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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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• Governing equations
• 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
3K.W. Cheng, h-adaptive XFEM
Evolving interfaces
Moving boundaries as discontinuities
Fluid 1
strong discontinuity
density viscosity
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Fluid 2
weak discontinuity
strong discontinuity
velocity pressure
4K.W. Cheng, h-adaptive XFEM
Evolving interfaces
Example: Bubble with surface tension
strong discontinuity
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pressure
strong discontinuity
5K.W. Cheng, h-adaptive XFEM
Overview
• Evolving interfaces as discontinuities
• Mesh refinement: h-XFEM
• Governing equations
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• Governing equations
• Discretization
• Numerical results
• Summary
6K.W. Cheng, h-adaptive XFEM
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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Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� 5 levels of refinement possible.
8/19/2010 8K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 1 refinement
8/19/2010 9K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 2 refinement
8/19/2010 10K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 2 refinement
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
11K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 2 refinement
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
At most one hanging
node exists on each
edge of an element.
12K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 2 refinement
2:1 Rule
Maximum difference
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Maximum difference
between the level of
refinement of adjacent
elements cannot be
more than one.
At most one hanging
node exists on each
edge of an element.
13K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 3 refinement
8/19/2010 14K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 4 refinement
8/19/2010 15K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Mesh is refined w.r.t. the interface.
� Level 5 refinement
8/19/2010 16K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
Refined Mesh at
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Interface at
17K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
Refined Mesh at
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Interface at
Interface at
18K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
Refined Mesh at
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Interface at
Interface at
19K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
Refined Mesh at Refined Mesh at
8/19/2010 20K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
Refined Mesh at Refined Mesh at
8/19/2010 21K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
� Project field values from old mesh to new mesh -> projection errors
Refined Mesh at Refined Mesh at
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known nodal values at
unknown nodal values at
Interpolation
22K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
� A band around the interface is refined.
Refined Mesh at
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Interface at
band
23K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Re-meshing for moving interfaces
� New interface still falls within refined mesh at
Refined Mesh at
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Interface at
Interface at
24K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� 5 topologically different hanging node elements
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Mesh refinement : h-XFEM
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
26K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Original regular shape functions
4 3
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21
5
1 2
34
5
27K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Modified regular shape functions (only 1 and 2)
4 3
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21
5
1 2
34
5
28K.W. Cheng, h-adaptive XFEM
Mesh refinement : h-XFEM
Hanging nodes
� Sum up
4 3
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21
5
29K.W. Cheng, h-adaptive XFEM
Overview
• Evolving interfaces as discontinuities
• Mesh refinement: h-XFEM
• Governing equations
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• Governing equations
• Discretization
• Numerical results
• Summary
30K.W. Cheng, h-adaptive XFEM
Governing equations
Incompressible Navier Stokes equations
� Momentum and continuity equations
� Constitutive equation (Newtonian fluids)
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� Constitutive equation (Newtonian fluids)
31K.W. Cheng, h-adaptive XFEM
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
32K.W. Cheng, h-adaptive XFEM
Governing equations
Level-set transport equation
� Level-set equation
� Initial condition
Velocity of the fluid
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� Initial condition
� no Dirichlet boundary conditions
33K.W. Cheng, h-adaptive XFEM
Overview
• Evolving interfaces as discontinuities
• Mesh refinement: h-XFEM
• Governing equations
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• Governing equations
• Discretization
• Numerical results
• Summary
34K.W. Cheng, h-adaptive XFEM
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
35K.W. Cheng, h-adaptive XFEM
Discretization
Temporal discretization
� Consider the u-comp of the NS equations
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� Crank-Nicolson (trapezoidal rule)
� 2nd order accurate
36K.W. Cheng, h-adaptive XFEM
Discretization
Spatial discretization
� XFEM approximation
8/19/2010 37K.W. Cheng, h-adaptive XFEM
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
38K.W. Cheng, h-adaptive XFEM
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
39K.W. Cheng, h-adaptive XFEM
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
40K.W. Cheng, h-adaptive XFEM
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
41K.W. Cheng, h-adaptive XFEM
Overview
• Evolving interfaces as discontinuities
• Mesh refinement: h-XFEM
• Governing equations
428/19/2010
• Governing equations
• Discretization
• Numerical results
• Summary
42K.W. Cheng, h-adaptive XFEM
Numerical results
Rising bubble
[Fries, IJNMF, 2008]
[Smolianski ,
PhD thesis, 2001]
8/19/2010L = 2d
H = 4d
d
43K.W. Cheng, h-adaptive XFEM
Numerical results
Rising bubble
pressure
set to zero
Slip condition for
velocity along the
boundary
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d
Initial condition
44K.W. Cheng, h-adaptive XFEM
Numerical results
Uniform refinement vs Adaptive refinement
� 48 X 96 mesh vs 6 X 12 mesh with level 3 refinement
8/19/2010 45K.W. Cheng, h-adaptive XFEM
Numerical results
Uniform refinement vs Adaptive refinement
� 48 X 96 mesh vs 6 X 12 mesh with level 3 refinement
8/19/2010 46K.W. Cheng, h-adaptive XFEM
Numerical results
Mass conservation improves with higher refinement levels
� 6 X 12 mesh with level 2/3/4 refinement
8/19/2010 47K.W. Cheng, h-adaptive XFEM
Numerical results
Rising bubble
� 6 X 12 mesh, level 3 refinement
8/19/2010 48K.W. Cheng, h-adaptive XFEM
Numerical results
Merging bubbles
[Tornberg, Comp. &
Visual. Sci., 2001]
8/19/2010L = 3
H = 4
d
Hw = 3
49K.W. Cheng, h-adaptive XFEM
Numerical results
Merging bubbles
� 6 X 8 mesh with level 4 refinement
8/19/2010 50K.W. Cheng, h-adaptive XFEM
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
51
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.
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Thank you for your attention.
Kwok-Wah Cheng
Thomas-Peter Fries
www.xfem.rwth-aachen.de
8/19/2010 52K.W. Cheng, h-adaptive XFEM