Computational Interface Dynamics (Selected Topics in Numerical Analysis) G63.2011.001/G22.2945.001
First Day of Class: Monday September 12, 2005.
Monday 1.25-3.15 p.m.
WWH Room 1013
Office: WWH 1119
Phone: 212 998 3299
email: tornberg (in the domain) cims.nyu.edu
For shorter questions, drop by at any time.
For longer consultations, please make an appointment.
Lectures: The class meets regularly on Mondays from September
12 to December 12. Exceptions: No class Monday Oct 10 (Columbus
day). Instead: Class on Wed Nov 23, same time and location. (All of
NYU is running on a Monday schedule that week).
Class on Mon Nov 21 is cancelled.
Grading: This course will be graded as a seminar course requiring a
Projects: There will be two relatively small computer
assignments, implementing and testing a simple version of a
front-tracking and a level-set method, respectively. Then, each
student is to select his or her own subject for a course project,
which is to be presented in class towards the end of the
semester. Suggestions on project topics are available, but other
topics are welcome (and encouraged). Oral presentations will be held
in class on Dec 5th and Dec 12th. A written report describing the work is due
Monday Dec 19th.
Description: Moving internal boundaries or interfaces
separating different regions are present in many problems in
nature. Two examples are interfaces separating two different fluids in
a multiphase flow, and between solid and liquid regions in a
solidification process. Such interfaces are in general dynamic,
i.e. they evolve in time. Frequently, singular forces are supported on
these interfaces (e.g. surface tension forces) and material parameters
might be discontinuous across the interface (e.g. density and
viscosity). Considering these features, the numerical approximation
and simulation of such problems is a challenging task and a field of
extensive ongoing research.
In this course, we will focus our attention on so called interface
tracking/capturing methods. Two important groups of numerical methods
that will be discussed are the front-tracking methods and the
level-set methods. First, we will discuss the formulations and
implementations for problems where the interface velocity either is
assumed to be given or where it depends solely on local properties of
the front. Then we will study applications where the intefaces
describes an material interface, and the dynamics are coupled to
variables in the bulk.
In interface tracking methods, the governing equations
are solved on an underlying grid, which is kept
fixed. Hence, this grid does not in general match the interfaces, which
are separately represented. The coupling of the interface tracking
method to the solution on the underlying grid will be discussed for
several different applications, including details of how to deal with
singular forces and discontinuous material coefficients.
In addition to the interface tracking methods, we will also discuss
some methods designed with a special application in mind, such as
boundary integral methods for fluid-structure interactions in low
Reynolds number flows.
- Introduction, including examples of applications
- Mathematical representation of interfaces:
- Explicit and implicit definitions.
- Numerical representation of interfaces:
- Point set, functions, implicit discretizations.
- Evolution of interfaces
- (equations for - assuming known interface velocity).
ODEs for point set, PDE for functional definition,
PDE for implicit definition.
- Numerical methods for interface evolution (assuming known interface velocity).
- Front-tracking methods
- Parameterization of curve/surface. Time-stepping of
ODEs. Redistribution of points. Computation of normal vectors and
- Level-set methods
- Definition of signed distance function. Solving evolution
PDE. Viscosity solutions. Extension of velocity from interface to domain (PDE approach and fast-marching approach).
Reinitialization of level-set function (PDE approach and fast-marching).
Computing curvature and normal vectors. Time-stepping schemes.
- Other methods (Segment projection method and Volume of Fluid etc.)
- Coupling to external physics (schematic idea)
- Grid based methods: Coupling between the grid and the interface.
- Multiphase flow - immiscible fluids.
- Governing equations, basic idea. Something about moving mesh
methods. Difficulty for fixed grid: discontinuous coefficients,
singular source terms. Volume of Fluid (VOF)
methods. Finite difference/finite element methods with front-tracking and level-set methods.
Placing the interface on the grid: Regularization of singular
surface tension forces and discontinuous
density/viscosity. Interpolation of velocities from grid to
interface (front-tracking). Immersed interface method.
- Solidification, dendritic growth.
Governing equations, basic idea.
Front-tracking methods, level-set methods, phase field methods.
- Geometrical optics (High frequency wave propagation)
First arrivals, Eikonal equation.
Ray tracing. Formulation for evolution of wave front in phase space.
- Boundary integral methods.
- Applicable for linear equations. Here, we will consider the
application of fluid-structure interactions for Stokes flow (zero
Reynolds number). Slender-body asymptotics for slender fibers (rigid
- Other applications (brief)
- Image processing, epitaxial growth, etching...
- Additional topics
Here is a list of references that will be updated as the course proceeds.
Below are a few suggested readings to start with.
- Regarding the level-set method, the following two books are useful:
- J.A. Sethian.
Level Set Methods and Fast Marching Methods: Evolving
interfaces in Computational Geometry, Fluid
Mechanics, Computer Vision and Materials Science.
Cambridge University Press, Cambridge, 1999.
- S. J. Osher and R. P. Fedkiw.
Level Set Methods and Dynamic Implicit Surfaces.
- There is an early paper about front-tracking for immiscible
multiphase flows (finite difference method) that is still a basic reference:
- S.O Unverdi and G. Tryggvason. A Front-Tracking Method for Viscous,
Incompressible, Multi-fluid Flows. Journal of Computational Physics,
- And likewise for solidification:
- D. Juric and G. Tryggvason.
A Front-Tracking Method for Dendritic Solidification.
Journal of Computational Physics. 123:127-148, 1996.
- There is also an Acta Numerica paper by
Peskin that describes the immersed boundary method. This application
is for fluids with immersed elastic boundaries.
- Peskin, C.S. The Immersed Boundary Method.
Acta Numerica, 2002. Pages 479-517.