h-adaptive XFEM for two-phase incompressible · PDF fileh-adaptive XFEM for two-phase...

h-adaptive XFEM for two-phase

incompressible flow

Kwok-Wah Cheng

Thomas-Peter Fries

RWTH Aachen University, Germany

IV ECCM, Paris

May 18th , 2010

Overview

• Evolving interfaces as discontinuities

• Mesh refinement: h-XFEM

• Governing equations

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• Discretization

• Numerical results

• Summary

2K.W. Cheng, h-adaptive XFEM

Evolving interfaces

Moving boundaries as discontinuities

density viscosity

Fluid 1

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velocity pressure

Fluid 2


Evolving interfaces

Moving boundaries as discontinuities

Fluid 1

strong discontinuity

density viscosity

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Fluid 2

weak discontinuity


velocity pressure


Evolving interfaces

Example: Bubble with surface tension


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pressure



Overview




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• Discretization


• Summary


Mesh refinement : h-XFEM

Motivation for h-XFEM

� In addition to jumps and kinks, there can be high gradients near the interface (e.g. Two-phase flows with large density and viscosity ratios).

� Adaptive mesh refinement in addition to enrichments for jumps/kinks.

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Hanging nodes

� Mesh is refined w.r.t. the interface.

� 5 levels of refinement possible.



Hanging nodes


� Level 1 refinement



Hanging nodes





Hanging nodes



2:1 Rule

Maximum difference Level 1

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Maximum difference

between the level of

refinement of adjacent

elements cannot be

more than one.

Level 2



Hanging nodes



2:1 Rule

Maximum difference Level 1

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Maximum difference



elements cannot be

more than one.

Level 2

At most one hanging

node exists on each

edge of an element.



Hanging nodes



2:1 Rule

Maximum difference

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Maximum difference



elements cannot be

more than one.

At most one hanging

node exists on each

edge of an element.



Hanging nodes





Re-meshing for moving interfaces

Refined Mesh at

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Interface at




Refined Mesh at

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Interface at

Interface at




Refined Mesh at Refined Mesh at




� Project field values from old mesh to new mesh -> projection errors


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known nodal values at

unknown nodal values at

Interpolation




� A band around the interface is refined.

Refined Mesh at

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Interface at

band




� New interface still falls within refined mesh at

Refined Mesh at

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Interface at

Interface at



Hanging nodes

� 5 topologically different hanging node elements



Construction of shape functions

� Ensure shape function at hanging node conforms to bilinear shape functions on adjacent elements

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Hanging node

Regular node



Hanging nodes

� Original regular shape functions

4 3

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21

5

1 2

34

5



Hanging nodes

� Modified regular shape functions (only 1 and 2)

4 3

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21

5

1 2

34

5



Hanging nodes

� Sum up

4 3

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21

5


Overview




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• Discretization


• Summary


Governing equations

Incompressible Navier Stokes equations

� Momentum and continuity equations

� Constitutive equation (Newtonian fluids)

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� Constitutive equation (Newtonian fluids)


Governing equations

Incompressible Navier Stokes equations

� Dirchlet and Neumann boundary conditions

� Interface conditions

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� Interface conditions

� Initial condition

curvature

surface tension coefficient

Dirichlet boundary

Fluid-fluid interface Neumann boundary


Governing equations

Level-set transport equation

� Level-set equation


Velocity of the fluid

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� no Dirichlet boundary conditions


Overview




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• Discretization


• Summary


Discretization

Time-dependence of enrichment function

� Enrichment functions depend on the time-dependent level-set function (even on fixed meshes).

If we discretize w.r.t. space first, -> semi-discrete approach

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� If we discretize w.r.t. space first, -> semi-discrete approach

� Therefore, need to discretize time before space.

Time-dependence accounted

for by nodal values


Discretization

Temporal discretization

� Consider the u-comp of the NS equations

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� Crank-Nicolson (trapezoidal rule)

� 2nd order accurate


Discretization

Spatial discretization

� XFEM approximation


Discretization

Sub-cell integration

� Existence of terms from time levels and in weak form; e.g.

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� Numerical integration of cut elements should consider interfaces

and Integration point


Discretization

Sub-cell integration

� Existence of terms from time levels and in weak form; e.g.

Fully implicit pressureVelocity space not

enriched

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� Numerical integration of cut elements considers only interface Integration point


Flow solver

Methodology

� XFEM with interface capturing (level-set method).

� Time-stepping with Crank-Nicolson method.

� Only pressure space is enriched (sign-enrichment).

� Fully implicit treatment of pressure.

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� Fully implicit treatment of pressure.

� Only one interface position needs to be considered at each time-step.

� Surface tension reformulated using the Laplace-Beltrami operator

� Picard (fixed-point) iterations for the nonlinear convective term.

� Reinitialization of the level-set done by recomputing signed-distance to new interpolated interface at each time-step.

� Primitive variable formulation


Flow solver

Procedure

� Strong coupling between the Navier Stokes and level-set equations.

Navier Stokes Navier Stokes

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Navier Stokes

equations

Level-set

equation

Navier Stokes

equations

Level-set

equation

Remesh

&

projection


Overview




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• Discretization


• Summary


Numerical results

Rising bubble

[Fries, IJNMF, 2008]

[Smolianski ,

PhD thesis, 2001]

8/19/2010L = 2d

H = 4d

d


Numerical results

Rising bubble

pressure

set to zero

Slip condition for

velocity along the

boundary

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d

Initial condition


Numerical results

Uniform refinement vs Adaptive refinement

� 48 X 96 mesh vs 6 X 12 mesh with level 3 refinement


Numerical results

Mass conservation improves with higher refinement levels

� 6 X 12 mesh with level 2/3/4 refinement


Numerical results

Rising bubble

� 6 X 12 mesh, level 3 refinement


Numerical results

Merging bubbles

[Tornberg, Comp. &

Visual. Sci., 2001]

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H = 4

d

Hw = 3


Numerical results

Merging bubbles

� 6 X 8 mesh with level 4 refinement


Conclusions

� A robust method of simulating evolving interfaces (i.e. 2-fluid flows) where both enrichment and adaptive h-refinement complement each other well.

� Enrichment of the pressure field alone together with adaptive h-refinement produce accurate results at a fraction of the computational cost compared to global

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fraction of the computational cost compared to global refinement or enrichment of the velocity as well as pressure fields.

� Adaptive h-refinement seems to be the most viable option when the analytical behaviour of the solution is unknown and therefore no suitable enrichment function can be realized easily.


Thank you for your attention.

Kwok-Wah Cheng

Thomas-Peter Fries

www.xfem.rwth-aachen.de


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