Computation of Three Dimensional Dendrites with Finite Elements

Schmidt, Alfred

doi:10.1006/jcph.1996.0095

Cited by 144 publications

(111 citation statements)

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“…Both multigrid methods and adaptive mesh refinement were used, and in [14] this nonsymmetric discretization was coupled to a volume of fluid front tracking method in order to solve the Stefan problem. In [27] the authors used adaptive finite element methods for both the heat equation and for the interface evolution producing stunning three dimensional results. Other remarkable three dimensional results can be found in the finite difference diffusion Monte Carlo method of [26].…”

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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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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“…The integration can then be performed in a standard way by calculating integrals of piecewise linear functions. See [34] for further details. …”

Section: Numerical Integrationmentioning

confidence: 99%

Finite element method for epitaxial growth with attachment–detachment kinetics

Bänsch

Haußer

Lakkis

et al. 2004

Journal of Computational Physics

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An adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth. Such problems consist of an adatom (adsorbed atom) diffusion equation on terraces of different height; boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment; and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional "surface" diffusion. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. The diffusion equation is discretized by the first-order implicit scheme in time and the linear finite element method in space. A technique of extension is used to avoid the complexity in the spatial discretization near boundaries. All the elements are marked, and the marking is updated in each time step, to trace the terrace height. The evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements. Simple adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem. Numerical tests on pure geometrical motion, mass balance, and the stability of a growing circular island demonstrate that the method is stable, efficient, and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period.

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“…Here, 12 , 13 , and 23 are positive parameters, rð/ i ; / j Þ ¼ / i r/ j À / j r/ i ði; jÞ 2 fð1; 2Þ; ð1; 3Þ; ð2; 3Þg; a is the real valued function defined as in [13] by…”

Section: The Modelmentioning

confidence: 99%

“…A high spatial resolution is, therefore, needed to describe the smooth transition, and consequently explicit methods formulated on regular grids require very large numbers of grid points and time steps. In order to reduce the computational time and the memory requirements, adaptive finite elements have been developed [22,23], using isotropic finite elements. Recently, a phase-field model based on adaptive finite elements with high aspect ratio and an implicit formulation of diffusion terms was proposed in order to further reduce the number of vertices and time steps required for a computation [24].…”

Section: Introductionmentioning

confidence: 99%