for each ( d, w) D h W h ( d n+1 d n ), w + ((u )d n, w) + γ nem (w n+1, w) = 0, k ) ( λ nem (I(c n ) d n+1, d + λ nem I(c n )g k (d n+1, d n ), d ) +λ anch (Λ d (d n+1, c n ), d ) (w n+1, d) = 0, where u := u n + 2 k ( d n ) t w n+1, g k (d n+1, d n ) is an approximation of g(d(t n+1 )) and Λ d (d n+1, c n ) is the discrete approximation of δe anch δd (d(t n+1), c(t n+1 )): Λ d (d n+1, c n ) := δ 1 c n 2 d n+1 + δ 2 (d n+1 c n ) c n 50 / 68

52 Nematic-Isotropic. Splitting schemes STEP 2: Find (c n+1, µ n+1 ) C h M h s. t., for ( c, µ) C h M h ( c n+1 c n ), µ (c n u, µ) + γ mix ( µ n+1, µ) = 0, k λ mix ( c n+1, c) + λ mix (f k (c n+1, c n ), c) [ 1 +λ nem (i k (c n+1, c n ) +λ anch (Λ c (d n+1, c n+1 ), c ] ) 2 d n G(d n+1 ), c ) (µ n+1, c) = 0, where u := u n 2 k c n µ n+1, f k (c n+1, c n ) and i k (c n+1, c n ) are approximations of f (c(t n+1 )) and i(c(t n+1 )), resp. and Λ c (d n+1, c n+1 ) is the discrete approximation of δe anch δc (d(t n+1), c(t n+1 )): Λ c (d n+1, c n+1 ) := δ 1 d n+1 2 c n+1 + δ 2 (d n+1 c n+1 ) d n+1 51 / 68

53 Nematic-Isotropic. Splitting schemes STEP 3: Find (u n+1, p n+1 ) V h P h s. t., for each (ū, p) V h P h ( u n+1 ) û, ū + c(u n, u n+1, ū) (p n+1, ū) k +(ν(c n+1 )Du n+1, Dū) = 0, ( u n+1, p) = 0, where and c(u, v, w) := û := u + u, 2 ( ) (u )v, w + 1 ( ) u, v w / 68

54 Nematic-Isotropic. Local (in time) discrete energy law Scheme given by STEPS 1-3 satisfies the following local discrete energy law: δ t E(d n+1, c n+1, u n+1 ) + γ nem w n+1 2 L 2 +γ mix µ n+1 2 L 2 + ν(c n+1 ) 1/2 Du n+1 2 L 2 +ND n+1 u + ND n+1 elast (cn ) + ND n+1 penal (cn ) +ND n+1 philic + NDn+1 phobic + NDn+1 interp + NDn+1 anch = / 68

55 Nematic-Isotropic. Local (in time) discrete energy law The numerical dissipation terms are: ( NDu n+1 = 1 u n+1 û 2 2k L + û u 2 + û u 2 L 2 L ) ND n+1 elast (cn ) = λ nem k 2 ND n+1 penal (cn ) = λ nem ND n+1 philic = λ k mix 2 ND n+1 phobic = λ mix + u u n 2 + u u n 2 L 2 L 2 2 i(c n ) δ t d n+1 2 dx, Ω Ω Ω Ω i(c n ) ( ) g k (d n+1, d n ) δ t d n+1 δ t G(d n+1 ) dx, δ t c n+1 2 dx, ( ) f k (c n+1, c n ) δ t c n+1 δ t F(c n+1 ) dx, 54 / 68

56 Nematic-Isotropic. Local (in time) discrete energy law and ND n+1 anch ( d ND n+1 n+1 interp = λ 2 ) nem + G(d n+1 ) Ω 2 ( i k (c n+1, c n ) δ t c n+1 δ t I(c n+1 ) ) dx, = λ anch k 2 Ω (δ 1 ( δ t d n+1 2 c n 2 + d n+1 2 δ t c n+1 2) +δ 2 ( δt d n+1 c n 2 + d n+1 δ t c n+1 2) ) dx. with (δ 1, δ 2 ) defined in (2) depending on the type of anchoring. 55 / 68

57 Nematic-Isotropic. The functions f k, g k and i k QUESTION: How to define f k (c n+1, c n ), g k (d n+1, d n ), i k (c n+1, c n ) to obtain linear unconditionally energy-stable schemes? That is, we want f k (c n+1, c n ), g k (d n+1, d n ), i k (c n+1, c n ) linear such that ND n+1 penal (cn ) 0, ND n+1 phobic 0, and NDn+1 interp / 68

58 Nematic-Isotropic. The function f k f k (c n+1, c n ) := f (c n ) f (c n+1 c n ), (9) in our case reduces to f k (c n+1, c n ) = f (c n ) + (c n+1 c n ) (10) where f (c) is the C 1 -truncation of F (c): 2 (c + 1) if c 1, ε2 1 f (c) = ε 2 (c2 1) c if c [ 1, 1], 2 (c 1) if c 1, ε2 (11) 57 / 68

