A gradient-augmented level set method with an optimally local, coherent advection scheme

Abstract

a b s t r a c tThe level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.

A Semi-implicit Gradient Augmented Level Set Method

“…al. demonstrated that for analytic flow fields the gradient augmented provides results comparable to higher-order WENO schemes at a lower computational cost [16]. Results for flows depending on derivatives of the level set function where not presented.…”

“…Take To ensure that the level set and gradient field remain coupled throughout time Eqs. (2.4) and (2.5) are advanced in a coherent and fully coupled manner [16]. Lagrangian techniques are used to trace characteristics back in time to determine departure locations.…”

“…The gradient augmented level set method is an extension of the standard level set method which advances the level set and the gradient of the level set field [16]. This is accomplished by advecting the level set and its gradient as independent quantities in a coupled manner, ensuring that the relationship between the quantities remains.…”

High-order methods for computing distances to implicitly defined surfaces

“…The resulting method is approximately third order in the distance function for smooth interfaces [10]. A similar approach can be used in gradient augmented level set methods [18], where both φ and its gradient are defined at each grid point, in which case a type of Hermite interpolation defines a high-order approximation of the interface. This again requires a nonlinear minimisation method to find closest points for query points adjacent to the interface -reinitialisation methods for gradient augmented level set methods include that of [4], which is based in part on Chopp's quasiNewton method, and the method of [7], which follows the principles of the fast marching method by using Huygens' principle and Newton's method restricted to individual tetrahedrons.…”

Simulation of vesicle using level set method solved by high order finite element

“…In [10], the authors used an augmented level set (as described in [31]) in which they solve by finite difference schemes an advection equation for φ and an other one for ∇φ. Moreover, three equations are solved at reinitialization step to take into account the higher derivatives of φ.…”