Adaptivity with Moving Grids


22 Φεβ 2014 (πριν από 3 χρόνια και 1 μήνα)

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Adaptivity with Moving Grids

By Santhanu Jana

Talk Overview


Techniques in Grid Movement

Physical and Numerical Implications in time

dependent PDE‘s

Outlook and Conclusions


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



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


Movement of the interface is unknown in advance

and must be determined as part of the solution

Computational Techniques in Moving

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

Solution: Local Grid


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


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

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

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

2) H.S.Udaykumar, H.C.Kan, W.Shyy, and R.Tran

Multiphase dynamics in arbitrary geometries on fixed cartesian grids,

Journal of Computational Physics, 137, (1997), 366

Eulerian Methods: Immersed Boundary
Method (3)

Marker Particles





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


Important Considerations:

Interface Representation

Assignment of Material Properties

(Change of Contants in PDE)

Immersed Boundary Treatment

Immersed Boundary Method: Interface

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

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

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

H. S. Udaykumar, R. Mittal, P. Rampunggoon and A. Khanna,

A Sharp Interface Cartesian Grid Method for Simulating Flows with Complex Moving

Journal of Computational Physics, Volume 174, Issue 1, 20 November 2001, Pages 345

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

Solver doesnot see discontinuity (Eulerian

Solver experiences distributed forces and
material properties on the vicinity of the

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

Outlook and Conclusion

Lagrangian Methods are physically consistent over
Eulerian Methods but suffers when grid distortion is

In Eulerian Methods mergers and break ups are tackled

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.