A PDE-Based Fast Local Level Set Method

Abstract

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.

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

“…Solving (7) can introduce numerical error into the level set function that perturbs it away from being a distance function, even for special choices of Ṽ that are constant in the normal direction from the interface [2,48] and thereby preserve distance functions. This is compensated for by reinitializing the level set function at regular intervals by solving (8) to steady state [45,51]. Here, τ is pseudo-time, and ϕ 0 is the original level set function prior to the reinitialization.…”

“…To find the best compromise between accuracy and computational efficiency, we seek to update ϕ only as much as is necessary to accurately advect the interface. This can be done using the narrow band/local level set technique [40,45]. Given an initialized level set function ϕ, only the points that fall within a fixed distance of the interface are updated during level set operations (e.g., velocity extensions and level set reinitialization).…”

“…, and (x i−1 , y j−2 ) are all in the same region, then we can improve our approximation of the off-grid derivative in (38) with (45) Putting all this together, the higher-order discretization of [D∇p · n] = h is given by (46) In principle, all these partial differences can be made more accurate still by using additional node points. In particular, one could use quadratic or cubic interpolations in the off-grid directional derivative differences.…”

“…Using a 3.3 GHz Pentium 4 workstation and a C++ implementation, the 501 × 501 simulation required under 24 hours to compute. Because our (mitosis) time scale ranges from approximately 18 to 36 hours for this problem, this corresponds to 45 The tumor continues to grow at a slower rate until the interior of the tumor becomes necrotic. (See t = 10.0.)…”

Level Set Method for Simulating the Interface Kinematics: Application of a Discontinuous Galerkin Method

“…But if the slope is much higher than 1, the smoothed Signum function becomes too steep that may result in the numerical instability. In order to overcome this problem, it is proposed in [27] that an infinitely smoothed formulation in terms of the updated level set function ϕ instead of the initial level set function ϕ 0 is used as,…”