Modelling of interfaces and free boundaries

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Brett Anderson
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1 University of Regensburg Regensburg, March 2009

2 Outline 1 Introduction 2 Obstacle problems 3 Stefan problem 4 Shape optimization

3 Introduction What is a free boundary problem? Solve a partial differential equation (PDE) in a domain whose boundary is unknown in advance.

4 Introduction What is a free boundary problem? Solve a partial differential equation (PDE) in a domain whose boundary is unknown in advance. Task: Determine both the free boundary and the solution of the PDE

5 boundary conditions: standard boundary conditions for PDEs additional conditions to specify free boundary

6 boundary conditions: standard boundary conditions for PDEs additional conditions to specify free boundary Example: Dirichlet problem: u = f in Ω u = 0 on Γ = Ω. If Γ is free: Specify u n = 0 on Γ

7 One main strategy for free boundary problems: Reformulate the problem in such a way that the free boundary disappears Methods to achieve this: Use variational inequalities Use discontinuous variables and weak formulations

8 Where free boundaries appear in applications? obstacle problems for a membrane contact problems for an elastic body finance - Black-Scholes models porous media flow tumor growth free surface flow coating flows epitaxial growth shape optimization problems (reduce drag by changing the shape shape is given with the help of a free boundary. PDE to be solved Navier-Stokes equation)

9 Obstacle problems Obstacle problems and variational inequalities Consider: elastic membrane as graph u : Ω R, Ω R d obstacle ϕ : Ω R boundary data u 0 : Ω R force f : Ω R Minimize such that E(u) := Ω ( ) 1 2 u 2 fu dx min u ϕ in Ω u = u 0 on Ω

10 boundary of {u > ϕ} is the free boundary on {u > ϕ} the PDE u = f holds

11 Let u be a minimum and v is assumed to fulfill the constraints. Then it holds 0 d dε E(u + ε(v u)) ( ) ε=0 = u (v u) f (v u) dx. It holds N := {u > ϕ} is open Choose v = u ± εζ with ζ C0 (N) [ ( u) f ] ζ dx = 0 for all ζ C0 (N) Ω Ω

12 Hence u f = 0 in N Furthermore: [ ( u) f ] ζ dx 0 for all ζ C0 (Ω) with ζ 0 Ω and hence u f 0 for almost all x Ω.

13 It can be shown (if data are smooth enough): u C 1 (Ω). Hence u = ϕ and u n = ϕ n (BE CAREFUL) on free boundary Γ := N. Define active set A = Ω \ N. obstacle ϕ 0

14 Reformulation as complementarity system u f 0 u ϕ ( u f )(u ϕ) = 0 in Ω, u = u 0 on Ω, u = ϕ on Γ, u n = ϕ n on Γ.

15 Reformulation with Lagrange multipliers Define Lagrange multiplicator { 0 if x N, µ(x) = u(x) f (x) if x A, We then obtain u f = µ u ϕ µ 0 µ(u ϕ) = 0 in Ω.

16 Reformulation as semi-smooth equation Rewrite u ϕ µ 0 µ(u ϕ) = 0 in Ω as µ = max(0, µ c(u ϕ)) in Ω. where c > 0

17 Formulation as free boundary problem Find N Ω, a free boundary Γ = N and u : N R, such that u f = 0 in N u = u 0 on Ω u = ϕ, u n = ϕ n on Γ u ϕ in N ϕ f 0 in Ω \ N

18 The free boundary problem for an elastic membrane can be rewritten as minimization problem variational inequality complementarity problem with the help of Lagrange multipliers with the help of a semi-smooth equation

19 Situations in which free boundary problems can be reformulated as a variational inequality: contact of an elastic body with an obstacle (vector valued variational inequality) dam problem. Filtration of water through a dam. Free boundary separates wet and dry parts. Black-Scholes variational inequalities (pricing of options) parabolic variational inequality solidification problems (Stefan problem) thin obstacle problem (Signorini problem) obstacle problems for plates

