A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems

2015 ◽  
Vol 18 (1) ◽  
pp. 203-229 ◽  
Author(s):  
Guoqiao You ◽  
Shingyu Leung
Keyword(s):  
Dynamical Systems ◽  
Level Set Method ◽  
Level Set ◽  
Limit Cycles ◽  
Time Discretization ◽  
Numerical Approach ◽  
Initial Guess ◽  
Implicit Time ◽  
Time Step ◽  
Image Regularization

AbstractWe propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.

GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

2004 ◽  
Vol 01 (04) ◽  
pp. 747-768
Author(s):  
CHRISTIAN ROHDE ◽  
MAI DUC THANH
Keyword(s):  
Phase Transition ◽  
Surface Tension ◽  
Initial Data ◽  
Time Discretization ◽  
Approximate Solutions ◽  
Implicit Time ◽  
Time Step ◽  
Dynamical Phase Transition ◽  
Transition Dynamics ◽  
Variational Scheme

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

scholarly journals Physical parametrisation of fire-spotting for operational fire spread models: response analysis with a model based on the Level Set Method

2018 ◽  
Author(s):  
Inderpreet Kaur ◽  
Anton Butenko ◽  
Gianni Pagnini
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Discrete Event ◽  
Fire Spread ◽  
Response Analysis ◽  
System Specification ◽  
Time Step ◽  
Processing Scheme ◽  
Model Based ◽  
The Many

Abstract. Fire-spotting is often responsible for a dangerous flare up in the wildfire and causes secondary ignitions isolated from the primary fire zone leading to perilous situations. In this paper a complete physical parametrisation of fire-spotting is presented within a formulation aimed to include random processes into operational fire spread models. This formulation can be implemented into existing operational models as a post-processing scheme at each time step, without calling for any major changes in the original framework. In particular, the efficacy of this formulation has already been shown for wildfire simulators based on an Eulerian moving interface method, namely the Level Set Method (LSM) that forms the baseline of the operational software WRF-SFIRE, and for wildfire simulators based on a Lagrangian front tracking technique, namely the Discrete Event System Specification (DEVS) that forms the baseline of the operational software FOREFIRE. The simple and computationally less expensive parametrisation includes the important parameters necessary for describing the landing behavior of the firebrands. The results from different simulations with a simple model based on the LSM highlight the response of the parametrisation to varying fire intensities, wind conditions and different firebrand radii. The contribution of the firebrands towards increasing the fire perimeter varies according to different concurrent conditions and the simulation results prove to be in agreement with the physical processes. Among the many rigorous approaches available in literature to model the firebrand transport and distribution, the approach presented here proves to be simple yet versatile for application to operational fire spread models.

scholarly journals Coupled atmosphere-wildland fire modeling with WRF-Fire version 3.3

2011 ◽  
Vol 4 (1) ◽  
pp. 497-545 ◽  
Author(s):  
J. Mandel ◽  
J. D. Beezley ◽  
A. K. Kochanski
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Wildland Fire ◽  
Surface Wind ◽  
Numerical Algorithms ◽  
Fire Spread ◽  
Weather Research And Forecasting ◽  
Time Step ◽  
Spread Model ◽  
Fire Spread Model

Abstract. We describe the physical model, numerical algorithms, and software structure of WRF-Fire. WRF-Fire consists of a fire-spread model, implemented by the level-set method, coupled with the Weather Research and Forecasting model. In every time step, the fire model inputs the surface wind, which drives the fire, and outputs the heat flux from the fire into the atmosphere, which in turn influences the atmosphere. The level-set method allows submesh representation of the burning region and flexible implementation of various kinds of ignition. WRF-Fire is distributed as a part of WRF and it uses the WRF parallel infrastructure for parallel computing.

scholarly journals Modelling dynamic ice-sheet boundaries and grounding line migration using the level set method

2020 ◽  
Vol 66 (259) ◽  
pp. 766-776
Author(s):  
M. Alamgir Hossain ◽  
Sam Pimentel ◽  
John M. Stockie
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Numerical Approach ◽  
Ice Surface ◽  
Grounding Line ◽  
High Degree ◽  
Ice Water ◽  
Benchmark Solutions

AbstractComputing predictions of future sea level that include well-defined uncertainty bounds requires models that are capable of robustly simulating the evolution of ice sheets and glaciers. Ice flow behaviour is known to be sensitive to the location and geometry of dynamic ice boundaries such as the grounding line (GRL), terminus position and ice surface elevation, so that any such model should track these interfaces with a high degree of accuracy. To address this challenge, we implement a numerical approach that uses the level-set method (LSM) that accurately models the evolution of the ice–air and ice–water interface as well as capturing topological changes in ice-sheet geometry. This approach is evaluated by comparing simulations of grounded and marine-terminating ice sheets to various analytical and numerical benchmark solutions. A particular advantage of the LSM is its ability to explicitly track the moving margin and GRL while using a fixed grid finite-difference scheme. Our results demonstrate that the LSM is an accurate and robust approach for tracking the ice surface interface and terminus for advancing and retreating ice sheets, including the transient marine ice-sheet interface and GRL positions.

