Space-time XFEM for two-phase mass transport

Space-time XFEM for two-phase mass transport Space-time XFEM for two-phase mass transport Christoph Lehrenfeld joint work with Arnold Reusken EFEF, Prague, June 5-6th 2015 Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 0/18

Problem setting - Models Two-phase flows Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 1/18

Problem setting - Models Two-phase flows Ω 2 Ω 1 Γ Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 2/18

Problem setting - Models Two-phase flows Fluid dynamics (within the phases Ω i ) incompressible Navier Stokes equations (w, p) (mass and momentum conservation) Ω 1 Γ Mass transport (within the phases Ω i ) convection diffusion equation (u) (species conservation) Ω 2 Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 2/18

Problem setting - Models Two-phase flows Fluid dynamics (within the phases Ω i ) incompressible Navier Stokes equations (w, p) (mass and momentum conservation) Ω 2 Ω 1 Γ Mass transport (within the phases Ω i ) convection diffusion equation (u) (species conservation) interface conditions (on Γ) mass and mom. cons. (capillary forces) (w, p) species conservation and thermodyn. equil. (u) Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 2/18

Problem setting - Models Mass transport model Mass transport equation t u + w u α u = 0 in Ω i (t), i = 1, 2, [ α u n] = 0 [βu] = 0 on Γ(t), on Γ(t), ( Henry condition) (+ initial + boundary conditions) α, β: piecewise constant diffusion / Henry coefficients. Compatibility conditions on the velocity: V n = w n on Γ(t), V : interface velocity., div(w) = 0 in Ω no relative velocity on Γ, incompressible flow Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 3/18

Problem setting - Representation of the interface (interface capturing) Description of the interface Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 4/18

Problem setting - Representation of the interface (interface capturing) Description of the interface Level set function φ = 0, x Γ(t), φ(x, t) < 0, x Ω 1 (t), > 0, x Ω 2 (t). Ω 2 Ω 1 Γ φ describes position of the interface Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 4/18

Problem setting - Representation of the interface (interface capturing) Description of the interface Level set function φ = 0, x Γ(t), φ(x, t) < 0, x Ω 1 (t), > 0, x Ω 2 (t). φ describes position of the interface Ω 2 Ω 1 Γ Level set equation t φ + w φ = 0 in Ω, + i.c. & b.c. ( ) describes the motion of the interface ( ) Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 4/18

Problem setting - Representation of the interface (interface capturing) Description of the interface Level set function φ = 0, x Γ(t), φ(x, t) < 0, x Ω 1 (t), > 0, x Ω 2 (t). φ describes position of the interface Ω 2 Ω 1 Γ Level set equation t φ + w φ = 0 in Ω, + i.c. & b.c. ( ) describes the motion of the interface ( ) Implicit description of the interface (Euler) unfitted meshes Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 4/18

Problem setting - Numerical challenges Numerical aspects Mass transport equation t u+ w u α u = 0 in Ω 1 (t) Ω 2 (t), [ α u] n = 0 on Γ(t), [βu] = 0 on Γ(t). Γ(t) n u Γ(t) unfitted meshes discontinuous conc. (Γ) u 1 /u 2 = β 2 /β 1 α 1 u 1 n = α 2 u 2 n Ω 1 (t) Ω 2 (t) n moving interface (moving discont.) Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 5/18

Problem setting - Numerical challenges Numerical aspects Mass transport equation t u+ w u α u = 0 in Ω 1 (t) Ω 2 (t), [ α u] n = 0 on Γ(t), [βu] = 0 on Γ(t). Γ(t) n Numerical approaches Extended finite element space (XFEM) Nitsche s method to enforce (weakly) the Henry condition Space-time formulation for moving discontinuity Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 5/18

Discretization - Extended finite element methods (XFEM) Approximation of discontinuities Approximation of discontinuous function, u H k+1 (Ω 1 Ω 2 ) space: V h (Standard FE space, degree k) inf u u h L u h V 2 (Ω) c h h Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 6/18

