On Multigrid for Phase Field

Transcription

1 On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004

2 Synopsis adaptivity in time and space fast, robust solution of discrete spatial problems vector-valued Allen Cahn equations (Kh. & Krause) obstacle potential: monotone multigrid (MMG) by successive 1-D minimization logarithmic potential: robustness by constrained Newton linearization numerical experiments diffusion induced grain boundary motion (Kh., Krause & Styles) phase field models multigrid algorithms for the phase field and the solute Cahn Hilliard equation with obstacle potential (Gräser & Kh.) minimization, saddle point formulation and Uzawa-type iteration numerical experiments

3 Ginzburg Landau Approach Ginzburg Landau free energy: E(u) = phase field: u [ 1, +1] Ω ε u ψ(u) dx ε ψ T logarithmic free energy: ψ T (u) = 1 2T((1 u) ln(1 u 2 ) + (1 + u) ln( 1+u 2 )) T c(1 u 2 ) temperature T, critical temperature T c u deep quench limit: T 0 ψ T ψ 0 robustness of numerical solvers for T 0: no classical Newton linearization

4 Phase Field Models isothermal case: T = const. phase transition: Allen-Cahn equation (non-conserving) εu t = d du E(u) = ε u 1 ε ψ (u) (Cahn 60, Allen & Cahn 77) phase separation: Cahn-Hilliard equations (conserving) εu t = w, w = d du E(u) = ε u + 1 ε ψ (u) (Cahn & Hilliard 58) Lyapunov functional: d dt E(u(t)) 0 deep quench limit: complementarity problems, variational inequalities

5 Phase Transition with N Phases concentrations: u1,...,un vector-valued phase field: u = (u1,...,un) R N Gibbs simplex: u(x, t) G = {v R N 0 vi, N i=1 vi = 1} (closed, convex) examples: N=2: u 1 u 2 G N=3: u 1 u 3 u 2 G

6 Vector-valued Allen-Cahn Equation: Obstacle Potential Ginzburg-Landau free energy: E(u) = Ω ε N 2 i=1 u i εψ(u) dx, ε > 0 obstacle potential: ψ = ψ 0 = T c N 2 N i=1 u i(1 u i ) u G parabolic variational inequality (Bronsard & Reitich 93, Garcke et al. 98, 99a, 99b): u(t) G : Gibbs constraints: G = ε(u t, v u) + ε( u, (v u)) T cn ε (u,v u) 0 v {v ( H 1 (Ω) ) N v(x) G a.e. in Ω } G special case N = 2: scalar free energy for u := u 1 u 2

7 Discretization semi-implicit Euler discretization (concave part explicit): stepsize τ (unconditionally stable) linear finite elements S N j : triangulation T j, meshsize h j = O(2 j ), nodes N j, nodal basis λ (j) p discrete spatial problems: u k j G j : u k j, v u k j + τ( u k j, (v u k j)) }{{} a(u k j,v uk j ) τ 1 + T c N ε 2uk 1 j, v u k j }{{} l(v u k j ) k uj G j discrete Gibbs constraints: G j = { v S N j v(p) G p N j }

8 Equivalent Formulation: Convex Minimization u j G j : J (u j ) J (v) v G j quadratic energy: J (v) = 1 2a(v, w) l(v) s.p.d. bilinear form: a(v, w) = v, w + τ( v, w) linear functional: l(v) = 1 + T c N τ u k 1 ε 2 j,v closed convex set: G j = { v S N j v(p) G p N j } Sj iterative solution by descent methods

9 Polygonal Gauß-Seidel Relaxation (Kh. & Krause 02) descent directions: λ (j) p E m, p N j, m = 1,...,M edge vectors of G: E 1,...,E M R N, M := N(N 1) 2 = O(N 2 ) E 2 E 3 G hyperplane: H j = span{λ (j) p E m p N j,m = 1,...,M} E 1 polygonal splitting: H j = p N j M m=1 V (j) p,m, V (j) p,m = span{λ (j) p E m } complexity: O(N 2 n j ) successive minimization of J = J + χ Gj on 1-D subspaces V (j) p,m for p N j and m = 1,...,M

