Redistancing by flow of time dependent eikonal equation

Abstract

Construction of signed distance to a given inteface is a topic of special interest to level set methods. There are currently, however, few algorithms that can efficiently produce highly accurate solutions. We introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations. We provide operation counts and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy. Comparison with the standard level set reinitialization algorithm shows ours is superior in terms of predictability and local construction, critical, for example, in local level set methods. We further apply the same ideas to extension of values off interfaces. Together, our proposed approaches can be used to advance the level set method for fast and accurate computations of the latest scientific problems.In a level set method [11], an interface is represented as the zero level set of a continuous realvalued function, called a level set function. Let φ denote this function. Then φ embeds the interface Γ as its zero level set: Γ = {x ∈ R m |φ(x) = 0}. This representation retains geometric information of the interface, which can be extracted using formulas involving derivatives and integrals of φ. Furthermore, by adding a time variable, the level set function can be used to capture a given dynamics of the interface using a time dependent PDE in φ. The location of the interface at time t in this case is the zero level set of φ at that time: Γ(t) = {x ∈ R m |φ(x, t) = 0}.The level set function φ is certainly not unique; for example αφ, α = 0, is also a level set function for the same interface. Thus one can ask for the best level set function according to some chosen criterion. In applications, even when a physically relevant level set function exists, that function may not satisfy that choice and so should not be used. Instead, the commonly desired criterion for the form of the level set function is one that leads to small errors when numerically solving the time dependent PDE for the dynamics of the interface, or when extracting the interface location. This form is most important to a level set method near the interface of interest.Analytic solutions to the time dependent PDE for φ do not exist in all but the most trivial cases. Thus, one needs to turn to numerical approximations and solutions. The errors of a large class of popular numerical methods that operate on a discretization of the function over a grid, called finite differencing schemes, are typically calculated from Taylor expansions. This implies the derivatives of φ have a bearing on the error, with large magnitudes correlated to large errors. In addition to smoothness in φ, one would thus desire that φ avoid derivatives with large magnitudes, most notably at or near the zero level set of interest.

A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces

“…In this setting, we may consider ∂U as an implicit interface, and extend the values of w outside of U following the approach which is called "velocity extension" in the level set method literature; see e.g. [OF01], or more specifically [CT08].…”

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

“…Another approach to obtaining a "time" dependent H-J equation from the static H-J equation is using the so called paraxial formulation in which a preferred spatial direction is assumed in the characteristic propagation [21,17,29,36,37]. High order numerical schemes are well developed for the time dependent H-J equation on structured and unstructured meshes [34,25,51,24,33,7,26,31,35,1,3,4,6,8]; see a recent review on high order numerical methods for time dependent H-J equations by Shu [46]. Due to the finite speed of propagation and the CFL condition for the discrete time step size, the number of time steps has to be of the same order as that for one of the spatial dimensions so that the solution converges in the entire domain.…”

Implicit boundary integral methods for the Helmholtz equation in exterior domains

“…The computation of the signed distance functions and the closest point mappings are by now considered standard routines in the level set methods [17] and can be done to high-order accuracy in many different ways, e.g., [1,5,21,24], where by extending the interface coordinates as constants along interface normals, P Γ can be computed easily to fourth-order in the grid spacing. Once an accurate distance function is computed on the grid, its gradient can be computed by standard finite differencing or by more accurate but wider WENO stencils [7].…”