Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Research articles
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- recent Research articles articles1471-2946Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Modulational instability in the full-dispersion Camassa-Holm equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170153?rss=1
We determine the stability and instability of a sufficiently small and periodic travelling wave to long-wavelength perturbations, for a nonlinear dispersive equation which extends a Camassa–Holm equation to include all the dispersion of water waves and the Whitham equation to include nonlinearities of medium-amplitude waves. In the absence of the effects of surface tension, the result qualitatively agrees with the Benjamin–Feir instability of a Stokes wave. In the presence of the effects of surface tension, it qualitatively agrees with those from formal asymptotic expansions of the physical problem and improves upon that for the Whitham equation, predicting the critical wave number at the strong surface tension limit. We discuss the modulational stability and instability in the Camassa–Holm equation and other related models.
]]>2017-07-19T00:08:13-07:00info:doi/10.1098/rspa.2017.0153hwp:master-id:royprsa;rspa.2017.01532017-07-19Research articles47322032017015320170153<![CDATA[Gyro-elastic beams for the vibration reduction of long flexural systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170136?rss=1
The paper presents a model of a chiral multi-structure incorporating gyro-elastic beams. Floquet–Bloch waves in periodic chiral systems are investigated in detail, with the emphasis on localization and the formation of standing waves. It is found that gyricity leads to low-frequency standing modes and generation of stop-bands. A design of an earthquake protection system is offered here, as an interesting application of vibration isolation. Theoretical results are accompanied by numerical simulations in the time-harmonic regime.
]]>2017-07-19T00:08:13-07:00info:doi/10.1098/rspa.2017.0136hwp:master-id:royprsa;rspa.2017.01362017-07-19Research articles47322032017013620170136<![CDATA[Effects of boundary conditions on bistable behaviour in axisymmetrical shallow shells]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170230?rss=1
Multistable shells are thin-walled structures that have more than one stable state of self-stress. We consider isotropic axisymmetrical shallow shells of arbitrary polynomial shapes using a Föppl–von Kármán analytical model. By employing a Rayleigh–Ritz approach, we identify stable shapes from local minima in the strain energy formulation, and we formally characterize the level of influence of the boundary conditions on the critical geometry for achieving bistable inversion—an effect not directly answered in the literature. Systematic insight is afforded by connecting the boundary to ground through sets of extensional and rotational linear springs. For typical cap-like shells, it is shown that bistability is generally enhanced when the extensional spring stiffness increases and when the rotational spring stiffness decreases, i.e. when boundary movements in-plane are resisted but when their rotations are not; however, for certain other shapes and large in-plane stiffness values, bistability can be enhanced by resisting but not entirely preventing edge rotations. Our predictions are furnished as detailed regime maps of the critical geometry, which are accurately correlated against finite-element analysis. Furthermore, the suitabilities of single degree-of-freedom models, for which solutions are achieved in closed form, are evaluated and compared to our more accurate predictions.
]]>2017-07-19T00:08:13-07:00info:doi/10.1098/rspa.2017.0230hwp:master-id:royprsa;rspa.2017.02302017-07-19Research articles47322032017023020170230<![CDATA[Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20160915?rss=1
A unified framework is presented for the solution structure of three-dimensional discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearizing transform, which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2016.0915hwp:master-id:royprsa;rspa.2016.09152017-07-12Research articles47322032016091520160915<![CDATA[Modelling the fear of crime]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170156?rss=1
How secure people feel in a particular region is obviously linked to the actual crime suffered in that region but the exact relationship between crime and its fear is quite subtle. Two regions may have the same crime rate but their local perception of security may differ. Equally, two places may have the same perception of security even though one may have a significantly lower crime rate. Furthermore, a negative perception might persist for many years, even when crime rates drop. Here, we develop a model for the dynamics of the perception of security of a region based on the distribution of crime suffered by the population using concepts similar to those used for opinion dynamics. Simulations under a variety of conditions illustrate different scenarios and help us determine the impact of suffering more, or less, crime. The inhomogeneous concentration of crime together with a memory loss process is incorporated into the model for the perception of security, and results explain why people are often more fearful than actually victimized; why a region is perceived as being insecure despite a low crime rate; and why a decrease in the crime rate might not significantly improve the perception of security.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0156hwp:master-id:royprsa;rspa.2017.01562017-07-12Research articles47322032017015620170156<![CDATA[A note on flow reversal in a wavy channel filled with anisotropic porous material]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170193?rss=1
Viscous flow through a symmetric wavy channel filled with anisotropic porous material is investigated analytically. Flow inside the porous bed is assumed to be governed by the anisotropic Brinkman equation. It is assumed that the ratio of the channel width to the wavelength is small (i.e. ^{2}<<1). The problem is solved up to O(^{2}) assuming that ^{2}^{2}<<1, where is the anisotropic ratio. The key purpose of this paper is to study the effect of anisotropic permeability on flow near the crests of the wavy channel which causes flow reversal. We present a detailed analysis of the flow reversal at the crests. The ratio of the permeabilities (anisotropic ratio) is responsible for the flow separation near the crests of the wall where viscous forces are effective. For a flow configuration (say, low amplitude parameter) in which there is no separation if the porous media is isotropic, introducing anisotropy causes flow separation. On the other hand, interestingly, flow separation occurs even in the case of isotropic porous medium if the amplitude parameter a is large.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0193hwp:master-id:royprsa;rspa.2017.01932017-07-12Research articles47322032017019320170193<![CDATA[Multi-fractal characterization of bacterial swimming dynamics: a case study on real and simulated Serratia marcescens]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170154?rss=1
To add to the current state of knowledge about bacterial swimming dynamics, in this paper, we study the fractal swimming dynamics of populations of Serratia marcescens bacteria both in vitro and in silico, while accounting for realistic conditions like volume exclusion, chemical interactions, obstacles and distribution of chemoattractant in the environment. While previous research has shown that bacterial motion is non-ergodic, we demonstrate that, besides the non-ergodicity, the bacterial swimming dynamics is multi-fractal in nature. Finally, we demonstrate that the multi-fractal characteristic of bacterial dynamics is strongly affected by bacterial density and chemoattractant concentration.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0154hwp:master-id:royprsa;rspa.2017.01542017-07-12Research articles47322032017015420170154<![CDATA[A non-local asymptotic theory for thin elastic plates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170249?rss=1
The three-dimensional dynamic non-local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential non-local kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for non-local behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to the first-order corrections to the bending and extensional stiffness in the classical two-dimensional plate equations.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0249hwp:master-id:royprsa;rspa.2017.02492017-07-12Research articles47322032017024920170249<![CDATA[A phase-plane analysis of localized frictional waves]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20160606?rss=1
Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.
]]>2017-07-05T00:08:16-07:00info:doi/10.1098/rspa.2016.0606hwp:master-id:royprsa;rspa.2016.06062017-07-05Research articles47322032016060620160606<![CDATA[The role of molybdenum in suppressing cold dwell fatigue in titanium alloys]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170189?rss=1
We test a hypothesis to explain why Ti-6242 is susceptible to cold dwell fatigue (CDF), whereas Ti-6246 is not. The hypothesis is that, in Ti-6246, substitutional Mo-atoms in α-Ti grains trap vacancies, thereby limiting creep relaxation. In Ti-6242, this creep relaxation enhances the loading of grains unfavourably oriented for slip and they subsequently fracture. Using density functional theory to calculate formation and binding energies between Mo-atoms and vacancies, we find no support for the hypothesis. In the light of this result, and experimental observations of the microstructures in these alloys, we agree with the recent suggestion (Qiu et al. 2014 Metall. Mater. Trans. A45, 6075–6087. (doi:10.1007/s11661-014-2541-5)) that Ti-6246 has a much smaller susceptibility to CDF because it has a smaller grain size and a more homogeneous distribution of grain orientations. We propose that the reduction of the susceptibility to CDF of Ti-6242 at temperatures above about 200°C is due to the activation of <c+a> slip in ‘hard’ grains, which reduces the loading of grain boundaries.
]]>2017-07-05T00:08:08-07:00info:doi/10.1098/rspa.2017.0189hwp:master-id:royprsa;rspa.2017.01892017-07-05Research articles47322032017018920170189<![CDATA[Piecewise affine stress-free martensitic inclusions in planar nonlinear elasticity]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170235?rss=1
We consider a partial differential inclusion problem which models stress-free martensitic inclusions in an austenitic matrix, based on the standard geometrically nonlinear elasticity theory. We show that for specific parameter choices there exist piecewise affine continuous solutions for the square-to-oblique and the hexagonal-to-oblique phase transitions. This suggests that for specific crystallographic parameters the hysteresis of the phase transformation will be particularly small.
