CS240A Final Project: Marc Griesemer, Joe Papac, and Kevin Sanft

A parallel implementation of the level set method

Final paper and presentation

Final paper, pdf.

Final presentation, ppt, pdf.

Videos

Here are some of the videos we created.

3-D videos

tearing of a sphere, avi
uniform translation of a sphere, avi
a sphere in stagnation flow, avi
flattening of a sphere, avi

2-D videos

distortion of a "Z", avi
stagnation and shear of a square, avi
uniform translation of a circle, avi
tearing of a circle, avi
another tearing of a circle, avi

Status update, May 23, 2007

Goals for the project:

Solve the 2D linear level set equation to obtain the motion and deformation of an object in an externally generated fluid flow
We want to test the following fluid flows: irrotational vortex flow, stagnation flow, and inviscid doublet.
Implement the nonlinear reinitialization equation for the level set function, phi_t + sign(phi_0) * [abs(grad(phi)) - 1 ] = 0.
Develop a 3-D code (if time permits). Extending the level set code to three dimensions is relatively straightforward, however we must find an easy way to visualize the data after the simulation.

Accomplishments:

We've discussed the mathematics behind the level set simulation and formed an initial strategy for a parallel algorithm. This essentially amounts to solving a linear hyperbolic partial differential equation. This will consist of dividing the domain into subdomains. On each subdomain, we will calculate the spacial gradient of the level set function with third order HJ ENO polynomial discretization. In order to perform this discretization we must incorporate 3 ghost nodes in each direction. We have discussed a strategy for communication of these "ghost points" and for carrying out the numerical method.

Current status:

We have created tasks for each group member. We have a serial implementation of the code written in C, and are currently developing the parallel version with C and MPI.

Description of the level set method

The level set method is a technique for implicitly tracking interface evolution. The goal of this project is to implement a parallel algorithm for the level set method in externally generated flows.

Consider a two-dimensional domain that contains two dissimilar fluids. The interface between these fluids can be described implicitly by defining a three dimensional scalar field, phi(x,y) such that the interface location lies at phi(x,y)=0, the points in the interior of the interface have a negative sign which corresponds to fluid 1, and the points outside of the interface have a positive sign which corresponds to fluid 2.

In this framework, the motion of the interface due to an external velocity field V, such is the case with fluid flow, can be tracked by the solution of the following partial differential equation.

[level set equation]

The level set framework allows for complex topological changes such as merging and pinching of fluid regions to be handled implicitly and efficiently.

Examples of fluid flow simulations

The physics of immiscible porous media flow is being investigated through a Navier-Stokes/level set numerical method. The following video shows one such 2-D simulation, mov.

A parallel algorithm for the level set equation

The goal of this project is to implement a parallel algorithm for the solution of the level set equation in two and three dimensions. We will simulate large domains and investigate the speedup behavior as a function of the number of processors.

The level set equation relies on accurate spacial gradients, therefore high order spacial discretizations such as the third order accurate Hamilton-Jacobi essentially non-oscillatory (HJ-ENO) interpolation are used. The HJ-ENO discretization requires at most three neighboring cells in either direction for calculation of the spacial gradient. Because of this requirement, we must define a band of ghost cells where the domain is divided.