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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946April, 2017Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![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[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[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[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[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[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[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[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[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[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[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[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[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[Some aspects of radical cascade and relay reactions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160859?rss=1
The ability to create carbon–carbon bonds is at the heart of organic synthesis. Radical processes are particularly apt at creating such bonds, especially in cascade or relay sequences where more than one bond is formed, allowing for a rapid assembly of complex structures. In the present brief overview, examples taken from the authors' laboratory will serve to illustrate the strategic impact of radical-based approaches on synthetic planning. Transformations involving nitrogen-centred radicals, electron transfer from metallic nickel and the reversible degenerative exchange of xanthates will be presented and discussed. The last method has proved to be a particularly powerful tool for the intermolecular creation of carbon–carbon bonds by radical additions even to unactivated alkenes. Various functional groups can be brought into the same molecule in a convergent manner and made to react together in order to further increase the structural complexity. One important benefit of this chemistry is the so-called RAFT/MADIX technology for the manufacture of block copolymers of almost any desired architecture.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0859hwp:master-id:royprsa;rspa.2016.08592017-04-05Perspective47322002016085920160859<![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[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[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[Multifunctional scanning ion conductance microscopy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160889?rss=1
Scanning ion conductance microscopy (SICM) is a nanopipette-based technique that has traditionally been used to image topography or to deliver species to an interface, particularly in a biological setting. This article highlights the recent blossoming of SICM into a technique with a much greater diversity of applications and capability that can be used either standalone, with advanced control (potential–time) functions, or in tandem with other methods. SICM can be used to elucidate functional information about interfaces, such as surface charge density or electrochemical activity (ion fluxes). Using a multi-barrel probe format, SICM-related techniques can be employed to deposit nanoscale three-dimensional structures and further functionality is realized when SICM is combined with scanning electrochemical microscopy (SECM), with simultaneous measurements from a single probe opening up considerable prospects for multifunctional imaging. SICM studies are greatly enhanced by finite-element method modelling for quantitative treatment of issues such as resolution, surface charge and (tip) geometry effects. SICM is particularly applicable to the study of living systems, notably single cells, although applications extend to materials characterization and to new methods of printing and nanofabrication. A more thorough understanding of the electrochemical principles and properties of SICM provides a foundation for significant applications of SICM in electrochemistry and interfacial science.
]]>2017-04-12T01:01:34-07:00info:doi/10.1098/rspa.2016.0889hwp:master-id:royprsa;rspa.2016.08892017-04-12Special feature47322002016088920160889<![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[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[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[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[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[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[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[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<![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[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[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[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[Uncertainty quantification and optimal decisions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170115?rss=1
A mathematical model can be analysed to construct policies for action that are close to optimal for the model. If the model is accurate, such policies will be close to optimal when implemented in the real world. In this paper, the different aspects of an ideal workflow are reviewed: modelling, forecasting, evaluating forecasts, data assimilation and constructing control policies for decision-making. The example of the oil industry is used to motivate the discussion, and other examples, such as weather forecasting and precision agriculture, are used to argue that the same mathematical ideas apply in different contexts. Particular emphasis is placed on (i) uncertainty quantification in forecasting and (ii) how decisions are optimized and made robust to uncertainty in models and judgements. This necessitates full use of the relevant data and by balancing costs and benefits into the long term may suggest policies quite different from those relevant to the short term.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2017.0115hwp:master-id:royprsa;rspa.2017.01152017-04-26Special feature47322002017011520170115