A traveling wave solution for coupled surface and grain boundary motion

Kanel, Jacob; Novick-Cohen, Amy; Vilenkin, Arkady

doi:10.1016/s1359-6454(02)00603-1

Cited by 26 publications

(27 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We observe that the evolutions in Figure 2 exhibit travelling wave solutions, where the profile of the travelling wave is highly dependent on the chosen surface energies ς. Such travelling wave solutions were first mentioned in Mullins (1958), see also Kanel et al (2003Kanel et al ( , 2004). Similar travelling wave solutions can be observed for a slightly simpler setup, where the initial curves form a letter "T", but where at the external boundary a non 90…”

Section: Isotropic Flowsmentioning

confidence: 74%

“…On the basis of our numerical studies we conjecture that the two dimensional travelling wave profile studied by Mullins (1958) and Kanel et al (2003) is stable also with respect to truly three dimensional perturbations.…”

Section: Introductionmentioning

confidence: 80%

“…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%

“…In particular, the question arose whether the appearance of the surface groove will slow down the velocity of the grain boundary. In this context travelling wave solutions have been studied in the literature, see Mullins (1958); Kanel et al (2003). But recent studies seem to indicate that the grain groove only has a minimal effect on the grain boundary motion.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

2010

View full text Add to dashboard Cite

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.

show abstract

Section: Isotropic Flowsmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2010

View full text Add to dashboard Cite

show abstract

“…Aspects of the model of this phenomenon described below were originally proposed in [18], and its present form was developed in [13] but rewritten in the PDAE formulation of parametrized curves developed in [20]. Let X 1 (σ, t) denote the grain boundary and X 2,3 (σ, t) denote the free surfaces to the left and right of the triple junction.…”

Section: Mixed-order Quarter Loop Modelmentioning

confidence: 99%

Stability of travelling wave solutions for coupled surface and grain boundary motion

Beck

Pan

Wetton

2010

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We investigate the spectral stability of the travelling wave solution for the coupled motion of a free surface and grain boundary that arises in materials science. In this problem a grain boundary, which separates two materials that are identical except for their crystalline orientation, evolves according to mean curvature. At a triple junction, this boundary meets the free surfaces of the two crystals, which move according to surface diffusion. The model is known to possess a unique travelling wave solution. We study the linearization about the wave, which necessarily includes a free boundary at the location of the triple junction. This makes the analysis more complex than that of standard travelling waves, and we discuss how existing theory applies in this context. Furthermore, we compute numerically the associated point spectrum by restricting the problem to a finite computational domain with appropriate physical boundary conditions. Numerical results strongly suggest that the two-dimensional wave is stable with respect to both two-and three-dimensional perturbations.

show abstract