Numerical Solution of Vector-Valued Phase Field Models Carsten Gr - PowerPoint PPT Presentation

Numerical Solution of Vector-Valued Phase Field Models Carsten Gr¨ aser, Ralf Kornhuber, and Uli Sack (FU Berlin) DIMO 2013 – Diffuse Interface Models Levico Terme, September 10 – 13, 2013 Matheon

Synopsis • phase transition and phase separation • vector-valued Allen-Cahn equations (Bronsard & Reitich 93, Garcke et al. 98, 99a, 99b) – robust formulation of discrete spatial problems (logarithmic � obstacle potential) – polygonal Gauß-Seidel relaxation (Kh. & Krause 03, 06) – truncated non-smooth Newton multigrid (TNNMG) (Gr¨ aser & Kh. 09, Gr¨ aser 11) – numerical experiments • vector-valued Cahn-Hilliard equations (Blowey et al. 96, Barret & Blowey 96, 97, Gr¨ aser, Kh. & Sack 13) – robust formulation of discrete spatial problems – truncated Schur Newton methods (Gr¨ aser & Kh. 06, 09, Gr¨ aser 08, 11) – numerical experiments

Ginzburg-Landau Approach to Phase Transition/Separation ε |∇ u | 2 + 1 � Ginzburg-Landau free energy: E ( u ) = ε Ψ θ ( u ) dx Ω order parameter (phase field): u ∈ [ − 1 , +1] diffuse interface: Γ ε = { x ∈ Ω | u ( x ) ∈ ( u a , u b ) } , binodal values u a , u b ∈ [ − 1 , 1] logarithmic free energy: (Giacomin & Lebowitz 98) Ψ ′ 500 θ 400 300 Ψ θ ( u ) = 1 2 θ ((1 − u ) ln( 1 − u 2 ) 200 100 + (1 + u ) ln( 1+ u 2 θ c (1 − u 2 ) 2 )) + 1 0 −100 −200 critical temperature θ c temperature θ , −300 −400 deep quench limit: θ → 0 ⇒ Ψ θ → Ψ 0 −500 −1.5 −1 −0.5 0 0.5 1 1.5 u

Phase Field Models isothermal case: θ = const . phase transition: Allen-Cahn equation (non-conserving) εu t = − d du E ( u ) = ε ∆ u − 1 ε Ψ ′ θ ( u ) (Cahn 60, Allen & Cahn 77) phase separation: Cahn-Hilliard equations (conserving) du E ( u ) = − ε ∆ u + 1 d ε Ψ ′ εu t = ∆ w, w = θ ( u ) (Cahn & Hilliard 58) d dt E ( u ( t )) ≤ 0 Lyapunov functional: deep quench limit θ = 0 : variational inequalities robustness of numerical solvers for θ → 0 (Kh. & Krause 03, 06, Gr¨ aser & Kh. 09, Gr¨ aser 11)

Phase Field Models isothermal case: θ = const . phase transition: Allen-Cahn equation (non-conserving) εu t = − d du E ( u ) = ε ∆ u − 1 ε Ψ ′ θ ( u ) (Cahn 60, Allen & Cahn 77) phase separation: Cahn-Hilliard equations (conserving) du E ( u ) = − ε ∆ u + 1 d ε Ψ ′ εu t = ∆ w, w = θ ( u ) (Cahn & Hilliard 58) d dt E ( u ( t )) ≤ 0 Lyapunov functional: deep quench limit θ = 0 : variational inequalities robustness of numerical solvers for θ → 0 (Kh. & Krause 03, 06, Gr¨ aser & Kh. 09, Gr¨ aser 11)

N Phases concentrations: u 1 , . . . , u N u = ( u 1 , . . . , u N ) ∈ R N vector-valued phase field: N u ( x, t ) ∈ G = { v ∈ R N | 0 ≤ v i , � Gibbs simplex: v i = 1 } (closed, convex) i =1 u 3 u 2 �������������� �������������� G �������������� �������������� �������������� �������������� �������������� �������������� u 2 �������������� �������������� �������������� �������������� G �������������� �������������� N=2: N=3: �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� u 1 �������������� �������������� �������������� �������������� �������������� �������������� u 1

Ginzburg-Landau Free Energy N ε |∇ u i | 2 + 1 � � E ( u ) = ε Ψ θ ( u ) dx, ε > 0 2 Ω i =1 Ψ θ ( u ) = Φ θ ( u ) + χ H 1 ( u ) + θ cN multi-well potential: 2 Cu · u  N �  θu i ln( u i ) + χ [0 , ∞ ) ( u i ) , θ > 0  Φ θ ( u ) = i =1  χ [0 , ∞ ) ( u i ) , θ = 0  H 1 = { v ∈ R N | � N i =1 v i = 1 } critical temperature: θ c = 1 C = (1 − δ ij ) N symmetric interaction matrix: i,j =1