20 Stefan problem Stefan problem models melting and solidification Derivation from energy conservation Variables: T : temperature u: internal energy density L: latent heat q: heat flux For simplicity We set specific heat c v = 1 mass density ρ = 1 q = T u(t ) = { T solid phase T + L liquid phase

21 Energy conservation: For all V R d we have d udx = q n ds x, n outer normal dt V If V lies in pure phases V V ( t u + q)dx = 0 t u + q = 0, q = T t T T = 0

22 Let Γ be the free boundary between liquid and solid. If Γ V we obtain using a transport theorem: d udx = t u [u] l dt svds x V V Γ V [u] l s : jump of u across free boundary V : normal velocity of the free boundary. x(t) Γ(t) then we define V (x(t), t) := ẋ(t) n. Remark: [u] l s = L

23 V q n = V q + Γ V [q] l s n ds x Combining both equations and using the fact that V is arbitrary gives Stefan condition LV = [ T ] n on Γ This is only one condition on Γ. How many conditions do we expect?

24 Possibilities for two more conditions I II T = 0 on Γ (two boundary conditions) βv = γκ T on Γ (Gibbs-Thomson law) where β: kinetic undercooling coefficient γ: capillarity coefficient κ = Γ n : mean curvature How to treat the free boundary Case I : We can hide the free boundary Case II : We discuss serveral possibilites to treat the free boundary

25 Classical Stefan problem is equivalent to t T T = 0 in liquid + solid T = 0 on Γ LV = [ T ] n on Γ t (T + Lχ {T >0} ) = T in distributional sense }{{} u(t )= where χ {T >0} is the characteristic function of the set {T > 0}.

26 invert Solve t u = β(u) in weak sense degenerate parabolic PDE analysis difficult but simple numerical methods available

27 Now βv = γκ T instead of T = 0 (avoids mushy regions, takes interface energy into account) How to represent the free boundary At least five possibilities: as a graph use distance function to a reference manifold using a parameterization use level set approach use phase field approach

28 Graph approach Write free boundary Γ(t) as Γ(t) = {(x, h(x, t)) x Ω } with height function h : Ω [0, T ) R 1+ h 2 V = th ( κ = ( β t h =γ 1+ h 2 h 1+ h 2 h 1+ h 2 highly nonlinear parabolic PDE in variable h ) ) T

29 Parametric approach I Choose (d 1)-dimensional reference surface M R d (e.g. free boundary at time t = 0). Search for maps X (., t) : M R d, t [0, T ) normal velocity: X t n, n has to be computed from X mean curvature: trace of second fundamental form important relation Γ X = κn Γ : Laplace Beltrami operator on Γ Γ X = 1 d 1 ( g θ i g ij ) g X θ j i,j=1

30 Parametric approach II Two possibilities to write the equation βv = γκ T a) βx t = γ Γ X Tn = βx t = (γκ T )n b) Velocity always in normal direction (numerical approaches based on this formulation: Dziuk, Schmidt) βx t n = γκ T Γ X = κn Allows also for solutions X which have a velocity in tangential direction (numerical approach based on this formulation: Barrett, Garcke, Nürnberg)

31 Often additional aspects have to be considered Examples: several phases triple junctions anisotropy replace κ by anistropy curvature κ γ Sharp interface computations for anisotropic Stefan problem (Barrett, G., Nürnberg)

32 For anisotropic energies the Wulff shape is the energy minimizers (Numerical computation: Barrett, G., Nürnberg) ) Coupling at triple junctions Young s law has to be satisfied (Numerical computation:barrett, G., Nürnberg) )

33 Level set approach I Look for Γ(t) as zero level set of φ : R d [0, ) R i.e. Γ(t) = {x R d φ(x, t) = 0}. Require: φ 0 on Γ. We obtain: n = φ φ, V = φ t φ, κ = φ φ βv = γκ T can be written as β φ t φ = γ φ φ T

34 Level set approach II highly nonlinear degenerate not defined if φ = 0 β φ t φ = γ φ φ T φ has to be computed away from the interface and φ R d \Γ has no interpretation Mathematical theory: use viscosity solutions (Crandall, Ishii, Lions; Evans, Spruck; Chen, Giga, Goto). Numerical approaches: Osher, Fedkiv and Sethian very efficient, many applications

