On Thermodynamically Consistent Stefan Problems with Variable Surface Energy

Prüß, Jan; Simonett, Gieri; Wilke, Mathias

doi:10.1007/s00205-015-0938-y

Cited by 8 publications

(8 citation statements)

References 37 publications

(66 reference statements)

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“…Applications range from modeling the freezing (or melting) of water to the solidification of crystals from a melt and dendritic growth [17,33,39,53,54,60]. The mathematical theory of the Stefan problem can be found in [14,27,37,41,[47][48][49][50] as well as for the related Mullins-Sekerka problem [20,23,26,42,51].…”

Section: Introductionmentioning

confidence: 99%

Semi-discrete error estimates and implementation of a mixed method for the Stefan problem

Davis

Walker

2017

ESAIM: M2AN

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We analyze a dual formulation and finite element method for simulating the Stefan problem with surface tension (originally presented in [C.B. Davis and S.W. Walker, Int. Free Bound. 17 (2015) 427-464]). The method uses a mixed form of the heat equation in the solid and liquid (bulk) domains, and imposes a weak formulation of the interface motion law (on the solid-liquid interface) as a constraint. The computational method uses a conforming mesh approach to accurately capture the jump conditions across the interface. Preliminary error estimates are derived, under reduced regularity assumptions, for the difference between the time semi-discrete solution and the fully discrete solution over one time step. Moreover, details of the implementation are discussed including mesh generation issues. Several simulations of interface growth (in two dimensions) are presented to illustrate the method.

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Section: Introductionmentioning

confidence: 99%

Semi-discrete error estimates and implementation of a mixed method for the Stefan problem

Davis

Walker

2017

ESAIM: M2AN

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“…Applications range from modeling the freezing (or melting) of water to the solidification of crystals from a melt and dendritic growth [15,30,38,51,52,59]. Mathematical theory for the Stefan problem with Gibbs-Thomson law is available for local and global in time solutions [13,25,36,39,[45][46][47][48]. Well-posedness results are also available if the heat equation in the bulk phases is replaced by a quasi-static approximation (i.e.…”

Section: Introductionmentioning

confidence: 99%

A mixed formulation of the Stefan problem with surface tension

Davis¹,

Walker²

2016

Interfaces Free Bound.

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A dual formulation and finite element method is proposed and analyzed for simulating the Stefan problem with surface tension. The method uses a mixed form of the heat equation in the solid and liquid (bulk) domains, and imposes a weak formulation of the interface motion law (on the solidliquid interface) as a constraint. The basic unknowns are the heat fluxes and temperatures in the bulk, and the velocity and temperature on the interface. The formulation, as well as its discretization, is viewed as a saddle point system. Well-posedness of the time semi-discrete and fully discrete formulations is proved in three dimensions, as well as an a priori (stability) bound and conservation law. Simulations of interface growth (in two dimensions) are presented to illustrate the method.Our paper presents a completely mixed formulation of the Stefan problem, including the bulk heat equations [8]. In other words, we formulate the problem in a saddle-point framework, where the heat equations are in mixed form, and the interface motion law appears as a constraint in the system of equations with a balancing Lagrange multiplier that represents the interface temperature. To the best of our knowledge, this is a new method for the Stefan problem with surface tension. Some highlights of our method are the following.We prove that both the time semi-discrete and fully discrete systems have a priori bounds (in time) that mimic the continuous model. This assumes the interface velocity is reasonably regular and that there are no topological changes. Moreover, we can prove that both the time semi-discrete and fully discrete systems maintain conservation of thermal energy. In [5], they only achieve this for their discrete in space, continuous in time, scheme. The interface is represented by a surface triangulation that conforms to the bulk mesh which deforms with the interface. Hence, occasional re-meshing is needed, which is done by the method in [63]. One advantage of this method is that all integrals in the finite element formulation can be computed exactly. In addition, we do not need to compute the intersection of meshes at adjacent time steps to transfer solution variables from one mesh to the next (e.g. for computing L 2 projections from one mesh to another). Our method can be modified to include anisotropic surface tension via [5], which is relevant to crystal growth. The well-posedness of the method remains unchanged, as well as the a priori bound and conservation law. Other variations of the Stefan problem (e.g. Mullins-Sekerka) can be formulated with our approach by straightforward modifications. One can even include moving contact line effects when the solid phase is attached to a rigid boundary [60, 64]. SummaryIn Section 2, we describe the governing equations. Section 3 describes the fully continuous weak formulation and derives a formal a priori bound and conservation law. Section 4 explains the timediscretization and how the interface motion is handled. A variational formulation of the time semidiscrete problem is given, its ...

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“…These situations appear frequently in applications, for example in problems with symmetries, and in problems with moving boundaries, see for instance [8,9,12,14,15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%