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[Reciprocal absorbing boundary condition for the time-domain numerical analysis of wave motion in unbounded layered media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160528?rss=1
A global absorbing boundary condition is introduced for the time-domain numerical analysis of wave motion in unbounded layered media. This condition is obtained by applying the reciprocity theorem for linearly viscoelastic media. Example problems are solved towards evaluation of the accuracy and effectiveness of the approach developed in this work.
]]>2017-03-22T00:05:49-07:00info:doi/10.1098/rspa.2016.0528hwp:master-id:royprsa;rspa.2016.05282017-03-22Research articles47321992016052820160528<![CDATA[Landslides and tsunamis predicted by incompressible smoothed particle hydrodynamics (SPH) with application to the 1958 Lituya Bay event and idealized experiment]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160674?rss=1
Tsunamis caused by landslides may result in significant destruction of the surroundings with both societal and industrial impact. The 1958 Lituya Bay landslide and tsunami is a recent and well-documented terrestrial landslide generating a tsunami with a run-up of 524 m. Although recent computational techniques have shown good performance in the estimation of the run-up height, they fail to capture all the physical processes, in particular, the landslide-entry profile and interaction with the water. Smoothed particle hydrodynamics (SPH) is a versatile numerical technique for describing free-surface and multi-phase flows, particularly those that exhibit highly nonlinear deformation in landslide-generated tsunamis. In the current work, the novel multi-phase incompressible SPH method with shifting is applied to the Lituya Bay tsunami and landslide and is the first methodology able to reproduce realistically both the run-up and landslide-entry as documented in a benchmark experiment. The method is the first paper to develop a realistic implementation of the physics that in addition to the non-Newtonian rheology of the landslide includes turbulence in the water phase and soil saturation. Sensitivity to the experimental initial conditions is also considered. This work demonstrates the ability of the proposed method in modelling challenging environmental multi-phase, non-Newtonian and turbulent flows.
]]>2017-03-22T00:05:49-07:00info:doi/10.1098/rspa.2016.0674hwp:master-id:royprsa;rspa.2016.06742017-03-22Research articles47321992016067420160674<![CDATA[Efficient generation of receiver operating characteristics for the evaluation of damage detection in practical structural health monitoring applications]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160736?rss=1
Permanently installed guided wave monitoring systems are attractive for monitoring large structures. By frequently interrogating the test structure over a long period of time, such systems have the potential to detect defects much earlier than with conventional one-off inspection, and reduce the time and labour cost involved. However, for the systems to be accepted under real operational conditions, their damage detection performance needs to be evaluated in these practical settings. The receiver operating characteristic (ROC) is an established performance metric for one-off inspections, but the generation of the ROC requires many test structures with realistic damage growth at different locations and different environmental conditions, and this is often impractical. In this paper, we propose an evaluation framework using experimental data collected over multiple environmental cycles on an undamaged structure with synthetic damage signatures added by superposition. Recent advances in computation power enable examples covering a wide range of practical scenarios to be generated, and for multiple cases of each scenario to be tested so that the statistics of the performance can be evaluated. The proposed methodology has been demonstrated using data collected from a laboratory pipe specimen over many temperature cycles, superposed with damage signatures predicted for a flat-bottom hole growing at different rates at various locations. Three damage detection schemes, conventional baseline subtraction, singular value decomposition (SVD) and independent component analysis (ICA), have been evaluated. It has been shown that in all cases, the component methods perform significantly better than the residual method, with ICA generally the better of the two. The results have been validated using experimental data monitoring a pipe in which a flat-bottom hole was drilled and enlarged over successive temperature cycles. The methodology can be used to evaluate the performance of an installed monitoring system and to show whether it is capable of detecting particular damage growth at any given location. It will enable monitoring results to be evaluated rigorously and will be valuable in the development of safety cases.
]]>2017-03-22T00:05:49-07:00info:doi/10.1098/rspa.2016.0736hwp:master-id:royprsa;rspa.2016.07362017-03-22Research articles47321992016073620160736<![CDATA[SPARSE--A subgrid particle averaged Reynolds stress equivalent model: testing with a priori closure]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160769?rss=1
A Lagrangian particle cloud model is proposed that accounts for the effects of Reynolds-averaged particle and turbulent stresses and the averaged carrier-phase velocity of the subparticle cloud scale on the averaged motion and velocity of the cloud. The SPARSE (subgrid particle averaged Reynolds stress equivalent) model is based on a combination of a truncated Taylor expansion of a drag correction function and Reynolds averaging. It reduces the required number of computational parcels to trace a cloud of particles in Eulerian–Lagrangian methods for the simulation of particle-laden flow. Closure is performed in an a priori manner using a reference simulation where all particles in the cloud are traced individually with a point-particle model. Comparison of a first-order model and SPARSE with the reference simulation in one dimension shows that both the stress and the averaging of the carrier-phase velocity on the cloud subscale affect the averaged motion of the particle. A three-dimensional isotropic turbulence computation shows that only one computational parcel is sufficient to accurately trace a cloud of tens of thousands of particles.
]]>2017-03-22T01:44:10-07:00info:doi/10.1098/rspa.2016.0769hwp:master-id:royprsa;rspa.2016.07692017-03-22Research articles47321992016076920160769<![CDATA[Biofriendly nanocomposite containers with inhibition properties for the protection of metallic surfaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160827?rss=1
An attempt to combine two ‘green’ compounds in nanocomposite microcontainers in order to increase protection properties of waterborne acryl-styrene copolymer (ASC) coatings has been made. N-lauroylsarcosine (NLS) served as a corrosion inhibitor, and linseed oil (LO) as a carrier-forming component. LO is compatible with this copolymer and can impart to the coating self-healing properties. For the evaluation of the protective performance, three types of coatings were compared. In the first two, NLS was introduced in the coating formulation in the forms of free powder and micro-containers filled with LO, correspondingly. The last one was a standard ASC coating without inhibitor at all. Low-carbon steel substrates were coated by these formulations by spraying and subjected subsequently to the neutral salt spray test according to DIN ISO 9227. Results of these tests as well as the data obtained by electrochemical study suggest that such containers can be used for the improvement of adhesion of ASC-based coatings to the substrate and for the enhancement of their protective performance upon integrity damage, whereas the barrier properties of intact coatings were decreased.
]]>2017-03-22T00:05:49-07:00info:doi/10.1098/rspa.2016.0827hwp:master-id:royprsa;rspa.2016.08272017-03-22Research articles47321992016082720160827<![CDATA[Screened Coulomb interactions with non-uniform surface charge]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160906?rss=1
The screened Coulomb interaction between a pair of infinite parallel planes with spatially varying surface charge is considered in the limit of small electrical potentials for arbitrary Debye lengths. A simple expression for the disjoining pressure is derived in terms of a two-dimensional integral in Fourier space. The integral is evaluated for periodic and random charge distributions and the disjoining pressure is expressed as a sum over Fourier–Bloch reciprocal lattice vectors or in terms of an integral involving the autocorrelation function, respectively. The force between planes with a finite area of uniform charge, a model for the DLVO interaction between finite surfaces, is also calculated. It is shown that the overspill of the charge cloud beyond the region immediately between the charged areas results in a reduction of the disjoining pressure, as reported by us recently in the long Debye length limit for planes of finite width.
]]>2017-03-22T00:05:49-07:00info:doi/10.1098/rspa.2016.0906hwp:master-id:royprsa;rspa.2016.09062017-03-22Research articles47321992016090620160906<![CDATA[A computational continuum model of poroelastic beds]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160932?rss=1
Despite the ubiquity of fluid flows interacting with porous and elastic materials, we lack a validated non-empirical macroscale method for characterizing the flow over and through a poroelastic medium. We propose a computational tool to describe such configurations by deriving and validating a continuum model for the poroelastic bed and its interface with the above free fluid. We show that, using stress continuity condition and slip velocity condition at the interface, the effective model captures the effects of small changes in the microstructure anisotropy correctly and predicts the overall behaviour in a physically consistent and controllable manner. Moreover, we show that the performance of the effective model is accurate by validating with fully microscopic resolved simulations. The proposed computational tool can be used in investigations in a wide range of fields, including mechanical engineering, bio-engineering and geophysics.
