Office: WWH 1119

Phone: 212 998 3299

email: tornberg (in the domain) cims.nyu.edu

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 curvature.

- 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.)

- Front-tracking methods
- 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. 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.

- 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 and flexible).

- Other applications (brief)
- Image processing, epitaxial growth, etching...

- Additional topics

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.