Space-timeXFEM for two-phase mass transport - TU Wienlehrenfeld/Talks/prague2015.pdf ·...

Space-time XFEM fortwo-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

Space-time XFEM for two-phase mass transport


Problem setting - Models

Two-phase flows


Γ

Ω1

Ω2


Two-phase flows


Γ

Ω1

Ω2

Fluid dynamics (within the phases Ωi)incompressible Navier Stokes equations (w, p)

(mass and momentum conservation)

Mass transport (within the phases Ωi)convection diffusion equation (u)

(species conservation)


Two-phase flows


Γ

Ω1

Ω2

Fluid dynamics (within the phases Ωi)incompressible Navier Stokes equations (w, p)

(mass and momentum conservation)

Mass transport (within the phases Ωi)convection diffusion equation (u)

(species conservation)

interface conditions (on Γ)

mass and mom. cons. (capillary forces) (w, p)species conservation and thermodyn. equil. (u)


Two-phase flows


Mass transport equation

∂tu+ w · ∇u− α∆u = 0 in Ωi(t), i = 1, 2,

[[−α∇u · n]] = 0 on Γ(t),

[[βu]] = 0 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


Mass transport model


Problem setting - Representation of the interface (interface capturing)

Description of the interface


Γ

Ω1

Ω2

Level set function φ

φ(x, t)

= 0, x ∈ Γ(t),

< 0, x ∈ Ω1(t),

> 0, x ∈ Ω2(t).

φ describes position of the interface




Γ

Ω1

Ω2


φ(x, t)

= 0, x ∈ Γ(t),

< 0, x ∈ Ω1(t),

> 0, x ∈ Ω2(t).


Level set equation

∂tφ+ w · ∇φ = 0 in Ω,+ i.c. & b.c. (∗)

(∗) describes the motion of the interface




Γ

Ω1

Ω2


φ(x, t)

= 0, x ∈ Γ(t),

< 0, x ∈ Ω1(t),

> 0, x ∈ Ω2(t).


Level set equation

∂tφ+ w · ∇φ = 0 in Ω,+ i.c. & b.c. (∗)

(∗) describes the motion of the interface

Implicit description of the interface (Euler)

unfitted meshes





∂tu+ w · ∇u− α∆u = 0 in Ω1(t) ∪ Ω2(t),

[[−α∇u]] · n = 0 on Γ(t),

[[βu]] = 0 on Γ(t).

Γ(t)

n

n

u Γ(t)

Ω2(t)Ω1(t)

unfitted meshes

discontinuous conc. (Γ)u1/u2 = β2/β1

α1∇u1 · n = α2∇u2 · n

moving interface(moving discont.)

Problem setting - Numerical challenges

Numerical aspects



∂tu+ w · ∇u− α∆u = 0 in Ω1(t) ∪ Ω2(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

Problem setting - Numerical challenges

Numerical aspects


Approximation of discontinuous function, u ∈ Hk+1(Ω1 ∪ Ω2)

space: Vh (Standard FE space, degree k)

infuh∈Vh

‖u− uh‖L2(Ω) ≤ c√h

Discretization - Extended finite element methods (XFEM)

Approximation of discontinuities


Approximation of discontinuous function, u ∈ Hk+1(Ω1 ∪ Ω2)

space: V Γh

= R1Vh ⊕ R2Vh

infuh∈V Γ

h

‖u− uh‖L2(Ω) ≤ chk+1


Approximation of discontinuities


V Γh = R1Vh ⊕R2Vh ⇐⇒ V Γ

h = Vh ⊕ V xh

Extend P1 basis with discontinuous basis functions near Γ:

V Γh = Vh ⊕ pΓ

j , pΓj := R∗pj , R∗ =

R1 if xj in Ω2,R2 if xj in Ω1

pj

Γ

pΓj

Γ

Henry condition not respected.FE Space is non-conforming!