]]>2017-03-22T00:05:49-07:00info:doi/10.1098/rspa.2016.0932hwp:master-id:royprsa;rspa.2016.09322017-03-22Research articles47321992016093220160932<![CDATA[Equilibrium shapes of a heterogeneous bubble in an electric field: a variational formulation and numerical verifications]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160494?rss=1
The equilibrium shape of a bubble/droplet in an electric field is important for electrowetting over dielectrics (EWOD), electrohydrodynamic (EHD) enhancement for heat transfer and electro-deformation of a single biological cell among others. In this work, we develop a general variational formulation in account of electro-mechanical couplings. In the context of EHD, we identify the free energy functional and the associated energy minimization problem that determines the equilibrium shape of a bubble in an electric field. Based on this variational formulation, we implement a fixed mesh level-set gradient method for computing the equilibrium shapes. This numerical scheme is efficient and validated by comparing with analytical solutions at the absence of electric field and experimental results at the presence of electric field. We also present simulation results for zero gravity which will be useful for space applications. The variational formulation and numerical scheme are anticipated to have broad applications in areas of EWOD, EHD and electro-deformation in biomechanics.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0494hwp:master-id:royprsa;rspa.2016.04942017-03-15Research articles47321992016049420160494<![CDATA[Device-independent tests of quantum channels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160721?rss=1
We develop a device-independent framework for testing quantum channels. That is, we falsify a hypothesis about a quantum channel based only on an observed set of input–output correlations. Formally, the problem consists of characterizing the set of input–output correlations compatible with any arbitrary given quantum channel. For binary (i.e. two input symbols, two output symbols) correlations, we show that extremal correlations are always achieved by orthogonal encodings and measurements, irrespective of whether or not the channel preserves commutativity. We further provide a full, closed-form characterization of the sets of binary correlations in the case of: (i) any dihedrally covariant qubit channel (such as any Pauli and amplitude-damping channels) and (ii) any universally-covariant commutativity-preserving channel in an arbitrary dimension (such as any erasure, depolarizing, universal cloning and universal transposition channels).
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0721hwp:master-id:royprsa;rspa.2016.07212017-03-15Research articles47321992016072120160721<![CDATA[Analytical results regarding electrostatic resonances of surface phonon/plasmon polaritons: separation of variables with a twist]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160796?rss=1
The boundary integral equation (BIE) method ascertains explicit relations between localized surface phonon and plasmon polariton resonances and the eigenvalues of its associated electrostatic operator. We show that group-theoretical analysis of the Laplace equation can be used to calculate the full set of eigenvalues and eigenfunctions of the electrostatic operator for shapes and shells described by separable coordinate systems. These results not only unify and generalize many existing studies, but also offer us the opportunity to expand the study of phenomena such as cloaking by anomalous localized resonance. Hence, we calculate the eigenvalues and eigenfunctions of elliptic and circular cylinders. We illustrate the benefits of using the BIE method to interpret recent experiments involving localized surface phonon polariton resonances and the size scaling of plasmon resonances in graphene nanodiscs. Finally, symmetry-based operator analysis can be extended from the electrostatic to the full-wave regime. Thus, bound states of light in the continuum can be studied for shapes beyond spherical configurations.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0796hwp:master-id:royprsa;rspa.2016.07962017-03-15Research articles47321992016079620160796<![CDATA[History dependence and the continuum approximation breakdown: the impact of domain growth on Turings instability]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160744?rss=1
A diffusively driven instability has been hypothesized as a mechanism to drive spatial self-organization in biological systems since the seminal work of Turing. Such systems are often considered on a growing domain, but traditional theoretical studies have only treated the domain size as a bifurcation parameter, neglecting the system non-autonomy. More recently, the conditions for a diffusively driven instability on a growing domain have been determined under stringent conditions, including slow growth, a restriction on the temporal interval over which the prospect of an instability can be considered and a neglect of the impact that time evolution has on the stability properties of the homogeneous reference state from which heterogeneity emerges. Here, we firstly relax this latter assumption and observe that the conditions for the Turing instability are much more complex and depend on the history of the system in general. We proceed to relax all the above constraints, making analytical progress by focusing on specific examples. With faster growth, instabilities can grow transiently and decay, making the prediction of a prospective Turing instability much more difficult. In addition, arbitrarily high spatial frequencies can destabilize, in which case the continuum approximation is predicted to break down.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0744hwp:master-id:royprsa;rspa.2016.07442017-03-15Research articles47321992016074420160744<![CDATA[The Fourier decomposition method for nonlinear and non-stationary time series analysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160871?rss=1
for many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory, and demonstrate its efficacy for the analysis of nonlinear and non-stationary time series. The proposed FDM decomposes any data into a small number of ‘Fourier intrinsic band functions’ (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM. The proposed MFDM generates a finite number of band-limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0871hwp:master-id:royprsa;rspa.2016.08712017-03-15Research articles47321992016087120160871<![CDATA[A laboratory study of nonlinear changes in the directionality of extreme seas]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160290?rss=1
This paper concerns the description of surface water waves, specifically nonlinear changes in the directionality. Supporting calculations are provided to establish the best method of directional wave generation, the preferred method of directional analysis and the inputs on which such a method should be based. These calculations show that a random directional method, in which the phasing, amplitude and direction of propagation of individual wave components are chosen randomly, has benefits in achieving the required ergodicity. In terms of analysis procedures, the extended maximum entropy principle, with inputs based upon vector quantities, produces the best description of directionality. With laboratory data describing the water surface elevation and the two horizontal velocity components at a single point, several steep sea states are considered. The results confirm that, as the steepness of a sea state increases, the overall directionality of the sea state reduces. More importantly, it is also shown that the largest waves become less spread or more unidirectional than the sea state as a whole. This provides an important link to earlier descriptions of deterministic wave groups produced by frequency focusing, helps to explain recent field observations and has important practical implications for the design of marine structures and vessels.
]]>2017-03-08T01:00:53-08:00info:doi/10.1098/rspa.2016.0290hwp:master-id:royprsa;rspa.2016.02902017-03-08Research articles47321992016029020160290<![CDATA[Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160681?rss=1
We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov’s theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.
]]>2017-03-08T00:05:22-08:00info:doi/10.1098/rspa.2016.0681hwp:master-id:royprsa;rspa.2016.06812017-03-08Research articles47321992016068120160681<![CDATA[Incompatibility-governed elasto-plasticity for continua with dislocations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160734?rss=1
In this paper, a novel model for elasto-plastic continua is presented and developed from the ground up. It is based on the interdependence between plasticity, dislocation motion and strain incompatibility. A generalized form of the equilibrium equations is provided, with as additional variables, the strain incompatibility and an internal thermodynamic variable called incompatibility modulus, which drives the plastic behaviour of the continuum. The traditional equations of elasticity are recovered as this modulus tends to infinity, while perfect plasticity corresponds to the vanishing limit. The overall nonlinear scheme is determined by the solution of these equations together with the computation of the topological derivative of the dissipation, in order to comply with the second principle of thermodynamics.
]]>2017-03-08T00:05:22-08:00info:doi/10.1098/rspa.2016.0734hwp:master-id:royprsa;rspa.2016.07342017-03-08Research articles47321992016073420160734<![CDATA[Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160798?rss=1
We propose a novel stable and efficient dissipation-preserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude. Through introducing the conformal multi-symplectic structure, the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation. The proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law. More specifically, two subsystems in conjunction with the original damped wave equation are derived. One is actually the conservative Hamiltonian wave equation and the other is a dissipative linear ordinary differential equation (ODE) system. Standard symplectic method is devoted to the conservative system, whereas the analytical solution is obtained for the ODE system. An explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the Strang splitting technique. Stability analysis and convergence tests are given thereafter. A benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation. Further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media.
]]>2017-03-08T00:05:22-08:00info:doi/10.1098/rspa.2016.0798hwp:master-id:royprsa;rspa.2016.07982017-03-08Research articles47321992016079820160798<![CDATA[G-Strands on symmetric spaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160795?rss=1
We study the G-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G, and we treat in more detail examples with symmetric space SU(2)/S^{1} and SO(4)/SO(3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G-strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1+2 Camassa–Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions.
