Adaptivity with Moving Grids

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Feb 22, 2014 (3 years and 3 months ago)

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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.