Tetrahedralization of Isosurfaces with Guaranteed-Quality by Edge Rearrangement (TIGER)

Walker, Shawn W.

doi:10.1137/120866075

Cited by 11 publications

(10 citation statements)

References 54 publications

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“…h . This procedure is done in FELICITY [62,66] using the mesh generation tool TIGER [64]. For the examples in this paper, Ω 0 h is a triangulation of the square (−L, L) 2 .…”

Section: Discretization and Initializationmentioning

confidence: 99%

“…there are rigorous (and reasonable!) bounds on the angles of the triangles [64]. Note: TIGER can also generate robust tetrahedral meshes of three dimensional domains.…”

Section: Discretization and Initializationmentioning

confidence: 99%

“…Given the interface mesh Γ i h , which is a closed curve, it is straightforward to determine what points are inside the curve and what points are outside. This is all that is needed by TIGER [64]; in essence, one can think of Γ i h as the zero-level set of a hypothetical level set function. The new mesh (of Ω h ) that is generated has a conforming interface Γ i h that interpolates Γ i h in a piecewise linear fashion.…”

Section: Algorithmmentioning

confidence: 99%

“…In our method, the interface is represented by a surface triangulation that conforms to the bulk mesh and deforms with the interface. Thus, we must do occasional re-meshing with the method in [64]. We emphasize that 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.…”

Section: Introductionmentioning

confidence: 99%

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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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“…h . This procedure is done in FELICITY [62,66] using the mesh generation tool TIGER [64]. For the examples in this paper, Ω 0 h is a triangulation of the square (−L, L) 2 .…”

Section: Discretization and Initializationmentioning

confidence: 99%

“…there are rigorous (and reasonable!) bounds on the angles of the triangles [64]. Note: TIGER can also generate robust tetrahedral meshes of three dimensional domains.…”

Section: Discretization and Initializationmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Davis

Walker

2017

ESAIM: M2AN

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“…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.…”

mentioning

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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Parametric finite element approximations of curvature-driven interface evolutions

Barrett

Garcke

Nürnberg

2020

Handbook of Numerical Analysis

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Tetrahedralization of Isosurfaces with Guaranteed-Quality by Edge Rearrangement (TIGER)

Cited by 11 publications

References 54 publications

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

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

A mixed formulation of the Stefan problem with surface tension

Parametric finite element approximations of curvature-driven interface evolutions

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