A PDE-Based Fast Local Level Set Method

1991

We develop a fast method to localize the level set method of Osher and Sethian (1988, J. Comput. Phys. 79, 12) and address two important issues that are intrinsic to the level set method: (a) how to extend a quantity that is given only on the interface to a neighborhood of the interface; (b) how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface. This fast local level set method reduces the computational effort by one order of magnitude, works in as much generality as the original one, and is conceptually simple and easy to implement. Our approach differs from previous related works in that we extract all the information needed from the level set function (or functions in multiphase flow) and do not need to find explicitly the location of the interface in the space domain. The complexity of our method to do tasks such as extension and distance reinitialization is O(N ), where N is the number of points in space, not O(N log N ) as in works by , Proc. Nat. Acad. Sci. 93, 1591 and Helmsen and co-workers (1996, SPIE Microlithography IX, p. 253). This complexity estimation is also valid for quite general geometrically based front motion for our localized method. 411 interface problem has been transformed into a two dimensional problem. In three space dimensions, considerable computational labor (O(n 3 )) is required per time step."

Including derivative information in the modelling of moving interfaces has been proposed as one method to increase the accuracy of numerical schemes with minimal additional cost. Here a new level set reinitialization technique using the fast marching method is presented. This augmented fast marching method will calculate the signed distance function and up to the secondorder derivatives of the signed distance function for arbitrary interfaces. In addition to enforcing the condition ∇φ 2 = 1, where φ is the level set function, the method ensures that ∇ ( ∇φ ) 2 = 0 and ∇∇ ( ∇φ ) 2 = 0 are also satisfied. Results indicate that for both two-and three-dimensional interfaces the resulting level set and curvature field are smooth even for coarse grids. Convergence results show that using first-order upwind derivatives and the augmented fast marching method result in a second-order accurate level set and gradient field and a first-order accurate curvature field.

2018

In this thesis, different level set techniques were developed to propagate the signed distance function values, from the cells adjacent to the body, in the rest of the computational domain. Furthermore, also an high-order transport scheme was implemented in order to modify the level set field through time. Regarding propagation, two method were designed, both based on the solution of the heat equation to obtain the distance field from an object. The former is an iterative fixedpoint approach where the final recovery of the distance field was achieved by solving an elliptic problem. In this case, the main dilemma was to find suitable boundary conditions for the final elliptic equation since the distance calculation, being an hyperbolic problem, should not be influenced by them. Two different type of boundary conditions were considered. The latter procedure starts from the solution of an initial heat problem followed by some Newton’s method iterations to refine the distance field acco...

2007

In this talk we present a brief introduction to level set methods. The basic mathematical formulation will be recalled, as well as the most common numerical methods used for the discretization of the equation. Some application to the computation of signed distance function, motion by mean curvature, and crystal gwowth will be illustrated.

2012

A hybrid scheme for reconstructing signed distance function in the frame work of augmented level set methods is proposed. This method is based on gradient augmented formulation of the re-initializing PDE for the distance function. In our hybrid approach, equations are solved by first dividing the domain into an interfacial and non-interfacial region. For nodes belonging to the interfacial regions, level set and its gradient values are updated by explicitly locating the interface, where as for the remaining nodes, the re-formulated equations are solved using a semi-Lagrangian approach. Two approaches are presented to locate the interface, one uses the underlying Hermite polynomial evaluated along the characteristic curve, and the other uses the variant of Newton method proposed by Chopp [1]. Results show 4 order and 3 order convergence rate in L1 norm for the level set function and its gradient respectively. The same spatial convergence rate is observed when equations are discretized...

Journal of Computational Physics, 2015

In this paper, a new re-initialization method for the conservative level-set function is put forward. First, it has been shown that the re-initialization and advection equations of the conservative level-set function are mathematically equivalent to the re-initialization and advection equations of the localized signed distance function. Next, a new discretization for the spatial derivatives of the conservative level-set function has been proposed. This new discretization is consistent with the re-initialization procedure and it guarantees a second-order convergence rate of the interface curvature on gradually refined grids. The new re-initialization method does not introduce artificial deformations to stationary and non-stationary interfaces, even when the number of re-initialization steps is large.

2013

The level set method was devised by Osher and Sethian [2] in as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field v [1]. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth function ϕ(x, t), i.e., Γ (t) = {x|ϕ(x, t) = 0}. ϕ is positive inside Ω, negative outside Ω and is zero on Γ (t) [1]. This paper presents a reaction-diffusion method used to describe a physicochemical phenomenon that comprises two elements, namely chemical reactions and diffusion for implicit active contours[21][37][39][40], which is completely free of the costly re-initialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in a diffusion-augmented level set method with efficien...

This paper proposes and implements a novel hybrid level set method which combines the numerical efficiency of the local level set approach with the temporal stability afforded by a semi-implicit technique. By introducing an extraction/insertion algorithm into the local level set approach, we can accurately capture complicated behaviors such as interface separation and coalescence. This technique solves a well known problem when treating a semi-implicit system with spectral methods, where spurious interface movements emerge when two interfaces are close to each other. Numerical experiments show that the proposed method is stable, efficient and scales up well into three dimensional problems.

2019

We present a novel interface-capturing scheme, THINC-scaling, to unify the VOF (volume of fluid) and the level set methods, which have been developed as two completely different approaches widely used in various applications. The THINC-scaling scheme preserves at the samectime the advantages of both VOF and level set methods, i.e. the mass/volume conservation of the VOF method and the geometrical faithfulness of the level set method. THINCscaling scheme allows to represent interface with high-order polynomials, and has algorithmic simplicity which eases its implementation in unstructured grids.

Proceedings of the National Academy of Sciences, 2001