Level set methods in Cactus Carbunescu Razvan Corneliu

Implementation of Level Set Methodsin Cactus Framework

By Carbunescu Razvan Corneliu

Level set methods in Cactus Carbunescu Razvan Corneliu

Introduction

•Computational Fluid Dynamics is used as avaluable engineering simulation tool to understandand design complex systems involving fluid flow,heat transfer, mixing and multiphase flows.

•a simple example of multiphase fluid flow wouldbe a drop of water falling into a glass

•these problems involve complex boundary andinterface conditions evolving in time and thereforelevel set methods can be a powerful tool to handlesuch problems computationally

Level set methods in Cactus Carbunescu Razvan Corneliu

Level Set Methods

•are a general and powerful technique to representan object's boundary by the means of an implicitfunction that has a specific value on the boundary

•allows for the easy modeling of interfaces betweendifferent fluids that interact by the use of partialdifferential equations to represent the interactions

•are used extensively in Computational FluidDynamics (CFD) because of these abilities

Level set methods in Cactus Carbunescu Razvan Corneliu

Cactus Framework

•“is a framework for developing portable, modularapplications, in particular, although not exclusively,high-performance simulation code”

•provides great flexibility for the written codeallowing for it to be easily modified to meet newneeds

•is an open source code so also allows for furtherdevelopment by any other interested parties

Level set methods in Cactus Carbunescu Razvan Corneliu

Zalesak’s problem

•problem discusses the movement of a notched discthat is rotating in a given velocity field

•Zalesak’s problem has often been used in papersto test the implementation of level set methods

•the disc should maintain constant volume after onefull rotation with the use of level set methodsalthough the shape is not preserved with simplelevel set methods

Level set methods in Cactus Carbunescu Razvan Corneliu

Implicit functions

•a function which has a value of 0 on the boundaryor interface of the object, negative inside, andpositive outside

•these functions are easily manipulated by the useof partial differential equations

•the best choice for level set methods is the SignedDistance Function (SDF)

–d if x is inside the object

SDF(x)= 0 if x is on the interface

+d if x if outside the object

Level set methods in Cactus Carbunescu Razvan Corneliu

Signed Distance Function

•the value of the function is the distance from theinterface (d) with a negative or positive signassigned depending on it’s location inside oroutside the object

Equation of movement

•the evolution of an object represented by an implicit functionin an external velocity field is given by the following partialdifferential equation:

Where:

Level set methods in Cactus Carbunescu Razvan Corneliu

∂Φ∂t + V ۰ ▼Φ = 0▼

→

V= ( u , v , w ) and→ → → → ▼Φ = ( , , )▼

∂Φ∂x∂Φ∂y∂Φ∂z

Level set methods in Cactus Carbunescu Razvan Corneliu

Discretization

•partial derivatives are represented by discretizedapproximations

•this schemes used are upwind differencing and are first orderaccurate

∂Φ∂t

Φ – Φ .n+1i,j,k

ni,j,k≈ Δt

∂Φ∂x

Φ – Φ .ni,j,k

ni-1,j,k

Δx

∂Φ∂y

Φ – Φ .ni,j,k

ni,j-1,k

Δy∂Φ∂z

Φ – Φ .ni,j,k

ni,j,k-1

Δz≈

≈

≈

Stability concerns

•the schemes are not stable for all timesteps and velocities

•a scheme’s stability can be determined by analysis of a factorknown as the Courant number

Level set methods in Cactus Carbunescu Razvan Corneliu

Level set methods in Cactus Carbunescu Razvan Corneliu

Reinitialization importance

•the equation presented before changes the nature ofthe function after a number of time steps

•constant reinitialization of the function to the signeddistance function is needed to preserve it’s qualities

•the reinitialization technique chosen for the example isa slight variant to the Fast Marching Method algorithm

•the FMM algorithm first adds the values closest to theinterface to a list and then continues to expand thatband

Level set methods in Cactus Carbunescu Razvan Corneliu

Reinitialization algorithm

•uses only the neighbors of any point and their sources on theinterface to determine the value at that point

where:

- points on the grid

- points near the interface

- currently selected point in the list

- currently selected neighbor

- other points around the neighbor

- points at which the neighbor looks

for determining the S.D.F.

Reinitialization results

•The following results are on a grid size of 100x100 and witha timestep of dt = 0.0005

Level set methods in Cactus Carbunescu Razvan Corneliu

Level set methods in Cactus Carbunescu Razvan Corneliu

Current status and future work

•The implementation has been partly moved to parallelbut there are still a few problems to work out with theparallel reinitialization algorithm

•The next move will be to move the thorns in Cactusfrom 2D to 3D and implement higher order accurateschemes (2nd order has been implemented but still needstesting)

•After that will be accomplished more advancedproblems like multiphase flow and curvature driven flowswill be implemented

Level set methods in Cactus Carbunescu Razvan Corneliu

Acknowledgements

•The Center for Computation and Technology at LSU andtheir excellent technical, academic and organizationalpersonnel which helped me with the research.

•I would especially like to thank Dr. Gabrielle Allen,Mr.Yaakoub el Khamra, Dr. Mayank Tyagi and Ms. KathyTraxler

•The Department of Computer Science, the ACM and theStudent Government organizations at LSU for helpingfund the presentation

Level set methods in Cactus Carbunescu Razvan Corneliu

Thank you !