]]>2017-07-05T00:08:16-07:00info:doi/10.1098/rspa.2017.0235hwp:master-id:royprsa;rspa.2017.02352017-07-05Research articles47322032017023520170235<![CDATA[Trigonal curves and algebro-geometric solutions to soliton hierarchies I]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170232?rss=1
This is the first part of a study, consisting of two parts, on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, explore general properties of meromorphic functions defined as ratios of the Baker–Akhiezer functions, and determine zeros and poles of the Baker–Akhiezer functions and their Dubrovin-type equations. We analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.
]]>2017-07-05T00:08:08-07:00info:doi/10.1098/rspa.2017.0232hwp:master-id:royprsa;rspa.2017.02322017-07-05Research articles47322032017023220170232<![CDATA[Trigonal curves and algebro-geometric solutions to soliton hierarchies II]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170233?rss=1
This is a continuation of a study on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, we straighten out all flows in soliton hierarchies under the Abel–Jacobi coordinates associated with Lax pairs, upon determining the Riemann theta function representations of the Baker–Akhiezer functions, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviours of the Baker–Akhiezer functions. We emphasize that we analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.
]]>2017-07-05T00:08:08-07:00info:doi/10.1098/rspa.2017.0233hwp:master-id:royprsa;rspa.2017.02332017-07-05Research articles47322032017023320170233<![CDATA[Error bounds for the asymptotic expansion of the Hurwitz zeta function]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170363?rss=1
In this paper, we reconsider the large-a asymptotic expansion of the Hurwitz zeta function (s,a). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes G-function and the s-derivative of the Hurwitz zeta function (s,a) are provided. A detailed discussion on the sharpness of our error bounds is also given.
]]>2017-07-05T00:08:16-07:00info:doi/10.1098/rspa.2017.0363hwp:master-id:royprsa;rspa.2017.03632017-07-05Research articles47322032017036320170363<![CDATA[Turbulent boundary layer under the control of different schemes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170038?rss=1
This work explores experimentally the control of a turbulent boundary layer over a flat plate based on wall perturbation generated by piezo-ceramic actuators. Different schemes are investigated, including the feed-forward, the feedback, and the combined feed-forward and feedback strategies, with a view to suppressing the near-wall high-speed events and hence reducing skin friction drag. While the strategies may achieve a local maximum drag reduction slightly less than their counterpart of the open-loop control, the corresponding duty cycles are substantially reduced when compared with that of the open-loop control. The results suggest a good potential to cut down the input energy under these control strategies. The fluctuating velocity, spectra, Taylor microscale and mean energy dissipation are measured across the boundary layer with and without control and, based on the measurements, the flow mechanism behind the control is proposed.
]]>2017-06-28T00:08:07-07:00info:doi/10.1098/rspa.2017.0038hwp:master-id:royprsa;rspa.2017.00382017-06-28Research articles47322022017003820170038<![CDATA[Non-reciprocal wave propagation in modulated elastic metamaterials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170188?rss=1
Time-reversal symmetry for elastic wave propagation breaks down in a resonant mass-in-mass lattice whose inner-stiffness is weakly modulated in space and in time in a wave-like fashion. Specifically, one-way wave transmission, conversion and amplification as well as unidirectional wave blocking are demonstrated analytically through an asymptotic analysis based on coupled mode theory and numerically thanks to a series of simulations in harmonic and transient regimes. High-amplitude modulations are then explored in the homogenization limit where a non-standard effective mass operator is recovered and shown to take negative values over unusually large frequency bands. These modulated metamaterials, which exhibit either non-reciprocal behaviours or non-standard effective mass operators, offer promise for applications in the field of elastic wave control in general and in one-way conversion/amplification in particular.
]]>2017-06-28T00:08:07-07:00info:doi/10.1098/rspa.2017.0188hwp:master-id:royprsa;rspa.2017.01882017-06-28Research articles47322022017018820170188<![CDATA[The effect of coherent stirring on the advection-condensation of water vapour]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170196?rss=1
Atmospheric water vapour is an essential ingredient of weather and climate. The key features of its distribution can be represented by kinematic models which treat it as a passive scalar advected by a prescribed flow and reacting through condensation. Condensation acts as a sink that maintains specific humidity below a prescribed, space-dependent saturation value. To investigate how the interplay between large-scale advection, small-scale turbulence and condensation controls moisture distribution, we develop simple kinematic models which combine a single circulating flow with a Brownian-motion representation of turbulence. We first study the drying mechanism of a water-vapour anomaly released inside a vortex at an initial time. Next, we consider a cellular flow with a moisture source at a boundary. The statistically steady state attained shows features reminiscent of the Hadley cell such as boundary layers, a region of intense precipitation and a relative humidity minimum. Explicit results provide a detailed characterization of these features in the limit of strong flow.
]]>2017-06-28T00:08:07-07:00info:doi/10.1098/rspa.2017.0196hwp:master-id:royprsa;rspa.2017.01962017-06-28Research articles47322022017019620170196<![CDATA[Is a time symmetric interpretation of quantum theory possible without retrocausality?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160607?rss=1
Huw Price has proposed an argument that suggests a time symmetric ontology for quantum theory must necessarily be retrocausal, i.e. it must involve influences that travel backwards in time. One of Price's assumptions is that the quantum state is a state of reality. However, one of the reasons for exploring retrocausality is that it offers the potential for evading the consequences of no-go theorems, including recent proofs of the reality of the quantum state. Here, we show that this assumption can be replaced by a different assumption, called -mediation, that plausibly holds independently of the status of the quantum state. We also reformulate the other assumptions behind the argument to place them in a more general framework and pin down the notion of time symmetry involved more precisely. We show that our assumptions imply a timelike analogue of Bell's local causality criterion and, in doing so, give a new interpretation of timelike violations of Bell inequalities. Namely, they show the impossibility of a (non-retrocausal) time symmetric ontology.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2016.0607hwp:master-id:royprsa;rspa.2016.06072017-06-21Research articles47322022016060720160607<![CDATA[Liquid toroidal drop under uniform electric field]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160633?rss=1
The problem of a stationary liquid toroidal drop freely suspended in another fluid and subjected to an electric field uniform at infinity is addressed analytically. Taylor’s discriminating function implies that, when the phases have equal viscosities and are assumed to be slightly conducting (leaky dielectrics), a spherical drop is stationary when Q=(2R^{2}+3R+2)/(7R^{2}), where R and Q are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, Ca_{E}, that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (Ca_{E}=0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some Ca_{E}>0. This work finds Q and Ca_{E} such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radius and volume 4/3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The found Q and Ca_{E} depend on R and , and for large , e.g. ≥3, they have simple approximations: Q~(R^{2}+R+1)/(3R^{2}) and CaE~33/2 (6 ln +2 ln[96]–9)/(12 ln +4 ln[96]–17) (R+1)2/(R–1)2.
]]>2017-06-21T03:05:36-07:00info:doi/10.1098/rspa.2016.0633hwp:master-id:royprsa;rspa.2016.06332017-06-21Research articles47322022016063320160633<![CDATA[Energy-based analysis of biomolecular pathways]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160825?rss=1
Decomposition of biomolecular reaction networks into pathways is a powerful approach to the analysis of metabolic and signalling networks. Current approaches based on analysis of the stoichiometric matrix reveal information about steady-state mass flows (reaction rates) through the network. In this work, we show how pathway analysis of biomolecular networks can be extended using an energy-based approach to provide information about energy flows through the network. This energy-based approach is developed using the engineering-inspired bond graph methodology to represent biomolecular reaction networks. The approach is introduced using glycolysis as an exemplar; and is then applied to analyse the efficiency of free energy transduction in a biomolecular cycle model of a transporter protein [sodium-glucose transport protein 1 (SGLT1)]. The overall aim of our work is to present a framework for modelling and analysis of biomolecular reactions and processes which considers energy flows and losses as well as mass transport.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2016.0825hwp:master-id:royprsa;rspa.2016.08252017-06-21Research articles47322022016082520160825<![CDATA[The rolling suitcase instability: a coupling between translation and rotation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170076?rss=1
A two-wheel suitcase or trolley can exhibit undamped rocking oscillations from one wheel to the other when pulled fast enough. We study this instability both experimentally—with a toy model of a suitcase rolling on a treadmill—and theoretically. The suitcase oscillates only if a finite perturbation is applied. This is because intrinsic dissipation occurs when the supporting wheel switches. When unstable, the suitcase either increasingly rocks until overturning or reaches a stable limit cycle. The friction force at the rolling wheels constrains wheels to roll without slipping. This constraint imposes a coupling between the translational motion and the three-dimensional rotational motion of the suitcase that drives the rocking instability. The same behaviours are observed in the experiments and in the simulations. The asymptotic scaling laws we observe in the simulations are explained by means of a simplified model where the coupling force is explicit.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0076hwp:master-id:royprsa;rspa.2017.00762017-06-21Research articles47322022017007620170076<![CDATA[A free-boundary model of diffusive valley growth: theory and observation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170159?rss=1
Valleys that form around a stream head often develop characteristic finger-like elevation contours. We study the processes involved in the formation of these valleys and introduce a theoretical model that indicates how shape may inform the underlying processes. We consider valley growth as the advance of a moving boundary travelling forward purely through linearly diffusive erosion, and we obtain a solution for the valley shape in three dimensions. Our solution compares well to the shape of slowly growing groundwater-fed valleys found in Bristol, Florida. Our results identify a new feature in the formation of groundwater-fed valleys: a spatially variable diffusivity that can be modelled by a fixed-height moving boundary.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0159hwp:master-id:royprsa;rspa.2017.01592017-06-21Research articles47322022017015920170159<![CDATA[New conformal mapping for adaptive resolving of the complex singularities of Stokes wave]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170198?rss=1
A new highly efficient method is developed for computation of travelling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for the Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0198hwp:master-id:royprsa;rspa.2017.01982017-06-21Research articles47322022017019820170198<![CDATA[Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160759?rss=1
In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude–frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken’s delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.
