A Simple Level Set Method for Solving Stefan Problems

Chen, S.; Merriman, Barry; Osher, Stanley; Smereka, Peter

doi:10.1006/jcph.1997.5721

Cited by 456 publications

(406 citation statements)

References 22 publications

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“…In [29], Chen et al had earlier proposed a similar methodology, except that the treatment of the boundary conditions was different. Kim et al applied that framework to the simulation of the solidification processes in [70].…”

Section: A Level-set Approach To the Stefan Problemmentioning

confidence: 99%

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

2012

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We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain's boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain's boundary and (iii) a quadtree/octree nodebased adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.

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Section: A Level-set Approach To the Stefan Problemmentioning

confidence: 99%

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

2012

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“…In addition, a Gibbs-Thomson condition relating the temperature of the interface to its local curvature and velocity may also be applied. The level set method has been applied to the dendrite growth problem by a number of authors [32,33] and the technique is potentially very exiting as it removes the problems associated with the diffuse interface used in phase-field methods. However, it is fair to say that the application of the technique is far less well developed than phase-field, with models generally being restricted to single phase, thermally controlled solidification.…”

Section: Discusionmentioning

confidence: 99%

Quantification of mesh induced anisotropy effects in the phase-field method

Mullis

2006

Computational Materials Science

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Phase-field modelling is one of the most powerful techniques currently available for the simulation from first principles the time dependant evolution of complex solidification microstructures. However, unless care is taken the computational mesh used to solve the set of partial differential equations that result from the phase-field formulation of the solidification problem may introduce a stray, or implicit, anisotropy, which would be highly undesirable in quantitative calculations. In this paper we quantify this effect as a function of various computational parameters and subsequently suggest techniques for mitigating the effect of this stray anisotropy. 81.30.Fb, PACS:

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“…T (e) = {T } ,Ṫ (e) = ∂T ∂t (29) where K (e) T is the thermal conductivity matrix, A (e) T -heat convection matrix, M (e)…”

Section: Numerical Modelmentioning

confidence: 99%

“…In this method, interfacial geometric quantities such as curvature and outward normal can be estimated with the use of level set field ϕ. The method was first applied to Stefan problems in [29].…”

Section: Introductionmentioning

confidence: 99%

Effect Of Natural Convection On Directional Solidification Of Pure Metal

Skrzypczak

Węgrzyn-Skrzypczak

Winczek

2015

Archives of Metallurgy and Materials

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The paper is focused on the modeling of the directional solidification process of pure metal. During the process the solidification front is sharp in the shape of the surface separating liquid from solid in three dimensional space or a curve in 2D. The position and shape of the solid-liquid interface change according to time. The local velocity of the interface depends on the values of heat fluxes on the solid and liquid sides. Sharp interface solidification belongs to the phase transition problems which occur due to temperature changes, pressure, etc. Transition from one state to another is discontinuous from the mathematical point of view. Such process can be identified during water freezing, evaporation, melting and solidification of metals and alloys, etc.The influence of natural convection on the temperature distribution and the solid-liquid interface motion during solidification of pure copper is studied. The mathematical model of the process is based on the differential equations of heat transfer with convection, Navier-Stokes equation and the motion of the interface. This system of equations is supplemented by the appropriate initial and boundary conditions. In addition the continuity conditions at the solidification interface must be properly formulated. The solution involves the determination of the temporary temperature and velocity fields and the position of the interface. Typically, it is impossible to obtain the exact solution of such problem. The numerical model of solidification of pure copper in a closed cavity is presented, the influence of the natural convection on the phase change is investigated. Mathematical formulation of the problem is based on the Stefan problem with moving internal boundaries. The equations are spatially discretized with the use of fixed grid by means of the Finite Element Method (FEM). Front advancing technique uses the Level Set Method (LSM). Chorin's projection method is used to solve Navier-Stokes equation. Such approach makes possible to uncouple velocities and pressure. The Petrov-Galerkin formulation is employed to stabilize numerical solutions of the equations. The results of numerical simulations in the 2D region are discussed and compared to the results obtained from the simulation where movement of the liquid phase was neglected.Keywords: Pure Metal, Solidification, Sharp Interface, Natural Convection, Finite Element Method, Level Set Method Praca porusza problematykę modelowania kierunkowego krzepnięcia czystego metalu. Podczas tego procesu obserwuje się formowanie ostrego frontu krzepnięcia w postaci powierzchni separującej ciecz i ciało stałe w przypadku trójwymiarowym lub krzywej w przypadku płaskim. Położenie oraz kształt interfejsu krzepnięcia zmieniają się w czasie a wartości prędkości lokalnych zależą od różnicy intensywności strumieni ciepła po stronie ciała stałego i cieczy. Krzepnięcie z ostrym frontem należy do grupy procesów z przemianami fazowymi, które warunkowane są zmianami temperatury, ciśnienia, itp. Przejście fazowe z jednego stanu w drugi...

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A Simple Level Set Method for Solving Stefan Problems

Abstract: tion. In particular, this model can be applied to the widely studied problem of dendritic crystal growth.

Cited by 456 publications

References 22 publications

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

Quantification of mesh induced anisotropy effects in the phase-field method

Effect Of Natural Convection On Directional Solidification Of Pure Metal

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