Reinitialization Scheme Research Articles

Second-Order Accurate Computation of Curvatures in a Level Set Framework Using Novel High-Order Reinitialization Schemes

We present a high-order accurate scheme for the reinitialization equation of Sussman et al.(J. Comput. Phys. 114:146---159, [1994]) that guarantees accurate computation of the interface's curvatures in the context of level set methods. This scheme is an extension of the work of Russo and Smereka (J. Comput. Phys. 163:51---67, [2000]). We present numerical results in two and three spatial dimensions to demonstrate fourth-order accuracy for the reinitialized level set function, third-order accuracy for the normals and second-order accuracy for the interface's mean curvature in the L 1- and L ?-norms. We also exploit the work of Min and Gibou (UCLA CAM Report (06-22), [2006]) to show second-order accurate scheme for the computation of the mean curvature on non-graded adaptive grids.

A second order accurate level set method on non-graded adaptive cartesian grids

We present a level set method on non-graded adaptive Cartesian grids, i.e. grids for which the ratio between adjacent cells is not constrained. We use quadtree and octree data structures to represent the grid and a simple algorithm to generate a mesh with the finest resolution at the interface. In particular, we present (1) a locally third order accurate reinitialization scheme that transforms an arbitrary level set function into a signed distance function, (2) a second order accurate semi-Lagrangian methods to evolve the linear level set advection equation under an externally generated velocity field, (3) a second order accurate upwind method to evolve the non-linear level set equation under a normal velocity as well as to extrapolate scalar quantities across an interface in the normal direction, and (4) a semi-implicit scheme to evolve the interface under mean curvature. Combined, we obtain a level set method on adaptive Cartesian grids with a negligible amount of mass loss. We propose numerical examples in two and three spatial dimensions to demonstrate the accuracy of the method.

1449 格子ボルツマン法による多孔体内流動解析のための境界面再構築法の開発(S38-4 混合・界面,S38 マイクロ・ナノフルイズ)

In this paper, a novel boundary reconstruction scheme for porous media flows is developed utilizing the reinitialization scheme of the level set method. The proposed scheme transfers usual digitally defined step-like boundary data to continuously smooth surface data. It is confirmed that the lattice Boltzmann simulation by the interpolated bounce-back boundary condition with the present scheme well reproduces the drag coefficients of a cylinder and a sphere. The results indicate that the present scheme improves predicition accuracy and it is more insensitive to grid resolutions than the standard scheme.

LES of premixed turbulent combustion using the level set approach

AbstractLES of premixed turbulent combustion is presented based on the level set approach. Level set method is a technique which is especially designed to deal with problems involving topological changes of interfaces, merging interface, singularities and formation of sharp cusps [8]. Our previous work [3] which was performed in RANS‐context contains the level set approach with reinitialization scheme of Sussman et al. [1]. For LES we have developed a completely new algorithm based on the Fast Marching method. It does not need the iterative reinitialization schemes, and, therefore, it quarantees more accurate solution and drastic speed‐up of the computation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A study of numerical methods for the level set approach

The computation of moving curves by the level set method typically requires reinitializations of the underlying level set function. Two types of reinitialization methods are studied: a high order “PDE” approach and a second order Fast Marching method. Issues related to the efficiency and implementation of both types of methods are discussed, with emphasis on the tube/narrow band implementation and accuracy considerations. The methods are also tested and compared. Fast Marching reinitialization schemes are faster but limited to second order, PDE based reinitialization schemes can easily be made more accurate but are slower, even with a tube/narrow band implementation.

Sharp interface tracking using the phase-field equation

A general interface tracking method based on the phase-field equation is presented. The zero phase-field contour is used to implicitly track the sharp interface on a fixed grid. The phase-field propagation equation is derived from an interface advection equation by expressing the interface normal and curvature in terms of a hyperbolic tangent phase-field profile across the interface. In addition to normal interface motion driven by a given interface speed or by interface curvature, interface advection by an arbitrary external velocity field is also considered. In the absence of curvature-driven interface motion, a previously developed counter term is used in the phase-field equation to cancel out such motion. Various modifications of the phase-field equation, including nonlinear preconditioning, are also investigated. The accuracy of the present method is demonstrated in several numerical examples for a variety of interface motions and shapes that include singularities, such as sharp corners and topology changes. Good convergence with respect to the grid spacing is obtained. Mass conservation is achieved without the use of separate re-initialization schemes or Lagrangian marker particles. Similarities with and differences to other interface tracking approaches are emphasized.

