Sharp Interface method and levelset function -Ghost Fluid Method Serie 3
Sharp interface method as one methods used in multi material simulation has the advantage of maintaining a sharp interface and swiftness in tracking the movement of the interface
Level Set methods
For Eulerian single-code model used in this study, A level set function is used to models the interface location and has the following purpose:
- It represents a sharp interface. item
- It identified the location of a material interface, and is used ti split the computational domain into multiple regions.item
There are three techniques that are available for reinitialisation – Iterative reinitialisation, fast marching and fast sweeping. iterative reinitialisation is an iterative scheme where the Eikonal equation is transformed to a hyperbolic PDE by introducing a time derivative :
\[\frac{\partial \phi}{\partial \tau} + sgn(\phi)(| \nabla \phi|-1) = 0\] The sign of the level set function depends on which side of the interface we are reinitialising. The variable \(\tau\) is known as 'fictitious time', in which the boundaries of the level set function are held fixed, thus this equation has a steady state solution. Due to inefficiency, and uncertainty in the number of iterations, I adopt the fast sweep reinitialisation approach purposed by. The principle of the fast sweeping method is the level set function, also a signed distance function always gives the shortest distance to an interface and therefore a numerical technique to compute a value \(\phi _i\) is dependent only on \(|\phi|<|\phi_i|\). The fast sweeping method carries out a series of sweep which compute values of level set function such that \(| \nabla \phi|=1 \) based on neighbouring values. Multiple sweep alternates in x, y, z, positive and negative directions . The movement of interface is modelled using a Hamilton-Jacobi equation : \[\frac{\partial \phi}{\partial t} + H( \nabla \phi) = 0\]where the operator H acts upon the gradient of the level set function. For the problem involved in this study, a simple advection equation is adequate to describe the evolution of interface :
\[\frac{\partial \phi}{\partial t} + \textbf{v} \cdotp \nabla \phi = 0\] $\textbf{v}$ is the velocity of the materials, which has no dependence on the level set function. Hence the discretisation can be derived: \[\frac{\phi_i^{n+1}-\phi_i^{n}}{\Delta t} + v_{x,i}^n(\frac{\partial \phi}{\partial x})_i^{n}+ v_{y,i}^n(\frac{\partial \phi}{\partial y})_i^{n}+v_{y,i}^n(\frac{\partial \phi}{\partial z})_i^{n} =0\]Several numerical scheme are available for the spatial derivative, for this study first order upwind scheme is adopted. The upwind and downwind approximation depends on the direction of the velocity.
References
- Fedkiw, R. P., T. Aslam, B. Merriman, and S. Osher (1999), A Non-oscillatory Eulerian Ap- proach to Interfaces in Multimaterial Flows (the Ghost Fluid Method), Journal of Compu- tational Physics, 152(2), 457–492.
- Osher, S., R. Fedkiw, and K. Piechor (2004), Level Set Methods and Dynamic Implicit Surfaces, Applied Mechanics Reviews 3.Zhao, H. (2004), A fast sweeping method for Eikonal equations, Mathematics of Computation,