10 convergence of subsequence: u ν k j Convergence u = u pλ (j) p G j u is fixed point: a(u,λ (j) p η i ) l(λ (j) p η i ) η i span{e i }, u p + η i G Lemma (Kh., Krause & Ziegler) u p, v p G there is a decomposition N 1 v p u p = η i, η i span{e mi }, u p + η i G i=1 v p u p u = u j is the solution: a(u,λ (j) p (v p u p)) l(λ (j) p (v p u p)) v p G a(u,v u ) l(v u ) v G j

11 A Variant: Block Gauß-Seidel nodal splitting: H j = p N j V p, V p = {λ (j) p E E span{e 1,...,E M }} successive minimization of J + χ Gj on V p : convergence proof similar (but simpler) solution of (N 1) D subproblems complicated for large N typical drawback of any Gauß-Seidel approach: convergence speed might deteriorate exponentially (i.e. ρ j = 1 O(2 j ))

12 Multilevel Version of Polygonal Gauß-Seidel adaptive refinement: T 0 T 1 T j nodes: N 0 N 1 N j nested finite element spaces: S 0 S 1 S j, λ (j 2) q S k = span{λ (k) p p N k } λ (j) p q p polygonal multilevel splitting: H j = j k=0 p N k M m=1 V (k) p,m V (k) p,m = span{λ (k) p E m }

13 Polygonal Monotone Multigrid (Kh. & Krause 02) sucessive minimization of J = J + χ Gj on subspaces V (k) p,m for p N k, m = 1,...,M and k = j,j 1,...,0 k = j polygonal Gauß-Seidel relaxation on S N j : fine grid smoother: M j (convergent descent method) k < j coarse grid correction: C j (monotone: J (C j w)) J (C j w)) technical modifications (monotone restriction, truncation) monotone multigrid: u ν+1 j = C j ū ν j ū ν j = M ju ν j properties: implementation as multigrid V -cycle: complexity O(N 2 n j ) globally convergent: u ν j u j ν asymptotic multigrid convergence rates for N = 2

14 logarithmic potential: Logarithmic Potential ψ T (u) = Tφ(u) + φ 0 (u) convex part: φ(u) = N i=1 u i ln(u i ) concave part: φ 0 (u) = T c N 2 N i=1 u i(1 u i ) ψ T has N distinct minima for 0 T< T c spatial problems: u j G j : a(u j,v) + τ ε 2 T φ (u j ),v = l(v) v H j minimization of convex energy J T (v) = 1 2 a(v, v) + τ ε 2 T φ(v), 1 l(v): u j G j : J T (u j ) J T (v) v G j family of problems: T > 0 logarithmic potential, T = 0 obstacle potential robustness: family of convergent algorithms T 0

15 What can we do? nonlinear polygonal Gauß-Seidel: robust but slow multilevel version: too expensive (no representation of φ on S k, k < j) Newton s method: not robust (not applicable for T = 0) constrained Newton linearization: small Newton corrections implicit local damping

16 Robust Multigrid for the Logarithmic Potential (Kh. & Krause 04) given iterate u ν j fine grid smoothing: M j (globally convergent descent method) nonlinear polygonal Gauß-Seidel iteration: smoothed iterate ū ν j = M ju ν j coarse grid correction: C j (monotone) Newton linearization of φ at ū ν j constrain corrections to smooth regime G u ν j of φ 1 step of polygonal MMG with local damping G Gūν j (p) ū ν j (p) new iterate u ν+1 j = C j ū ν j family of multigrid algorithms T 0 : V -cycle with complexity O(N 2 n j ), (asymptotic) multigrid convergence

17 Numerical Experiments: The Scalar Case N = 2 time discretization: backward Euler parameter: ε = 0.005, T = 1 10 T c, T c = 1, τ = ε2, j = 7 (h j = 1/256 ε/2) initial condition: u(x, 0) 1, random evolution: t = 100τ t = 500τ t = 2000τ t = 3000τ