Wave front evolution in strongly heterogeneous layered media using the fast marching method

SUMMARY The fast marching method (FMM) is a grid based numerical scheme for tracking the evolution of monotonically advancing interfaces via finite-difference solution of the eikonal equation. Like many other grid based techniques, FMM is only capable of finding the first-arriving phase in continuous media; however, it distinguishes itself by combining both unconditional stability and rapid computation, making it a truly practical scheme for velocity fields of arbitrary complexity. The aim of this paper is to investigate the potential of FMM for finding later arriving phases in layered media. In particular, we focus on reflections from smooth subhorizontal interfaces that separate regions of continuous velocity variation. The method we adopt for calculating reflected phases involves two stages: the first stage initializes FMM at the source and tracks the incident wave front to all points on the reflector surface; the second stage tracks the reflected wave front by reinitializing FMM from the interface point with minimum traveltime. Layer velocities are described by a regular grid of velocity nodes and layer boundaries are described by a set of interface nodes that may be irregularly distributed. A triangulation routine is used to locally suture interface nodes to neighbouring velocity nodes in order to facilitate the tracking of wave fronts to and from the reflector. A number of synthetic tests are carried out to assess the accuracy, speed and robustness of the new scheme. These include comparisons with analytic solutions and with solutions obtained from a shooting method of ray tracing. The convergence of traveltimes as grid spacing is reduced is also examined. Results from these tests indicate that wave fronts can be accurately tracked with minimal computational effort, even in the presence of complex velocity fields and layer boundaries with high curvature. Incident wave fronts containing gradient discontinuities or shocks also pose no difficulty. Further development of the wave front reinitialization scheme should allow other later arrivals such as multiples to be successfully located.

Adaptation and learning of a fuzzy system by nearest neighbor clustering

Conventional hill-climbing type adaptive algorithms suffer from large transient error in abrupt changing environments. In this paper, we suggest a novel idea to address this problem. A data re-initialization (DR) scheme is proposed using multiple models approach with special initialization technique to smooth transition between different environments. The design is based on adaptive fuzzy nearest neighbor clustering algorithm with on-line enhancements. The fast adaptation is realized by the data re-initialization and switching between different models; learning is realized by recording and retrieving the trained up models. The algorithm is conceptually simple and feasible for real-time applications. The performance of the algorithm is tested with simulation studies.

An efficient direct BEM numerical scheme for phase change problems using Fourier series

Due to the moving boundary nature of the solidification and melting problems, the application of the standard time domain boundary element formulation of the problem presents the difficulty of the numerical evaluation of the history dependence, representing more and more work as time proceeds. In that regard, we present an algorithm based on a Green's function which allows a Fourier representation of the domain integral in the reinitialization scheme of the direct boundary element method formulation of the problem, from which it is possible to carry out the history integration in a recursive manner. To obtain the reinitialization approach, in the integral representation formula, the Green's function corresponding to zero temperature in a box containing the original domain will be used, instead of using the classical free space fundamental solution. This Green's function is given in terms of the original fundamental solution plus a regular solution of the heat equation inside the domain in consideration. It can therefore be used in the integral representation formula of the heat equation (direct formulation) to obtain the solution of a heat problem in such domain. The mentioned Green's function can be obtained by the images method, and the resulting source series can also be rewritten in terms of a double Fourier series, that will be used in the domain integral of the integral representation formula to transform such integral into equivalent surface integrals. In this work a new reinitialization scheme has been applied to solve moving boundary problems, to detect the location of the phase-change interface in solidification and melting problems. The numerical results were compared with analytical solutions (in the case, where they were available) showing a good agreement between the results, giving special emphasis to the interface location and the temperature distribution.

Efficient reinitialization of the prewhitened constant modulus algorithm

We present a reinitialization scheme for blind equalizers adapted via the constant modulus algorithm (CMA) when an all-pole prefilter is included to whiten the received signal. The mechanism exploits the special structure of the minimum mean squared error (MMSE) equalizers and their relation with CMA equalizers. A heuristic rule for blind determination of the best equalization delay is also provided. Using these guidelines, the equalizer is capable of finding the optimal setting in an online and computationally efficient fashion. In particular, estimation and inversion of the channel output autocorrelation matrix is not needed, in contrast with previous approaches.

Adaptation and Learning with Fuzzy Clustering Algorithms: A Data Re-Initialization (DR) Scheme

Abstract This paper presents a novel idea to avoid large transient error of hill-climbing type adaptive algorithms in abrupt changing environments. A data re-initialization (DR) scheme is proposed using multiple models approach with special initialization technique for smooth transition between different environments. The design is based on adaptive fuzzy clustering algorithm with on-line modifications; furthermore, it is equipped with learning capabilities. The fast adaptation is realized by DR and switching between different models; learning is realized by recording and retrieving the trained up models. The algorithm is conceptually simple and simulation results are appealing. Finally, conclusion and remarks are given.

Channel surfing reinitialization for the constant modulus algorithm

The proper initialization of the constant modulus algorithm (CMA) is a critical problem that has not been solved. This paper presents a new reinitialization scheme for CMA. The proposed scheme searches in the channel space for the constant modulus equalizer with minimum mean-square error by exploiting the structure of the minimum mean square error (MMSE) equalizer and connections between the constant modulus and the MMSE equalizers.