]]>2017-06-14T00:07:50-07:00info:doi/10.1098/rspa.2016.0759hwp:master-id:royprsa;rspa.2016.07592017-06-14Research articles47322022016075920160759<![CDATA[On the regularization of impact without collision: the Painleve paradox and compliance]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160773?rss=1
We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.
]]>2017-06-14T00:07:50-07:00info:doi/10.1098/rspa.2016.0773hwp:master-id:royprsa;rspa.2016.07732017-06-14Research articles47322022016077320160773<![CDATA[Regularly configured structures with polygonal prisms for three-dimensional auxetic behaviour]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160926?rss=1
We report here structures, constructed with regular polygonal prisms, that exhibit negative Poisson’s ratios. In particular, we show how we can construct such a structure with regular n-gonal prism-shaped unit cells that are again built with regular n-gonal component prisms. First, we show that the only three possible values for n are 3, 4 and 6 and then discuss how we construct the unit cell again with regular n-gonal component prisms. Then, we derive Poisson’s ratio formula for each of the three structures and show, by analysis and numerical verification, that the structures possess negative Poisson’s ratio under certain geometric conditions.
]]>2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2016.0926hwp:master-id:royprsa;rspa.2016.09262017-06-14Research articles47322022016092620160926<![CDATA[Windowed Green function method for the Helmholtz equation in the presence of multiply layered media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170161?rss=1
This paper presents a new methodology for the solution of problems of two- and three-dimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in the presence of an arbitrary number of penetrable layers. Relying on the use of certain slow-rise windowing functions, the proposed windowed Green function approach efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (Bruno et al. 2016 SIAM J. Appl. Math.76, 1871–1898. (doi:10.1137/15M1033782)), is fast, accurate, flexible and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper, the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on the use of Sommerfeld integrals.
]]>2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2017.0161hwp:master-id:royprsa;rspa.2017.01612017-06-14Research articles47322022017016120170161<![CDATA[The cutting of metals via plastic buckling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160863?rss=1
The cutting of metals has long been described as occurring by laminar plastic flow. Here we show that for metals with large strain-hardening capacity, laminar flow mode is unstable and cutting instead occurs by plastic buckling of a thin surface layer. High speed in situ imaging confirms that the buckling results in a small bump on the surface which then evolves into a fold of large amplitude by rotation and stretching. The repeated occurrence of buckling and folding manifests itself at the mesoscopic scale as a new flow mode with significant vortex-like components—sinuous flow. The buckling model is validated by phenomenological observations of flow at the continuum level and microstructural characteristics of grain deformation and measurements of the folding. In addition to predicting the conditions for surface buckling, the model suggests various geometric flow control strategies that can be effectively implemented to promote laminar flow, and suppress sinuous flow in cutting, with implications for industrial manufacturing processes. The observations impinge on the foundations of metal cutting by pointing to the key role of stability of laminar flow in determining the mechanism of material removal, and the need to re-examine long-held notions of large strain deformation at surfaces.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2016.0863hwp:master-id:royprsa;rspa.2016.08632017-06-07Research articles47322022016086320160863<![CDATA[Knotted fields and explicit fibrations for lemniscate knots]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160829?rss=1
We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme–Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2016.0829hwp:master-id:royprsa;rspa.2016.08292017-06-07Research articles47322022016082920160829<![CDATA[Nonlinear evolution of a thin anodic film]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160930?rss=1
The formation of pores in anodic aluminium oxide films is treated with a model equation. The model treats the oxide layer as a thin viscous liquid in two dimensions. Surface tension on the top boundary, electrostriction due to the external electric field and mass flow through the bottom boundary due to oxide formation are all included. Viscous flow is treated with the creeping flow assumption. The model equation is solved numerically using a Fourier spectral method in space and Adams–Bashforth/Adams–Moulton methods in time. Initial conditions include sinusoidal shapes as well as random shapes. The results show that pores form at the trough of the initial sinusoidal shape. Random shapes get smoothed before forming pore structures with spacing different than predicted by linear theory.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2016.0930hwp:master-id:royprsa;rspa.2016.09302017-06-07Research articles47322022016093020160930<![CDATA[Addressing the discrepancy of finding the equilibrium melting point of silicon using molecular dynamics simulations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170084?rss=1
We performed molecular dynamics simulations to study the equilibrium melting point of silicon using (i) the solid–liquid coexistence method and (ii) the Gibbs free energy technique, and compared our novel results with the previously published results obtained from the Monte Carlo (MC) void-nucleated melting method based on the Tersoff-ARK interatomic potential (Agrawal et al. Phys. Rev. B72, 125206. (doi:10.1103/PhysRevB.72.125206)). Considerable discrepancy was observed (approx. 20%) between the former two methods and the MC void-nucleated melting result, leading us to question the applicability of the empirical MC void-nucleated melting method to study a wide range of atomic and molecular systems. A wider impact of the study is that it highlights the bottleneck of the Tersoff-ARK potential in correctly estimating the melting point of silicon.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2017.0084hwp:master-id:royprsa;rspa.2017.00842017-06-07Research articles47322022017008420170084<![CDATA[Wave reflection and transmission in multiply stented blood vessels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170015?rss=1
Closed circulatory systems display an exquisite balance between vascular elasticity and viscous fluid effects, to induce pulse-smoothing and avoid resonance during the cardiac cycle. Stents in the arterial tree alter this balance through stiffening and because a periodic structure is introduced, capable of interacting with the fluid in a complex way. While the former feature has been investigated, the latter received no attention so far. But periodic structures are the building blocks of metamaterials, known for their ‘non-natural’ behaviour. Thus, the investigation of a stent's periodic microstructure dynamical interactions is crucial to assess possible pathological responses. A one-dimensional fluid–structure interaction model, simple enough to allow an analytical solution for situations of interest involving one or two interacting stents, is introduced. It is determined: (i) whether or not frequency bands exist in which reflected blood pulses are highly increased and (ii) if these bands are close to the characteristic frequencies of arteries and finally, (iii) if the internal structure of the stent can sensibly affect arterial blood dynamics. It is shown that, while the periodic structure of an isolated stent can induce anomalous reflection only in pathological conditions, the presence of two interacting stents is more critical, and high reflection can occur at frequencies not far from the physiological values.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0015hwp:master-id:royprsa;rspa.2017.00152017-06-07Research articles47322022017001520170015<![CDATA[Ultrasonic defect characterization using parametric-manifold mapping]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170056?rss=1
The aim of ultrasonic non-destructive evaluation includes the detection and characterization of defects, and an understanding of the nature of defects is essential for the assessment of structural integrity in safety critical systems. In general, the defect characterization challenge involves an estimation of defect parameters from measured data. In this paper, we explore the extent to which defects can be characterized by their ultrasonic scattering behaviour. Given a number of ultrasonic measurements, we show that characterization information can be extracted by projecting the measurement onto a parametric manifold in principal component space. We show that this manifold represents the entirety of the characterization information available from far-field harmonic ultrasound. We seek to understand the nature of this information and hence provide definitive statements on the defect characterization performance that is, in principle, extractable from typical measurement scenarios. In experiments, the characterization problem of surface-breaking cracks and the more general problem of elliptical voids are studied, and a good agreement is achieved between the actual parameter values and the characterization results. The nature of the parametric manifold enables us to explain and quantify why some defects are relatively easy to characterize, whereas others are inherently challenging.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0056hwp:master-id:royprsa;rspa.2017.00562017-06-07Research articles47322022017005620170056<![CDATA[Analytical solutions for two-dimensional Stokes flow singularities in a no-slip wedge of arbitrary angle]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170134?rss=1
An analytical method to find the flow generated by the basic singularities of Stokes flow in a wedge of arbitrary angle is presented. Specifically, we solve a biharmonic equation for the stream function of the flow generated by a point stresslet singularity and satisfying no-slip boundary conditions on the two walls of the wedge. The method, which is readily adapted to any other singularity type, takes full account of any transcendental singularities arising at the corner of the wedge. The approach is also applicable to problems of plane strain/stress of an elastic solid where the biharmonic equation also governs the Airy stress function.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2017.0134hwp:master-id:royprsa;rspa.2017.01342017-06-07Research articles47322022017013420170134<![CDATA[Extensional channel flow revisited: a dynamical systems perspective]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170151?rss=1
Extensional self-similar flows in a channel are explored numerically for arbitrary stretching–shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. Previous studies focused on the analysis of branches of steady solutions for particular stretching–shrinking rates, although recent studies focused also on the dynamical aspects of the problems. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space. For large Reynolds numbers, the restriction to self-similarity results in simple flows with no realistic behaviour, but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic Navier–Stokes solutions.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0151hwp:master-id:royprsa;rspa.2017.01512017-06-07Research articles47322022017015120170151<![CDATA[From arteries to boreholes: steady-state response of a poroelastic cylinder to fluid injection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160753?rss=1
The radially outward flow of fluid into a porous medium occurs in many practical problems, from transport across vascular walls to the pressurization of boreholes. As the driving pressure becomes non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid has an increasingly strong impact on the flow. For very large pressures or very soft materials, as is the case for hydraulic fracturing and arterial flows, this coupling can lead to large deformations and, hence, to strong deviations from a classical, linear-poroelastic response. Here, we study this problem by analysing the steady-state response of a poroelastic cylinder to fluid injection. We consider the qualitative and quantitative impacts of kinematic and constitutive nonlinearity, highlighting the strong impact of deformation-dependent permeability. We show that the wall thickness (thick versus thin) and the outer boundary condition (free versus constrained) play a central role in controlling the mechanics.
