Crystal growth and dendritic solidification

Sethian, James A.; Strain, John

doi:10.1016/0021-9991(92)90140-t

Cited by 253 publications

(146 citation statements)

References 31 publications

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“…Kim et al applied that framework to the simulation of the solidification processes in [70]. We note that the first level-set approach to solve the Stefan problem was given in Sethian and Strain [126]. In that work, the diffusion equation was solved using a boundary integral approach.…”

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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“…The earliest level set method for solidification type problems was presented in [31] where the authors recast the equations of motion into a boundary integral equation and used the level set method to update the location of the interface. In [2], the boundary integral equations were avoided by using a finite difference method to solve the diffusion equation on a Cartesian grid with Dirichlet boundary conditions imposed on the interface.…”

Section: Introductionmentioning

confidence: 99%

A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem

Gibou

Fedkiw

2005

Journal of Computational Physics

231

236

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In this paper, we first describe a fourth order accurate finite difference discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent time step restrictions associated with explicit schemes. We then turn our focus to the Stefan problem and construct a third order accurate method that also includes an implicit time discretization. Multidimensional computational results are presented to demonstrate the order accuracy of these numerical methods.

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“…Level set methods have been applied to a wide array of domains in the last decades, from computational fluid dynamics (e.g., [31,37]), groundwater flow [12], crystal growth [30] to image processing [20]. This study uses the level set method to investigate the issue of deformation of the geometry of a sedimentary basin due to mechanical compaction.…”

Section: Introductionmentioning

confidence: 99%

Novel basin modelling concept for simulating deformation from mechanical compaction using level sets

et al. 2017

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As sedimentation progresses in the formation and evolution of a depositional geologic basin, the rock strata are subject to various stresses. With increasing lithostatic pressure, compressional forces act to compact the porous rock matrix, leading to overpressure buildup, changes in the fluid pore pressure and fluid flow. In the context of petroleum systems modelling, the present study concerns the geometry changes that a compacting basin experiences subject to deposition. The purpose is to track the positions of the rock layer interfaces as compaction occurs. To handle the challenge of potentially large geometry deformations, a new modelling concept is proposed that couples the pore pressure equation with a level set method to determine the movement of lithostratigraphic interfaces. The level set method propagates an interface according to a prescribed speed. The coupling term for the pore pressure and level-set equations consists of this speed function, which is dependent on the compaction law. The two primary features of this approach are the simplicity of the grid and the flexibility of the speed function. A first evaluation of the model concept is presented based on an implementation for one spatial dimension accounting for vertical effective stress. Isothermal conditions with a constant fluid density and viscosity were assumed. The accuracy of the implemented numerical solution for the case of a single stratigraphic unit with a linear compaction law was compared to the available analytical solution [38]. The multi-layer setup and the nonlinear case were tested for plausibility.

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Crystal growth and dendritic solidification

Cited by 253 publications

References 31 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

A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem

Novel basin modelling concept for simulating deformation from mechanical compaction using level sets

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