“Extended” FEM (XFEM)


Nitsche (Henry condition in a weak sense)

stationary problem (Γ(t) = Γ):

−α∆u+ w · ∇u = f in Ω1∪Ω2,

[[−α∇u · n]] = 0 on Γ,

[[βu]] = 0 on Γ.

Discrete variational formulation

a(u, v) :=∑i=1,2

∫Ωi

β(α∇u∇v+w·∇u v)dx

+ Nitsche bilinear form Nh(·, ·)(Henry-condition in a weak sense)

Nh(u, v) =−∫

Γ

α∇u · n[[βv]] ds−∫

Γ

α∇v · n[[βu]] ds + αλh−1T

∫Γ

[[βu]][[βv]] ds

λ: stab. parameter. Nitsche term for consistency , symmetry and stability .

Discretization - Nitsche method

Implementation of the Henry condition


Remarks on Nitsche-XFEM

discrete Nitsche-XFEM problem1,2:Find uh ∈ V Γ

h , such that

a(uh, vh) +Nh(uh, vh) = f(vh) ∀ vh ∈ V Γh .

With standard ideas from the analysis of non-conforming FEM:

‖u− uh‖L2(Ω) ≤ chk+1‖u‖Hk+1(Ω1,2) (as in the one-phase case)

1A. Hansbo, P. Hansbo, 2002, Comp. Meth. Appl. Mech. Eng.

2A. Reusken, T.H. Nguyen, 2009, J. Four. Anal. Appl.


Nitsche-XFEM for diffusion


Nitsche-XFEM and convection stabilization

Nitsche-XFEM with dominant convection with Streamline-Diffusion stabilization3:

Nitsche-XFEM SD-Nitsche-XFEM

Treatment of moving interfaces?

3C.L., A. Reusken, 2012, SIAM J. Sci. Comp.


Nitsche-XFEM for convection-diffusion


x

t

∂tu ≈un − un−1

∆t

∂tu 6≈un − un−1

∆t

Γ∗

Avoid taking differences across the interface!

Divide phases by means of a space-time variational formulation

Discretization - Space-time formulation

Formulation in space-time


Space-time FE space (no XFEM)

Divide space-time domain into slabs (discont. FE,⇒ time stepping scheme)

In = (tn−1, tn], Qn = Ω× In, W := v : Q→ R | v|Qn ∈Wn

x

t

tn

tn−1

Qn−1

Qn

Qn+1


Space-time finite element space


Space-time FE space (no XFEM)

Divide space-time domain into slabs (discont. FE,⇒ time stepping scheme)

In = (tn−1, tn], Qn = Ω× In, W := v : Q→ R | v|Qn ∈Wn

x

t

tn

tn−1

Qn−1

Qn

Qn+1

Finite element space with tensor-product-structure

Wn := v : Qn → R | v(x, t) = φ0(x) + tφ1(x), φ0, φ1 ∈ Vn .

x

ttn

tn−1

Qn


Space-time finite element space


Enrichment of the space-time FE space: WΓ∗n = Wn ⊕W x

n

Γn∗ ⊕ Γn

∗

xj−1 xj+1

tn

tn−1Γ∗

xj−1 xj+1

tn

tn−1Γ∗

By design

Good approximation properties

FE space is non-conforming w.r.t.Henry condition


Space-time XFEM


components (sketch)

tn

tn−1

Qn

3

1 1 1

u+(tn−1)u−(tn−1) 2

1 space-time interior (consistency to PDE inside the phases)

2 continuity (consistency in time)⇒ upwind in time

3 Henry condition⇒ Nitsche


Discrete formulation


Discrete formulation: Find u ∈WΓ∗n , s.t. 1 + 2 + 3 = 0 ∀ v ∈WΓ∗

n

1 consistency to PDE within the phases

2∑i=1

∫Qn

i

(∂tui + w · ∇ui

)βivi + αiβi∇ui · ∇vi − βifvi dx dt

2 upwind in time∫Ωβ(u+(tn−1)− u(tn−1)

