Approximation of Helfrich's Functional via Diffuse Interfaces

Bellettini, Giovanni; Mugnai, Luca

doi:10.1137/09077549x

Cited by 31 publications

(56 citation statements)

References 32 publications

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“…Our proofs are based on the previous works [2, 13, 34–36, 41, 42]. For completeness, we give all the necessary details.…”

Section: Proof Of Lemma 41mentioning

confidence: 99%

Variational Implicit Solvation with Solute Molecular Mechanics: From Diffuse-Interface to Sharp-Interface Models

Li¹,

Zhao²

2013

SIAM J. Appl. Math.

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Central in a variational implicit-solvent description of biomolecular solvation is an effective free-energy functional of the solute atomic positions and the solute-solvent interface (i.e., the dielectric boundary). The free-energy functional couples together the solute molecular mechanical interaction energy, the solute-solvent interfacial energy, the solute-solvent van der Waals interaction energy, and the electrostatic energy. In recent years, the sharp-interface version of the variational implicit-solvent model has been developed and used for numerical computations of molecular solvation. In this work, we propose a diffuse-interface version of the variational implicit-solvent model with solute molecular mechanics. We also analyze both the sharp-interface and diffuse-interface models. We prove the existence of free-energy minimizers and obtain their bounds. We also prove the convergence of the diffuse-interface model to the sharp-interface model in the sense of Γ-convergence. We further discuss properties of sharp-interface free-energy minimizers, the boundary conditions and the coupling of the Poisson–Boltzmann equation in the diffuse-interface model, and the convergence of forces from diffuse-interface to sharp-interface descriptions. Our analysis relies on the previous works on the problem of minimizing surface areas and on our observations on the coupling between solute molecular mechanical interactions with the continuum solvent. Our studies justify rigorously the self consistency of the proposed diffuse-interface variational models of implicit solvation.

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“…Our proofs are based on the previous works [2, 13, 34–36, 41, 42]. For completeness, we give all the necessary details.…”

Section: Proof Of Lemma 41mentioning

confidence: 99%

Variational Implicit Solvation with Solute Molecular Mechanics: From Diffuse-Interface to Sharp-Interface Models

Li¹,

Zhao²

2013

SIAM J. Appl. Math.

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“…In this energetic variational framework, the regions { x : φ ( x ) > 0}and { x : φ ( x ) < 0} represent, respectively, the inside and outside of the vesicle, whereas the level set { x : φ ( x ) = 0}gives the position of the membrane. Formal asymptotics as well as rigorous mathematical analysis provide the connection between the sharp interface model and this regularized theory when the parameter ε tends to zero. As this limit is never achieved in the numerical calculations, an error originated by the model is always present on the computed results.…”

Section: Applications Of the Second‐order Max‐ent Approximantsmentioning

confidence: 99%

Second‐order convex maximum entropy approximants with applications to high‐order PDE

Rosolen

Millán

Arroyo

2012

Numerical Meth Engineering

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SUMMARYWe present a new approach for second-order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for nonuniform node sets and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of PDEs, including structural vibrations, elliptic second-order PDEs, and fourth-order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with nonuniform node distributions and the high accuracy of the solutions. Surprisingly, the first-order meshfree max-ent approximants with large supports are competitive when compared with the proposed second-order approach in all the tested examples, even in the higher order PDEs.

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“…The Helfrich energy also arises as the Γ -limit of some phase field models, see e.g. [2,9]. Such models are used to approximate the Willmore energy in image processing and our work may appear as an alternative in this field.…”

Section: Introductionmentioning

confidence: 99%

A Highly Anisotropic Nonlinear Elasticity Model for Vesicles I. Eulerian Formulation, Rigidity Estimates and Vanishing Energy Limit

Merlet

2015

Arch Rational Mech Anal

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Approximation of Helfrich's Functional via Diffuse Interfaces

Abstract: Abstract. We give a rigorous proof of the approximability of the so-called Helfrich's functional via diffuse interfaces, under a constraint on the ratio between the bending rigidity and the Gauss-rigidity.

Cited by 31 publications

References 32 publications

Variational Implicit Solvation with Solute Molecular Mechanics: From Diffuse-Interface to Sharp-Interface Models

Variational Implicit Solvation with Solute Molecular Mechanics: From Diffuse-Interface to Sharp-Interface Models

Second‐order convex maximum entropy approximants with applications to high‐order PDE

A Highly Anisotropic Nonlinear Elasticity Model for Vesicles I. Eulerian Formulation, Rigidity Estimates and Vanishing Energy Limit

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