]]>2017-03-08T01:00:53-08:00info:doi/10.1098/rspa.2016.0795hwp:master-id:royprsa;rspa.2016.07952017-03-08Research articles47321992016079520160795<![CDATA[A biochemo-mechano coupled, computational model combining membrane transport and pericellular proteolysis in tissue mechanics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160812?rss=1
We present a computational model for the interaction of surface- and volume-bound scalar transport and reaction processes with a deformable porous medium. The application in mind is pericellular proteolysis, i.e. the dissolution of the solid phase of the extracellular matrix (ECM) as a response to the activation of certain chemical species at the cell membrane and in the vicinity of the cell. A poroelastic medium model represents the extra cellular scaffold and the interstitial fluid flow, while a surface-bound transport model accounts for the diffusion and reaction of membrane-bound chemical species. By further modelling the volume-bound transport, we consider the advection, diffusion and reaction of sequestered chemical species within the extracellular scaffold. The chemo-mechanical coupling is established by introducing a continuum formulation for the interplay of reaction rates and the mechanical state of the ECM. It is based on known experimental insights and theoretical work on the thermodynamics of porous media and degradation kinetics of collagen fibres on the one hand and a damage-like effect of the fibre dissolution on the mechanical integrity of the ECM on the other hand. The resulting system of partial differential equations is solved via the finite-element method. To the best of our knowledge, it is the first computational model including contemporaneously the coupling between (i) advection–diffusion–reaction processes, (ii) interstitial flow and deformation of a porous medium, and (iii) the chemo-mechanical interaction impelled by the dissolution of the ECM. Our numerical examples show good agreement with experimental data. Furthermore, we outline the capability of the methodology to extend existing numerical approaches towards a more comprehensive model for cellular biochemo-mechanics.
]]>2017-03-08T00:05:22-08:00info:doi/10.1098/rspa.2016.0812hwp:master-id:royprsa;rspa.2016.08122017-03-08Research articles47321992016081220160812<![CDATA[Identifying the significance of nonlinear normal modes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160789?rss=1
Nonlinear normal modes (NNMs) are widely used as a tool for understanding the forced responses of nonlinear systems. However, the contemporary definition of an NNM also encompasses a large number of dynamic behaviours which are not observed when a system is forced and damped. As such, only a few NNMs are required to understand the forced dynamics. This paper firstly demonstrates the complexity that may arise from the NNMs of a simple nonlinear system—highlighting the need for a method for identifying the significance of NNMs. An analytical investigation is used, alongside energy arguments, to develop an understanding of the mechanisms that relate the NNMs to the forced responses. This provides insight into which NNMs are pertinent to understanding the forced dynamics, and which may be disregarded. The NNMs are compared with simulated forced responses to verify these findings.
]]>2017-03-01T00:05:20-08:00info:doi/10.1098/rspa.2016.0789hwp:master-id:royprsa;rspa.2016.07892017-03-01Research articles47321992016078920160789<![CDATA[Rapidly convergent quasi-periodic Green functions for scattering by arrays of cylinders--including Wood anomalies]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160802?rss=1
This paper presents a full-spectrum Green-function methodology (which is valid, in particular, at and around Wood-anomaly frequencies) for evaluation of scattering by periodic arrays of cylinders of arbitrary cross section—with application to wire gratings, particle arrays and reflectarrays and, indeed, general arrays of conducting or dielectric bounded obstacles under both transverse electric and transverse magnetic polarized illumination. The proposed method, which, for definiteness, is demonstrated here for arrays of perfectly conducting particles under transverse electric polarization, is based on the use of the shifted Green-function method introduced in a recent contribution (Bruno & Delourme 2014 J. Computat. Phys.262, 262–290 (doi:10.1016/j.jcp.2013.12.047)). A certain infinite term arises at Wood anomalies for the cylinder-array problems considered here that is not present in the previous rough-surface case. As shown in this paper, these infinite terms can be treated via an application of ideas related to the Woodbury–Sherman–Morrison formulae. The resulting approach, which is applicable to general arrays of obstacles even at and around Wood-anomaly frequencies, exhibits fast convergence and high accuracies. For example, a few hundreds of milliseconds suffice for the proposed approach to evaluate solutions throughout the resonance region (wavelengths comparable to the period and cylinder sizes) with full single-precision accuracy.
]]>2017-03-01T00:05:20-08:00info:doi/10.1098/rspa.2016.0802hwp:master-id:royprsa;rspa.2016.08022017-03-01Research articles47321992016080220160802<![CDATA[Efficient computation of null geodesics with applications to coherent vortex detection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160807?rss=1
Recent results suggest that boundaries of coherent fluid vortices (elliptic coherent structures) can be identified as closed null geodesics of appropriate Lorentzian metrics defined on the flow domain. Here we derive an automated method for computing such null geodesics based on the geometry of the underlying geodesic flow. Our approach simplifies and improves existing procedures for computing variationally defined Eulerian and Lagrangian vortex boundaries. As an illustration, we compute objective vortex boundaries from satellite-inferred ocean velocity data. A MATLAB implementation of our method is available at https://github.com/MattiaSerra/Closed-Null-Geodesics-2D.
]]>2017-03-01T00:43:08-08:00info:doi/10.1098/rspa.2016.0807hwp:master-id:royprsa;rspa.2016.08072017-03-01Research articles47321992016080720160807<![CDATA[1/f Noise and multifractality from bristlecone pine growth explained by the statistical convergence of random data]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160586?rss=1
Tree-ring growth records from bristlecone pines reveal an irregular pattern of fluctuations that have been linked to climatic change but otherwise have remained poorly understood. We find within these records evidence for a temporally related variance to mean power law, 1/f noise and multifractality that empirically resembles a fractal stochastic process and could be attributed to self-organized criticality. These growth records, however, also conformed to a non-Gaussian statistical distribution (the Tweedie compound Poisson distribution) characterized by an inherent variance to mean power law, that by itself implies 1/f noise. This distribution has a fundamental role in statistical theory as a focus of convergence for many types of random data, much like the Gaussian distribution has with the central limit theorem. The growth records were also multifractal, with the dimensional exponent of the Tweedie distribution critically balanced near the transition point between fractal stochastic processes and gamma distributed data, possibly consequent to a related convergence effect. Non-Gaussian random systems, like those related to bristlecone pine tree growth, may express 1/f noise and multifractality through mathematical convergence effects alone, without the dynamical assumptions of self-organized criticality.
]]>2017-02-22T00:44:15-08:00info:doi/10.1098/rspa.2016.0586hwp:master-id:royprsa;rspa.2016.05862017-02-22Research articles47321982016058620160586<![CDATA[Stokes phenomena in discrete Painleve II]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160539?rss=1
We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.
]]>2017-02-22T04:44:30-08:00info:doi/10.1098/rspa.2016.0539hwp:master-id:royprsa;rspa.2016.05392017-02-22Research articles47321982016053920160539<![CDATA[Effective equations governing an active poroelastic medium]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160755?rss=1
In this work, we consider the spatial homogenization of a coupled transport and fluid–structure interaction model, to the end of deriving a system of effective equations describing the flow, elastic deformation and transport in an active poroelastic medium. The ‘active’ nature of the material results from a morphoelastic response to a chemical stimulant, in which the growth time scale is strongly separated from other elastic time scales. The resulting effective model is broadly relevant to the study of biological tissue growth, geophysical flows (e.g. swelling in coals and clays) and a wide range of industrial applications (e.g. absorbant hygiene products). The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscale growth, coupled to transport of solute, that explicitly incorporates details of the structure and dynamics of the microscopic system, and, moreover, admits finite growth and deformation at the pore scale. The resulting macroscale model comprises a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection–reaction–diffusion equation. The resultant system of effective equations is then compared with other recent models under a selection of appropriate simplifying asymptotic limits.
]]>2017-02-22T00:44:15-08:00info:doi/10.1098/rspa.2016.0755hwp:master-id:royprsa;rspa.2016.07552017-02-22Research articles47321982016075520160755<![CDATA[A vortex model for forces and moments on low-aspect-ratio wings in side-slip with experimental validation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160760?rss=1
This paper studies low-aspect-ratio () rectangular wings at high incidence and in side-slip. The main objective is to incorporate the effects of high angle of attack and side-slip into a simplified vortex model for the forces and moments. Experiments are also performed and are used to validate assumptions made in the model. The model asymptotes to the potential flow result of classical aerodynamics for an infinite aspect ratio. The -> 0 limit of a rectangular wing is considered with slender body theory, where the side-edge vortices merge into a vortex doublet. Hence, the velocity fields transition from being dominated by a spanwise vorticity monopole ( >> 1) to a streamwise vorticity dipole ( ~ 1). We theoretically derive a spanwise loading distribution that is parabolic instead of elliptic, and this physically represents the additional circulation around the wing that is associated with reattached flow. This is a fundamental feature of wings with a broad-facing leading edge. The experimental measurements of the spanwise circulation closely approximate a parabolic distribution. The vortex model yields very agreeable comparison with direct measurement of the lift and drag, and the roll moment prediction is acceptable for ≤ 1 prior to the roll stall angle and up to side-slip angles of 20°.