Discretization - Extended finite element methods (XFEM) Approximation of discontinuities Approximation of discontinuous function, u H k+1 (Ω 1 Ω 2 ) space: V Γ h = R 1 V h R 2 V h inf u u h L u h Vh Γ 2 (Ω) ch k+1 Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 6/18

Discretization - Extended finite element methods (XFEM) Extended FEM (XFEM) Vh Γ = R 1V h R 2 V h Vh Γ = V h Vh x Extend P 1 basis with discontinuous basis functions near Γ: { Vh Γ = V h {p Γ j }, p Γ j := R p j, R R1 if x = j in Ω 2, R 2 if x j in Ω 1 Γ Γ p j p Γ j Henry condition not respected. FE Space is non-conforming! Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 7/18

Discretization - Extended finite element methods (XFEM) Extended FEM (XFEM) Vh Γ = R 1V h R 2 V h Vh Γ = V h Vh x Extend P 1 basis with discontinuous basis functions near Γ: { Vh Γ = V h {p Γ j }, p Γ j := R p j, R R1 if x = j in Ω 2, R 2 if x j in Ω 1 Γ Γ p j p Γ j Henry condition not respected. FE Space is non-conforming! Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 7/18

Discretization - Nitsche method Implementation of the Henry condition Nitsche stationary problem (Γ(t) = Γ): α u + w u = f in Ω 1 Ω 2, [ α u n] = 0 on Γ, [βu] = 0 on Γ. (Henry condition in a weak sense) Discrete variational formulation a(u, v):= β(α u v+w u v)dx Ω i i=1,2 + Nitsche bilinear form N h (, ) (Henry-condition in a weak sense) N h (u, v) = {α u n }[βv ] ds {α v n }[βu] ds + ᾱλh 1 T [βu][[βv ] ds Γ Γ Γ λ: stab. parameter. Nitsche term for consistency, symmetry and stability. Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 7/18

Discretization - Nitsche method Nitsche-XFEM for diffusion discrete Nitsche-XFEM problem 1,2 : Find u h Vh Γ, such that Remarks on Nitsche-XFEM a(u h, v h ) + N h (u h, v h ) = f(v h ) v h V Γ h. With standard ideas from the analysis of non-conforming FEM: u u h L 2 (Ω) ch k+1 u H k+1 (Ω 1,2) (as in the one-phase case) 1 A. Hansbo, P. Hansbo, 2002, Comp. Meth. Appl. Mech. Eng. 2 A. Reusken, T.H. Nguyen, 2009, J. Four. Anal. Appl. Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 8/18

Discretization - Nitsche method Nitsche-XFEM for convection-diffusion Nitsche-XFEM and convection stabilization Nitsche-XFEM with dominant convection with Streamline-Diffusion stabilization 3 : Nitsche-XFEM SD-Nitsche-XFEM Treatment of moving interfaces? 3 C.L., A. Reusken, 2012, SIAM J. Sci. Comp. Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 9/18

Discretization - Nitsche method Nitsche-XFEM for convection-diffusion Nitsche-XFEM and convection stabilization Nitsche-XFEM with dominant convection with Streamline-Diffusion stabilization 3 : Nitsche-XFEM SD-Nitsche-XFEM Treatment of moving interfaces? 3 C.L., A. Reusken, 2012, SIAM J. Sci. Comp. Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 9/18

Discretization - Space-time formulation Formulation in space-time t t u un u n 1 t Γ t u un u n 1 t x Avoid taking differences across the interface! Divide phases by means of a space-time variational formulation Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 10/18

Discretization - Space-time formulation Space-time finite element space Space-time FE space (no XFEM) Divide space-time domain into slabs (discont. FE, time stepping scheme) I n = (t n 1, t n ], Q n = Ω I n, W := { v : Q R v Q n W n } t Q n+1 t n t n 1 Q n Q n 1 x Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 11/18