]]>2017-05-31T00:07:48-07:00info:doi/10.1098/rspa.2016.0753hwp:master-id:royprsa;rspa.2016.07532017-05-31Research articles47322012016075320160753<![CDATA[Formula graded discrete Lax pairs and Yang-Baxter maps]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160946?rss=1
We recently introduced a class of ZN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper, we introduce the corresponding Yang–Baxter maps. Many well-known examples belong to this scheme for N=2, so, for N≥3, our systems may be regarded as generalizations of these. In particular, for each N we introduce a class of multi-component Yang–Baxter maps, which include H^{B}_{III} (of Papageorgiou et al. 2010 SIGMA 6, 003 (9 p). (doi:10.3842/SIGMA.2010.033)), when N=2, and that associated with the discrete modified Boussinesq equation, for N=3. For N≥5 we introduce a new family of Yang–Baxter maps, which have no lower dimensional analogue. We also present new multi-component versions of the Yang–Baxter maps F_{IV} and F_{V} (given in the classification of Adler et al. 2004 Commun. Anal. Geom. 12, 967–1007. (doi:10.4310/CAG.2004.v12.n5.a1)).
]]>2017-05-31T00:07:48-07:00info:doi/10.1098/rspa.2016.0946hwp:master-id:royprsa;rspa.2016.09462017-05-31Research articles47322012016094620160946<![CDATA[Passive vibration control: a structure-immittance approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170011?rss=1
Linear passive vibration absorbers, such as tuned mass dampers, often contain springs, dampers and masses, although recently there has been a growing trend to employ or supplement the mass elements with inerters. When considering possible configurations with these elements broadly, two approaches are normally used: one structure-based and one immittance-based. Both approaches have their advantages and disadvantages. In this paper, a new approach is proposed: the structure–immittance approach. Using this approach, a full set of possible series–parallel networks with predetermined numbers of each element type can be represented by structural immittances, obtained via a proposed general formulation process. Using the structural immittances, both the ability to investigate a class of absorber possibilities together (advantage of the immittance-based approach), and the ability to control the complexity, topology and element values in resulting absorber configurations (advantages of the structure-based approach) are provided at the same time. The advantages of the proposed approach are demonstrated through two case studies on building vibration suppression and automotive suspension design, respectively.
]]>2017-05-31T00:07:48-07:00info:doi/10.1098/rspa.2017.0011hwp:master-id:royprsa;rspa.2017.00112017-05-31Research articles47322012017001120170011<![CDATA[Nonlinear waves in solids with slow dynamics: an internal-variable model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170024?rss=1
In heterogeneous solids such as rocks and concrete, the speed of sound diminishes with the strain amplitude of a dynamic loading (softening). This decrease, known as ‘slow dynamics’, occurs at time scales larger than the period of the forcing. Also, hysteresis is observed in the steady-state response. The phenomenological model by Vakhnenko et al. (2004 Phys. Rev. E 70, 015602. (doi:10.1103/PhysRevE.70.015602)) is based on a variable that describes the softening of the material. However, this model is one dimensional and it is not thermodynamically admissible. In the present article, a three-dimensional model is derived in the framework of the finite-strain theory. An internal variable that describes the softening of the material is introduced, as well as an expression of the specific internal energy. A mechanical constitutive law is deduced from the Clausius–Duhem inequality. Moreover, a family of evolution equations for the internal variable is proposed. Here, an evolution equation with one relaxation time is chosen. By construction, this new model of the continuum is thermodynamically admissible and dissipative (inelastic). In the case of small uniaxial deformations, it is shown analytically that the model reproduces qualitatively the main features of real experiments.
]]>2017-05-31T00:07:48-07:00info:doi/10.1098/rspa.2017.0024hwp:master-id:royprsa;rspa.2017.00242017-05-31Research articles47322012017002420170024<![CDATA[Curvature-driven morphing of non-Euclidean shells]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170087?rss=1
We investigate how thin structures change their shape in response to non-mechanical stimuli that can be interpreted as variations in the structure’s natural curvature. Starting from the theory of non-Euclidean plates and shells, we derive an effective model that reduces a three-dimensional stimulus to the natural fundamental forms of the mid-surface of the structure, incorporating expansion, or growth, in the thickness. Then, we apply the model to a variety of thin bodies, from flat plates to spherical shells, obtaining excellent agreement between theory and numerics. We show how cylinders and cones can either bend more or unroll, and eventually snap and rotate. We also study the nearly isometric deformations of a spherical shell and describe how this shape change is ruled by the geometry of a spindle. As the derived results stem from a purely geometrical model, they are general and scalable.
]]>2017-05-31T00:07:48-07:00info:doi/10.1098/rspa.2017.0087hwp:master-id:royprsa;rspa.2017.00872017-05-31Research articles47322012017008720170087<![CDATA[Causal dissipation for the relativistic dynamics of ideal gases]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160729?rss=1
We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier–Stokes equations.
]]>2017-05-24T00:07:43-07:00info:doi/10.1098/rspa.2016.0729hwp:master-id:royprsa;rspa.2016.07292017-05-24Research articles47322012016072920160729<![CDATA[Probing quantum state space: does one have to learn everything to learn something?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160866?rss=1
Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement set-up that reveals this property with as little effort as possible. Here, we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the task as a membership problem related to a partitioning of the quantum state space and, in doing so, connect it to the geometry of the state space. For a general membership problem, we prove various sufficient criteria that force informational completeness, and we explicitly treat several physically relevant examples. For the specific cases that do not require informational completeness, we also determine bounds on the minimal number of measurement outcomes needed to ensure success in the task.
]]>2017-05-24T00:07:44-07:00info:doi/10.1098/rspa.2016.0866hwp:master-id:royprsa;rspa.2016.08662017-05-24Research articles47322012016086620160866<![CDATA[Exact semi-separation of variables in waveguides with non-planar boundaries]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170017?rss=1
Series expansions of unknown fields =nZn in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions Z_{n} are determined by solving local Sturm–Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to Z_{n} cannot be compatible with the physical boundary conditions of , leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in Athanassoulis & Belibassakis (Athanassoulis & Belibassakis 1999 J. Fluid Mech. 389, 275–301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly convergent representation of the field , valid for any smooth, non-planar boundaries and any smooth enough . This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.
]]>2017-05-24T00:07:43-07:00info:doi/10.1098/rspa.2017.0017hwp:master-id:royprsa;rspa.2017.00172017-05-24Research articles47322012017001720170017<![CDATA[Relative geodesics in bi-invariant Lie groups]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160619?rss=1
Motivated by registration problems, this paper deals with a curve matching problem in homogeneous spaces. Let G be a connected finite-dimensional bi-invariant Lie group and K a closed subgroup. A smooth curve g in G is said to be admissible if it can transform two smooth curves f_{1} and f_{2} in G/K from one to the other. An (f_{1}, f_{2})-relative geodesic (Holm et al. 2013 Proc. R. Soc. A469, 20130297. (doi:10.1098/rspa.2013.0297)) is defined as a critical point of the total energy E(g) as g varies in the set of all (f_{1}, f_{2})-admissible curves. We obtain the Euler–Lagrange equation, a first-order differential equation, satisfied by a relative geodesic. Furthermore, the Euler–Lagrange equation is simplified for the case where G/K is globally symmetric. As a concrete example, relative geodesics are found for special cases where G is SO(3) and K is SO(2). As an application of discrepancy for curves in S^{2}, we construct and study a new measure of non-congruency for constant speed curves in Euclidean 3-space. Numerical examples are given to illustrate results.
