On energy-stable schemes for two Vesicle Membrane phase-field models - PowerPoint PPT Presentation

Bending energy and constraints Lagrange multiplier problem Penalized problem Numerical simulations Conclusions and Future work On energy-stable schemes for two Vesicle Membrane phase-field models Francisco Guillén-González (Universidad de Sevilla) guillen@us.es Joint work with: Giordano Tierra (University of Notre Dame, USA). Depto. EDAN and IMUS. Universidad de Sevilla SciCADE2013, Valladolid16-20 September 2013 F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem Penalized problem Numerical simulations Conclusions and Future work Bending energy and constraints 1 Lagrange multiplier problem 2 The problem Linear and unconditionally energy-stable scheme. Penalized problem 3 The problem Nonlinear unconditionally energy-stable scheme Numerical simulations 4 Conclusions and Future work 5 F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem Penalized problem Numerical simulations Conclusions and Future work A diffuse interface model Hydrodynamic system modeling the deformation of vesicle membranes in incompressible viscous fluids. The system consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem Penalized problem Numerical simulations Conclusions and Future work Sharp interface equilibrium model The equilibrium configurations of vesicle membranes can be characterized by the Helfrich bending elasticity energy of the surface [W. Helfrich 73, Elastic properties of lipid bilayers: theory and possible experiments] such that they are minimizers of the bending energy under possible constraints like prescribed surface area (incompressibility of the membrane) and bulk volume (the change in volume is normally a much slower process in comparison with the shape change). Let Γ be a smooth, surface representing the membrane of the vesicle. The most simplified form of the interfacial energy is � k 2 ( H − H 0 ) 2 ds E elastic = Γ where H is the mean curvature of Γ , k is the bending rigidity and H 0 is the spontaneous curvature that describes certain physical/chemical difference between the inside and the outside of the membrane. For the simplicity, we assume that k is a positive constant and H 0 = 0. F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem Penalized problem Numerical simulations Conclusions and Future work Diffuse interface model φ takes the value 1 inside of the vesicle membrane and − 1 outside. The phase-field approximation of the Helfrich bending elasticity energy is given by a modified Willmore Bending energy : � � 2 � E ε ( φ ) = 1 − ε ∆ φ + 1 f ( φ ) = ( φ 2 − 1 ) φ ε f ( φ ) dx with 2 ε Ω ε > 0 is a small positive parameter (compared to the vesicle size) that characterizes the transition layer of the phase function. [Du, Liu, Wang 04], [Wang 08] Convergence of the phase-field model to the original sharp interface model as the transition width of the diffuse interface ε → 0 [Du, Liu, Ryham, Wang 05], [Wang 08] Diffuse interface models simplify numerical approximations because it suffices to consider a fixed computational grid rather than tracking the position of the interface F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem Penalized problem Numerical simulations Conclusions and Future work Dynamics model Model: Interaction of a vesicle membrane with the fluid field, which describes the evolution of vesicles immersed in an incompressible, Newtonian fluid. PDE system (Navier-Stokes + Allen-Cahn): For ν, λ, γ > 0 (constants):  � δ E ε �   ∂ t u + ( u · ∇ ) u − ν ∆ u + ∇ p − λ ∇ φ = 0 ,    δφ   ∇ · u = 0 ,   � δ E ε �      ∂ t φ + u · ∇ φ = − γ . δφ System can be obtained via an energetic variation approach [Yue, Feng, Liu, Shen 04], [Hyon, Kwak, Liu 10] Energy law (Lyapunov functional) : Calling E tot ( u , φ ) = E kin ( u ) + λ E ε ( φ ) : � � 2 � � d δ E ε � � dt E tot ( u , φ ) + ν �∇ u � 2 L 2 (Ω) + λγ = 0 . � � δφ L 2 (Ω) For simplicity, we