]]>2017-02-22T04:44:30-08:00info:doi/10.1098/rspa.2016.0760hwp:master-id:royprsa;rspa.2016.07602017-02-22Research articles47321982016076020160760<![CDATA[From the elastica compass to the elastica catapult: an essay on the mechanics of soft robot arm]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160870?rss=1
An elastic rod is clamped at one end and has a dead load attached to the other (free) end. The rod is then slowly rotated using the clamp. When the load is smaller than the buckling value, the rod describes a continuous set of quasi-static forms and its end traces a (smooth, convex and simple) closed curve, which would be a circle if the rod were rigid. The closed curve is analytically determined through the integration of the Euler’s elastica, so that for sufficiently small loads the mechanical system behaves as an ‘elastica compass’. For loads higher than that of buckling, the elastica reaches a configuration from which a snap-back instability occurs, realizing a sort of ‘elastica catapult’. The whole quasi-static evolution leading to the critical configuration for snapping is calculated through the elastica and the subsequent dynamic motion simulated using two numerical procedures, one ad hoc developed and another based on a finite-element scheme. The theoretical results are then validated on a specially designed and built apparatus. An obvious application of the present model would be in the development of soft robotic limbs, but the results are also of interest for the optimization analysis in pole vaulting.
]]>2017-02-22T00:44:15-08:00info:doi/10.1098/rspa.2016.0870hwp:master-id:royprsa;rspa.2016.08702017-02-22Research articles47321982016087020160870<![CDATA[A construction of a large family of commuting pairs of integrable symplectic birational four-dimensional maps]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160535?rss=1
We give a construction of completely integrable four-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan–Hirota–Kimura discretization scheme, we arrive at pairs of birational four-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on R4, and possess two independent integrals of motion, which are perturbations of the original Hamilton functions and which are in involution with respect to the perturbed symplectic structure. Thus, these maps are completely integrable in the Liouville–Arnold sense. Moreover, under a suitable normalization of the original pairs of vector fields, the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0535hwp:master-id:royprsa;rspa.2016.05352017-02-15Research articles47321982016053520160535<![CDATA[A wave theory of heat transport with applications to Kapitsa resistance and thermal rectification]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160584?rss=1
We develop a theory of thermal transport in nanoscale-layered structures based on wave processes. The theory incorporates two fundamental principles, first, that the spectra of thermally excited waves are determined by the temperature differential and the heat flux, and second, that the wave fields in the heat exchanging domains are coupled. The developed method includes classical theories as special cases that are valid in larger scales, and it naturally explains such phenomena as interface thermal resistance (Kapitsa resistance) and thermal rectification (asymmetry of thermal transport). Numerical examples demonstrate the feasibility of the approach, and they show good agreement with measurements of Kapitsa resistance reported in the literature.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0584hwp:master-id:royprsa;rspa.2016.05842017-02-15Research articles47321982016058420160584<![CDATA[An experimental study of ultrasonic vibration and the penetration of granular material]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160673?rss=1
This work investigates the potential use of direct ultrasonic vibration as an aid to penetration of granular material. Compared with non-ultrasonic penetration, required forces have been observed to reduce by an order of magnitude. Similarly, total consumed power can be reduced by up to 27%, depending on the substrate and ultrasonic amplitude used. Tests were also carried out in high-gravity conditions, displaying a trend that suggests these benefits could be leveraged in lower gravity regimes.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0673hwp:master-id:royprsa;rspa.2016.06732017-02-15Research articles47321982016067320160673<![CDATA[Bioinspired turbine blades offer new perspectives for wind energy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160726?rss=1
Wind energy is becoming a significant alternative solution for future energy production. Modern turbines now benefit from engineering expertise, and a large variety of different models exists, depending on the context and needs. However, classical wind turbines are designed to operate within a narrow zone centred around their optimal working point. This limitation prevents the use of sites with variable wind to harvest energy, involving significant energetic and economic losses. Here, we present a new type of bioinspired wind turbine using elastic blades, which passively deform through the air loading and centrifugal effects. This work is inspired from recent studies on insect flight and plant reconfiguration, which show the ability of elastic wings or leaves to adapt to the wind conditions and thereby to optimize performance. We show that in the context of energy production, the reconfiguration of the elastic blades significantly extends the range of operating regimes using only passive, non-consuming mechanisms. The versatility of the new turbine model leads to a large increase of the converted energy rate, up to 35%. The fluid/elasticity mechanisms involved for the reconfiguration capability of the new blades are analysed in detail, using experimental observations and modelling.
]]>2017-02-15T00:05:25-08:00info:doi/10.1098/rspa.2016.0726hwp:master-id:royprsa;rspa.2016.07262017-02-15Research articles47321982016072620160726<![CDATA[Eshelby's problem of a spherical inclusion eccentrically embedded in a finite spherical body]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160808?rss=1
Resorting to the superposition principle, the solution of Eshelby's problem of a spherical inclusion located eccentrically inside a finite spherical domain is obtained in two steps: (i) the solution to the problem of a spherical inclusion in an infinite space; (ii) the solution to the auxiliary problem of the corresponding finite spherical domain subjected to appropriate boundary conditions. Moreover, a set of functions called the sectional and harmonic deviators are proposed and developed to work out the auxiliary solution in a series form, including the displacement and Eshelby tensor fields. The analytical solutions are explicitly obtained and illustrated when the geometric and physical parameters and the boundary condition are specified.
]]>2017-02-15T01:16:11-08:00info:doi/10.1098/rspa.2016.0808hwp:master-id:royprsa;rspa.2016.08082017-02-15Research articles47321982016080820160808<![CDATA[Field patterns: a new mathematical object]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160819?rss=1
Field patterns occur in space–time microstructures such that a disturbance propagating along a characteristic line does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space–time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multi-component potential, with as many components as the number of field patterns. Presumably, if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multi-dimensional space that one can consider them to live, or between the different field components of the multi-component potential if one views them that way. As a result of PT-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time.
]]>2017-02-15T01:16:11-08:00info:doi/10.1098/rspa.2016.0819hwp:master-id:royprsa;rspa.2016.08192017-02-15Research articles47321982016081920160819<![CDATA[Stability of vertical magnetic chains]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160703?rss=1
A linear stability analysis is performed for a pair of coaxial vertical chains made from permanently magnetized balls under the influence of gravity. While one chain rises from the ground, the other hangs from above, with the remaining ends separated by a gap of prescribed length. Various boundary conditions are considered, as are situations in which the magnetic dipole moments in the two chains are parallel or antiparallel. The case of a single chain attached to the ground is also discussed. The stability of the system is examined with respect to three quantities: the number of balls in each chain, the length of the gap between the chains, and a single dimensionless parameter which embodies the competition between magnetic and gravitational forces. Asymptotic scaling laws involving these parameters are provided. The Hessian matrix is computed in exact form, allowing the critical parameter values at which the system loses stability and the respective eigenmodes to be determined up to machine precision. A comparison with simple experiments for a single chain attached to the ground shows good agreement.
]]>2017-02-08T00:05:14-08:00info:doi/10.1098/rspa.2016.0703hwp:master-id:royprsa;rspa.2016.07032017-02-08Research articles47321982016070320160703<![CDATA[Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160751?rss=1
Multi-fidelity modelling enables accurate inference of quantities of interest by synergistically combining realizations of low-cost/low-fidelity models with a small set of high-fidelity observations. This is particularly effective when the low- and high-fidelity models exhibit strong correlations, and can lead to significant computational gains over approaches that solely rely on high-fidelity models. However, in many cases of practical interest, low-fidelity models can only be well correlated to their high-fidelity counterparts for a specific range of input parameters, and potentially return wrong trends and erroneous predictions if probed outside of their validity regime. Here we put forth a probabilistic framework based on Gaussian process regression and nonlinear autoregressive schemes that is capable of learning complex nonlinear and space-dependent cross-correlations between models of variable fidelity, and can effectively safeguard against low-fidelity models that provide wrong trends. This introduces a new class of multi-fidelity information fusion algorithms that provide a fundamental extension to the existing linear autoregressive methodologies, while still maintaining the same algorithmic complexity and overall computational cost. The performance of the proposed methods is tested in several benchmark problems involving both synthetic and real multi-fidelity datasets from computational fluid dynamics simulations.
