Energy-driven pattern formation

Kohn, Robert V.

doi:10.4171/022-1/15

Cited by 60 publications

(69 citation statements)

References 49 publications

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“…This implies that there will be no mechanism in any energy-gradient dynamics model that could select a preferred slope of the mounds. See the relevant discussions in [8,9,11,12,21]. The chemical potential is defined to be the variational derivative of the energy (1), i.e.,…”

Section: Introductionmentioning

confidence: 99%

A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

Chen

Wang

et al. 2013

J Sci Comput

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We present a linear iteration algorithm to implement a second-order energy stable numerical scheme for a model of epitaxial thin film growth without slope selection. The PDE, which is a nonlinear, fourth-order parabolic equation, is the L 2 gradient flow of the energyThe energy stability is preserved by a careful choice of the second-order temporal approximation for the nonlinear term, as reported in recent work [18]. The resulting scheme is highly nonlinear, and its implementation is non-trivial. In this paper, we propose a linear iteration algorithm to solve the resulting nonlinear system. To accomplish this we introduce an O(s 2 ) (with s the time step size) artificial diffusion term, a Douglas-Dupont-type regularization, that leads to a contraction mapping property. As a result, the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be very efficiently implemented with the help of FFT in a collocation Fourier spectral setting. We present a careful analysis showing convergence for the numerical scheme in a discreteSome numerical simulation results are presented to demonstrate the efficiency of the linear iteration solver and the convergence of the scheme as a whole.

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Section: Introductionmentioning

confidence: 99%

A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

Chen

Wang

et al. 2013

J Sci Comput

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“…The highly-cited article of Seul and Andelman [25] addresses exactly this issue and provides a partial list of relevant physical systems in Table 1. Problem (1.2) is also very close to variational problems arising in the study of magnetic domains and walls (see for example [15], [12] and the references therein).…”

Section: G(x Y) (U(x) − M) (U(y) − M) DX Dymentioning

confidence: 99%

Uniform energy distribution for an isoperimetric problem with long-range interactions

Alberti

Choksi

Otto³

2008

J. Amer. Math. Soc.

104

205

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We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and long-range interactions. The short-range interaction is attractive and comes from an interfacial energy, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale. However, proving any periodicity result turns out to be a formidable task in dimensions larger than one. In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are sufficiently large with respect to the intrinsic length scale

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“…martensitic materials), biomembranes and block copolymers (cf. [32] and the references therein). In this article we will address energy-driven pattern formation induced by competing short and long-range interactions (cf.…”

Section: Introduction: Energy-driven Pattern Formationmentioning

confidence: 99%

On global minimizers for a variational problem with long-range interactions

Choksi

2012

Quart. Appl. Math.

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Abstract. Energy-driven pattern formation induced by competing short and longrange interactions is common in many physical systems. In these proceedings we report on certain rigorous asymptotic results concerning global minimizers of a nonlocal perturbation to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. We also discuss two hybrid numerical methods for accessing the ground states of these functionals.

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Energy-driven pattern formation

Cited by 60 publications

References 49 publications

A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

Uniform energy distribution for an isoperimetric problem with long-range interactions

On global minimizers for a variational problem with long-range interactions

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