Three Dimensional Numerical Modelling of Pier Scour Under Current and Waves Using Level Set Method

2014 ◽  
Author(s):  
Mohammad Saud Afzal ◽  
Hans Bihs ◽  
Arun Kamath ◽  
Øivind A. Arntsen
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Free Surface ◽  
Level Set Method ◽  
Level Set ◽  
Three Dimensional ◽  
Critical Shear Stress ◽  
Offshore Structures ◽  
Time Step ◽  
Pier Scour

Stability of offshore structures can be threatened by local scouring which could ultimately lead to their failure. As a consequence, knowledge of the scouring mechanism and the accurate prediction of the characteristic scour geometry is very important for the design of such structures. A three-dimensional computational fluid dynamics model is used to calculate the scour and the deposition pattern around a pier for two different boundary conditions: constant discharge and regular waves. The computational fluid dynamics (CFD) model solves Reynolds-Averaged Navier-Stokes (RANS) equations in all three dimensions. The location of the free surface is represented using the level set method, which calculates the complex motion of the free surface in a very realistic manner. For the implementation of waves, the CFD code is used as a numerical wave tank. For the geometric representation of the moveable sediment bed, the level set method is used. The numerical results for the local scour prediction are compared with physical experiments. The performance of the turbulence models, the formulations of the critical shear stress for the sloping bed and the effect of the variation of the sediment time stepping are investigated. The decoupling of the hydrodynamic and the morphodynamic time step is tested and found to be a reasonable assumption. For the two situations of local pier scour under current and wave conditions, the numerical model predicts the general evolution (geometry, location and maximum scour depth) and time development of the scour hole accurately.

A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

2014 ◽  
Vol 15 (4) ◽  
pp. 1091-1107 ◽  
Author(s):  
Yinhua Xia ◽  
Yan Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Time Discretization ◽  
Local Discontinuous Galerkin Method ◽  
Implicit Time ◽  
Discrete Version ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Spatial Derivatives ◽  
Number Of Particles

AbstractIn this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

A Novel Approach for Color Tongue Image Fast Segmentation Based on Level Sets

2011 ◽  
Vol 341-342 ◽  
pp. 714-719 ◽  
Author(s):  
Ming Feng Zhu ◽  
Jian Qiang Du
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Level Sets ◽  
Step Method ◽  
Time Step ◽  
Novel Approach ◽  
Feature Weight ◽  
Variational Level Set ◽  
Variable Time Step ◽  
Variational Level Set Method

To make variational level set method applicable to the application of tongue segmentation, this article introduced a kind of color tongue image fast segmentation method based on level sets. The early variational level set method is improved from the following aspects: the boundary feature weight function is improved to make this method adaptable to color tongue image segmentation, and a kind of variable time step method is introduced to increase the efficiency and accuracy of segmentation. As the comparison experiment results show, our method is about 1 time faster than the early method, and its accuracy is higher than the early one.

scholarly journals An Interface-Fitted Finite Element Level Set Method with Application to Solidification and Solvation

2011 ◽  
Vol 10 (1) ◽  
pp. 32-56 ◽  
Author(s):  
Bo Li ◽  
John Shopple
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Level Set Method ◽  
Level Set ◽  
Finite Element Mesh ◽  
Local Geometry ◽  
Normal Velocity ◽  
Time Step ◽  
Element Level ◽  
Element Mesh

AbstractA new finite element level set method is developed to simulate the interface motion. The normal velocity of the moving interface can depend on both the local geometry, such as the curvature, and the external force such as that due to the flux from both sides of the interface of a material whose concentration is governed by a diffusion equation. The key idea of the method is to use an interface-fitted finite element mesh. Such an approximation of the interface allows an accurate calculation of the solution to the diffusion equation. The interface-fitted mesh is constructed from a base mesh, a uniform finite element mesh, at each time step to explicitly locate the interface and separate regions defined by the interface. Several new level set techniques are developed in the framework of finite element methods. These include a simple finite element method for approximating the curvature, a new method for the extension of normal velocity, and a finite element least-squares method for the reinitialization of level set functions. Application of the method to the classical solidification problem captures the dendrites. The method is also applied to the molecular solvation to determine optimal solute-solvent interfaces of solvation systems.

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