A Multipoint Flux Approximation Finite Volume Scheme for Solving Anisotropic Reaction–Diffusion Systems in 3D

Strachota, Pavel; Beneš, Michal

doi:10.1007/978-3-642-20671-9_78

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…To incorporate anisotropic surface energies into the model, we follow the approach of Finsler geometry as introduced by [46] and later used in a number of our previous works, e.g. [36,38,39,[47][48][49][50]. Given the phase field p, the evolution of the sharp phase interface Γ can be determined implicitly by the relation…”

Section: Surface Energy Anisotropymentioning

confidence: 99%

Focusing the latent heat release in 3D phase field simulations of dendritic crystal growth

Strachota¹,

Wodecki²,

Beneš³

2021

Modelling Simul. Mater. Sci. Eng.

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We investigate a family of phase field models for simulating dendritic growth of a pure supercooled substance. The central object of interest is the reaction term in the Allen-Cahn equation, which is responsible for the spatial distribution of latent heat release during solidification. In this context, several existing forms of the reaction term are analyzed. Inspired by the known conclusions of matched asymptotic analysis, we propose a new variant (the 'ΣP1-P' model) that is simple enough to allow mathematical and numerical analysis and robust enough to be applicable to solidification under very large supercooling. The important component of the model (the Σ limiter) can also be incorporated into the original models to extend the range of their applicability. The individual models are tested in a number of numerical simulations focusing on mesh-dependence and model parameter settings. When the phase interface thickness is kept large with respect to the microscopic capillary length to make numerical computations feasible, the parameters of the Σ limiter can be tuned to improve agreement with previous models. The results obtained using the ΣP1-P model exhibit a good quantitative agreement with experimental data from rapid solidification of nickel melts.

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Section: Surface Energy Anisotropymentioning

confidence: 99%

Focusing the latent heat release in 3D phase field simulations of dendritic crystal growth

Strachota¹,

Wodecki²,

Beneš³

2021

Modelling Simul. Mater. Sci. Eng.

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“…To incorporate anisotropy, we follow the approach of Finsler geometry as introduced by [44] and later used in a number of our previous works, e.g. [41,30,40,45,33,32,46].…”

Section: The Anisotropic Phase Field Modelmentioning

confidence: 99%

Focusing the Latent Heat Release in 3D Phase Field Simulations of Dendritic Crystal Growth

Strachota,

Wodecki,

Beneš

2020

Preprint

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We investigate a family of phase field models for simulating dendritic growth of a pure supercooled substance. Our aim is to remove limitations inherent to some existing models both in terms of the applicability to physically realistic situations and the feasibility of mathematical and numerical analysis. The central object of interest is the reaction term in the Allen-Cahn equation, which is responsible for spatial distribution of latent heat release during solidification. Several existing forms of the reaction term are analyzed and new variants are proposed, with consistent asymptotic behavior as the interface thickness tends to zero. The resulting models are tested in a number of numerical simulations focusing on mesh-dependence, model parameter settings, and the applicability to solidification under very large supercooling.

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“…[3,4,11,14,19] and Section 2.2). Such a flow also has a special importance in anisotropic diffusion image segmentation and edge detection models (see [23,26,32,34]). Knowing underlying image anisotropy one can construct an efficient algorithm to segment important boundaries in the image or even denoising it by means of a anisotropic variant of Perona-Malik model [26,34].…”

Section: Introductionmentioning

confidence: 99%

Solution to the inverse Wulff problem by means of the enhanced semidefinite relaxation method

Ševčovič¹,

Trnovská²

2014

Journal of Inverse and Ill-Posed Problems

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We propose a novel method of resolving the optimal anisotropy function. The idea is to construct the optimal anisotropy function as a solution to the inverse Wul problem, i.e. as a minimizer for the anisoperimetric ratio for a given Jordan curve in the plane. It leads to a nonconvex quadratic optimization problem with linear matrix inequalities. In order to solve it we propose the so-called enhanced semide nite relaxation method which is based on a solution to a convex semide nite problem obtained by a semide nite relaxation of the original problem augmented by quadratic-linear constraints. We show that the sequence of nite-dimensional approximations of the optimal anisoperimetric ratio converges to the optimal anisoperimetric ratio which is a solution to the inverse Wul problem. Several computational examples, including those corresponding to boundaries of real snow akes, and discussion on the rate of convergence of numerical method are also presented in this paper.

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A Multipoint Flux Approximation Finite Volume Scheme for Solving Anisotropic Reaction–Diffusion Systems in 3D

Cited by 4 publications

References 8 publications

Focusing the latent heat release in 3D phase field simulations of dendritic crystal growth

Focusing the latent heat release in 3D phase field simulations of dendritic crystal growth

Focusing the Latent Heat Release in 3D Phase Field Simulations of Dendritic Crystal Growth

Solution to the inverse Wulff problem by means of the enhanced semidefinite relaxation method

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