18 Robust Convergence parameter: ε = 0.005, T c = 1, τ = 1 2 ε2 initial condition: u(x, 0) 1, random algorithm: multigrid V (1, 1)-cycle with nested iteration, uniform refinement various time steps (T = 0.001, h j = 1/256) meshsize: h j 0 (T = 0.001, 1. time step) temperature T 0 (1. time step, h j = 1/256) time steps k refinement level j inverse temperature 1/T

19 N = 5 Phases in 2D parameter: logarithmic potential ψ T, 0 T T c = 1, ε = 0.08 semi-implicit Euler method: 1. time step, τ 0 = ε 2 /80, linear finite elements: uniform refinement j = 6 (h j ε/5) multigrid algorithm: truncated MMG V (3, 3)-cycle with nested iteration time step τ (T = 0.0, h j = 1/64) mesh size: h j 0 (T = 0.0, τ = τ 0 ) temperature T 0 (τ = τ 0 h j = 1/128) time step size τ refinement level j inverse temperature 1/T

20 N = 3 Phases in 3D parameter: logarithmic potential ψ 0, T c = 1, ε = 0.16 semi-implicit Euler method: 1. time step, τ 0 = ε 2 /2, linear finite elements: uniform refinement j = 5 (h j ε/5) multigrid algorithm: truncated MMG V (3, 3)-cycle with nested iteration timesteps: τ k (T = 0.0, h j = 1/32) meshsize: h j 0 (T = 0.0, τ = τ 0 ) temperature T 0 (τ = τ 0, h j = 1/32) time step size τ refinement level j inverse temperature 1/T

21 Conclusion fast, robust solver for arbitrary N and 2 or 3 space dimensions convergence slows down for T > 0 and increasing τ: local damping strategy too pessimistic Current Work anisotropic interfacial energy (cf. Garcke, Nestler & Stoth 99) adaptivity in time and space heuristic refinement indicators, a posteriori error estimates (cf. Kessler, Nochetto & Schmidt 03)

22 Diffusion Induced Grain Boundary Motion (Garcke & Styles 04) un-alloyed and alloyed grains: ϕ i and ϕ i+n, i = 1,...,N order parameter: ϕ = (ϕ 1,...,ϕ 2N ), free energy with obstacle potential: E(ϕ,u) = ( 1 Ω 2ε u2 + ε N 2 i=1 (ϕ i + ϕ i+n ) ψ 0(ϕ) + g(u) ) N i=1 ϕ i dx alloying solute: u semi-linear parabolic equation: u t (D(ϕ) u) = q(u) grain 3 grain 2 degenerate diffusion: D(ϕ) = 2N i=1 (1 ϕ i) grain 1

23 Multigrid for the Phase Field variational inequality: ϕ(t) G : ε(ϕ t, η ϕ) + ε( ϕ, ( ϕ ṽ)) ( 1 εϕ + g(u)e, ϕ ṽ) v G ṽ = (v i + v i+n ) N i=1 RN, e = (1,...,1,0,...,0) R 2N semi-implicit Euler method, finite elements: ϕ k+1 j, v ϕ k+1 j + τ( ϕ k+1 j, (ṽ ϕ k+1 j )) }{{} a(ϕ k+1 j,v ϕ k+1 j ) ( τ + 1)ϕ k ε 2 j + τ ε g(ϕk j )e, v ϕk+1 j }{{} l(v ϕ k+1 j ) fast solution by monotone multigrid complexity: O((2N) 2 n j ), alternative model (Elliott & Styles 03): N phases

24 Multigrid for the Solute semi-implicit Euler method, finite elements: u k+1 j S j : ϕ k+1,v + τ(d(ϕ k j ) uk+1 j, v) = u k j, v + τ(q(uk j ), v) v S j j observation: u k+1 j (p) = u k j (p) p N j = {p N j D(ϕ k j ) suppλ p (j) 0} reduced solution space: S j = {v Ω v S j} Ω = Ω \ Ω, Ω = p N j suppλ (j) p reduced Neumann problem: u k+1 j S j : b(uk+1 j,v) = f(v) v S j no representation of Ω on coarse grids: truncated multigrid (Kh. & Yserentant 94)

25 Current work numerical results comparison of thin/thick film models with 3D realistic geometries improved models effects of continuum mechanics