Discretization - Space-time formulation Space-time finite element space Space-time FE space (no XFEM) Divide space-time domain into slabs (discont. FE, time stepping scheme) I n = (t n 1, t n ], Q n = Ω I n, W := { v : Q R v Q n W n } t Q n+1 t n t n 1 Q n Q n 1 x Finite element space with tensor-product-structure W n := { v : Q n R v(x, t) = φ 0 (x) + tφ 1 (x), φ 0, φ 1 V n }. t t n t n 1 Q n x Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 11/18

Discretization - Space-time formulation Space-time XFEM Enrichment of the space-time FE space: W Γ n = W n W x n Γ n Γ n t n t n 1 x j 1 Γ x j+1 t n By design Good approximation properties FE space is non-conforming w.r.t. Henry condition t n 1 x j 1 Γ x j+1 Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 12/18

Discretization - Space-time formulation Discrete formulation components (sketch) 3 t n t n 1 1 1 1 Q n u +(t n 1) u (t n 1) 2 1 space-time interior (consistency to PDE inside the phases) 2 continuity (consistency in time) upwind in time 3 Henry condition Nitsche Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 13/18

Discretization - Space-time formulation Discrete formulation Discrete formulation: Find u Wn Γ Γ, s.t. 1 + 2 + 3 = 0 v Wn 1 consistency to PDE within the phases 2 ( t u i + w u i ) βi v i + α i β i u i v i β i fv i dx dt i=1 Q n i 2 upwind in time β ( u + (t n 1 ) u(t n 1 ) ) v + (t n 1 ) dx Ω t n t n 1 3 1 1 1 Q n u+(tn 1) u (tn 1) 2 3 Nitsche (Henry condition in a weak sense) tn t n 1 ( {α u n }[βv ] ds {α v n }[βu] ds + ᾱλh 1 T Γ(t) Γ(t) Γ(t) [βu][[βv ] ds ) dt Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 13/18

Discretization - Space-time Nitsche-XFEM DG Error bounds Space-Time Nitsche-XFEM DG 4 (u u h )(, T ) 1 c(h 2 + t 2 ). Numerical experiments (1D) 4 and (3D) 6 (Test cases with piecewise smooth solutions) (u u h )(, T ) L 2 c(h 2 + t 3 ). 10 0 Error analysis (linear in space and time) 10 1 10 1 10 2 10 3 10 4 nx = 8 nx = 16 nx = 32 nx = 64 nx = 128 order 1,2,3 1 2 4 8 16 32 64 10 2 10 3 nt = 4 nt = 8 nt = 16 nt = 32 nt = 64 order 1,2 8 16 32 64 128 nt nx 4 C.L., A. Reusken, 2013, SIAM J. Num. Ana. 5 C.L., 2015, SIAM J. Sci. Comp. Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 14/18

Discretization - Example Example two-phase flow Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 15/18

Discretization - Linear solvers Linear systems Condition of (space-time) Nitsche-XFEM (κ(a)) support of XFEM basis functions arbitrarily small. = system matrix A arbitrarily ill-conditioned. Diagonal preconditioning (κ(d 1 A A)) Observation: κ(d 1 A A) ch 2, Proven for stationary case 6. c c(γ) Optimal preconditioning for stationary case 6 After transformation u β 1 u: ( A A ss ) ( 0 A ss 0 0 A xx xx 0 D A ) κ(w 1 A) 1 ( W ss MG 0 ) xx 0 D A }{{} W 6 C.L., A. Reusken, 2014, subm. to Num. Math Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 16/18