]]>2017-05-17T00:07:29-07:00info:doi/10.1098/rspa.2016.0619hwp:master-id:royprsa;rspa.2016.06192017-05-17Research articles47322012016061920160619<![CDATA[Convective instabilities of Maxwell-Cattaneo fluids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160712?rss=1
Motivated by the need to understand better the dynamics of non-Fourier fluids, we examine the linear and weakly nonlinear stabilities of a horizontal layer of fluid obeying the Maxwell–Cattaneo relationship of heat flux and temperature using three different forms of the time derivative of the heat flux. Linear stability mode regime diagrams in the parameter plane have been established and used to summarize the linear instabilities. The energy balance of the system is used to identify the mechanism by which the Maxwell–Cattaneo effect (i) introduces overstability, (ii) leads to preferred stationary modes with the critical Rayleigh and wavelengths either both increasing or both decreasing, (iii) gives rise to instabilities in a layer heated from above, and (iv) enhances heat transfer. A formal weakly nonlinear analysis leads to evolution equations for the amplitudes of linear instability modes. It is shown that the amplitude of the stationary mode obeys an equation of the Landau–Stuart type. The two equally excitable overstable modes obey two equations of the same type coupled by an interaction term. The evolution of the different amplitudes leads to supercritical stability, supercritical instability or subcritical instability depending on the model and parameter values. The results are presented in regime diagrams.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2016.0712hwp:master-id:royprsa;rspa.2016.07122017-05-17Research articles47322012016071220160712<![CDATA[Aeroacoustic catastrophes: upstream cusp beaming in Lilley's equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160880?rss=1
The downstream propagation of high-frequency acoustic waves from a point source in a subsonic jet obeying Lilley's equation is well known to be organized around the so-called ‘cone of silence’, a fold catastrophe across which the amplitude may be modelled uniformly using Airy functions. Here we show that acoustic waves not only unexpectedly propagate upstream, but also are organized at constant distance from the point source around a cusp catastrophe with amplitude modelled locally by the Pearcey function. Furthermore, the cone of silence is revealed to be a cross-section of a swallowtail catastrophe. One consequence of these discoveries is that the peak acoustic field upstream is not only structurally stable but also at a similar level to the known downstream field. The fine structure of the upstream cusp is blurred out by distributions of symmetric acoustic sources, but peak upstream acoustic beaming persists when asymmetries are introduced, from either arrays of discrete point sources or perturbed continuum ring source distributions. These results may pose interesting questions for future novel jet-aircraft engine designs where asymmetric source distributions arise.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2016.0880hwp:master-id:royprsa;rspa.2016.08802017-05-17Research articles47322012016088020160880<![CDATA[Crossing the quasi-threshold manifold of a noise-driven excitable system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170058?rss=1
We consider the noise-induced escapes in an excitable system possessing a quasi-threshold manifold, along which there exists a certain point of minimal quasi-potential. In the weak noise limit, the optimal escaping path turns out to approach this particular point asymptotically, making it analogous to an ordinary saddle. Numerical simulations are performed and an elaboration on the effect of small but finite noise is given, which shows that the ridges where the prehistory probability distribution peaks are located mainly within the region where the quasi-potential increases gently. The cases allowing anisotropic noise are discussed and we found that varying the noise term in the slow variable would dramatically raise the whole level of quasi-potentials, leading to significant changes in both patterns of optimal paths and exit locations.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2017.0058hwp:master-id:royprsa;rspa.2017.00582017-05-17Research articles47322012017005820170058<![CDATA[Uncertainty relations on nilpotent Lie groups]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170082?rss=1
We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic Rn, and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2017.0082hwp:master-id:royprsa;rspa.2017.00822017-05-17Research articles47322012017008220170082<![CDATA[A note on the self-similar solutions to the spontaneous fragmentation equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160740?rss=1
We provide a method to compute self-similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours. Our procedure is applied to specific cases: (i) the case of mitosis, where fragmentation results into two identical fragments, (ii) fragmentation limited to the formation of sufficiently large fragments, and (iii) processes with fragmentation kernel presenting a power-like behaviour.
]]>2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2016.0740hwp:master-id:royprsa;rspa.2016.07402017-05-10Research articles47322012016074020160740<![CDATA[Dispersionless (3+1)-dimensional integrable hierarchies]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160857?rss=1
In this paper, we introduce a multi-dimensional version of the R-matrix approach to the construction of integrable hierarchies. Applying this method to the case of the Lie algebra of functions with respect to the contact bracket, we construct integrable hierarchies of (3+1)-dimensional dispersionless systems of the type recently introduced in Sergyeyev (2014 (http://arxiv.org/abs/1401.2122)).
]]>2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2016.0857hwp:master-id:royprsa;rspa.2016.08572017-05-10Research articles47322012016085720160857<![CDATA[An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170080?rss=1
In Parnell & Abrahams (2008 Proc. R. Soc. A464, 1461–1482. (doi:10.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.
]]>2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2017.0080hwp:master-id:royprsa;rspa.2017.00802017-05-10Research articles47322012017008020170080<![CDATA[Elementary exact calculations of degree growth and entropy for discrete equations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160831?rss=1
Second-order discrete equations are studied over the field of rational functions C(z), where z is a variable not appearing in the equation. The exact degree of each iterate as a function of z can be calculated easily using the standard calculations that arise in singularity confinement analysis, even when the singularities are not confined. This produces elementary yet rigorous entropy calculations.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2016.0831hwp:master-id:royprsa;rspa.2016.08312017-05-03Research articles47322012016083120160831<![CDATA[Well-posed continuum equations for granular flow with compressibility and {mu}(I)-rheology]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160846?rss=1
Continuum modelling of granular flow has been plagued with the issue of ill-posed dynamic equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent μ(I)-rheology is ill-posed when the non-dimensional inertial number I is too high or too low. Here, incorporating ideas from critical-state soil mechanics, we derive conditions for well-posedness of partial differential equations that combine compressibility with I-dependent rheology. When the I-dependence comes from a specific friction coefficient μ(I), our results show that, with compressibility, the equations are well-posed for all deformation rates provided that μ(I) satisfies certain minimal, physically natural, inequalities.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2016.0846hwp:master-id:royprsa;rspa.2016.08462017-05-03Research articles47322012016084620160846<![CDATA[Simulating tubulin-associated unit transport in an axon: using bootstrapping for estimating confidence intervals of best-fit parameter values obtained from indirect experimental data]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170045?rss=1
In this paper, we first develop a model of axonal transport of tubulin-associated unit (tau) protein. We determine the minimum number of parameters necessary to reproduce published experimental results, reducing the number of parameters from 18 in the full model to eight in the simplified model. We then address the following questions: Is it possible to estimate parameter values for this model using the very limited amount of published experimental data? Furthermore, is it possible to estimate confidence intervals for the determined parameters? The idea that is explored in this paper is based on using bootstrapping. Model parameters were estimated by minimizing the objective function that simulates the discrepancy between the model predictions and experimental data. Residuals were then identified by calculating the differences between the experimental data and model predictions. New, surrogate ‘experimental’ data were generated by randomly resampling residuals. By finding sets of best-fit parameters for a large number of surrogate data the histograms for the model parameters were produced. These histograms were then used to estimate confidence intervals for the model parameters, by using the percentile bootstrap. Once the model was calibrated, we applied it to analysing some features of tau transport that are not accessible to current experimental techniques.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2017.0045hwp:master-id:royprsa;rspa.2017.00452017-05-03Research articles47322012017004520170045<![CDATA[The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170050?rss=1
New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2017.0050hwp:master-id:royprsa;rspa.2017.00502017-05-03Research articles47322012017005020170050<![CDATA[Geometric decompositions of collective motion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160571?rss=1
Collective motion in nature is a captivating phenomenon. Revealing the underlying mechanisms, which are of biological and theoretical interest, will require empirical data, modelling and analysis techniques. Here, we contribute a geometric viewpoint, yielding a novel method of analysing movement. Snapshots of collective motion are portrayed as tangent vectors on configuration space, with length determined by the total kinetic energy. Using the geometry of fibre bundles and connections, this portrait is split into orthogonal components each tangential to a lower dimensional manifold derived from configuration space. The resulting decomposition, when interleaved with classical shape space construction, is categorized into a family of kinematic modes—including rigid translations, rigid rotations, inertia tensor transformations, expansions and compressions. Snapshots of empirical data from natural collectives can be allocated to these modes and weighted by fractions of total kinetic energy. Such quantitative measures can provide insight into the variation of the driving goals of a collective, as illustrated by applying these methods to a publicly available dataset of pigeon flocking. The geometric framework may also be profitably employed in the control of artificial systems of interacting agents such as robots.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0571hwp:master-id:royprsa;rspa.2016.05712017-04-26Research articles47322002016057120160571<![CDATA[Flow through a very porous obstacle in a shallow channel]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160672?rss=1
A theoretical model, informed by numerical simulations based on the shallow water equations, is developed to predict the flow passing through and around a uniform porous obstacle in a shallow channel, where background friction is important. This problem is relevant to a number of practical situations, including flow through aquatic vegetation, the performance of arrays of turbines in tidal channels and hydrodynamic forces on offshore structures. To demonstrate this relevance, the theoretical model is used to (i) reinterpret core flow velocities in existing laboratory-based data for an array of emergent cylinders in shallow water emulating aquatic vegetation and (ii) reassess the optimum arrangement of tidal turbines to generate power in a tidal channel. Comparison with laboratory-based data indicates a maximum obstacle resistance (or minimum porosity) for which the present theoretical model is valid. When the obstacle resistance is above this threshold the shallow water equations do not provide an adequate representation of the flow, and the theoretical model over-predicts the core flow passing through the obstacle. The second application of the model confirms that natural bed resistance increases the power extraction potential for a partial tidal fence in a shallow channel and alters the optimum arrangement of turbines within the fence.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0672hwp:master-id:royprsa;rspa.2016.06722017-04-26Research articles47322002016067220160672<![CDATA[Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160765?rss=1