]]>2017-02-08T00:05:14-08:00info:doi/10.1098/rspa.2016.0751hwp:master-id:royprsa;rspa.2016.07512017-02-08Research articles47321982016075120160751<![CDATA[Methodical fitting for mathematical models of rubber-like materials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160811?rss=1
A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.
]]>2017-02-08T00:05:14-08:00info:doi/10.1098/rspa.2016.0811hwp:master-id:royprsa;rspa.2016.08112017-02-08Research articles47321982016081120160811<![CDATA[On the role of micro-inertia in enriched continuum mechanics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160722?rss=1
In this paper, the role of gradient micro-inertia terms {macron}|| u,t||2 and free micro-inertia terms ||P,t||2 is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the term {macron}|| u,t||2 alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the term ||P,t||2 alone describes the full complex behaviour of band-gap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertia {macron}|| u,t||2, in the sense of Cartan–Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertia {macron}|| u,t||2 on more classical enriched models such as the Mindlin–Eringen and the internal variable ones. We find that the addition of such a gradient micro-inertia allows for the onset of one band-gap in the Mindlin–Eringen model and three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for, which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the best performing one, among the considered enriched models, for the description of non-local band-gap metamaterials.
]]>2017-02-01T01:05:46-08:00info:doi/10.1098/rspa.2016.0722hwp:master-id:royprsa;rspa.2016.07222017-02-01Research articles47321982016072220160722<![CDATA[Efficient least angle regression for identification of linear-in-the-parameters models]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160775?rss=1
Least angle regression, as a promising model selection method, differentiates itself from conventional stepwise and stagewise methods, in that it is neither too greedy nor too slow. It is closely related to L_{1} norm optimization, which has the advantage of low prediction variance through sacrificing part of model bias property in order to enhance model generalization capability. In this paper, we propose an efficient least angle regression algorithm for model selection for a large class of linear-in-the-parameters models with the purpose of accelerating the model selection process. The entire algorithm works completely in a recursive manner, where the correlations between model terms and residuals, the evolving directions and other pertinent variables are derived explicitly and updated successively at every subset selection step. The model coefficients are only computed when the algorithm finishes. The direct involvement of matrix inversions is thereby relieved. A detailed computational complexity analysis indicates that the proposed algorithm possesses significant computational efficiency, compared with the original approach where the well-known efficient Cholesky decomposition is involved in solving least angle regression. Three artificial and real-world examples are employed to demonstrate the effectiveness, efficiency and numerical stability of the proposed algorithm.
]]>2017-02-01T01:05:46-08:00info:doi/10.1098/rspa.2016.0775hwp:master-id:royprsa;rspa.2016.07752017-02-01Research articles47321982016077520160775<![CDATA[Origin of the scaling laws of sediment transport]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160785?rss=1
In this paper, we discover the origin of the scaling laws of sediment transport under turbulent flow over a sediment bed, for the first time, from the perspective of the phenomenological theory of turbulence. The results reveal that for the incipient motion of sediment particles, the densimetric Froude number obeys the ‘(1 + )/4’ scaling law with the relative roughness (ratio of particle diameter to approach flow depth), where is the spectral exponent of turbulent energy spectrum. However, for the bedforms, the densimetric Froude number obeys a ‘(1 + )/6’ scaling law with the relative roughness in the enstrophy inertial range and the energy inertial range. For the bedload flux, the bedload transport intensity obeys the ‘3/2’ and ‘(1 + )/4’ scaling laws with the transport stage parameter and the relative roughness, respectively. For the suspended load flux, the non-dimensional suspended sediment concentration obeys the ‘–Z’ scaling law with the non-dimensional vertical distance within the wall shear layer, where Z is the Rouse number. For the scour in contracted streams, the non-dimensional scour depth obeys the ‘4/(3 – )’, ‘–4/(3 – )’ and ‘–(1 + )/(3 – )’ scaling laws with the densimetric Froude number, the channel contraction ratio (ratio of contracted channel width to approach channel width) and the relative roughness, respectively.
]]>2017-01-25T00:05:42-08:00info:doi/10.1098/rspa.2016.0785hwp:master-id:royprsa;rspa.2016.07852017-01-25Research articles47321972016078520160785<![CDATA[Numerical and analytical results for the two discs capacitor problem]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160792?rss=1
In this paper, we study the two discs capacitor, for equal and different radii. The new results obtained allow a complete characterization of capacity coefficients and forces at short distances. An extensive numerical calculation confirms the theoretical results. The study shows the existence of a hierarchy in the divergent behaviour of the capacitance coefficients and this implies some unusual behaviour of the forces, strictly related to the dimensionality of the near-contact zone between electrodes.
]]>2017-01-25T00:05:42-08:00info:doi/10.1098/rspa.2016.0792hwp:master-id:royprsa;rspa.2016.07922017-01-25Research articles47321972016079220160792<![CDATA[Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160705?rss=1
Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.
]]>2017-01-25T00:05:42-08:00info:doi/10.1098/rspa.2016.0705hwp:master-id:royprsa;rspa.2016.07052017-01-25Research articles47321972016070520160705<![CDATA[Why do students quit school? Implications from a dynamical modelling study]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160204?rss=1
In 2012, more than three million students dropped out from high school. At this pace, we will have more than 30 million Americans without a high school degree by 2022 and relatively high dropout rates among Hispanic and African American students. We have developed and analysed a data-driven mathematical model that includes multiple interacting mechanisms and estimates of parameters using data from a specifically designed survey applied to a certain group of students of a high school in Chicago to understand the dynamics of dropouts. Our analysis suggests students' academic achievement is directly related to the level of parental involvement more than any other factors in our study. However, if the negative peer influence (leading to lower academic grades) increases beyond a critical value, the effect of parental involvement on the dynamics of dropouts becomes negligible.
]]>2017-01-18T00:05:26-08:00info:doi/10.1098/rspa.2016.0204hwp:master-id:royprsa;rspa.2016.02042017-01-18Research articles47321972016020420160204<![CDATA[On compression and damage evolution in two thermoplastics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160495?rss=1
The well-known Taylor cylinder impact test, which follows the impact of a flat-ended cylindrical rod onto a rigid stationary anvil, is conducted over a range of impact speeds for two polymers, polytetrafluoroethylene (PTFE) and polyetheretherketone (PEEK). In previous work, experiments and a model were developed to capture the deformation behaviour of the cylinder after impact. These works showed a region in which spatial and temporal variation of both longitudinal and radial deformation provided evidence of changes in phase within the material. In this further series of experiments, this region is imaged in a range of impacted targets at the Diamond synchrotron. Further techniques were fielded to resolve compressed regions within the recovered polymer cylinders that showed a fracture zone in the impact region. The combination of macroscopic high-speed photography and three-dimensional X-ray imaging has identified the development of failure with these polymers and shown that there is no abrupt transition in behaviours but rather a continuous range of responses to competing operating mechanisms. The behaviours noted in PEEK in these polymers show critical gaps in understanding of polymer high strain-rate response.
]]>2017-01-18T00:05:27-08:00info:doi/10.1098/rspa.2016.0495hwp:master-id:royprsa;rspa.2016.04952017-01-18Research articles47321972016049520160495<![CDATA[Learning partial differential equations via data discovery and sparse optimization]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160446?rss=1
We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0446hwp:master-id:royprsa;rspa.2016.04462017-01-18Research articles47321972016044620160446<![CDATA[A study on the temperature field of a bronze-bonded diamond wheel dressed using a laser/ultrasonic vibration combined method]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160597?rss=1
To solve problems in dressings for metal-bonded super-hard abrasive wheels, such as low efficiency and rapid wear, this article introduces a laser-assisted ultrasonic vibration dressing technique. Firstly, finite-element simulations were conducted on the dressing process of a bronze-bonded diamond wheel, and the wheel's temperature field distributions under different laser parameters were simulated. By analysing the simulation results of temperature fields and the melting point of the bronze bond, the laser parameters for laser-assisted ultrasonic vibration dressing tests were optimized, and then actual tests were carried out on the bronze-bonded diamond wheel. Results showed that, with appropriate technology parameters, the laser-assisted ultrasonic vibration dressing technique achieved desirable dressing results; specifically, the dressing force was low, the abrasive particles had high protrusions and the wheel had a large chip space and favourable surface topography.
]]>2017-01-18T00:05:26-08:00info:doi/10.1098/rspa.2016.0597hwp:master-id:royprsa;rspa.2016.05972017-01-18Research articles47321972016059720160597<![CDATA[The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160585?rss=1
Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber KG), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E84, 016301 (doi:10.1103/PhysRevE.84.016301)), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time c at which thermalization is triggered and show that c~KG, with =–49. Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.