59 Nematic-Isotropic. The functions g k and i k g k (d n+1, d n ) = g(d n ) (d n+1 d n ), (12) where g(d) is the C 1 -truncation of g(d): 2 ( d 1) d if d 1, g(d) = d ( d 2 1) d if d 1, and we also take i k (c n+1, c n ) = i(c n ) (cn+1 c n ). (13) 58 / 68

60 Nematic-Isotropic. Well-Posedness of the Schemes Lemma If D h W h, then there exist a unique solution (d n+1, w n+1 ) of STEP 1 using the potential approximation (12) for g k (d n+1, d n ). Lemma If 1 C h, then there exist a unique solution (c n+1, µ n+1 ) of STEP 2 using the potential approximations (10) and (13) for f k (c n+1, c n ) and i k (c n+1, c n ), respectively. Lemma If the pair of FE spaces (V h, P h ) satisfies the discrete inf-sup condition (p, ū) β > 0 such that p L 2 β sup ū V h \{Θ} ū H 1 p P h, (14) then there exist a unique solution (u n+1, p n+1 ) of STEP 3. We propose the following choice for the discrete spaces: (u, p) P 2 P 1, (c, µ) P 1 P 1 and (d, w) P 1 P 1, (15) that satisfy the assumptions of Lemmas 6, 7 and / 68

61 Nematic-Isotropic. Numerical simulations The newtonian fluid is represented by blue color while the nematic fluid is represented by red one. For simplicity we are considering constant viscosity ν(c) = ν 0. Ω [0, T ] h dt ν 0 η [ 1, 1] 2 [0, 10] 2/90 0,001 1,0 0,075 λ nem λ mix λ anch γ nem γ mix ε 0,1 0,01 0,1 0,5 0,01 0,05 60 / 68

62 Nematic-Isotropic. Circular droplet and director field parallel to the y-axis 61 / 68

63 Nematic-Isotropic. Circular droplet and director field parallel to the y-axis TOTAL ENERGY CASE 1 No Anchoring Parallel Homeotropic Time 62 / 68

64 Nematic-Isotropic. Elliptic droplet with two points defects at (±1/2, 0) A Hedgehog defect at (1/2, 0) and an Antihedgehog defect at ( 1/2, 0) d 0 (x) = d/ d 2 + 0,05 2, with d = (x 2 + y 2 0,25, y). Defect annihilation in Nematic Liquid Crystals Defect annihilation in Nematic Liquid Crystals Drops 63 / 68

65 Nematic-Isotropic. Circular droplet and director field parallel to the y-axis TOTAL ENERGY CASE 2 No Anchoring Parallel Homeotropic Time 64 / 68

66 Nematic-Isotropic. Spinodal Decomposition Random initial data for c, i.e., c [ 10 2, 10 2 ] in Ω = [0, 1] [0, 1], t [0, 1] and dt = The initial director vector is computed using the function: ( ) d = I(c) sin(x y) sin(x y), cos(x y) cos(x y). 65 / 68

67 Nematic-Isotropic. Spinodal Decomposition TOTAL ENERGY CASE 3 No Anchoring Parallel Homeotropic Time 66 / 68

68 References van Bijnen RMW, Otten RHJ, van der Schoot P. Texture and shape of two-dimensional domains of nematic liquid crystals. Physical Review E 2012; 86 : Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics 1958; 28 : Climent-Ezquerra B, Guillén-González F. A review of mathematical analysis of nematic and smectic-a liquid crystal models. European Journal of Applied Mathematics 2013; 10 : 1-21 Collings PJ. Liquid Crystals:Nature s Delicate Phase of Matter. Princeton University Press, Guillén-González F, Rodríguez-Bellido MA, Tierra G. Linear unconditional energy-stable splitting schemes for a phase-field model for nematic-isotropic flows with anchoring effects. Int. J. Numer. Meth. Engng 2016; Shen J, Yang X. Decoupled energy stable schemes for phase field models of two-phase complex fluids. SIAM Journal of Scientific Computing 2014; 36 : Tierra G, Guillén-González F. Numerical methods for solving the Cahn-Hilliard equation and its applicability to related Energy based models. Archives of Computational Methods in Engineering 2015; 22 : Yang X, Forest MG, Li H, Liu C, Shen J, Wang Q, Chen F. Modeling and simulations of drop pinch-off from liquid crystal filaments and the leaky liquid crystal faucet immersed in viscous fluids. Journal of Computational Physics 2013; 236 : 1-14, Yue P, Feng JJ, Liu C, Shen J. A diffuse-interface method for simulating two-phase flows of complex fluids. Journal of Fluid Mechanics 2004; 515 : / 68

69 THANK YOU FOR YOUR ATTENTION! 68 / 68