All metamaterial applications are based upon the idea that extreme material properties can be achieved through appropriate dynamic homogenization of composites. This homogenization is almost always done for infinite domains and the results are then applied to finite samples. This process ignores the evanescent waves which appear at the boundaries of such finite samples. In this paper, we first clarify the emergence and purpose of these evanescent waves in a model problem consisting of an interface between a layered composite and a homogeneous medium. We show that these evanescent waves form boundary layers on either side of the interface beyond which the composite can be represented by appropriate infinite domain homogenized relations. We show that if one ignores the boundary layers, then the displacement and stress fields are discontinuous across the interface. Therefore, the scattering coefficients at such an interface cannot be determined through the conventional continuity conditions involving only propagating modes. Here, we propose an approximate variational approach for sidestepping these boundary layers. The aim is to determine the scattering coefficients without the knowledge of evanescent modes. Through various numerical examples we show that our technique gives very good estimates of the actual scattering coefficients beyond the long wavelength limit.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0765hwp:master-id:royprsa;rspa.2016.07652017-04-26Research articles47322002016076520160765<![CDATA[Design and simulation of origami structures with smooth folds]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160716?rss=1
Origami has enabled new approaches to the fabrication and functionality of multiple structures. Current methods for origami design are restricted to the idealization of folds as creases of zeroth-order geometric continuity. Such an idealization is not proper for origami structures of non-negligible fold thickness or maximum curvature at the folds restricted by material limitations. For such structures, folds are not properly represented as creases but rather as bent regions of higher-order geometric continuity. Such fold regions of arbitrary order of continuity are termed as smooth folds. This paper presents a method for solving the following origami design problem: given a goal shape represented as a polygonal mesh (termed as the goal mesh), find the geometry of a single planar sheet, its pattern of smooth folds, and the history of folding motion allowing the sheet to approximate the goal mesh. The parametrization of the planar sheet and the constraints that allow for a valid pattern of smooth folds are presented. The method is tested against various goal meshes having diverse geometries. The results show that every determined sheet approximates its corresponding goal mesh in a known folded configuration having fold angles obtained from the geometry of the goal mesh.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0716hwp:master-id:royprsa;rspa.2016.07162017-04-26Research articles47322002016071620160716<![CDATA[Wave directional spreading from point field measurements]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160781?rss=1
Ocean waves have multidirectional components. Most wave measurements are taken at a single point, and so fail to capture information about the relative directions of the wave components directly. Conventional means of directional estimation require a minimum of three concurrent time series of measurements at different spatial locations in order to derive information on local directional wave spreading. Here, the relationship between wave nonlinearity and directionality is utilized to estimate local spreading without the need for multiple concurrent measurements, following Adcock & Taylor (Adcock & Taylor 2009 Proc. R. Soc. A465, 3361–3381. (doi:10.1098/rspa.2009.0031)), with the assumption that directional spreading is frequency independent. The method is applied to measurements recorded at the North Alwyn platform in the northern North Sea, and the results compared against estimates of wave spreading by conventional measurement methods and hindcast data. Records containing freak waves were excluded. It is found that the method provides accurate estimates of wave spreading over a range of conditions experienced at North Alwyn, despite the noisy chaotic signals that characterize such ocean wave data. The results provide further confirmation that Adcock and Taylor's method is applicable to metocean data and has considerable future promise as a technique to recover estimates of wave spreading from single point wave measurement devices.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0781hwp:master-id:royprsa;rspa.2016.07812017-04-26Research articles47322002016078120160781<![CDATA[Element analysis: a wavelet-based method for analysing time-localized events in noisy time series]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160776?rss=1
A method is derived for the quantitative analysis of signals that are composed of superpositions of isolated, time-localized ‘events’. Here, these events are taken to be well represented as rescaled and phase-rotated versions of generalized Morse wavelets, a broad family of continuous analytic functions. Analysing a signal composed of replicates of such a function using another Morse wavelet allows one to directly estimate the properties of events from the values of the wavelet transform at its own maxima. The distribution of events in general power-law noise is determined in order to establish significance based on an expected false detection rate. Finally, an expression for an event’s ‘region of influence’ within the wavelet transform permits the formation of a criterion for rejecting spurious maxima due to numerical artefacts or other unsuitable events. Signals can then be reconstructed based on a small number of isolated points on the time/scale plane. This method, termed element analysis, is applied to the identification of long-lived eddy structures in ocean currents as observed by along-track measurements of sea surface elevation from satellite altimetry.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0776hwp:master-id:royprsa;rspa.2016.07762017-04-26Research articles47322002016077620160776<![CDATA[Wrinkle surface instability of an inhomogeneous elastic block with graded stiffness]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160882?rss=1
Surface instabilities have been studied extensively for both homogeneous materials and film/substrate structures but relatively less for materials with continuously varying properties. This paper studies wrinkle surface instability of a graded neo-Hookean block with exponentially varying modulus under plane strain by using the linear bifurcation analysis. We derive the first variation condition for minimizing the potential energy functional and solve the linearized equations of equilibrium to find the necessary conditions for surface instability. It is found that for a homogeneous block or an inhomogeneous block with increasing modulus from the surface, the critical stretch for surface instability is 0.544 (0.456 strain), which is independent of the geometry and the elastic modulus on the surface of the block. This critical stretch coincides with that reported by Biot (1963 Appl. Sci. Res.12, 168–182. (doi:10.1007/BF03184638)) 53 years ago for the onset of wrinkle instabilities in a half-space of homogeneous neo-Hookean materials. On the other hand, for an inhomogeneous block with decreasing modulus from the surface, the critical stretch for surface instability ranges from 0.544 to 1 (0–0.456 strain), depending on the modulus gradient, and the length and height of the block. This sheds light on the effects of the material inhomogeneity and structural geometry on surface instability.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0882hwp:master-id:royprsa;rspa.2016.08822017-04-26Research articles47322002016088220160882<![CDATA[Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160809?rss=1
In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0809hwp:master-id:royprsa;rspa.2016.08092017-04-26Research articles47322002016080920160809<![CDATA[Stochastic modelling of membrane filtration]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160948?rss=1
Membrane fouling during particle filtration occurs through a variety of mechanisms, including internal pore clogging by contaminants, coverage of pore entrances and deposition on the membrane surface. In this paper, we present an efficient method for modelling the behaviour of a filter, which accounts for different retention mechanisms, particle sizes and membrane geometries. The membrane is assumed to be composed of a series of, possibly interconnected, pores. The central feature is a conductivity function, which describes the blockage of each individual pore as particles arrive, which is coupled with a mechanism to account for the stochastic nature of the arrival times of particles at the pore. The result is a system of ordinary differential equations based on the pore-level interactions. We demonstrate how our model can accurately describe a wide range of filtration scenarios. Specifically, we consider a case where blocking via multiple mechanisms can occur simultaneously, which have previously required the study through individual models; the filtration of a combination of small and large particles by a track-etched membrane and particle separation using interconnected pore networks. The model is significantly faster than comparable stochastic simulations for small networks, enabling its use as a tool for efficient future simulations.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0948hwp:master-id:royprsa;rspa.2016.09482017-04-26Research articles47322002016094820160948<![CDATA[The deferred limit method for long waves in a curved waveguide]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160900?rss=1
This paper presents a technique, based on a deferred approach to a limit, for analysing the dispersion relation for propagation of long waves in a curved waveguide. The technique involves the concept of an analytically satisfactory pair of Bessel functions, which is different from the concept of a numerically satisfactory pair, and simplifies the dispersion relations for curved waveguide problems. Details are presented for long elastic waves in a curved layer, for which symmetric and antisymmetric waves are strongly coupled. The technique gives high-order corrections to a widely used approximate dispersion relation based a kinematic hypothesis, and determines rigorously which of its coefficients are exact.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0900hwp:master-id:royprsa;rspa.2016.09002017-04-26Research articles47322002016090020160900<![CDATA[Deformed quons and bi-coherent states]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170049?rss=1
We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This deformation involves interesting mathematical problems and suggests possible applications to pseudo-hermitian quantum mechanics. We construct bi-coherent states associated to D-pseudo-quons, and we show that they share many of their properties with ordinary coherent states. In particular, we find conditions for these states to exist, to be eigenstates of suitable annihilation operators and to give rise to a resolution of the identity. Two examples are discussed in details, one connected to an unbounded similarity map, and the other to a bounded map.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2017.0049hwp:master-id:royprsa;rspa.2017.00492017-04-26Research articles47322002017004920170049<![CDATA[Exact solution for the Poisson field in a semi-infinite strip]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160908?rss=1
The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.
