ME 226: Level Set Methods and Their Applications

 

·       Professor

·       Aims

·       Lecture Schedule

·       Syllabus

·       Teaching Assistant

·       Textbook

·       Grading Policy

·       Homework

·       Handouts

 

Professor

Name: Frederic G. Gibou
Office: Engineering II Bldg. room 2334
Phone: 893-7152
e-mail: fgibou@engineering.ucsb.edu

Office Hours: MF 4-5:30pm

Lectures: Every MWF at 12:00 - 12:50. Room: Psych 1802

Aims

 

The level set method is a general technique to keep track of a moving boundary that can undergo complex topological changes such as the merging and the breaking of complex flows. Since its introduction by Osher and Sethian in 1987 it has received considerable attention and applications have ranged from traditional fluid dynamics to materials science, computer vision and computer graphics. Some results obtained with this technique are given below.

 

This class will be an introduction to level set methods and their applications. In particular, we will first present the mathematical description of the level set method and then focus on the development of the numerical methods necessary to its implementation. For example, we will provide a detailed exposition of the numerical schemes traditionally used in the discretization of the general level set evolution equation and the reinitialization equation (ENO-WENO, Godunov Method, Lax-Friedrich Methods, etc.). We will also introduce the Ghost Fluid Method as a mean to impose boundary conditions at the interface. Finally, we will cover some applications taken from the fields of Computational Fluid Dynamics, Materials Sciences, Computer Vision and Computer Graphics.

 

                       

 

Movies: Ellipse traveling through a shallow pool of water (left) – Formation of a milk crown (middle) - Simulation of a growing crystal (right).

 

Lecture Schedule

There will be no class Monday January 10 and Wednesday January 13. These classes will be rescheduled Tuesday March 8 and Thursday March 10.

 

Syllabus

A copy of the syllabus can be found here.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Teaching Assistants

There are no Teaching Assistants for this class.

 

Textbook

The textbook for the course is Level Set Methods and Dynamic Implicit Surfaces, by Osher and Fedkiw. Other interesting materials can be found in:

·       Level Set Methods and Fast Marching Methods, J.A. Sethian

·       Finite Difference Schemes and Partial Differential Equations, J. Strikwerda

·       Numerical Methods for Conservation Laws, R. Leveque

·       Riemann Solvers and Numerical Methods for Fluid Dynamics, E. Toro

·       Numerical Analysis, R. Burden and J. Faires.

·       Elementary Applied Partial Differential Equations, R. Haberman

·       Relevant research articles will be handed out in class.

Grading Policy

Your grade will be based on homework (50%) and a final project (50%). The goal of the project is to demonstrate the ability to use the level set method in a practical application. Examples include Free Surface Flows, Multiphase Flows, Image Segmentation, Graphics, Differential Geometry, etc.

Homework

1.      Homework 1

 

2.      Homework 2

 

Working together in groups on homework is strongly encouraged!

 

Handouts

1.     Hamilton-Jacobi ENO

 

2.     Godunov Schemes