Analysis of Carrier's Problem

Chapman, S. Jonathan; Farrell, Patrick E.

doi:10.1137/16m1096074

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…The method is generalisable, robust and computationally efficient for large-scale applications. It has been successfully applied to a diverse set of nonlinear problems including nonlinear partial differential equations (PDEs), singularly perturbed problems, the analysis of Bose-Einstein condensates, and the computation of disconnected bifurcation diagrams [17,20,22,23]. This paper is structured as follows: in Section II, we briefly describe the problem; in Section III, we introduce the deflation technique with an illustrative toy example.…”

Section: Introductionmentioning

confidence: 99%

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

et al. 2017

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We study the problem of a cholesteric liquid crystal confined to an elliptical channel. The system is geometrically frustrated because the cholesteric prefers to adopt a uniform rate of twist deformation, but the elliptical domain precludes this. The frustration is resolved by deformation of the layers or introduction of defects, leading to a particularly rich family of equilibrium configurations. To identify the solution set, we adapt and apply a new family of algorithms, known as deflation methods, that iteratively modify the free energy extremisation problem by removing previously known solutions. A second algorithm, deflated continuation, is used to track solution branches as a function of the aspect ratio of the ellipse and preferred pitch of the cholesteric.

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Section: Introductionmentioning

confidence: 99%

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

et al. 2017

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“…To cope with both nonlinearity and wavelength modulation we use the method introduced by Kuzmak (1959) (see also Ablowitz 2011;Chapman & Farrell 2017) -a generalised multiple-scales analysis also known as the principle of adiabatic invariants (Landau & Lifshitz 1976). Thus we introduce the independent fast variable…”

Section: Introducing the Fast Variablementioning

confidence: 99%

Multiple-scales analysis of wave evolution in the presence of rigid vegetation

et al. 2022

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The study of free-surface flows over vegetative structures presents a challenging setting for theoretical, computational and experimental analysis. In this work, we develop a multiple-scales asymptotic framework for the evolution of free-surface waves over rigid vegetation and a slowly varying substrate. The analysis quantifies the balance between the competing effects of vegetation and shoaling, and provides a prediction of the amplitude as the wave approaches a coastline. Our analysis unifies and extends existing theories that study these effects individually. The asymptotic predictions are shown to provide good agreement with full numerical simulations (varying depth) and published experimental results (constant depth).

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“…In this paper, we employ a recent algorithm called deflation for computing multiple solutions to nonlinear partial differential equations [20,21], which has been successfully applied to compute bifurcations to a wide range of physical problems such as the deformation of a hyperelastic beam [21], Carrier's problem [22], cholesteric liquid crystals [23], and the nonlinear Schrödinger equation in two and three dimensions [24][25][26]. We emphasise that deflated continuation and its extensions offer some advantages that could be combined with the standard bifurcation analysis tools.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of two-dimensional Rayleigh--Bénard convection using deflation

Boullé,

Dallas,

Farrell

2021

Preprint

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We perform a bifurcation analysis of the steady state solutions of Rayleigh-Bénard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialisation strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this non-linear problem, including disconnected branches of the bifurcation diagram, without the need of any prior knowledge of the dynamics. One of the disconnected branches we find contains a s-shape bifurcation with hysteresis, which is the origin of the flow pattern that may be related to the dynamics of flow reversals in the turbulent regime. Linear stability analysis is also performed to analyse the steady and unsteady regimes of the solutions in the parameter space and to characterise the type of instabilities.

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Analysis of Carrier's Problem

Cited by 5 publications

References 12 publications

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

Multiple-scales analysis of wave evolution in the presence of rigid vegetation

Bifurcation analysis of two-dimensional Rayleigh--Bénard convection using deflation

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