]]>2017-04-19T00:39:10-07:00info:doi/10.1098/rspa.2016.0908hwp:master-id:royprsa;rspa.2016.09082017-04-19Research articles47322002016090820160908<![CDATA[Coupling strength assumption in statistical energy analysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160927?rss=1
This paper is a discussion of the hypothesis of weak coupling in statistical energy analysis (SEA). The examples of coupled oscillators and statistical ensembles of coupled plates excited by broadband random forces are discussed. In each case, a reference calculation is compared with the SEA calculation. First, it is shown that the main SEA relation, the coupling power proportionality, is always valid for two oscillators irrespective of the coupling strength. But the case of three subsystems, consisting of oscillators or ensembles of plates, indicates that the coupling power proportionality fails when the coupling is strong. Strong coupling leads to non-zero indirect coupling loss factors and, sometimes, even to a reversal of the energy flow direction from low to high vibrational temperature.
]]>2017-04-19T00:39:10-07:00info:doi/10.1098/rspa.2016.0927hwp:master-id:royprsa;rspa.2016.09272017-04-19Research articles47322002016092720160927<![CDATA[A scale-entropy diffusion equation to explore scale-dependent fractality]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170054?rss=1
In the last three decades, fractal geometry became a mathematical tool widely used in physics. Nevertheless, it has been observed that real multi-scale phenomena display a departure to fractality that implies an impossibility to define the multi-scale features with an unique fractal dimension, leading to variations in the scale-space. The scale-entropy diffusion equation theorizes the organization of the scale dynamics involving scale-dependent fractals. A study of the theory is possible through the scale-entropy sink term in the equation and corresponds to precise behaviours in scale-space. In the first part of the paper, we study the scale space features when the scale-entropy sink term is modified. The second part is a numerical investigation and analysis of several solutions of the scale-entropy diffusion equation. By a precise measurement of the transition scales tested on truncated deterministic fractals, we developed a new simple method to estimate fractal dimension which appears to be much better than a classical method. Furthermore, we show that deterministic fractals display intrinsic log-periodic oscillations of the fractal dimension. In order to represent this complex behaviour, we introduce a departure to fractal diagram linking scale-space, scale-dependent fractal dimension and scale-entropy sink. Finally, we construct deterministic scale-dependent fractals and verify the results predicted by the scale-entropy diffusion equation.
]]>2017-04-19T00:05:36-07:00info:doi/10.1098/rspa.2017.0054hwp:master-id:royprsa;rspa.2017.00542017-04-19Research articles47322002017005420170054<![CDATA[Uncertainty transformation via Hopf bifurcation in fast-slow systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160346?rss=1
Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. We show that a random initial condition distribution can be transformed during the passage near a delayed/dynamic Hopf bifurcation: (i) to certain classes of symmetric copies, (ii) to an almost deterministic output, (iii) to a mixture distribution with differing moments and (iv) to a very restricted class of general distributions. We prove under which conditions the cases (i)–(iv) occur in certain classes vector fields.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0346hwp:master-id:royprsa;rspa.2016.03462017-04-12Research articles47322002016034620160346<![CDATA[Group theoretical derivation of the minimal coupling principle]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160629?rss=1
The group theoretical methods worked out by Bargmann, Mackey and Wigner, which deductively establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular first-order invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.
]]>2017-04-12T01:01:34-07:00info:doi/10.1098/rspa.2016.0629hwp:master-id:royprsa;rspa.2016.06292017-04-12Research articles47322002016062920160629<![CDATA[The roles of impact and inertia in the failure of a shoelace knot]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160770?rss=1
The accidental untying of a shoelace while walking often occurs without warning. In this paper, we discuss the series of events that lead to a shoelace knot becoming untied. First, the repeated impact of the shoe on the floor during walking serves to loosen the knot. Then, the whipping motions of the free ends of the laces caused by the leg swing produce slipping of the laces. This leads to eventual runaway untangling of the knot. As demonstrated using slow-motion video footage and a series of experiments, the failure of the knot happens in a matter of seconds, often without warning, and is catastrophic. The controlled experiments showed that increasing inertial effects of the swinging laces leads to increased rate of knot untying, that the directions of the impact and swing influence the rate of failure, and that the knot structure has a profound influence on a knot's tendency to untie under cyclic impact loading.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0770hwp:master-id:royprsa;rspa.2016.07702017-04-12Research articles47322002016077020160770<![CDATA[Robust identification of harmonic oscillator parameters using the adjoint Fokker-Planck equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160894?rss=1
We present a model-based output-only method for identifying from time series the parameters governing the dynamics of stochastically forced oscillators. In this context, suitable models of the oscillator’s damping and stiffness properties are postulated, guided by physical understanding of the oscillatory phenomena. The temporal dynamics and the probability density function of the oscillation amplitude are described by a Langevin equation and its associated Fokker–Planck equation, respectively. One method consists in fitting the postulated analytical drift and diffusion coefficients with their estimated values, obtained from data processing by taking the short-time limit of the first two transition moments. However, this limit estimation loses robustness in some situations—for instance when the data are band-pass filtered to isolate the spectral contents of the oscillatory phenomena of interest. In this paper, we use a robust alternative where the adjoint Fokker–Planck equation is solved to compute Kramers–Moyal coefficients exactly, and an iterative optimization yields the parameters that best fit the observed statistics simultaneously in a wide range of amplitudes and time scales. The method is illustrated with a stochastic Van der Pol oscillator serving as a prototypical model of thermoacoustic instabilities in practical combustors, where system identification is highly relevant to control.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0894hwp:master-id:royprsa;rspa.2016.08942017-04-12Research articles47322002016089420160894<![CDATA[Robust H{infty} state-feedback control for linear systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160934?rss=1
This paper investigates the problem of robust H_{} control for linear systems. First, the state-feedback closed-loop control algorithm is designed. Second, by employing the geometric progression theory, a modified augmented Lyapunov–Krasovskii functional (LKF) with the geometric integral interval is established. Then, parameter uncertainties and the derivative of the delay are flexibly described by introducing the convex combination skill. This technique can eliminate the unnecessary enlargement of the LKF derivative estimation, which gives less conservatism. In addition, the designed controller can ensure that the linear systems are globally asymptotically stable with a guaranteed H_{} performance in the presence of a disturbance input and parameter uncertainties. A liquid monopropellant rocket motor with a pressure feeding system is evaluated in a simulation example. It shows that this proposed state-feedback control approach achieves the expected results for linear systems in the sense of the prescribed H_{} performance.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0934hwp:master-id:royprsa;rspa.2016.09342017-04-12Research articles47322002016093420160934<![CDATA[Large gyres as a shallow-water asymptotic solution of Eulers equation in spherical coordinates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170063?rss=1
Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2017.0063hwp:master-id:royprsa;rspa.2017.00632017-04-12Research articles47322002017006320170063<![CDATA[Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170016?rss=1
This paper presents a mathematical framework for the design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns that can simultaneously fit two target surfaces with rotational symmetry about a common axis. The geometric parameters of the crease pattern and the folding angles of the target folded state are determined through a set of combined geometric and constraint equations. An algorithm to simulate the folding motion of the designed crease pattern is provided. Furthermore, the conditions and procedures to design folded ring structures that are both developable and flat-foldable and stacked folded structures consisting of two layers that can fold independently or compatibly are discussed. The proposed framework has potential applications in designing engineering doubly curved structures such as deployable domes and folded cores for doubly curved sandwich structures on the aircraft.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2017.0016hwp:master-id:royprsa;rspa.2017.00162017-04-12Research articles47322002017001620170016<![CDATA[Statistical emulation of landslide-induced tsunamis at the Rockall Bank, NE Atlantic]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170026?rss=1
Statistical methods constitute a useful approach to understand and quantify the uncertainty that governs complex tsunami mechanisms. Numerical experiments may often have a high computational cost. This forms a limiting factor for performing uncertainty and sensitivity analyses, where numerous simulations are required. Statistical emulators, as surrogates of these simulators, can provide predictions of the physical process in a much faster and computationally inexpensive way. They can form a prominent solution to explore thousands of scenarios that would be otherwise numerically expensive and difficult to achieve. In this work, we build a statistical emulator of the deterministic codes used to simulate submarine sliding and tsunami generation at the Rockall Bank, NE Atlantic Ocean, in two stages. First we calibrate, against observations of the landslide deposits, the parameters used in the landslide simulations. This calibration is performed under a Bayesian framework using Gaussian Process (GP) emulators to approximate the landslide model, and the discrepancy function between model and observations. Distributions of the calibrated input parameters are obtained as a result of the calibration. In a second step, a GP emulator is built to mimic the coupled landslide-tsunami numerical process. The emulator propagates the uncertainties in the distributions of the calibrated input parameters inferred from the first step to the outputs. As a result, a quantification of the uncertainty of the maximum free surface elevation at specified locations is obtained.