)v+(tn−1) dx

tn

tn−1

Qn

3

1 1 1

u+(tn−1)u−(tn−1) 2

3 Nitsche (Henry condition in a weak sense)∫ tn

tn−1

(−∫

Γ(t)α∇u · n[[βv]] ds −

∫Γ(t)α∇v · n[[βu]] ds + αλh−1

T

∫Γ(t)

[[βu]][[βv]] ds

)dt


Discrete formulation


Error bounds Space-Time Nitsche-XFEM DG4

‖(u− uh)(·, T )‖−1 ≤ c(h2 + ∆t2).

Numerical experiments (1D)4 and (3D)6 (Test cases with piecewise smooth solutions)

‖(u− uh)(·, T )‖L2 ≤ c(h2 + ∆t3).

1 2 4 8 16 32 6410−4

10−3

10−2

10−1

100

nt

nx = 8nx = 16nx = 32nx = 64nx = 128order 1,2,3

8 16 32 64 128

10−3

10−2

10−1

nx

nt = 4nt = 8nt = 16nt = 32nt = 64order 1,2

4C.L., A. Reusken, 2013, SIAM J. Num. Ana.

5C.L., 2015, SIAM J. Sci. Comp.

Discretization - Space-time Nitsche-XFEM DG

Error analysis (linear in space and time)


Discretization - Example

Example two-phase flow


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−1A A))

Observation: κ(D−1A A) ≤ ch−2, c 6= c(Γ)

Proven for stationary case6.

Optimal preconditioning for stationary case6

After transformation u→ β−1u:

A'(

Ass 00 Axx

)'(

Ass 00 DA

xx

)'(

WssMG 00 DA

xx

)︸︷︷︸

Wκ(W−1A) ' 1

6C.L., A. Reusken, 2014, subm. to Num. Math

Discretization - Linear solvers

Linear systems


Γ

Ω1

Ω2

Two-phase mass transport: solutions are discontinuous across mo-ving 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

A'(Wss

MG 00 DA

xx

)

Conclusion - Conclusion/Outlook

Conclusion


Γ

Ω1

Ω2






3

1 1 1

2

A'(Wss

MG 00 DA

xx

)


Conclusion


Γ

Ω1

Ω2






3

1 1 1

2

A'(Wss

MG 00 DA

xx

)

Numerical integration on implicitly described domains withnew decomposition rules in 4D.

Preconditioning of Nitsche-XFEM:diagonal precond., optimal precond. (Block+MG)


Conclusion


Γ

Ω1

Ω2

Higher order time integration for two-phase Stokes with moving inter-faces:I Jumps in pressure (no time derivative),I kinks in velocity.

Extended FE space for velocity and pressure

Nitsche’s method for velocity


3

1 1 1

2

A'(Wss

MG 00 DA

xx

)

(Higher order) quadrature strategies for implicit domains.

Linear solvers for (Nitsche)-XFEM:Space-time discr., Higher order discr., Stokes, ...


Outlook


Questions?

Conclusion - Questions

Thank you for your attention!


Integration on one element (prism)

QnT,1

QnT,2

Γ∗,nT

T ∈ Th

InRequire integrals of the form:∫

QnT,1

f dx

∫Qn

T,2

f dx

∫Γ∗,nT

f dx

Implicit representation of Γ∗ (zero level of φ), no explicit parametrization

strategy: Find approximation Γ∗h with explicit parametrizationrequires new decomposition rules in four dimensions6.

x1

x3

x2

x4

6C.L., 2015, SIAM J. Sci. Comp.

Backup-Slides - Numerical integration on implicit space-time domains

Numerical integration on implicit domains


Space-timeXFEM for two-phase mass transport - TU Wienlehrenfeld/Talks/prague2015.pdf ·...

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