A numerical method for coupled surface and grain boundary motion

Based on earlier work by the authors, in this paper we introduce novel fully discrete, fully practical parametric finite element approximations for geometric evolution equations of curves in the plane. The fully implicit approximations are unconditionally stable and intrinsically equidistribute the vertices at each time level. We present iterative solution methods for the systems of nonlinear equations arising at each time level and present several numerical results. The ideas easily generalize to the evolution of curve networks and to anisotropic surface energies.

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“…Finally, the authors in [25], see also [26], supplement the flow equations (1.1a,b) with the side constraint…”

Section: Introductionmentioning

confidence: 99%

“…On approximating (1.7) fully implicitly with the help of centred finite differences, the authors obtain an equidistribution property on the discrete level, although this is not explicitly stated in their work. In addition, their numerical method is extended to the case of evolving curve networks, see [25] for details.…”

Section: Introductionmentioning

confidence: 99%

The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute

Barrett

Garcke

Nürnberg

2010

Numerical Methods Partial

103

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“…Along with prescribing the curve always be scaled arclength (|z σ | = L), this defines a tangential velocity implicitly [32]. This is in contrast to formulations which prescribe a tangential velocity explicitly to preserve the scaled arclength [41].…”

Section: Scaled Arclength Formulationmentioning

confidence: 99%

A numerical framework for singular limits of a class of reaction diffusion problems

Moyles

Wetton

2015

Journal of Computational Physics

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We present a numerical framework for solving localized pattern structures of reaction-diffusion type far from the Turing regime. We exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems. The singular model involves the motion of a curve on which one of the chemical species is concentrated. The curve motion is non-local with an integral equation that has a logarithmic singularity. We generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures. One such model is the 2D Mullins-Sekerka flow which we implement as a test case of the method. We then analyze a specific model problem, the saturated Gierer-Meinhardt problem, where we demonstrate dynamic patterns for a variety of parameters and curve geometries.

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“…Pan and Cocks (1995); Tritscher (1999); Kanel et al (2003Kanel et al ( , 2004; Ch'ng and Pan (2004); Pan and Wetton (2008) and the references therein. In particular, the question arose whether the appearance of the surface groove will slow down the velocity of the grain boundary.…”

Section: Introductionmentioning

confidence: 99%

“…Mathematical approaches for the two dimensional case include finite element and finite difference approximations for parametric formulations, see e.g. Pan and Cocks (1995); Pan et al (1997); ; ; Kucherenko et al (2000); Ch'ng and Pan (2004,2005,2007); Kanel et al (2005); Pan and Wetton (2008) and Barrett, Garcke, and Nürnberg (2007a), numerical approximations of graph formulations, see e.g. Vilenkin et al (1997); Zhang and Wong (2002a,b), as well as finite difference and finite element methods for phase field models, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Finite-element approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering

2010

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We study the coupled surface and grain boundary motion in bi-and tricrystals in three space dimensions, building on previous work by the authors on the simplified two dimensional case. The motion of the interfaces, which in this paper are presented by two-dimensional hypersurfaces, is described by two types of normal velocities: motion by mean curvature and motion by surface diffusion. Three hypersurfaces meet at triple junction lines, where junction conditions need to hold. Similarly, boundary conditions are prescribed where an interface meets an external boundary, and these conditions naturally give rise to contact angles. We present a variational formulation of the flows, which leads to a fully practical finite element approximation that exhibits excellent mesh properties, with no mesh smoothing or remeshing required in practice. For the introduced parametric finite element approximation we show well-posedness and, in general, unconditional stability, i.e. there is no restriction on the chosen time step size. Moreover, the induced discrete equations are linear and easy to solve. A generalization to anisotropic surface energies is straightforward. Several numerical results in two and three space dimensions are presented, including simulations for thermal grooving and sintering. Three dimensional simulations featuring quadruple junction points, nonstandard boundary contact angles and fully anisotropic surface energies are also presented.

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A numerical method for coupled surface and grain boundary motion

Cited by 18 publications

References 29 publications

The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute

The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute

A numerical framework for singular limits of a class of reaction diffusion problems

Finite-element approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering

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