]]>2017-04-12T01:01:34-07:00info:doi/10.1098/rspa.2017.0026hwp:master-id:royprsa;rspa.2017.00262017-04-12Research articles47322002017002620170026<![CDATA[Octonions in random matrix theory]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160800?rss=1
The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N=2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N=3. We then proceed to consider the matrix structure X^{}X, when X has random octonion entries. Analytic results are obtained from N=2, but are observed to break down in the 3x3 case.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0800hwp:master-id:royprsa;rspa.2016.08002017-04-05Research articles47322002016080020160800<![CDATA[Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160817?rss=1
The evolution of the free surface of a three-dimensional conducting fluid in the presence of gravity, surface tension and vertical electric field due to parallel electrodes, is considered. Based on the analysis of the Dirichlet–Neumann operators, a series of fully nonlinear models is derived systematically from the Euler equations in the Hamiltonian framework without assumptions on competing length scales can therefore be applied to systems of arbitrary fluid depth and to disturbances with arbitrary wavelength. For special cases, well-known weakly nonlinear models in shallow and deep fluids can be generalized via introducing extra electric terms. It is shown that the electric field has a great impact on the physical system and can change the qualitative nature of the free surface: (i) when the separation distance between two electrodes is small compared with typical wavelength, the Boussinesq, Benney–Luke (BL) and Kadomtsev–Petviashvili (KP) equations with modified coefficients are obtained, and electric forces can turn KP-I to KP-II and vice versa; (ii) as the parallel electrodes are of large separation distance but the thickness of the liquid is much smaller than typical wavelength, we generalize the BL and KP equations by adding pseudo-differential operators resulting from the electric field; (iii) for a quasi-monochromatic plane wave in deep fluid, we derive the cubic nonlinear Schrödinger (NLS) equation, but its type (focusing or defocusing) is strongly influenced by the value of the electric parameter. For sufficient surface tension, numerical studies reveal that lump-type solutions exist in the aforementioned three regimes. Particularly, even when the associated NLS equation is defocusing for a wave train, lumps can exist in fully nonlinear models.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0817hwp:master-id:royprsa;rspa.2016.08172017-04-05Research articles47322002016081720160817<![CDATA[Soft phononic crystals with deformation-independent band gaps]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160865?rss=1
Soft phononic crystals have the advantages over their stiff counterparts of being flexible and reconfigurable. Normally, the band gaps of soft phononic crystals will be modified after deformation due to both geometric and constitutive nonlinearity. Indeed these are important properties that can be exploited to tune the dynamic properties of the material. However, in some instances, it may be that one wishes to deform the medium while retaining the band gap structure. A special class of soft phononic crystals is described here with band gaps that are independent or almost-independent of the imposed mechanical deformation, which enables the design of phononic crystals with robust performance. This remarkable behaviour originates from transformation elasticity theory, which leaves the wave equation and the eigenfrequencies invariant after deformation. The necessary condition to achieve such a property is that the Lagrangian elasticity tensor of the hyperelastic material should be constant, i.e. independent of deformation. It is demonstrated that incompressible neo-Hookean materials exhibit such a unique property. Semilinear materials also possess this property under special loading conditions. Phononic crystals composed of these two materials are studied theoretically and the predictions of invariance, or the manner in which the response deviates from invariance, are confirmed via numerical simulation.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0865hwp:master-id:royprsa;rspa.2016.08652017-04-05Research articles47322002016086520160865<![CDATA[Does a better model yield a better argument? An info-gap analysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160890?rss=1
Theories, models and computations underlie reasoned argumentation in many areas. The possibility of error in these arguments, though of low probability, may be highly significant when the argument is used in predicting the probability of rare high-consequence events. This implies that the choice of a theory, model or computational method for predicting rare high-consequence events must account for the probability of error in these components. However, error may result from lack of knowledge or surprises of various sorts, and predicting the probability of error is highly uncertain. We show that the putatively best, most innovative and sophisticated argument may not actually have the lowest probability of error. Innovative arguments may entail greater uncertainty than more standard but less sophisticated methods, creating an innovation dilemma in formulating the argument. We employ info-gap decision theory to characterize and support the resolution of this problem and present several examples.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0890hwp:master-id:royprsa;rspa.2016.08902017-04-05Research articles47322002016089020160890<![CDATA[Wave energy absorption by a submerged air bag connected to a rigid float]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160861?rss=1
A new wave energy device features a submerged ballasted air bag connected at the top to a rigid float. Under wave action, the bag expands and contracts, creating a reciprocating air flow through a turbine between the bag and another volume housed within the float. Laboratory measurements are generally in good agreement with numerical predictions. Both show that the trajectory of possible combinations of pressure and elevation at which the device is in static equilibrium takes the shape of an S. This means that statically the device can have three different draughts, and correspondingly three different bag shapes, for the same pressure. The behaviour in waves depends on where the mean pressure-elevation condition is on the static trajectory. The captured power is highest for a mean condition on the middle section.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0861hwp:master-id:royprsa;rspa.2016.08612017-04-05Research articles47322002016086120160861<![CDATA[Helicity conservation and twisted Seifert surfaces for superfluid vortices]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160853?rss=1
Starting from the continuum definition of helicity, we derive from first principles its different contributions for superfluid vortices. Our analysis shows that an internal twist contribution emerges naturally from the mathematical derivation. This reveals that the spanwise vector that is used to characterize the twist contribution must point in the direction of a surface of constant velocity potential. An immediate consequence of the Seifert framing is that the continuum definition of helicity for a superfluid is trivially zero at all times. It follows that the Gauss-linking number is a more appropriate definition of helicity for superfluids. Despite this, we explain how a quasi-classical limit can arise in a superfluid in which the continuum definition for helicity can be used. This provides a clear connection between a microscopic and a macroscopic description of a superfluid as provided by the Hall–Vinen–Bekarevich–Khalatnikov equations. This leads to consistency with the definition of helicity used for classical vortices.
]]>2017-04-05T01:29:13-07:00info:doi/10.1098/rspa.2016.0853hwp:master-id:royprsa;rspa.2016.08532017-04-05Research articles47322002016085320160853<![CDATA[Quantifying the bending of bilayer temperature-sensitive hydrogels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170092?rss=1
Stimuli-responsive hydrogels can serve as manipulators, including grippers, sensors, etc., where structures can undergo significant bending. Here, a finite-deformation theory is developed to quantify the evolution of the curvature of bilayer temperature-sensitive hydrogels when subjected to a temperature change. Analysis of the theory indicates that there is an optimal thickness ratio to acquire the largest curvature in the bilayer and also suggests that the sign or the magnitude of the curvature can be significantly affected by pre-stretches or small pores in the bilayer. This study may provide important guidelines in fabricating temperature-responsive bilayers with desirable mechanical performance.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2017.0092hwp:master-id:royprsa;rspa.2017.00922017-04-05Research articles47322002017009220170092<![CDATA[Microstructure-based hyperelastic models for closed-cell solids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170036?rss=1
For cellular bodies involving large elastic deformations, mesoscopic continuum models that take into account the interplay between the geometry and the microstructural responses of the constituents are developed, analysed and compared with finite-element simulations of cellular structures with different architecture. For these models, constitutive restrictions for the physical plausibility of the material responses are established, and global descriptors such as nonlinear elastic and shear moduli and Poisson’s ratio are obtained from the material characteristics of the constituents. Numerical results show that these models capture well the mechanical responses of finite-element simulations for three-dimensional periodic structures of neo-Hookean material with closed cells under large tension. In particular, the mesoscopic models predict the macroscopic stiffening of the structure when the stiffness of the cell-core increases.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2017.0036hwp:master-id:royprsa;rspa.2017.00362017-04-05Research articles47322002017003620170036