The phase field method for geometric moving interfaces and their numerical approximations

Du, Qiang; Feng, Xiaobing

doi:10.1016/bs.hna.2019.05.001

Cited by 72 publications

(50 citation statements)

References 427 publications

(667 reference statements)

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“…Conventional phase field (diffuse interface) models take on a 50 M. D'Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian and Z. Zhou free energy of the type(Du and Feng 2020)…”

mentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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“…Conventional phase field (diffuse interface) models take on a 50 M. D'Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian and Z. Zhou free energy of the type(Du and Feng 2020)…”

mentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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“…For completeness, let us mention that there are also interface capturing approaches that avoid the need P. Pozzi & B. Stinner to look after the mesh quality [7,23,28,5,17]. Such approaches comprise phase field models and level set methods, for overviews we refer to [11,4,13,27].…”

Section: P Pozzi and B Stinnermentioning

confidence: 99%

On motion by curvature of a network with a triple junction

Pozzi¹,

Stinner²

2021

The SMAI Journal of Computational Mathematics

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We numerically study the planar evolution by curvature flow of three parametrised curves that are connected by a triple junction in which conditions are imposed on the angles at which the curves meet. One of the key problems in analysing motion of networks by curvature law is the choice of a tangential velocity that allows for motion of the triple junction, does not lead to mesh degeneration, and is amenable to an error analysis. Our approach consists in considering a perturbation of a classical smooth formulation. The problem we propose admits a natural variational formulation that can be discretized with finite elements. The perturbation can be made arbitrarily small when a regularisation parameter shrinks to zero. Convergence of the new semi-discrete finite element scheme including optimal error estimates are proved. These results are supported by some numerical tests. Finally, the influence of the small regularisation parameter on the properties of scheme and the accuracy of the results is numerically investigated.

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“…For instance, numerical level set methods occasionally require reinitialisation of the level set function for accurate simulations [27,28], which is not the case for numerical phase field methods. We refer to [39] for a recent review of computational methods for phase field equations, where also the relation to level set approaches is discussed.…”

Section: Interface Capturingmentioning

confidence: 99%

“…One way to make simulations cheaper is using finite element methods with adaptive refinement as discussed in [39]. Figure 3C gives an impression of a mesh that is refined only in the interfacial layer for an ellipsoidal droplet in 2D.…”

Section: Interface Capturingmentioning

confidence: 99%

Mathematical modelling in cell migration: tackling biochemistry in changing geometries

Stinner

Bretschneider

2020

Biochemical Society Transactions

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Directed cell migration poses a rich set of theoretical challenges. Broadly, these are concerned with (1) how cells sense external signal gradients and adapt; (2) how actin polymerisation is localised to drive the leading cell edge and Myosin-II molecular motors retract the cell rear; and (3) how the combined action of cellular forces and cell adhesion results in cell shape changes and net migration. Reaction–diffusion models for biological pattern formation going back to Turing have long been used to explain generic principles of gradient sensing and cell polarisation in simple, static geometries like a circle. In this minireview, we focus on recent research which aims at coupling the biochemistry with cellular mechanics and modelling cell shape changes. In particular, we want to contrast two principal modelling approaches: (1) interface tracking where the cell membrane, interfacing cell interior and exterior, is explicitly represented by a set of moving points in 2D or 3D space and (2) interface capturing. In interface capturing, the membrane is implicitly modelled analogously to a level line in a hilly landscape whose topology changes according to forces acting on the membrane. With the increased availability of high-quality 3D microscopy data of complex cell shapes, such methods will become increasingly important in data-driven, image-based modelling to better understand the mechanochemistry underpinning cell motion.

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The phase field method for geometric moving interfaces and their numerical approximations

Cited by 72 publications

References 427 publications

Numerical methods for nonlocal and fractional models

Numerical methods for nonlocal and fractional models

On motion by curvature of a network with a triple junction

Mathematical modelling in cell migration: tackling biochemistry in changing geometries

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