A phase field model for the electromigration of intergranular voids

Barrett, John W.; Garcke, Harald; Nürnberg, Robert

doi:10.4171/ifb/161

Cited by 32 publications

(25 citation statements)

References 36 publications

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“…We also want to compare the four computed equal area steady state solutions with the exact energy minimizer. To this end, we determine the two surface energies ς 13 and ς 12 numerically, similarly to the technique used in [7]. That is, for the two pairs (i, j) = (1, 2) and (1, 3) we split the domain Ω into two pure phases e i , e j , with a vertically aligned straight interface between them.…”

Section: Double Bubblesmentioning

confidence: 99%

Numerical simulations of immiscible fluid clusters

Nürnberg

2009

Applied Numerical Mathematics

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We present a phase field model for multi-component surface diffusion, which can be used to describe the evolution of clusters of immiscible fluids such as soap bubble clusters. The model is given by a Cahn-Hilliard system with a non-smooth obstacle free energy and a degenerate mobility matrix. On stating the considered finite element approximation, we describe the iterative solver used to solve the nonlinear discrete system at each time step and present several numerical experiments for N = 3, 4, 5 and 6 components in two and three space dimensions, including simulations with topological changes.

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Section: Double Bubblesmentioning

confidence: 99%

Numerical simulations of immiscible fluid clusters

Nürnberg

2009

Applied Numerical Mathematics

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“…The next experiment is motivated by the considerations on a traveling wave solution to (1.23) that was first mentioned in [36] and which plays a major role in the study of grain boundary motion; see also [33] and [7,Figure 7]. The computations shown in Figure 28 start with three curves meeting at a single triple junction, of which the two horizontal ones experience motion by surface diffusion, while the third curve undergoes motion by mean curvature.…”

Section: 32mentioning

confidence: 99%

“…A level set approach for mean curvature flow of curve networks has been considered in [34,44,41]. A phase field model for the combined motion of mean curvature flow and surface diffusion, as well as SALK, was considered in [7]. A phase field model for a mean curvature flow system is considered in [27], and a surface diffusion flow system in [4].…”

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confidence: 99%

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On the Variational Approximation of Combined Second and Fourth Order Geometric Evolution Equations

Barrett¹,

Garcke²,

̈rnberg³

2007

SIAM J. Sci. Comput.

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121

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Abstract. We present a variational formulation of combined motion by minus the Laplacian of curvature and mean curvature flow, as well as related flows. The proposed scheme covers both the closed curve case and the case of curves that are connected via triple or quadruple junction points or intersect the external boundary. On introducing a parametric finite element approximation, we prove stability bounds and compare our scheme with existing approaches. The presented scheme has very good properties with respect to the equidistribution of mesh points and, if applicable, area conservation.Key words. surface diffusion, (inverse) mean curvature flow, surface attachment limited kinetics, nonlinear curve evolution, triple junctions, parametric finite elements, Schur complement, tangential movement AMS subject classifications. 65M60, 65M12, 35K55, 53C44 DOI. 10.1137/0606539741. Introduction. The motion of curves or surfaces driven by second or fourth order geometric evolution equations arises in many applications in materials science and, of course, in differential geometry. Well known is the mean curvature flow, where a hypersurface moves in the direction of its mean curvature vector. Also frequently arising are evolution laws where the normal speed of an evolving hypersurface is given as a function of mean curvature, in applications, such as image processing or mathematical physics. In mathematical physics the inverse mean curvature flow plays a role in the context of the positive mass conjecture. In materials science the motion of networks of curves or surfaces is also often important. The evolution of grain networks-arising in polycrystalline materials-is also given by mean curvature flow, where at junctions angle conditions have to hold.Fourth order geometric evolution equations also frequently are found in geometry and materials science. The situation that the normal velocity of a hypersurface is given by minus the Laplacian of mean curvature is called surface diffusion. This evolution law arises in situations where the diffusion of material is restricted to interfacial regions; see, e.g., [36]. When many phases appear, networks also have to be taken into account. As the evolution law is of fourth order, additional conditions, which act as boundary conditions for the evolution law on the curves or surfaces, have to hold.The goal of this paper is to propose and analyze a new approach to numerically solving geometric evolution laws of second and fourth order. The numerical method is variational and very flexible. In particular, it is possible to couple fourth and second order laws on the surfaces or at triple junctions in a straightforward way.

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“…This is of relevance e.g. in thermal grooving ( [57]), in interface motion in polycrystalline two-phase materials ( [21]), in sintering processes ( [60]), and in the evolution of boundaries in the electromigration of intergranular voids (see [8]). A parametric finite element approximation of such flows for curve networks in the plane has been considered in [6], while its extension to surface clusters is the subject of the forthcoming article [13].…”

Section: Geometric Evolution Equations For Surface Clustersmentioning

confidence: 99%

Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies

Barrett¹,

Garcke²,

Nürnberg³

2010

Interfaces Free Bound.

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We present a variational formulation for the evolution of surface clusters in R 3 by mean curvature flow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including computations for regularized crystalline surface energy densities.

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A phase field model for the electromigration of intergranular voids

Cited by 32 publications

References 36 publications

Numerical simulations of immiscible fluid clusters

Numerical simulations of immiscible fluid clusters

On the Variational Approximation of Combined Second and Fourth Order Geometric Evolution Equations

Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies

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