Computational Interface Dynamics (Selected Topics in Numerical Analysis) G63.2011.001/G22.2945.001

Fall 2005

First Day of Class: Monday September 12, 2005.
Time: Monday 1.25-3.15 p.m.
Location: WWH Room 1013
Instructor: Anna-Karin Tornberg
Office: WWH 1119
Phone: 212 998 3299
email: tornberg (in the domain) cims.nyu.edu

Office hours: 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 presentation.
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.
Tentative outline:
  1. Introduction, including examples of applications
  2. Mathematical representation of interfaces:
    Explicit and implicit definitions.
  3. Numerical representation of interfaces:
    Point set, functions, implicit discretizations.
  4. Evolution of interfaces
    (equations for - assuming known interface velocity). ODEs for point set, PDE for functional definition, PDE for implicit definition.
  5. Numerical methods for interface evolution (assuming known interface velocity).
    1. Front-tracking methods
      Parameterization of curve/surface. Time-stepping of ODEs. Redistribution of points. Computation of normal vectors and curvature.
    2. 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.
    3. Other methods (Segment projection method and Volume of Fluid etc.)
  6. Coupling to external physics (schematic idea)
    Grid based methods: Coupling between the grid and the interface.
  7. 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. Hybrid 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.
  8. Solidification, dendritic growth. Governing equations, basic idea. Front-tracking methods, level-set methods, phase field methods.
  9. Geometrical optics (High frequency wave propagation) First arrivals, Eikonal equation. Ray tracing. Formulation for evolution of wave front in phase space.
  10. 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 and flexible).
  11. Other applications (brief)
    Image processing, epitaxial growth, etching...
  12. Additional topics
Suggested reading: 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. Springer, 2002.
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, 100:25-37, 1992.
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.