35 Phase field approach I Replace V = κ + T β = γ = 1 by ε t ϕ = (ε ϕ 1 ε ψ (ϕ)) T 1 2 V = 1 2 κ T ψ(ϕ) = 9 32 (ϕ2 1) 2, ε > 0 small asymptotic limit ε 0

36 Phase field approach II In a phase field approach the free boundary (a hypersurface) is replaced by a diffuse interface }{{} interface size proportional to ε phase field simulation for solidification (M. Krä)

37 Phase field approach III Solve ε t ϕ = ε ϕ 1 ε ψ (ϕ) T t (T + ϕ) = T, L = 1 in order to approximate the Stefan problem with Gibbs-Thomson law. Phase field methods have been used for many interface problems: Examples: solidification interfaces in binary and multicomponent alloys grain growth (Allen-Cahn equation) coarsening phenomena (Cahn-Hilliard equation) epitaxial growth in thin films (quantum dot formation) fluid flow problems with interfaces electromigration topology optimization image processing

38 Advantages / Disadvantages of the approaches graph and parametric approach do not allow for topology changes level set and phase field approach handle topology changes implicitely graph and parametric approach need only (d 1)-dimensional variables computational advantage phase field and level set approaches need d-dimensional fields. Involve regularization parameters which might not have a physical significance

39 Phase field simulations I several scales and phenomena in material modelling experimentally observed dendrites

40 Phase field simulations II eutectic growth with multiple scales (phase field simulation, G. + Nestler) dendritic growth (phase field simulation, G. + Nestler)

41 Phase field simulations III eutectic growth in 3d with isotropic energies (phase field simulation, G. + Nestler) eutetic growth in 3d with anisotropic energies (phase-field simulation, G. + Nestler)

42 Introduction Obstacle problems Stefan problem Shape optimization Phase field simulations IV alignement of interfaces driven by anisotropic elasticity (G., Weikard) monotectic growth (Nestler, Wheeler)

43 Phase field simulations V surface diffusion in heteroepitaxial thin film growth Island formation in thin film epitaxial growth

Introduction Obstacle problems Stefan problem

G., Sarbu, Styles) t = 0.001 t = 0.002 t = 0.

44 Introduction Obstacle problems Stefan problem Shape optimization Phase field simulations VI t = t = t = t = Vector-valued Allen-Cahn equation (Blank, G., Sarbu, Styles) t = t = t = t = topology optimization (Blank, G., Sarbu, Styles)

45 Shape optimization THIS PART WILL BE ON THE BLACKBOARD!!!!!!!!!!!!!!!!!!!

46 Literatur Deckelnick, K., Dziuk, G., and Elliott, C.M., Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005), Eck, C., Garcke, H., and Knabner, P., Mathematische Modellierung. Springer-Lehrbuch. Springer-Verlag 2008 Elliott, C.M., Ockendon, J.R., Weak and variational methods for moving boundary problems. Research Notes in Mathematics 59. Pitman Boston, Mass.-London, 1982, iii+213 pp. Friedman, A., Variational principles and free-boundary problems, Wiley 1982 Journal Interfaces and Free Boundaries, EMS Publishing House

47 Literatur Kinderlehrer, D. and Stampacchia, G., An introduction to variational inequalities and their applications, Pure and Applied Mathematics 88. Academic Press, Inc., New York-London (1980), xx+313 pp. Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag New York, Applied Mathematical Sciences, 153, 2002, BOOK. Sethian, J.A., Level Set Methods. Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge University Press, (First edition) Sokolowski, J and Zolesio, J.-P., Introduction to Shape Optimization, Springer, 1991.

A sharp diffuse interface tracking method for approximating evolving interfaces

A sharp diffuse interface tracking method for approximating evolving interfaces Vanessa Styles and Charlie Elliott University of Sussex Overview Introduction Phase field models Double well and double obstacle