Discretization - Linear solvers Linear systems Condition of (space-time) Nitsche-XFEM (κ(a)) support of XFEM basis functions arbitrarily small. = system matrix A arbitrarily ill-conditioned. Diagonal preconditioning (κ(d 1 A A)) Observation: κ(d 1 A A) ch 2, Proven for stationary case 6. c c(γ) Optimal preconditioning for stationary case 6 After transformation u β 1 u: ( A A ss ) ( 0 A ss 0 0 A xx xx 0 D A ) κ(w 1 A) 1 ( W ss MG 0 ) xx 0 D A }{{} W 6 C.L., A. Reusken, 2014, subm. to Num. Math Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 16/18

Discretization - Linear solvers Linear systems Condition of (space-time) Nitsche-XFEM (κ(a)) support of XFEM basis functions arbitrarily small. = system matrix A arbitrarily ill-conditioned. Diagonal preconditioning (κ(d 1 A A)) Observation: κ(d 1 A A) ch 2, Proven for stationary case 6. c c(γ) Optimal preconditioning for stationary case 6 After transformation u β 1 u: ( A A ss ) ( 0 A ss 0 0 A xx xx 0 D A ) κ(w 1 A) 1 ( W ss MG 0 ) xx 0 D A }{{} W 6 C.L., A. Reusken, 2014, subm. to Num. Math Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 16/18

Conclusion - Conclusion/Outlook Conclusion Ω 2 Ω 1 Γ Two-phase mass transport: solutions are discontinuous across moving interfaces (Henry condition). Interface is described implicitly (level set method). Grid is not aligned to the interface. Extended FE space for the approx. of discontinuities Nitsche s method for Henry condition Space-time formulation for moving interfaces ( W ss ) A MG 0 Christoph Lehrenfeld EFEF, Prague, xx June 5-6th 2015 17/18

Conclusion - Conclusion/Outlook Conclusion Ω 2 Ω 1 Γ Two-phase mass transport: solutions are discontinuous across moving interfaces (Henry condition). Interface is described implicitly (level set method). Grid is not aligned to the interface. Extended FE space for the approx. of discontinuities Nitsche s method for Henry condition Space-time formulation for moving interfaces 3 1 1 1 2 ( W ss ) A MG 0 Christoph Lehrenfeld EFEF, Prague, xx June 5-6th 2015 17/18

Conclusion - Conclusion/Outlook Conclusion Ω 2 Ω 1 Γ Two-phase mass transport: solutions are discontinuous across moving interfaces (Henry condition). Interface is described implicitly (level set method). Grid is not aligned to the interface. Extended FE space for the approx. of discontinuities Nitsche s method for Henry condition Space-time formulation for moving interfaces 3 1 1 1 2 ( W ss ) A MG 0 xx 0 D A Numerical integration on implicitly described domains with new decomposition rules in 4D. Preconditioning of Nitsche-XFEM: diagonal precond., optimal precond. (Block+MG) Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 17/18

Conclusion - Conclusion/Outlook Outlook Ω 2 Ω 1 Γ Higher order time integration for two-phase Stokes with moving interfaces: Jumps in pressure (no time derivative), kinks in velocity. Extended FE space for velocity and pressure Nitsche s method for velocity Space-time formulation for moving interfaces 3 1 1 1 2 (Higher order) quadrature strategies for implicit domains. ( W ss ) A MG 0 xx 0 D A Linear solvers for (Nitsche)-XFEM: Space-time discr., Higher order discr., Stokes,... Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 17/18

Conclusion - Questions Thank you for your attention! Questions? Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 18/18

Backup-Slides - Numerical integration on implicit space-time domains Numerical integration on implicit domains Integration on one element (prism) Q n T,1 Γ,n T Q n T,2 I n Require integrals of the form: f dx f dx f dx Q n T,1 Q n T,2 Γ,n T T T h Implicit representation of Γ (zero level of φ), no explicit parametrization strategy: Find approximation Γ h with explicit parametrization requires new decomposition rules in four dimensions 6. x3 x1 x2 x4 6 C.L., 2015, SIAM J. Sci. Comp. Christoph Lehrenfeld EFEF, Prague, June 5-6th 2015 18/18