]]>2017-01-18T00:05:27-08:00info:doi/10.1098/rspa.2016.0585hwp:master-id:royprsa;rspa.2016.05852017-01-18Research articles47321972016058520160585<![CDATA[Identification of internal properties of fibres and micro-swimmers]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160517?rss=1
In this paper, we address the identifiability of constitutive parameters of passive or active micro-swimmers. We first present a general framework for describing fibres or micro-swimmers using a bead-model description. Using a kinematic constraint formulation to describe fibres, flagellum or cilia, we find explicit linear relationship between elastic constitutive parameters and generalized velocities from computing contact forces. This linear formulation then permits one to address explicitly identifiability conditions and solve for parameter identification. We show that both active forcing and passive parameters are both identifiable independently but not simultaneously. We also provide unbiased estimators for generalized elastic parameters in the presence of Langevin-like forcing with Gaussian noise using a Bayesian approach. These theoretical results are illustrated in various configurations showing the efficiency of the proposed approach for direct parameter identification. The convergence of the proposed estimators is successfully tested numerically.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0517hwp:master-id:royprsa;rspa.2016.05172017-01-18Research articles47321972016051720160517<![CDATA[The problem of missing terms in term by term integration involving divergent integrals]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160567?rss=1
Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity conditions, the problem of missing terms for the incomplete Stieltjes transform, 0af(x)(+x)–1 dx, and the Stieltjes transform itself, 0f(x)(+x)–1 dx, is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex-valued function f(z)(+z)^{–1}, with the divergent integrals arising from term by term integration interpreted as finite part integrals.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0567hwp:master-id:royprsa;rspa.2016.05672017-01-18Research articles47321972016056720160567<![CDATA[Recycling of waste gasket rubber granules by bulk CuCl2 and nano CuCl2: removal of Hg(II) ions by recycled rubber granules]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160771?rss=1
Environmental problems arise due to the millions of tons of waste rubber that are thrown away in the natural environment. Management of this waste rubber is a big environmental challenge. So, a new, simple and cost-effective recycling method for obtaining recycled waste rubber should be developed. In this study, we found that waste gasket rubber can be desulfurized by means of bulk and nano-sized transition metal halides in the presence of solvents. The recycled product of desulfurized waste gasket rubber granules that is obtained can be used as the cheapest adsorbent in the removal of mercury(II) ions from aqueous solution. Comparative batch studies have been conducted to elucidate the adsorption efficiency of desulfurized rubber using bulk copper chloride and also using nano-sized copper chloride under optimum conditions with commercial activated carbon.
]]>2017-01-18T00:05:26-08:00info:doi/10.1098/rspa.2016.0771hwp:master-id:royprsa;rspa.2016.07712017-01-18Research articles47321972016077120160771<![CDATA[Whitham modulation theory for the Ostrovsky equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160709?rss=1
This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localized cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg–de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localized, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0709hwp:master-id:royprsa;rspa.2016.07092017-01-18Research articles47321972016070920160709<![CDATA[Observation-based correction of dynamical models using thermostats]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160730?rss=1
Models used in simulation may give accurate short-term trajectories but distort long-term (statistical) properties. In this work, we augment a given approximate model with a control law (a ‘thermostat’) that gently perturbs the dynamical system to target a thermodynamic state consistent with a set of prescribed (possibly evolving) observations. As proof of concept, we provide an example involving a point vortex fluid model on the sphere, for which we show convergence of equilibrium quantities (in the stationary case) and the ability of the thermostat to dynamically track a transient state.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0730hwp:master-id:royprsa;rspa.2016.07302017-01-18Research articles47321972016073020160730<![CDATA[Topological origin and not purely antisymmetric wave functions of many-body states in the lowest Landau level]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160758?rss=1
In this paper, we recall the topological approach to quantum Hall effects. We note that, in the presence of a magnetic field, trajectories representing elements of the system’s braid group are of cyclotron orbit type. In two-dimensional spaces, this leads to the restriction of the full braid group, _{1}()—loopless generators (exchanges of M^{N} coordinates or classical particles) are unenforceable. As a result, the identification of a possible Hall-like state comes down to the identification of a possible subgroup of _{1}(). The latter follows from the connection between the one-dimensional unitary representation of the system’s braid group and particle statistics (unavoidable for any correlated state). In this work, we implement the topological approach to derive the lowest Landau-level pyramid of fillings. We point out that it contains all mysterious odd-denominator filling factors—like 411, 413 or 617—not trivial to explain within the standard picture. We also introduce, explicitly, cyclotron subgroup generators for all derived fractions. Preliminary results on wave functions, supported by several Monte Carlo calculations, are presented. It is worth emphasizing that not all proposed many-body functions are purely antisymmetric—they, however, transform in agreement with the scalar representations of the system’s braid group. The latter is enforced by standard quantization methods.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0758hwp:master-id:royprsa;rspa.2016.07582017-01-18Research articles47321972016075820160758<![CDATA[Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160461?rss=1
We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0461hwp:master-id:royprsa;rspa.2016.04612017-01-11Research articles47321972016046120160461<![CDATA[An investigation into inflection-point instability in the entrance region of a pulsating pipe flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160590?rss=1
This paper investigates the inflection-point instability that governs the flow disturbance initiated in the entrance region of a pulsating pipe flow. Under such a flow condition, the flow instability grows within a certain phase region in a pulsating cycle, during which the inflection point in the unsteady mean flow lifts away from the viscous effect-dominated region known as the Stokes layer. The characteristic frequency of the instability is found to be in agreement with that predicted by the mixing-layer model. In comparison with those cases not falling in this category, it is further verified that the flow phenomenon will take place only if the inflection point lifts away sufficiently from the Stokes layer.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0590hwp:master-id:royprsa;rspa.2016.05902017-01-11Research articles47321972016059020160590<![CDATA[Entropy in sound and vibration: towards a new paradigm]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160602?rss=1
This paper describes a discussion on the method and the status of a statistical theory of sound and vibration, called statistical energy analysis (SEA). SEA is a simple theory of sound and vibration in elastic structures that applies when the vibrational energy is diffusely distributed. We show that SEA is a thermodynamical theory of sound and vibration, based on a law of exchange of energy analogous to the Clausius principle. We further investigate the notion of entropy in this context and discuss its meaning. We show that entropy is a measure of information lost in the passage from the classical theory of sound and vibration and SEA, its thermodynamical counterpart.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0602hwp:master-id:royprsa;rspa.2016.06022017-01-11Research articles47321972016060220160602<![CDATA[Reconfiguration of a smart surface using heteroclinic connections]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160614?rss=1
A reconfigurable smart surface with multiple equilibria is presented, modelled using discrete point masses and linear springs with geometric nonlinearity. An energy-efficient reconfiguration scheme is then investigated to connect equal-energy unstable (but actively controlled) equilibria. In principle, zero net energy input is required to transition the surface between these unstable states, compared to transitions between stable equilibria across a potential barrier. These transitions between equal-energy unstable states, therefore, form heteroclinic connections in the phase space of the problem. Moreover, the smart surface model developed can be considered as a unit module for a range of applications, including modules which can aggregate together to form larger distributed smart surface systems.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0614hwp:master-id:royprsa;rspa.2016.06142017-01-11Research articles47321972016061420160614<![CDATA[Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160666?rss=1
Recent advances in nanofabrication make it possible to produce multilayer nanostructures composed of ultrathin film materials with thickness down to a few monolayers of atoms and lateral extent of several tens of nanometers. At these scales, ferromagnetic materials begin to exhibit unusual properties, such as perpendicular magnetocrystalline anisotropy and antisymmetric exchange, also referred to as Dzyaloshinskii–Moriya interaction (DMI), because of the increased importance of interfacial effects. The presence of surface DMI has been demonstrated to fundamentally alter the structure of domain walls. Here we use the micromagnetic modelling framework to analyse the existence and structure of chiral domain walls, viewed as minimizers of a suitable micromagnetic energy functional. We explicitly construct the minimizers in the one-dimensional setting, both for the interior and edge walls, for a broad range of parameters. We then use the methods of -convergence to analyse the asymptotics of the two-dimensional magnetization patterns in samples of large spatial extent in the presence of weak applied magnetic fields.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0666hwp:master-id:royprsa;rspa.2016.06662017-01-11Research articles47321972016066620160666<![CDATA[Real wave propagation in the isotropic-relaxed micromorphic model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160790?rss=1
For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.
