LEVEL SET METHODS FOR FLUID INTERFACES

Abstract

We provide an overview of level set methods, introduced by Osher and Sethian, for computing the solution to fluid interface problems. These are computational techniques which rely on an implicit formulation of the interface, represented through a time-dependent initial value partial differential equation. We discuss the essential ideas behind the techniques, the coupling of these techniques to finite difference methods for incompressible and compressible flow, and a collection of applications including two phase flow, ship hydrodynamics, and inkjet printhead design.

Nonlinear simulation of the effect of microenvironment on tumor growth

“…See Figure A.2. For more information on the level set method and its application to fluid mechanics, please see Osher and Sethian (1988), Sussman et al (1994), Malladi et al (1995Malladi et al ( , 1996, Adalsteinsson and Sethian (1999), Sethian (1999), Fedkiw (2001, 2002), and Sethian and Smereka (2003).…”

“…to steady-state, where τ is pseudo-time and ϕ 0 is the level set function prior to reinitialization (Osher and Sethian, 1988;Malladi et al, 1995Malladi et al, , 1996Adalsteinsson and Sethian, 1999;Sethian, 1999;Fedkiw, 2001, 2002;Sethian and Smereka, 2003). We solve the PDE's in (48) and (49) with the third-order total variation-diminishing Runge-Kutta method (Gottlieb and Shu, 1997;Gottlieb et al, 2001) and the fifth-order WENO method (Jiang and Shu, 1996;Jiang and Peng, 2000).…”

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

“…(See the books [43,48] and references [42,44,49].) In the level set method, the location of a region Ω is captured implicitly by introducing an auxilliary signed distance function ϕ that satisfies (6) In the level set approach, instead of explicitly tracking the position of interface Σ and manually handling topology changes, the level set function is updated by solving a PDE, which automatically accounts for the interface motion and all topology changes.…”

“…(47), we solve the related equation (48) to steady state, where τ is pseudo-time. For numerical stability, we begin by implicitly discretizing pseudo-time with a backwards Euler difference, lagging the dependence of D on p, and applying the standard second-order centered difference to the diffusional term: (49) Here, , x i = a + iΔx, y j = c + jΔy, τ n = nΔτ, and Δx, Δy and Δτ are spatial and pseudo-temporal discretization step sizes, respectively. This system has the form (50) which can be solved to steady state by constructing the operator A(x, p n ) and right-hand side b(x, p n ) and solving the linear system in (50) with an iterative method (e.g., BiCGStab(ℓ) from [50]) at every pseudo-time step until the system reaches steady state.…”

Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method

“…One option is Level Set Methods, introduced by Osher and Sethian [23]. A large collection of simulations have been performed coupling level set methods to Chorin's projection method ( [9]) to compute the solution of incompressible, viscous and inviscid two-phase flow, often in the presence of interface surface tension and considerable density variation between the two fluids, see ( [7,31,33,30,35]). In this approach, boundary conditions are required for both fluids.…”