Adaptivity with Moving Grids

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Adaptivity with Moving Grids

• By Santhanu Jana

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Talk Overview

• Motivation
• Techniques in Grid Movement
• Physical and Numerical Implications in time
• dependent PDEs
• Outlook and Conclusions

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Motivation

• Applications in Physics
• Fluid Structure
• Aerostructures and Aeroacoustics
• Moving Elastic Structures
• eg. Simulation of Heart
• Thermodynamical Considerations
• Phase Change Phenomena
• Free Surfaces
• Material Deformations
• Multiphase Flows

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Some Examples- Fluid Structure Interactions
Sourcehttp//www.onera.fr/ddss-en/aerthetur/aernu
mmai.html http//www.erc.msstate.edu/s
imcenter/04/april04.html
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Some Examples- Phase Change Phenomena
Crystal Growth
Source Work At LSTM
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Czochralski Crystal Growth
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Simulation of Free Surface
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What Is Moving Grid ?

• PDEs must be satisfied on each side of the
• interface (often different equations on each
  side)
• Solutions coupled through relationships or jump
• conditions that must hold at the interface
• These conditions may be in the form of
  differential
• equations
• Movement of the interface is unknown in advance
• and must be determined as part of the solution
  

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Computational Techniques in Moving Grids

• Lagrangian Methods
• Eulerian Methods
• Mixed Lagrangian and
• Eulerian Methods

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Lagrangian Methods(1)

• Explicit Tracking of the Interface Boundary
• No Smearing of Information at the boundary
• No Modeling is necessary to define the interface
• Un/structured boundary Conforming Grids
• No modelling to define the interface
• Grid Regeneration
• Grid Adaption
• Requires redistribution of field information

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Problems in Lagrangian Methods(2)

• Grid Distortion
• Solution Grid Sliding

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Problems in Lagrangian Methods(3)

• Resolving Complex Structures near the interface
• Solution Local Grid
• Refinement
• Increase the
• Convergence order

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Necessary Modifications in the Conservation
Equations(4)

• Eg Solution of Navier Stokes Equations

1 ) Momentum Equation
2 ) Energy Equation
3 ) Mass Conservation Equation
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Necessary Modifications in the Conservation
Equations(5)
4) Geometric Conservation
NOTE Grid Velocities should satisfy Geometric
Conservation Equation
References 1) Thomas, P.D., and Lombard, C.K.,
Geometric Conservation Law and Its Applications
to Flow Computations on Moving Grids," AIAA
Journal, Vol. 17, No. 10, pp. 1030-1037. 2)
Weiming Caso, Weizhang Huang and Robert D.
Russel A Moving mesh Method based on the
Geometric Conservation Law, SIAM J. SCI.
COMPUTING Vol24, No1, pp.118-142
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Eulerian Methods(1)

• Boundary is derived from a Field Variable
• eg VOF, Level Set
• Interface is diffused and occupies a few grid
  cells in
• practical calculations
• Strategies are necessary to sharpen and
  physically
• reconstruct the interface
• Boundary Conditions are incorporated in the
• governing PDE.
• Grid Generation Grid is created once

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Basic Features of Eulerian Methods(2)

• Grid Topology remains simple even though the
  interface
• may undergo large deformations
• Two Basic Approaches
• Immersed Boundary Method
• Without explicit tracking
• Interface Cut-Cell Method
• Interface tracked explicitly
• (Reconstruction procedures to calculate
  coefficients
• in the Solution Matrix)
• Ref 1) C.S.Peskin, Numerical Analysis of blood
  flow in the heart,
• Journal of Computational Physics, 25, (1977),
  220-252
• 2) H.S.Udaykumar, H.C.Kan, W.Shyy, and
  R.Tran-Son-Tay,
• Multiphase dynamics in arbitrary geometries on
  fixed cartesian grids,
• Journal of Computational Physics, 137, (1997),
  366-405

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Eulerian Methods Immersed Boundary Method (3)

The Interface between Fluid 1 and Fluid 2 is
represented by curve C is marked by particles
(dots) that do not coincide with the grid nodes
C

• Important Considerations
• Interface Representation
• Assignment of Material Properties
• (Change of Contants in PDE)
• Immersed Boundary Treatment

Marker Particles
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Immersed Boundary Method Interface
Representation(4)

• Immersed boundary represented by C(t)
• Curve in 2D and Surface in 3D.
• Markers or interfacial points of coordinates
• Markers are regularly distributed along C(t) at
  a
• fraction of grid spacing (ds).
• The interface is parameterised as a function of
• arclength by fitting a quadratic polynomial.
• The normal vector and curvature
• (divergence of normal vector) is evaluated.

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Immersed Boundary Method Material Properties(5)

• Assign in each fluid based on some step function
• Should handle the transition zone.
• Treatment handles improved Numerical
• Stability and solution smothness

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Immersed Boundary Method Boundary Treatment(6)

• Facilitates Communication between the moving
  markers
• (interface) and the fixed grid.
• Evaluation of the forces acting on the interface
• Estimation of interface velocity
• Advection of the interface.
• To improve accuracy of the interface tracking, a
  local
• grid refinement aroung the interface can be
  used.

Ref H. S. Udaykumar, R. Mittal, P. Rampunggoon
and A. Khanna,A Sharp Interface Cartesian Grid
Method for Simulating Flows with Complex Moving
Boundaries Journal of Computational Physics,
Volume 174, Issue 1, 20 November 2001, Pages
345-380
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Interface Cut Cell Method(6)

• Improvement over Immersed Boundary
• method
• Summary of the Procedure
• Location of Interface Marker.
• The interfacial marker closest to mesh point.
• Material parameters.
• Interface Cell Reconstruction
• Geometric details.
• Intersection of the immersed boundary with the
• Fixed grid mesh.
• Suitable stencil and evaluate coefficients

Example Stencil to evaluate variables
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Mixed Eulerian-Lagrangian methods

• Combines features of Eulerian and Lagrangian
  methods.
• Solver doesnot see discontinuity (Eulerian
  Methods)
• Solver experiences distributed forces and
  material properties on the vicinity of the
  interface
• No smearing of interface

Ref S. Kwak and C. Pozrikidis Adaptive
Triangulation of Evolving, Closed, or Open
Surfaces by the Advancing-Front Method Journal
of Computational Physics, Volume 145, Issue 1, 1
September 1998, Pages 61-88
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Outlook and Conclusion

• Lagrangian Methods are physically consistent over
  Eulerian Methods but suffers when grid distortion
  is severe.
• In Eulerian Methods mergers and break ups are
  tackled automatically.
• Interface Reconstruction in Eulerian Methods may
  be very complicated on nonorthogonal
  un/structured grid. Extension to 3D might be a
  problem.
• Local Refinement may be used to the capture the
  interface more accurately.