]]>2017-01-11T00:28:48-08:00info:doi/10.1098/rspa.2016.0790hwp:master-id:royprsa;rspa.2016.07902017-01-11Research articles47321972016079020160790<![CDATA[Beyond linear fields: the Lie-Taylor expansion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160525?rss=1
The work extends the linear fields’ solution of compressible nonlinear magnetohydrodynamics (MHD) to the case where the magnetic field depends on superlinear powers of position vector, usually, but not always, expressed in Cartesian components. Implications of the resulting Lie–Taylor series expansion for physical applicability of the Dolzhansky–Kirchhoff (D–K) equations are found to be positive. It is demonstrated how resistivity may be included in the D–K model. Arguments are put forward that the D–K equations may be regarded as illustrating properties of nonlinear MHD in the same sense that the Lorenz equations inform about the onset of convective turbulence. It is suggested that the Lie–Taylor series approach may lead to valuable insights into other fluid models.
]]>2017-01-04T00:05:20-08:00info:doi/10.1098/rspa.2016.0525hwp:master-id:royprsa;rspa.2016.05252017-01-04Research articles47321972016052520160525<![CDATA[Finite-element modelling of elastic wave propagation and scattering within heterogeneous media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160738?rss=1
The scattering treated here arises when elastic waves propagate within a heterogeneous medium defined by random spatial fluctuation of its elastic properties. Whereas classical analytical studies are based on lower-order scattering assumptions, numerical methods conversely present no such limitations by inherently incorporating multiple scattering. Until now, studies have typically been limited to two or one dimension, however, owing to computational constraints. This article seizes recent advances to realize a finite-element formulation that solves the three-dimensional elastodynamic scattering problem. The developed methodology enables the fundamental behaviour of scattering in terms of attenuation and dispersion to be studied. In particular, the example of elastic waves propagating within polycrystalline materials is adopted, using Voronoi tessellations to randomly generate representative models. The numerically observed scattering is compared against entirely independent but well-established analytical scattering theory. The quantitative agreement is found to be excellent across previously unvisited scattering regimes; it is believed that this is the first quantitative validation of its kind which provides significant support towards the existence of the transitional scattering regime and facilitates future deployment of numerical methods for these problems.
]]>2017-01-04T00:05:20-08:00info:doi/10.1098/rspa.2016.0738hwp:master-id:royprsa;rspa.2016.07382017-01-04Research articles47321972016073820160738<![CDATA[Effect of wall-mediated hydrodynamic fluctuations on the kinetics of a Brownian nanoparticle]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160397?rss=1
The reactive flux formalism (Chandler 1978 J. Chem. Phys.68, 2959–2970. (doi:10.1063/1.436049)) and the subsequent development of methods such as transition path sampling have laid the foundation for explicitly quantifying the rate process in terms of microscopic simulations. However, explicit methods to account for how the hydrodynamic correlations impact the transient reaction rate are missing in the colloidal literature. We show that the composite generalized Langevin equation (Yu et al. 2015 Phys. Rev. E91, 052303. (doi:10.1103/PhysRevE.91.052303)) makes a significant step towards solving the coupled processes of molecular reactions and hydrodynamic relaxation by examining how the wall-mediated hydrodynamic memory impacts the two-stage temporal relaxation of the reaction rate for a nanoparticle transition between two bound states in the bulk, near-wall and lubrication regimes.
]]>2016-12-21T00:05:17-08:00info:doi/10.1098/rspa.2016.0397hwp:master-id:royprsa;rspa.2016.03972016-12-21Research articles47221962016039720160397<![CDATA[A new self-propelled magnetic bearing with helical windings]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160579?rss=1
In this work, a design is proposed for an active, permanent magnet based, self-propelled magnetic bearing, i.e. levitating motor having the following features: (i) simple winding structure, (ii) high load supporting capacity, (iii) no eccentricity sensors, (iv) stable confinement in all translational dimensions, (v) stable confinement in all rotational dimensions, and (vi) high efficiency. This design uses an architecture consisting of a helically wound three-phase stator, and a rotor with the magnets also arranged in a helical manner. Active control is used to excite the rotor at a torque angle lying in the second quadrant. This torque angle is independent of the rotor's position inside the stator cavity; hence the control algorithm is similar to that of a conventional permanent magnet synchronous motor. It is motivated through a physical argument that the bearing rotor develops a lift force proportional to the output torque and that it remains stably confined in space. These assertions are then proved rigorously through a calculation of the magnetic fields, forces and torques. The stiffness matrix of the system is presented and a discussion of stable and unstable operating regions is given.
]]>2016-12-21T00:05:18-08:00info:doi/10.1098/rspa.2016.0579hwp:master-id:royprsa;rspa.2016.05792016-12-21Research articles47221962016057920160579<![CDATA[Using wire shaping techniques and holographic optics to optimize deposition characteristics in wire-based laser cladding]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160603?rss=1
In laser cladding, the potential benefits of wire feeding are considerable. Typical problems with the use of powder, such as gas entrapment, sub-100% material density and low deposition rate are all avoided with the use of wire. However, the use of a powder-based source material is the industry standard, with wire-based deposition generally regarded as an academic curiosity. This is because, although wire-based methods have been shown to be capable of superior quality results, the wire-based process is more difficult to control. In this work, the potential for wire shaping techniques, combined with existing holographic optical element knowledge, is investigated in order to further improve the processing characteristics. Experiments with pre-placed wire showed the ability of shaped wire to provide uniformity of wire melting compared with standard round wire, giving reduced power density requirements and superior control of clad track dilution. When feeding with flat wire, the resulting clad tracks showed a greater level of quality consistency and became less sensitive to alterations in processing conditions. In addition, a 22% increase in deposition rate was achieved. Stacking of multiple layers demonstrated the ability to create fully dense, three-dimensional structures, with directional metallurgical grain growth and uniform chemical structure.
]]>2016-12-21T00:05:18-08:00info:doi/10.1098/rspa.2016.0603hwp:master-id:royprsa;rspa.2016.06032016-12-21Research articles47221962016060320160603<![CDATA[A computational framework for ptychographic reconstructions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160640?rss=1
Ptychography is now a well-established X-ray microscopy tool for synchrotron end-stations equipped with a scanning stage and a pixelated detector. Ptychographic phasing algorithms use information from coherent diffraction to deliver quantitative images of the specimen at a resolution higher than the scanning resolution. These algorithms have traditionally been implemented in software on a per-instrument basis in various degrees of user-friendliness and sophistication. Here, we present Ptypy, a ptychography software written with the intention to serve as a framework across the diverse sets of available instruments and usage cases. A distinctive feature of the software is its formalism, which provides a convenient abstraction of the physical model, thus allowing for concise algorithmic implementations and portability across set-up geometries. We give an overview of the supported usage cases, explain the abstraction layer and design principles, and provide a step-by-step guide describing how an algorithm may be realized in a concise and readable manner. The software capabilities are illustrated with reconstructions from visible light and X-ray data.
]]>2016-12-21T00:05:18-08:00info:doi/10.1098/rspa.2016.0640hwp:master-id:royprsa;rspa.2016.06402016-12-21Research articles47221962016064020160640<![CDATA[Closed-form rocking overturning conditions for a family of pulse ground motions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160662?rss=1
This paper characterizes the stability of a rigid rocking block subjected to a family of multi-lobe pulse ground motions. It unveils a counter to intuition plurality of overturning (OT) modes despite the short duration and bounded energy of the examined ground motions. Accordingly, it describes with original closed-form expressions the critical conditions of all OT modes involving a finite number of impacts. It also proposes pertinent semi-analytical, exact analytical and approximate analytical solutions with respect to the determination of the (unknown) times of impact, as appropriate. The analysis reveals that the first, or lower bound, critical OT mode is in most cases toppling during free rocking after one impact specifically before the end of the pulse. For this case, it elucidates the physical mechanism behind the timing of impact that produces minimum amplitude and maximum amplitude critical OT, respectively, and proposes pertinent closed-form approximations. Finally, the study derives, in ‘universal’ terms, global ‘safety walls’ against rocking OT considering a large number of different pulse shapes.
]]>2016-12-21T00:05:18-08:00info:doi/10.1098/rspa.2016.0662hwp:master-id:royprsa;rspa.2016.06622016-12-21Research articles47221962016066220160662<![CDATA[Modulations of viscous fluid conduit periodic waves]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160533?rss=1
Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlinear Schrödinger (NLS) equation are derived, depending on the carrier wavenumber. Dark and bright envelope solitons are found to persist in long-time numerical solutions of the conduit equation, providing numerical evidence for the existence of strongly nonlinear, large-amplitude envelope solitons. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted and observed. In the large-amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. The viscous fluid conduit system is a mathematically tractable, experimentally viable model system for wide-ranging nonlinear, dispersive wave dynamics.