take ν, λ, γ = 1 F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem Penalized problem Numerical simulations Conclusions and Future work Two global constraints of conservation for the vesicle volume and surface area: � ε � � � 2 |∇ φ | 2 + 1 A ( φ ) = φ dx and B ( φ ) = ε F ( φ ) dx , Ω Ω 4 ( φ 2 − 1 ) 2 (Note that f ( φ ) = F ′ ( φ ) ) where F ( φ ) = 1 Introducing the auxiliary variable ω = − ε ∆ φ + 1 ε f ( φ ) , then � E ε ( φ ) = E ε ( ω ) = 1 ω 2 dx 2 ε Ω Some variational computations gives: δ A δ B δφ = 1 , δφ = ω and δ E ε δφ = − ∆ ω + 1 ε 2 ω f ′ ( φ ) F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem The problem Penalized problem Linear and unconditionally energy-stable scheme. Numerical simulations Conclusions and Future work Lagrange multiplier problem Idea: Modify the generic model to enforce the two physical constraints by Lagrange multipliers ( λ 1 ( t ) , λ 2 ( t ) ) and introduce an extra unknown z :  ∂ t u − ∆ u + ( u · ∇ ) u + ∇ p − z ∇ φ = 0 ,       ∇ · u = 0 ,    ∂ t φ + u · ∇ φ + z = 0 ,      A ( φ ) = α (= A ( φ 0 )) , B ( φ ) = β (= B ( φ 0 )) ,     + I . C . and B . C . where z = δ E ε δφ + λ 1 ( t ) δ A δφ + λ 2 ( t ) δ B δφ = − ∆ ω + 1 ε 2 ω f ′ ( φ ) + λ 1 ( t ) + λ 2 ( t ) ω, F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem The problem Penalized problem Linear and unconditionally energy-stable scheme. Numerical simulations Conclusions and Future work Reformulation of the model (I): time derivatives Taking the time derivative of the ω -equation:  ∂ t ω = − ε ∆ ∂ t φ + 1   ε f ′ ( φ ) ∂ t φ, t ∈ ( 0 , T ) , ω | t = 0 = ω 0 := − ε ∆ φ 0 + 1   ε f ( φ 0 ) Taking the time derivative of the two constraints:  � �  ∂ t φ = 0 , ω ∂ t φ = 0 , t ∈ ( 0 , T ) , Ω Ω  A ( φ 0 ) = α, B ( φ 0 ) = β F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem The problem Penalized problem Linear and unconditionally energy-stable scheme. Numerical simulations Conclusions and Future work Reformulation of the model (II): dissipation of free energy Then  ∂ t u − ∆ u + ( u · ∇ ) u + ∇ p − z ∇ φ = 0 ,  u      ∇ · u = 0 , p        ∂ t φ + u · ∇ φ + z = 0 , z       − ∆ ω + 1  ε 2 ω f ′ ( φ ) + λ 1 ( t ) + λ 2 ( t ) ω − z = 0 , ∂ t φ    ε ∂ t ω = − ∆ ∂ t φ + 1 1  ε 2 f ′ ( φ ) ∂ t φ,  ω     � �      ∂ t φ = 0 , ω ∂ t φ = 0 ,     Ω Ω   + I . C . and B . C . Modified Energy Law: d dt E tot ( u , ω ) + �∇ u � 2 L 2 + � z � 2 L 2 = 0 , with E tot ( u , ω ) = E kin ( u ) + E ε ( ω ) . F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem The problem Penalized problem Linear and unconditionally energy-stable scheme. Numerical simulations Conclusions and Future work First order, linear and unconditionally energy-stable scheme. Given u n , φ n , ω n , find u n + 1 , p n + 1 , φ n + 1 , ω n + 1 , λ n + 1 , λ n + 1 s.t. 1 2 � �  δ t u n + 1 , ¯ + c ( u n , u n + 1 , ¯ u ) + ( ∇ u n + 1 , ∇ ¯  u )  u  � �   − ( p n + 1 , ∇ · ¯ z n + 1 ∇ φ n , u u n + 1  u ) − = 0 ,       ( ∇ · u n + 1 , p ) = 0 , p n + 1     � �   + ( u n + 1 · ∇ φ n , ¯  δ t φ n + 1 , ¯ z ) + ( z n + 1 , ¯ z n + 1  z z ) = 0 ,     φ ) + 1 ( ∇ ω n + 1 , ∇ ¯ ε 2 ( f ′ ( φ n ) ω n + 1 , ¯ φ )+ λ n + 1 ( 1 , ¯ φ ) + λ n + 1 ( ω n , ¯ φ )  1 2       − ( z n + 1 , ¯ δ t φ n + 1  φ ) = 0 ,     � � � � � �  1 − 1   δ t ω n + 1 , ¯ ∇ δ t φ n + 1 , ∇ ¯ f ′ ( φ n ) δ t φ n + 1 , ¯ ω n + 1 ω − ω ω = 0 ,    ε ε 2   � �    δ t φ n + 1 = 0 ω n δ t φ n + 1 = 0 .  and Ω Ω F. Guillén-González Stable schemes for Vesicle Membrane

Bending energy and constraints Lagrange multiplier problem The problem Penalized problem Linear and unconditionally energy-stable scheme. Numerical simulations Conclusions and Future work Unconditional energy-stability , L 2 + ND n + 1 = 0 , δ t E tot ( u n + 1 , ω n + 1 ) + �∇ u n + 1 � 2 L 2 + � z n + 1 � 2 where ND n + 1 = k L 2 + k 2 � δ t u n + 1 � 2 2 ε � δ t ω n + 1 � 2 L 2 ≥ 0 Moreover, this scheme is well-defined. F. Guillén-González Stable schemes for Vesicle Membrane