Adaptivity with Moving Grids
By Santhanu Jana
Talk Overview
Motivation
Techniques in Grid Movement
Physical and Numerical Implications in time
dependent PDE‘s
Outlook and Conclusions
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
Some Examples

Fluid Structure Interactions
Source:
http://www.onera.fr/ddss

en/aerthetur/aernummai.html
:http://www.erc.msstate.edu/simcenter/04/april04.html
Some Examples

Phase Change Phenomena
Source: Work At LSTM
Crystal Growth
Czochralski Crystal Growth
Simulation of Free Surface
What Is Moving Grid ?
PDE’s 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
Computational Techniques in Moving
Grids
Lagrangian Methods
Eulerian Methods
Mixed Lagrangian and
Eulerian Methods
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
Problems in Lagrangian Methods(2)
Grid Distortion
Solution: Grid Sliding
Problems in Lagrangian Methods(3)
Resolving Complex
Structures near the
interface
Solution: Local Grid
Refinement
Increase the
Convergence order
Necessary Modifications in the
Conservation Equations(4)
Eg: Solution of Navier Stokes Equations
1 ) Momentum Equation:
2 ) Energy Equation
:
3 ) Mass Conservation Equation:
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
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
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
Eulerian Methods: Immersed Boundary
Method (3)
Marker Particles
FLUID 2
ds
n
FLUID I
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
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.
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
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
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
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
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.
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