]]>2016-12-14T00:47:45-08:00info:doi/10.1098/rspa.2016.0533hwp:master-id:royprsa;rspa.2016.05332016-12-14Research articles47221962016053320160533<![CDATA[The effect of a longitudinal density gradient on electron plasma wake field acceleration]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160630?rss=1
Three-dimensional, particle-in-cell, fully electromagnetic simulations of electron plasma wake field acceleration in the blow-out regime are presented. Earlier results are extended by (i) studying the effect of a longitudinal density gradient, (ii) avoiding the use of a co-moving simulation box, (iii) inclusion of ion motion, and (iv) studying fully electromagnetic plasma wake fields. It is established that injecting driving and trailing electron bunches into a positive density gradient of 10-fold increasing density over 10 cm long lithium vapour plasma results in spatially more compact and three times larger, compared with the uniform density case, electric fields (–6.4x10^{10} V m^{–1}), leading to acceleration of the trailing bunch up to 24.4 GeV (starting from an initial 20.4 GeV), with energy transfer efficiencies from the leading to trailing bunch of 75%. In the uniform density case, a –2.5x10^{10} V m^{–1} wake is created leading to acceleration of the trailing bunch up to 22.4 GeV, with energy transfer efficiencies of 65%. It is also established that injecting the electron bunches into a negative density gradient of 10-fold decreasing density over 10 cm long plasma results in spatially more spread and two and a half smaller electric fields (–1.0x10^{10} V m^{–1}), leading to a weaker acceleration of the trailing bunch up to 21.4 GeV, with energy transfer efficiencies of 45%. Taking ion motions into consideration shows that in the plasma wake ion number density can increase over a few times the background value. It is also shown that transverse electromagnetic fields in a plasma wake are of the same order as the longitudinal (electrostatic) ones.
]]>2016-12-14T00:05:25-08:00info:doi/10.1098/rspa.2016.0630hwp:master-id:royprsa;rspa.2016.06302016-12-14Research articles47221962016063020160630<![CDATA[The anelastic Ericksen problem: universal eigenstrains and deformations in compressible isotropic elastic solids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160690?rss=1
The elastic Ericksen problem consists of finding deformations in isotropic hyperelastic solids that can be maintained for arbitrary strain-energy density functions. In the compressible case, Ericksen showed that only homogeneous deformations are possible. Here, we solve the anelastic version of the same problem, that is, we determine both the deformations and the eigenstrains such that a solution to the anelastic problem exists for arbitrary strain-energy density functions. Anelasticity is described by finite eigenstrains. In a nonlinear solid, these eigenstrains can be modelled by a Riemannian material manifold whose metric depends on their distribution. In this framework, we show that the natural generalization of the concept of homogeneous deformations is the notion of covariantly homogeneous deformations—deformations with covariantly constant deformation gradients. We prove that these deformations are the only universal deformations and that they put severe restrictions on possible universal eigenstrains. We show that, in a simply-connected body, for any distribution of universal eigenstrains the material manifold is a symmetric Riemannian manifold and that in dimensions 2 and 3 the universal eigenstrains are zero-stress.
]]>2016-12-14T00:05:25-08:00info:doi/10.1098/rspa.2016.0690hwp:master-id:royprsa;rspa.2016.06902016-12-14Research articles47221962016069020160690<![CDATA[Lax pairs of discrete Painleve equations: (A2+A1)(1) case]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160696?rss=1
In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlevé equations by using a reduced hypercube structure. In particular, we consider the A5(1)-surface q-Painlevé system, which has the affine Weyl group symmetry of type (A_{2}+A_{1})^{(1)}. Two new Lax pairs are found.
]]>2016-12-14T00:47:45-08:00info:doi/10.1098/rspa.2016.0696hwp:master-id:royprsa;rspa.2016.06962016-12-14Research articles47221962016069620160696<![CDATA[Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160456?rss=1
Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.
]]>2016-12-07T01:14:17-08:00info:doi/10.1098/rspa.2016.0456hwp:master-id:royprsa;rspa.2016.04562016-12-07Research articles47221962016045620160456<![CDATA[Measurement of the Earth's rotation: 720 BC to AD 2015]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160404?rss=1
New compilations of records of ancient and medieval eclipses in the period 720 BC to AD 1600, and of lunar occultations of stars in AD 1600–2015, are analysed to investigate variations in the Earth’s rate of rotation. It is found that the rate of rotation departs from uniformity, such that the change in the length of the mean solar day (lod) increases at an average rate of +1.8 ms per century. This is significantly less than the rate predicted on the basis of tidal friction, which is +2.3 ms per century. Besides this linear change in the lod, there are fluctuations about this trend on time scales of decades to centuries. A power spectral density analysis of fluctuations in the range 2–30 years follows a power law with exponent –1.3, and there is evidence of increased power at a period of 6 years. There is some indication of an oscillation in the lod with a period of roughly 1500 years. Our measurements of the Earth’s rotation for the period 720 BC to AD 2015 set firm boundaries for future work on post-glacial rebound and core–mantle coupling which are invoked to explain the departures from tidal friction.
]]>2016-12-07T02:15:14-08:00info:doi/10.1098/rspa.2016.0404hwp:master-id:royprsa;rspa.2016.04042016-12-07Research articles47221962016040420160404<![CDATA[Quaternionic-like manifolds and homogeneous twistor spaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160598?rss=1
Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the ‘quaternionic-like manifolds’. These contain, as particular subclasses, the CR quaternionic and the -quaternionic manifolds. Moreover, the notion of ‘heaven space’ finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a -quaternionic manifold. We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.
]]>2016-12-07T01:14:17-08:00info:doi/10.1098/rspa.2016.0598hwp:master-id:royprsa;rspa.2016.05982016-12-07Research articles47221962016059820160598<![CDATA[Lamb's problem on random mass density fields with fractal and Hurst effects]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160638?rss=1
This paper reports on a generalization of Lamb's problem to a linear elastic, infinite half-space with random fields (RFs) of mass density, subject to a normal line load. Both, uncorrelated and correlated (with fractal and Hurst characteristics) RFs without any weak noise restrictions, are proposed. Cellular automata (CA) is used to simulate the wave propagation. CA is a local computational method which, for rectangular discretization of spatial domain, is equivalent to applying the finite difference method to the governing equations of classical elasticity. We first evaluate the response of CA to an uncorrelated mass density field, more commonly known as white-noise, of varying coarseness as compared to CA's node density. We then evaluate the response of CA to multiscale mass density RFs of Cauchy and Dagum type; these fields are unique in that they are able to model and decouple the field's fractal dimension and Hurst parameter. We focus on stochastic imperfection sensitivity; we determine to what extent the fractal or the Hurst parameter is a significant factor in altering the solution to the planar stochastic Lamb's problem by evaluating the coefficient of variation of the response when compared with the coefficient of variation of the RF.
]]>2016-12-07T01:14:17-08:00info:doi/10.1098/rspa.2016.0638hwp:master-id:royprsa;rspa.2016.06382016-12-07Research articles47221962016063820160638<![CDATA[Initiation and directional control of period-1 rotation for a parametric pendulum]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/472/2196/20160719?rss=1
We study a time-delayed feedback control for initiating period-1 rotations of a vertically excited parametric pendulum from arbitrary initial conditions. The possibility of controlling the direction of rotation has also been explored. We start with a simple linear time-delayed control for which the control gain corresponding to the most stable period-1 rotation has been obtained using the Floquet theory. This control increases the basins of attraction of rotations, but they do not encompass the full initial condition space. We modify our control law by using a switched control gain that destabilizes all the oscillatory solutions, and the entire initial condition space becomes the basin of attraction of either the clockwise or the anticlockwise rotation. By a suitable modification of the switching condition, we can choose a preferential stable direction of rotation. Hence, we can initiate either clockwise or anticlockwise rotation for a parametric pendulum from arbitrary initial conditions. Performance of our controller in achieving this objective has been demonstrated for different sets of parameters to establish its effectiveness.
]]>2016-12-07T01:14:17-08:00info:doi/10.1098/rspa.2016.0719hwp:master-id:royprsa;rspa.2016.07192016-12-07Research articles47221962016071920160719