An Overview of Network Bifurcations in the Functionalized Cahn-Hilliard Free Energy

Kraitzman, Noa; Promislow, Keith

doi:10.1007/978-3-319-16121-1_8

Cited by 9 publications

(6 citation statements)

References 26 publications

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“…where and η are given positive constants. This problem is motivated by the functionalized Cahn-Hilliard (FCH) equation [18] which models interfacial energy in amphiphilic phaseseparated mixtures. We follow a similar approach as before, and introduce the transformed variable v = u + 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

Causley¹,

Cho²,

Christlieb³

2017

SIAM J. Sci. Comput.

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Abstract. In this work, we develop an O(N ) implicit real space method in 1D and 2D for the Cahn Hilliard (CH) and vector Cahn Hilliard (VCH) equations, based on the Method Of Lines Transpose (MOL T ) formulation. This formulation results in a semi-discrete time stepping algorithm, which we prove is gradient stable in the H −1 norm. The spatial discretization follows from dimensional splitting, and an O(N ) matrix-free solver, which applies fast convolution to the modified Helmholtz equation. We propose a novel factorization technique, in which fourth order spatial derivatives are incorporated into the solver. The splitting error is included in the nonlinear fixed point iteration, resulting in a high order, logically Cartesian (line-by-line) update. Our method is fast, but not restricted to periodic boundaries like the fast Fourier transform (FFT).The basic solver is implemented using the Backward Euler formulation, and we extend this to both backward difference (BDF) stencils, implicit Runge Kutta (SDIRK) and spectral deferred correction (SDC) frameworks to achieve high orders of temporal accuracy. We demonstrate with numerical results that the CH, and VCH equations maintain gradient stability in one and two spatial dimensions. We also explore time-adaptivity, so that metastable states and ripening events can be simulated both quickly and efficiently.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Fully-discrete solution to 1D CH equation. It remains to discretize the convolution integral (18), and obtain a fully discrete algorithm. In our previous works [4,7,5,6], we accomplish this with fast convolution; for convenience, we will summarize the fast algorithm here.…”

Section: 2mentioning

confidence: 99%

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

Causley¹,

Cho²,

Christlieb³

2017

SIAM J. Sci. Comput.

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“…It to be pointed out is that the two fields in this description do not stand for cell types themselves, but any cell type can be described as a suitable linear combination of the two fields. The phase field literature is replete with formulations that model more complex structures (Choksi, 2012), including lipid bilayers (Dai and Promislow, 2013) and tubular structures (Kraitzman and Promislow, 2015). Rather than review those works this perspective has chosen to dwell on a few broader observations.…”

Section: Discussionmentioning

confidence: 99%

Perspectives on the mathematics of biological patterning and morphogenesis

Garikipati

2017

Journal of the Mechanics and Physics of Solids

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A central question in developmental biology is how size and position are determined. The genetic code carries instructions on how to control these properties in order to regulate the pattern and morphology of structures in the developing organism. Transcription and protein translation mechanisms implement these instructions. However, this cannot happen without some manner of sampling of epigenetic information on the current patterns and morphological forms of structures in the organism. Any rigorous description of space-and time-varying patterns and morphological forms reduces to one among various classes of spatio-temporal partial differential equations. Reaction-transport equations represent one such class. Starting from simple Fickian diffusion, the incorporation of reaction, phase segregation and advection terms can represent many of the patterns seen in the animal and plant kingdoms. Morphological form, requiring the development of three-dimensional structure, also can be represented by these equations of mass transport, albeit to a limited degree. The recognition that physical forces play controlling roles in shaping tissues leads to the conclusion that (nonlinear) elasticity governs the development of morphological form. In this setting, inhomogeneous growth drives the elasticity problem. The combination of reaction-transport equations with those of elasto-growth makes accessible a potentially unlimited spectrum of patterning and morphogenetic phenomena in developmental biology. This perspective communication is a survey of the partial differential equations of mathematical physics that have been proposed to govern patterning and morphogenesis in developmental biology. Several numerical examples are included to illustrate these equations and the corresponding physics, with the intention of providing physical insight wherever possible.

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“…Finally, our attempt in building a unifying framework is connected to the Functionalized Cahn-Hilliard (FCH) model proposed in [20,21,23,41,56,57]. The FCH energy describes how amphiphilic molecules self-assemble into complex network morphologies that feature spheres, tubules, sheets, and mixtures of them.…”

Section: Introductionmentioning

confidence: 99%

Generation of Tubular and Membranous Shape Textures with Curvature Functionals

Song

2021

J Math Imaging Vis

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Tubular and membranous shapes display a wide range of morphologies that are difficult to analyze within a common framework. By generalizing the classical Helfrich energy of biomembranes, we model them as solutions to a curvature optimization problem in which the principal curvatures may play asymmetric roles. We then give a novel phase-field formulation to approximate this geometric problem, and study its Gamma-limsup convergence. This results in an efficient GPU algorithm that we validate on well-known minimizers of the Willmore energy; the software for the implementation of our algorithm is freely available online. Exploring the space of parameters reveals that this comprehensive framework leads to a wide continuum of shape textures. This first step towards a unifying theory will have several implications, in biology for quantifying tubular shapes or designing bio-mimetic scaffolds, but also in computer graphics, materials science, or architecture.

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An Overview of Network Bifurcations in the Functionalized Cahn-Hilliard Free Energy

Cited by 9 publications

References 26 publications

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

Perspectives on the mathematics of biological patterning and morphogenesis

Generation of Tubular and Membranous Shape Textures with Curvature Functionals

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