Vector-Valued Allen-Cahn Equations projected L 2 -gradient flow of E : N 11 ⊤ ∈ R N × N , εu t = ε ∆ u − 1 ε P Ψ ′ P = I − 1 θ ( u ) , θ > 0 parabolic variational inequality (Bronsard & Reitich 93, Garcke et al. 98, 99a, 99b) : u ( t ) ∈ G : ε ( u t , v − u ) + ε ( ∇ u, ∇ ( v − u )) θ ≥ 0 + 1 ε ( φ θ ( v ) − φ θ ( u )) − N 1 ε ( u, v − u ) ≥ 0 ∀ v ∈ G � proper convex, lower semi-continuous functional: φ θ ( v ) = Ω Φ θ ( v ( x )) dx � N | v ( x ) ∈ G a.e. in Ω � � H 1 (Ω) � Gibbs constraints: G = v ∈

Discretization implicit Euler discretization: stepsize τ linear finite elements S N : j triangulation T j , meshsize h j = O (2 − j ) , nodes N j , nodal basis λ ( j ) of S j p discrete spatial problems: u j ∈ G j : ( u j , v − u j ) + τ ( ∇ u j , ∇ ( v − u j )) + τ ε 2 ( φ θ,j ( v ) − φ θ,j ( u )) − N τ ε 2 ( u j , v − u j ) ≥ ( u old j , v − u j ) ∀ v ∈ G j � quadrature rule (lumping): φ θ,j = Ω I S N j Φ θ ( v ) dx v ∈ S N � � G j = j | v ( p ) ∈ G ∀ p ∈ N j discrete Gibbs constraints:

Discretization implicit Euler discretization: stepsize τ linear finite elements S N : j triangulation T j , meshsize h j = O (2 − j ) , nodes N j , nodal basis λ ( j ) of S j p discrete spatial problems: u j ∈ G j : ( u j , v − u j ) + τ ( ∇ u j , ∇ ( v − u j )) + τ ε 2 ( φ θ,j ( v ) − φ θ,j ( u )) − N τ ε 2 ( u j , v − u j ) ≥ ( u old j , v − u j ) ∀ v ∈ G j � quadrature rule (lumping): φ θ,j = Ω I S N j Φ θ ( v ) dx v ∈ S N � � G j = j | v ( p ) ∈ G ∀ p ∈ N j discrete Gibbs constraints:

Discretization stepsize τ < ε 2 implicit Euler discretization: N linear finite elements S N : j triangulation T j , meshsize h j = O (2 − j ) , nodes N j , nodal basis λ ( j ) of S j p discrete spatial problems: u j ∈ G j : ( u j , v − u j ) + τ ( ∇ u j , ∇ ( v − u j )) + τ ε 2 ( φ θ,j ( v ) − φ θ,j ( u )) − N τ ε 2 ( u j , v − u j ) ≥ ( u old j , v − u j ) ∀ v ∈ G j � quadrature rule (lumping): φ θ,j = Ω I S N j Φ θ ( v ) dx v ∈ S N � � G j = j | v ( p ) ∈ G ∀ p ∈ N j discrete Gibbs constraints:

Convex Minimization variational inequality: a ( u j , v − u j ) + τ u j ∈ G j : ε 2 ( φ j ( v ) − φ j ( u )) ≥ ℓ ( v − u j ) ∀ v ∈ G j a ( v, w ) = (1 − N τ ( H 1 (Ω)) N -elliptic bilinear form: ε 2 )( v, w ) + τ ( ∇ v, ∇ w ) ℓ ( v ) = ( u old linear functional: j , v ) equivalent reformulation: u j ∈ G j : J ( u j ) ≤ J ( v ) ∀ v ∈ G j J ( v ) = 1 2 a ( v, v ) + τ ε 2 φ j ( v ) − ℓ ( v ) strictly convex energy:

Polygonal Gauß-Seidel Relaxation (Kh. & Krause 03) descent directions: λ ( j ) p E m , p ∈ N j , m = 1 , . . . , M edge vectors of G : E 2 E 3 G E 1 , . . . , E M ∈ R N , M := N ( N − 1) = O ( N 2 ) 2 E 1 complexity: O ( N 2 n j ) global convergence (Kh., Krause & Ziegler 06) ρ j = 1 − O (2 − j ) exponentially deteriorating convergence speed:

Polygonal Gauß-Seidel Relaxation (Kh. & Krause 03) descent directions: λ ( j ) p E m , p ∈ N j , m = 1 , . . . , M edge vectors of G : E 2 E 3 G E 1 , . . . , E M ∈ R N , M := N ( N − 1) = O ( N 2 ) 2 E 1 successive minimization of J + χ G j on 1-D subspaces span { λ ( j ) p E m } , p ∈ N j , m = 1 , . . . , M complexity: O ( N 2 n j ) global convergence (Kh., Krause & Ziegler 06) ρ j = 1 − O (2 − j ) exponentially deteriorating convergence speed:

Polygonal Gauß-Seidel Relaxation (Kh. & Krause 03) descent directions: λ ( j ) p E m , p ∈ N j , m = 1 , . . . , M edge vectors of G : E 2 E 3 G E 1 , . . . , E M ∈ R N , M := N ( N − 1) = O ( N 2 ) 2 E 1 successive minimization of J + χ G j on 1-D subspaces span { λ ( j ) p E m } , p ∈ N j , m = 1 , . . . , M complexity: O ( N 2 n j ) global convergence (Kh., Krause & Ziegler 06) ρ j = 1 − O (2 − j ) exponentially deteriorating convergence speed: