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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- recent issues1471-2946Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Relative geodesics in bi-invariant Lie groups]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160619?rss=1
Motivated by registration problems, this paper deals with a curve matching problem in homogeneous spaces. Let G be a connected finite-dimensional bi-invariant Lie group and K a closed subgroup. A smooth curve g in G is said to be admissible if it can transform two smooth curves f_{1} and f_{2} in G/K from one to the other. An (f_{1}, f_{2})-relative geodesic (Holm et al. 2013 Proc. R. Soc. A469, 20130297. (doi:10.1098/rspa.2013.0297)) is defined as a critical point of the total energy E(g) as g varies in the set of all (f_{1}, f_{2})-admissible curves. We obtain the Euler–Lagrange equation, a first-order differential equation, satisfied by a relative geodesic. Furthermore, the Euler–Lagrange equation is simplified for the case where G/K is globally symmetric. As a concrete example, relative geodesics are found for special cases where G is SO(3) and K is SO(2). As an application of discrepancy for curves in S^{2}, we construct and study a new measure of non-congruency for constant speed curves in Euclidean 3-space. Numerical examples are given to illustrate results.
]]>2017-05-17T00:07:29-07:00info:doi/10.1098/rspa.2016.0619hwp:master-id:royprsa;rspa.2016.06192017-05-17Research articles47322012016061920160619<![CDATA[Convective instabilities of Maxwell-Cattaneo fluids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160712?rss=1
Motivated by the need to understand better the dynamics of non-Fourier fluids, we examine the linear and weakly nonlinear stabilities of a horizontal layer of fluid obeying the Maxwell–Cattaneo relationship of heat flux and temperature using three different forms of the time derivative of the heat flux. Linear stability mode regime diagrams in the parameter plane have been established and used to summarize the linear instabilities. The energy balance of the system is used to identify the mechanism by which the Maxwell–Cattaneo effect (i) introduces overstability, (ii) leads to preferred stationary modes with the critical Rayleigh and wavelengths either both increasing or both decreasing, (iii) gives rise to instabilities in a layer heated from above, and (iv) enhances heat transfer. A formal weakly nonlinear analysis leads to evolution equations for the amplitudes of linear instability modes. It is shown that the amplitude of the stationary mode obeys an equation of the Landau–Stuart type. The two equally excitable overstable modes obey two equations of the same type coupled by an interaction term. The evolution of the different amplitudes leads to supercritical stability, supercritical instability or subcritical instability depending on the model and parameter values. The results are presented in regime diagrams.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2016.0712hwp:master-id:royprsa;rspa.2016.07122017-05-17Research articles47322012016071220160712<![CDATA[Causal dissipation for the relativistic dynamics of ideal gases]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160729?rss=1
We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier–Stokes equations.
]]>2017-05-24T00:07:43-07:00info:doi/10.1098/rspa.2016.0729hwp:master-id:royprsa;rspa.2016.07292017-05-24Research articles47322012016072920160729<![CDATA[A note on the self-similar solutions to the spontaneous fragmentation equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160740?rss=1
We provide a method to compute self-similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours. Our procedure is applied to specific cases: (i) the case of mitosis, where fragmentation results into two identical fragments, (ii) fragmentation limited to the formation of sufficiently large fragments, and (iii) processes with fragmentation kernel presenting a power-like behaviour.
]]>2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2016.0740hwp:master-id:royprsa;rspa.2016.07402017-05-10Research articles47322012016074020160740<![CDATA[Elementary exact calculations of degree growth and entropy for discrete equations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160831?rss=1
Second-order discrete equations are studied over the field of rational functions C(z), where z is a variable not appearing in the equation. The exact degree of each iterate as a function of z can be calculated easily using the standard calculations that arise in singularity confinement analysis, even when the singularities are not confined. This produces elementary yet rigorous entropy calculations.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2016.0831hwp:master-id:royprsa;rspa.2016.08312017-05-03Research articles47322012016083120160831<![CDATA[Well-posed continuum equations for granular flow with compressibility and {mu}(I)-rheology]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160846?rss=1
Continuum modelling of granular flow has been plagued with the issue of ill-posed dynamic equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent μ(I)-rheology is ill-posed when the non-dimensional inertial number I is too high or too low. Here, incorporating ideas from critical-state soil mechanics, we derive conditions for well-posedness of partial differential equations that combine compressibility with I-dependent rheology. When the I-dependence comes from a specific friction coefficient μ(I), our results show that, with compressibility, the equations are well-posed for all deformation rates provided that μ(I) satisfies certain minimal, physically natural, inequalities.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2016.0846hwp:master-id:royprsa;rspa.2016.08462017-05-03Research articles47322012016084620160846<![CDATA[Dispersionless (3+1)-dimensional integrable hierarchies]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160857?rss=1
In this paper, we introduce a multi-dimensional version of the R-matrix approach to the construction of integrable hierarchies. Applying this method to the case of the Lie algebra of functions with respect to the contact bracket, we construct integrable hierarchies of (3+1)-dimensional dispersionless systems of the type recently introduced in Sergyeyev (2014 (http://arxiv.org/abs/1401.2122)).
]]>2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2016.0857hwp:master-id:royprsa;rspa.2016.08572017-05-10Research articles47322012016085720160857<![CDATA[Probing quantum state space: does one have to learn everything to learn something?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160866?rss=1
Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement set-up that reveals this property with as little effort as possible. Here, we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the task as a membership problem related to a partitioning of the quantum state space and, in doing so, connect it to the geometry of the state space. For a general membership problem, we prove various sufficient criteria that force informational completeness, and we explicitly treat several physically relevant examples. For the specific cases that do not require informational completeness, we also determine bounds on the minimal number of measurement outcomes needed to ensure success in the task.
]]>2017-05-24T00:07:44-07:00info:doi/10.1098/rspa.2016.0866hwp:master-id:royprsa;rspa.2016.08662017-05-24Research articles47322012016086620160866<![CDATA[Aeroacoustic catastrophes: upstream cusp beaming in Lilley's equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20160880?rss=1
The downstream propagation of high-frequency acoustic waves from a point source in a subsonic jet obeying Lilley's equation is well known to be organized around the so-called ‘cone of silence’, a fold catastrophe across which the amplitude may be modelled uniformly using Airy functions. Here we show that acoustic waves not only unexpectedly propagate upstream, but also are organized at constant distance from the point source around a cusp catastrophe with amplitude modelled locally by the Pearcey function. Furthermore, the cone of silence is revealed to be a cross-section of a swallowtail catastrophe. One consequence of these discoveries is that the peak acoustic field upstream is not only structurally stable but also at a similar level to the known downstream field. The fine structure of the upstream cusp is blurred out by distributions of symmetric acoustic sources, but peak upstream acoustic beaming persists when asymmetries are introduced, from either arrays of discrete point sources or perturbed continuum ring source distributions. These results may pose interesting questions for future novel jet-aircraft engine designs where asymmetric source distributions arise.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2016.0880hwp:master-id:royprsa;rspa.2016.08802017-05-17Research articles47322012016088020160880<![CDATA[Exact semi-separation of variables in waveguides with non-planar boundaries]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170017?rss=1
Series expansions of unknown fields =nZn in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions Z_{n} are determined by solving local Sturm–Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to Z_{n} cannot be compatible with the physical boundary conditions of , leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in Athanassoulis & Belibassakis (Athanassoulis & Belibassakis 1999 J. Fluid Mech. 389, 275–301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly convergent representation of the field , valid for any smooth, non-planar boundaries and any smooth enough . This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.
]]>2017-05-24T00:07:43-07:00info:doi/10.1098/rspa.2017.0017hwp:master-id:royprsa;rspa.2017.00172017-05-24Research articles47322012017001720170017<![CDATA[Simulating tubulin-associated unit transport in an axon: using bootstrapping for estimating confidence intervals of best-fit parameter values obtained from indirect experimental data]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170045?rss=1
In this paper, we first develop a model of axonal transport of tubulin-associated unit (tau) protein. We determine the minimum number of parameters necessary to reproduce published experimental results, reducing the number of parameters from 18 in the full model to eight in the simplified model. We then address the following questions: Is it possible to estimate parameter values for this model using the very limited amount of published experimental data? Furthermore, is it possible to estimate confidence intervals for the determined parameters? The idea that is explored in this paper is based on using bootstrapping. Model parameters were estimated by minimizing the objective function that simulates the discrepancy between the model predictions and experimental data. Residuals were then identified by calculating the differences between the experimental data and model predictions. New, surrogate ‘experimental’ data were generated by randomly resampling residuals. By finding sets of best-fit parameters for a large number of surrogate data the histograms for the model parameters were produced. These histograms were then used to estimate confidence intervals for the model parameters, by using the percentile bootstrap. Once the model was calibrated, we applied it to analysing some features of tau transport that are not accessible to current experimental techniques.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2017.0045hwp:master-id:royprsa;rspa.2017.00452017-05-03Research articles47322012017004520170045<![CDATA[The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170050?rss=1
New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2017.0050hwp:master-id:royprsa;rspa.2017.00502017-05-03Research articles47322012017005020170050<![CDATA[Crossing the quasi-threshold manifold of a noise-driven excitable system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170058?rss=1
We consider the noise-induced escapes in an excitable system possessing a quasi-threshold manifold, along which there exists a certain point of minimal quasi-potential. In the weak noise limit, the optimal escaping path turns out to approach this particular point asymptotically, making it analogous to an ordinary saddle. Numerical simulations are performed and an elaboration on the effect of small but finite noise is given, which shows that the ridges where the prehistory probability distribution peaks are located mainly within the region where the quasi-potential increases gently. The cases allowing anisotropic noise are discussed and we found that varying the noise term in the slow variable would dramatically raise the whole level of quasi-potentials, leading to significant changes in both patterns of optimal paths and exit locations.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2017.0058hwp:master-id:royprsa;rspa.2017.00582017-05-17Research articles47322012017005820170058<![CDATA[An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170080?rss=1
In Parnell & Abrahams (2008 Proc. R. Soc. A464, 1461–1482. (doi:10.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.
]]>2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2017.0080hwp:master-id:royprsa;rspa.2017.00802017-05-10Research articles47322012017008020170080<![CDATA[Uncertainty relations on nilpotent Lie groups]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170082?rss=1
We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic Rn, and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.
]]>2017-05-17T00:07:30-07:00info:doi/10.1098/rspa.2017.0082hwp:master-id:royprsa;rspa.2017.00822017-05-17Research articles47322012017008220170082<![CDATA[Innovative and collaborative industrial mathematics in Europe]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170083?rss=1
This paper presents a brief review of how industrial mathematics, inspired by the Oxford Study Group activity, organized itself in Europe, gave rise to the European Consortium for Mathematics in Industry, the series of European Study Groups with Industry, and to new modes of productive contacts between industry and applied mathematicians in academia.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2017.0083hwp:master-id:royprsa;rspa.2017.00832017-05-03Special feature47322012017008320170083<![CDATA[Biological imaging using light-addressable potentiometric sensors and scanning photo-induced impedance microscopy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170130?rss=1
Light-addressable potentiometric sensors (LAPS) and scanning photo-induced impedance microscopy (SPIM) use photocurrent measurements at electrolyte–insulator–semiconductor substrates for spatio-temporal imaging of electrical potentials and impedance. The techniques have been used for the interrogation of sensor arrays and the imaging of biological systems. Sensor applications range from the detection of different types of ions and the label-free detection of charged molecules such as DNA and proteins to enzyme-based biosensors. Imaging applications include the temporal imaging of extracellular potentials and dynamic concentration changes in microfluidic channels and the lateral imaging of cell surface charges and cell metabolism. This paper will investigate the current state of the art of the measurement technology with a focus on spatial and temporal resolution and review the biological applications, these techniques have been used for. An outlook on future developments in the field will be given.
]]>2017-05-03T00:05:19-07:00info:doi/10.1098/rspa.2017.0130hwp:master-id:royprsa;rspa.2017.01302017-05-03Special feature47322012017013020170130<![CDATA[Reviewers in 2016]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170248?rss=1
2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2017.0248hwp:master-id:royprsa;rspa.2017.02482017-05-10Editorial47322012017024820170248<![CDATA[Correction to 'Microprocessors: the engines of the digital age]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2201/20170304?rss=1
2017-05-10T00:05:14-07:00info:doi/10.1098/rspa.2017.0304hwp:master-id:royprsa;rspa.2017.03042017-05-10Correction47322012017030420170304<![CDATA[Uncertainty transformation via Hopf bifurcation in fast-slow systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160346?rss=1
Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. We show that a random initial condition distribution can be transformed during the passage near a delayed/dynamic Hopf bifurcation: (i) to certain classes of symmetric copies, (ii) to an almost deterministic output, (iii) to a mixture distribution with differing moments and (iv) to a very restricted class of general distributions. We prove under which conditions the cases (i)–(iv) occur in certain classes vector fields.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0346hwp:master-id:royprsa;rspa.2016.03462017-04-12Research articles47322002016034620160346<![CDATA[Geometric decompositions of collective motion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160571?rss=1
Collective motion in nature is a captivating phenomenon. Revealing the underlying mechanisms, which are of biological and theoretical interest, will require empirical data, modelling and analysis techniques. Here, we contribute a geometric viewpoint, yielding a novel method of analysing movement. Snapshots of collective motion are portrayed as tangent vectors on configuration space, with length determined by the total kinetic energy. Using the geometry of fibre bundles and connections, this portrait is split into orthogonal components each tangential to a lower dimensional manifold derived from configuration space. The resulting decomposition, when interleaved with classical shape space construction, is categorized into a family of kinematic modes—including rigid translations, rigid rotations, inertia tensor transformations, expansions and compressions. Snapshots of empirical data from natural collectives can be allocated to these modes and weighted by fractions of total kinetic energy. Such quantitative measures can provide insight into the variation of the driving goals of a collective, as illustrated by applying these methods to a publicly available dataset of pigeon flocking. The geometric framework may also be profitably employed in the control of artificial systems of interacting agents such as robots.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0571hwp:master-id:royprsa;rspa.2016.05712017-04-26Research articles47322002016057120160571<![CDATA[Group theoretical derivation of the minimal coupling principle]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160629?rss=1
The group theoretical methods worked out by Bargmann, Mackey and Wigner, which deductively establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular first-order invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.
]]>2017-04-12T01:01:34-07:00info:doi/10.1098/rspa.2016.0629hwp:master-id:royprsa;rspa.2016.06292017-04-12Research articles47322002016062920160629<![CDATA[Flow through a very porous obstacle in a shallow channel]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160672?rss=1
A theoretical model, informed by numerical simulations based on the shallow water equations, is developed to predict the flow passing through and around a uniform porous obstacle in a shallow channel, where background friction is important. This problem is relevant to a number of practical situations, including flow through aquatic vegetation, the performance of arrays of turbines in tidal channels and hydrodynamic forces on offshore structures. To demonstrate this relevance, the theoretical model is used to (i) reinterpret core flow velocities in existing laboratory-based data for an array of emergent cylinders in shallow water emulating aquatic vegetation and (ii) reassess the optimum arrangement of tidal turbines to generate power in a tidal channel. Comparison with laboratory-based data indicates a maximum obstacle resistance (or minimum porosity) for which the present theoretical model is valid. When the obstacle resistance is above this threshold the shallow water equations do not provide an adequate representation of the flow, and the theoretical model over-predicts the core flow passing through the obstacle. The second application of the model confirms that natural bed resistance increases the power extraction potential for a partial tidal fence in a shallow channel and alters the optimum arrangement of turbines within the fence.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0672hwp:master-id:royprsa;rspa.2016.06722017-04-26Research articles47322002016067220160672<![CDATA[Design and simulation of origami structures with smooth folds]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160716?rss=1
Origami has enabled new approaches to the fabrication and functionality of multiple structures. Current methods for origami design are restricted to the idealization of folds as creases of zeroth-order geometric continuity. Such an idealization is not proper for origami structures of non-negligible fold thickness or maximum curvature at the folds restricted by material limitations. For such structures, folds are not properly represented as creases but rather as bent regions of higher-order geometric continuity. Such fold regions of arbitrary order of continuity are termed as smooth folds. This paper presents a method for solving the following origami design problem: given a goal shape represented as a polygonal mesh (termed as the goal mesh), find the geometry of a single planar sheet, its pattern of smooth folds, and the history of folding motion allowing the sheet to approximate the goal mesh. The parametrization of the planar sheet and the constraints that allow for a valid pattern of smooth folds are presented. The method is tested against various goal meshes having diverse geometries. The results show that every determined sheet approximates its corresponding goal mesh in a known folded configuration having fold angles obtained from the geometry of the goal mesh.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0716hwp:master-id:royprsa;rspa.2016.07162017-04-26Research articles47322002016071620160716<![CDATA[Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160765?rss=1
All metamaterial applications are based upon the idea that extreme material properties can be achieved through appropriate dynamic homogenization of composites. This homogenization is almost always done for infinite domains and the results are then applied to finite samples. This process ignores the evanescent waves which appear at the boundaries of such finite samples. In this paper, we first clarify the emergence and purpose of these evanescent waves in a model problem consisting of an interface between a layered composite and a homogeneous medium. We show that these evanescent waves form boundary layers on either side of the interface beyond which the composite can be represented by appropriate infinite domain homogenized relations. We show that if one ignores the boundary layers, then the displacement and stress fields are discontinuous across the interface. Therefore, the scattering coefficients at such an interface cannot be determined through the conventional continuity conditions involving only propagating modes. Here, we propose an approximate variational approach for sidestepping these boundary layers. The aim is to determine the scattering coefficients without the knowledge of evanescent modes. Through various numerical examples we show that our technique gives very good estimates of the actual scattering coefficients beyond the long wavelength limit.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0765hwp:master-id:royprsa;rspa.2016.07652017-04-26Research articles47322002016076520160765<![CDATA[The roles of impact and inertia in the failure of a shoelace knot]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160770?rss=1
The accidental untying of a shoelace while walking often occurs without warning. In this paper, we discuss the series of events that lead to a shoelace knot becoming untied. First, the repeated impact of the shoe on the floor during walking serves to loosen the knot. Then, the whipping motions of the free ends of the laces caused by the leg swing produce slipping of the laces. This leads to eventual runaway untangling of the knot. As demonstrated using slow-motion video footage and a series of experiments, the failure of the knot happens in a matter of seconds, often without warning, and is catastrophic. The controlled experiments showed that increasing inertial effects of the swinging laces leads to increased rate of knot untying, that the directions of the impact and swing influence the rate of failure, and that the knot structure has a profound influence on a knot's tendency to untie under cyclic impact loading.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0770hwp:master-id:royprsa;rspa.2016.07702017-04-12Research articles47322002016077020160770<![CDATA[Element analysis: a wavelet-based method for analysing time-localized events in noisy time series]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160776?rss=1
A method is derived for the quantitative analysis of signals that are composed of superpositions of isolated, time-localized ‘events’. Here, these events are taken to be well represented as rescaled and phase-rotated versions of generalized Morse wavelets, a broad family of continuous analytic functions. Analysing a signal composed of replicates of such a function using another Morse wavelet allows one to directly estimate the properties of events from the values of the wavelet transform at its own maxima. The distribution of events in general power-law noise is determined in order to establish significance based on an expected false detection rate. Finally, an expression for an event’s ‘region of influence’ within the wavelet transform permits the formation of a criterion for rejecting spurious maxima due to numerical artefacts or other unsuitable events. Signals can then be reconstructed based on a small number of isolated points on the time/scale plane. This method, termed element analysis, is applied to the identification of long-lived eddy structures in ocean currents as observed by along-track measurements of sea surface elevation from satellite altimetry.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0776hwp:master-id:royprsa;rspa.2016.07762017-04-26Research articles47322002016077620160776<![CDATA[Wave directional spreading from point field measurements]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160781?rss=1
Ocean waves have multidirectional components. Most wave measurements are taken at a single point, and so fail to capture information about the relative directions of the wave components directly. Conventional means of directional estimation require a minimum of three concurrent time series of measurements at different spatial locations in order to derive information on local directional wave spreading. Here, the relationship between wave nonlinearity and directionality is utilized to estimate local spreading without the need for multiple concurrent measurements, following Adcock & Taylor (Adcock & Taylor 2009 Proc. R. Soc. A465, 3361–3381. (doi:10.1098/rspa.2009.0031)), with the assumption that directional spreading is frequency independent. The method is applied to measurements recorded at the North Alwyn platform in the northern North Sea, and the results compared against estimates of wave spreading by conventional measurement methods and hindcast data. Records containing freak waves were excluded. It is found that the method provides accurate estimates of wave spreading over a range of conditions experienced at North Alwyn, despite the noisy chaotic signals that characterize such ocean wave data. The results provide further confirmation that Adcock and Taylor's method is applicable to metocean data and has considerable future promise as a technique to recover estimates of wave spreading from single point wave measurement devices.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0781hwp:master-id:royprsa;rspa.2016.07812017-04-26Research articles47322002016078120160781<![CDATA[Octonions in random matrix theory]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160800?rss=1
The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N=2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N=3. We then proceed to consider the matrix structure X^{}X, when X has random octonion entries. Analytic results are obtained from N=2, but are observed to break down in the 3x3 case.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0800hwp:master-id:royprsa;rspa.2016.08002017-04-05Research articles47322002016080020160800<![CDATA[Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160809?rss=1
In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0809hwp:master-id:royprsa;rspa.2016.08092017-04-26Research articles47322002016080920160809<![CDATA[Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160817?rss=1
The evolution of the free surface of a three-dimensional conducting fluid in the presence of gravity, surface tension and vertical electric field due to parallel electrodes, is considered. Based on the analysis of the Dirichlet–Neumann operators, a series of fully nonlinear models is derived systematically from the Euler equations in the Hamiltonian framework without assumptions on competing length scales can therefore be applied to systems of arbitrary fluid depth and to disturbances with arbitrary wavelength. For special cases, well-known weakly nonlinear models in shallow and deep fluids can be generalized via introducing extra electric terms. It is shown that the electric field has a great impact on the physical system and can change the qualitative nature of the free surface: (i) when the separation distance between two electrodes is small compared with typical wavelength, the Boussinesq, Benney–Luke (BL) and Kadomtsev–Petviashvili (KP) equations with modified coefficients are obtained, and electric forces can turn KP-I to KP-II and vice versa; (ii) as the parallel electrodes are of large separation distance but the thickness of the liquid is much smaller than typical wavelength, we generalize the BL and KP equations by adding pseudo-differential operators resulting from the electric field; (iii) for a quasi-monochromatic plane wave in deep fluid, we derive the cubic nonlinear Schrödinger (NLS) equation, but its type (focusing or defocusing) is strongly influenced by the value of the electric parameter. For sufficient surface tension, numerical studies reveal that lump-type solutions exist in the aforementioned three regimes. Particularly, even when the associated NLS equation is defocusing for a wave train, lumps can exist in fully nonlinear models.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0817hwp:master-id:royprsa;rspa.2016.08172017-04-05Research articles47322002016081720160817<![CDATA[Helicity conservation and twisted Seifert surfaces for superfluid vortices]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160853?rss=1
Starting from the continuum definition of helicity, we derive from first principles its different contributions for superfluid vortices. Our analysis shows that an internal twist contribution emerges naturally from the mathematical derivation. This reveals that the spanwise vector that is used to characterize the twist contribution must point in the direction of a surface of constant velocity potential. An immediate consequence of the Seifert framing is that the continuum definition of helicity for a superfluid is trivially zero at all times. It follows that the Gauss-linking number is a more appropriate definition of helicity for superfluids. Despite this, we explain how a quasi-classical limit can arise in a superfluid in which the continuum definition for helicity can be used. This provides a clear connection between a microscopic and a macroscopic description of a superfluid as provided by the Hall–Vinen–Bekarevich–Khalatnikov equations. This leads to consistency with the definition of helicity used for classical vortices.
]]>2017-04-05T01:29:13-07:00info:doi/10.1098/rspa.2016.0853hwp:master-id:royprsa;rspa.2016.08532017-04-05Research articles47322002016085320160853<![CDATA[Some aspects of radical cascade and relay reactions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160859?rss=1
The ability to create carbon–carbon bonds is at the heart of organic synthesis. Radical processes are particularly apt at creating such bonds, especially in cascade or relay sequences where more than one bond is formed, allowing for a rapid assembly of complex structures. In the present brief overview, examples taken from the authors' laboratory will serve to illustrate the strategic impact of radical-based approaches on synthetic planning. Transformations involving nitrogen-centred radicals, electron transfer from metallic nickel and the reversible degenerative exchange of xanthates will be presented and discussed. The last method has proved to be a particularly powerful tool for the intermolecular creation of carbon–carbon bonds by radical additions even to unactivated alkenes. Various functional groups can be brought into the same molecule in a convergent manner and made to react together in order to further increase the structural complexity. One important benefit of this chemistry is the so-called RAFT/MADIX technology for the manufacture of block copolymers of almost any desired architecture.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0859hwp:master-id:royprsa;rspa.2016.08592017-04-05Perspective47322002016085920160859<![CDATA[Wave energy absorption by a submerged air bag connected to a rigid float]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160861?rss=1
A new wave energy device features a submerged ballasted air bag connected at the top to a rigid float. Under wave action, the bag expands and contracts, creating a reciprocating air flow through a turbine between the bag and another volume housed within the float. Laboratory measurements are generally in good agreement with numerical predictions. Both show that the trajectory of possible combinations of pressure and elevation at which the device is in static equilibrium takes the shape of an S. This means that statically the device can have three different draughts, and correspondingly three different bag shapes, for the same pressure. The behaviour in waves depends on where the mean pressure-elevation condition is on the static trajectory. The captured power is highest for a mean condition on the middle section.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0861hwp:master-id:royprsa;rspa.2016.08612017-04-05Research articles47322002016086120160861<![CDATA[Soft phononic crystals with deformation-independent band gaps]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160865?rss=1
Soft phononic crystals have the advantages over their stiff counterparts of being flexible and reconfigurable. Normally, the band gaps of soft phononic crystals will be modified after deformation due to both geometric and constitutive nonlinearity. Indeed these are important properties that can be exploited to tune the dynamic properties of the material. However, in some instances, it may be that one wishes to deform the medium while retaining the band gap structure. A special class of soft phononic crystals is described here with band gaps that are independent or almost-independent of the imposed mechanical deformation, which enables the design of phononic crystals with robust performance. This remarkable behaviour originates from transformation elasticity theory, which leaves the wave equation and the eigenfrequencies invariant after deformation. The necessary condition to achieve such a property is that the Lagrangian elasticity tensor of the hyperelastic material should be constant, i.e. independent of deformation. It is demonstrated that incompressible neo-Hookean materials exhibit such a unique property. Semilinear materials also possess this property under special loading conditions. Phononic crystals composed of these two materials are studied theoretically and the predictions of invariance, or the manner in which the response deviates from invariance, are confirmed via numerical simulation.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0865hwp:master-id:royprsa;rspa.2016.08652017-04-05Research articles47322002016086520160865<![CDATA[Wrinkle surface instability of an inhomogeneous elastic block with graded stiffness]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160882?rss=1
Surface instabilities have been studied extensively for both homogeneous materials and film/substrate structures but relatively less for materials with continuously varying properties. This paper studies wrinkle surface instability of a graded neo-Hookean block with exponentially varying modulus under plane strain by using the linear bifurcation analysis. We derive the first variation condition for minimizing the potential energy functional and solve the linearized equations of equilibrium to find the necessary conditions for surface instability. It is found that for a homogeneous block or an inhomogeneous block with increasing modulus from the surface, the critical stretch for surface instability is 0.544 (0.456 strain), which is independent of the geometry and the elastic modulus on the surface of the block. This critical stretch coincides with that reported by Biot (1963 Appl. Sci. Res.12, 168–182. (doi:10.1007/BF03184638)) 53 years ago for the onset of wrinkle instabilities in a half-space of homogeneous neo-Hookean materials. On the other hand, for an inhomogeneous block with decreasing modulus from the surface, the critical stretch for surface instability ranges from 0.544 to 1 (0–0.456 strain), depending on the modulus gradient, and the length and height of the block. This sheds light on the effects of the material inhomogeneity and structural geometry on surface instability.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0882hwp:master-id:royprsa;rspa.2016.08822017-04-26Research articles47322002016088220160882<![CDATA[Multifunctional scanning ion conductance microscopy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160889?rss=1
Scanning ion conductance microscopy (SICM) is a nanopipette-based technique that has traditionally been used to image topography or to deliver species to an interface, particularly in a biological setting. This article highlights the recent blossoming of SICM into a technique with a much greater diversity of applications and capability that can be used either standalone, with advanced control (potential–time) functions, or in tandem with other methods. SICM can be used to elucidate functional information about interfaces, such as surface charge density or electrochemical activity (ion fluxes). Using a multi-barrel probe format, SICM-related techniques can be employed to deposit nanoscale three-dimensional structures and further functionality is realized when SICM is combined with scanning electrochemical microscopy (SECM), with simultaneous measurements from a single probe opening up considerable prospects for multifunctional imaging. SICM studies are greatly enhanced by finite-element method modelling for quantitative treatment of issues such as resolution, surface charge and (tip) geometry effects. SICM is particularly applicable to the study of living systems, notably single cells, although applications extend to materials characterization and to new methods of printing and nanofabrication. A more thorough understanding of the electrochemical principles and properties of SICM provides a foundation for significant applications of SICM in electrochemistry and interfacial science.
]]>2017-04-12T01:01:34-07:00info:doi/10.1098/rspa.2016.0889hwp:master-id:royprsa;rspa.2016.08892017-04-12Special feature47322002016088920160889<![CDATA[Does a better model yield a better argument? An info-gap analysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160890?rss=1
Theories, models and computations underlie reasoned argumentation in many areas. The possibility of error in these arguments, though of low probability, may be highly significant when the argument is used in predicting the probability of rare high-consequence events. This implies that the choice of a theory, model or computational method for predicting rare high-consequence events must account for the probability of error in these components. However, error may result from lack of knowledge or surprises of various sorts, and predicting the probability of error is highly uncertain. We show that the putatively best, most innovative and sophisticated argument may not actually have the lowest probability of error. Innovative arguments may entail greater uncertainty than more standard but less sophisticated methods, creating an innovation dilemma in formulating the argument. We employ info-gap decision theory to characterize and support the resolution of this problem and present several examples.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2016.0890hwp:master-id:royprsa;rspa.2016.08902017-04-05Research articles47322002016089020160890<![CDATA[Robust identification of harmonic oscillator parameters using the adjoint Fokker-Planck equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160894?rss=1
We present a model-based output-only method for identifying from time series the parameters governing the dynamics of stochastically forced oscillators. In this context, suitable models of the oscillator’s damping and stiffness properties are postulated, guided by physical understanding of the oscillatory phenomena. The temporal dynamics and the probability density function of the oscillation amplitude are described by a Langevin equation and its associated Fokker–Planck equation, respectively. One method consists in fitting the postulated analytical drift and diffusion coefficients with their estimated values, obtained from data processing by taking the short-time limit of the first two transition moments. However, this limit estimation loses robustness in some situations—for instance when the data are band-pass filtered to isolate the spectral contents of the oscillatory phenomena of interest. In this paper, we use a robust alternative where the adjoint Fokker–Planck equation is solved to compute Kramers–Moyal coefficients exactly, and an iterative optimization yields the parameters that best fit the observed statistics simultaneously in a wide range of amplitudes and time scales. The method is illustrated with a stochastic Van der Pol oscillator serving as a prototypical model of thermoacoustic instabilities in practical combustors, where system identification is highly relevant to control.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0894hwp:master-id:royprsa;rspa.2016.08942017-04-12Research articles47322002016089420160894<![CDATA[The deferred limit method for long waves in a curved waveguide]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160900?rss=1
This paper presents a technique, based on a deferred approach to a limit, for analysing the dispersion relation for propagation of long waves in a curved waveguide. The technique involves the concept of an analytically satisfactory pair of Bessel functions, which is different from the concept of a numerically satisfactory pair, and simplifies the dispersion relations for curved waveguide problems. Details are presented for long elastic waves in a curved layer, for which symmetric and antisymmetric waves are strongly coupled. The technique gives high-order corrections to a widely used approximate dispersion relation based a kinematic hypothesis, and determines rigorously which of its coefficients are exact.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2016.0900hwp:master-id:royprsa;rspa.2016.09002017-04-26Research articles47322002016090020160900<![CDATA[Exact solution for the Poisson field in a semi-infinite strip]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160908?rss=1
The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.
]]>2017-04-19T00:39:10-07:00info:doi/10.1098/rspa.2016.0908hwp:master-id:royprsa;rspa.2016.09082017-04-19Research articles47322002016090820160908<![CDATA[Coupling strength assumption in statistical energy analysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160927?rss=1
This paper is a discussion of the hypothesis of weak coupling in statistical energy analysis (SEA). The examples of coupled oscillators and statistical ensembles of coupled plates excited by broadband random forces are discussed. In each case, a reference calculation is compared with the SEA calculation. First, it is shown that the main SEA relation, the coupling power proportionality, is always valid for two oscillators irrespective of the coupling strength. But the case of three subsystems, consisting of oscillators or ensembles of plates, indicates that the coupling power proportionality fails when the coupling is strong. Strong coupling leads to non-zero indirect coupling loss factors and, sometimes, even to a reversal of the energy flow direction from low to high vibrational temperature.
]]>2017-04-19T00:39:10-07:00info:doi/10.1098/rspa.2016.0927hwp:master-id:royprsa;rspa.2016.09272017-04-19Research articles47322002016092720160927<![CDATA[Robust H{infty} state-feedback control for linear systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160934?rss=1
This paper investigates the problem of robust H_{} control for linear systems. First, the state-feedback closed-loop control algorithm is designed. Second, by employing the geometric progression theory, a modified augmented Lyapunov–Krasovskii functional (LKF) with the geometric integral interval is established. Then, parameter uncertainties and the derivative of the delay are flexibly described by introducing the convex combination skill. This technique can eliminate the unnecessary enlargement of the LKF derivative estimation, which gives less conservatism. In addition, the designed controller can ensure that the linear systems are globally asymptotically stable with a guaranteed H_{} performance in the presence of a disturbance input and parameter uncertainties. A liquid monopropellant rocket motor with a pressure feeding system is evaluated in a simulation example. It shows that this proposed state-feedback control approach achieves the expected results for linear systems in the sense of the prescribed H_{} performance.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2016.0934hwp:master-id:royprsa;rspa.2016.09342017-04-12Research articles47322002016093420160934<![CDATA[Stochastic modelling of membrane filtration]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20160948?rss=1
Membrane fouling during particle filtration occurs through a variety of mechanisms, including internal pore clogging by contaminants, coverage of pore entrances and deposition on the membrane surface. In this paper, we present an efficient method for modelling the behaviour of a filter, which accounts for different retention mechanisms, particle sizes and membrane geometries. The membrane is assumed to be composed of a series of, possibly interconnected, pores. The central feature is a conductivity function, which describes the blockage of each individual pore as particles arrive, which is coupled with a mechanism to account for the stochastic nature of the arrival times of particles at the pore. The result is a system of ordinary differential equations based on the pore-level interactions. We demonstrate how our model can accurately describe a wide range of filtration scenarios. Specifically, we consider a case where blocking via multiple mechanisms can occur simultaneously, which have previously required the study through individual models; the filtration of a combination of small and large particles by a track-etched membrane and particle separation using interconnected pore networks. The model is significantly faster than comparable stochastic simulations for small networks, enabling its use as a tool for efficient future simulations.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2016.0948hwp:master-id:royprsa;rspa.2016.09482017-04-26Research articles47322002016094820160948<![CDATA[Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170016?rss=1
This paper presents a mathematical framework for the design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns that can simultaneously fit two target surfaces with rotational symmetry about a common axis. The geometric parameters of the crease pattern and the folding angles of the target folded state are determined through a set of combined geometric and constraint equations. An algorithm to simulate the folding motion of the designed crease pattern is provided. Furthermore, the conditions and procedures to design folded ring structures that are both developable and flat-foldable and stacked folded structures consisting of two layers that can fold independently or compatibly are discussed. The proposed framework has potential applications in designing engineering doubly curved structures such as deployable domes and folded cores for doubly curved sandwich structures on the aircraft.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2017.0016hwp:master-id:royprsa;rspa.2017.00162017-04-12Research articles47322002017001620170016<![CDATA[Statistical emulation of landslide-induced tsunamis at the Rockall Bank, NE Atlantic]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170026?rss=1
Statistical methods constitute a useful approach to understand and quantify the uncertainty that governs complex tsunami mechanisms. Numerical experiments may often have a high computational cost. This forms a limiting factor for performing uncertainty and sensitivity analyses, where numerous simulations are required. Statistical emulators, as surrogates of these simulators, can provide predictions of the physical process in a much faster and computationally inexpensive way. They can form a prominent solution to explore thousands of scenarios that would be otherwise numerically expensive and difficult to achieve. In this work, we build a statistical emulator of the deterministic codes used to simulate submarine sliding and tsunami generation at the Rockall Bank, NE Atlantic Ocean, in two stages. First we calibrate, against observations of the landslide deposits, the parameters used in the landslide simulations. This calibration is performed under a Bayesian framework using Gaussian Process (GP) emulators to approximate the landslide model, and the discrepancy function between model and observations. Distributions of the calibrated input parameters are obtained as a result of the calibration. In a second step, a GP emulator is built to mimic the coupled landslide-tsunami numerical process. The emulator propagates the uncertainties in the distributions of the calibrated input parameters inferred from the first step to the outputs. As a result, a quantification of the uncertainty of the maximum free surface elevation at specified locations is obtained.
]]>2017-04-12T01:01:34-07:00info:doi/10.1098/rspa.2017.0026hwp:master-id:royprsa;rspa.2017.00262017-04-12Research articles47322002017002620170026<![CDATA[Microstructure-based hyperelastic models for closed-cell solids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170036?rss=1
For cellular bodies involving large elastic deformations, mesoscopic continuum models that take into account the interplay between the geometry and the microstructural responses of the constituents are developed, analysed and compared with finite-element simulations of cellular structures with different architecture. For these models, constitutive restrictions for the physical plausibility of the material responses are established, and global descriptors such as nonlinear elastic and shear moduli and Poisson’s ratio are obtained from the material characteristics of the constituents. Numerical results show that these models capture well the mechanical responses of finite-element simulations for three-dimensional periodic structures of neo-Hookean material with closed cells under large tension. In particular, the mesoscopic models predict the macroscopic stiffening of the structure when the stiffness of the cell-core increases.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2017.0036hwp:master-id:royprsa;rspa.2017.00362017-04-05Research articles47322002017003620170036<![CDATA[Deformed quons and bi-coherent states]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170049?rss=1
We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This deformation involves interesting mathematical problems and suggests possible applications to pseudo-hermitian quantum mechanics. We construct bi-coherent states associated to D-pseudo-quons, and we show that they share many of their properties with ordinary coherent states. In particular, we find conditions for these states to exist, to be eigenstates of suitable annihilation operators and to give rise to a resolution of the identity. Two examples are discussed in details, one connected to an unbounded similarity map, and the other to a bounded map.
]]>2017-04-26T01:28:57-07:00info:doi/10.1098/rspa.2017.0049hwp:master-id:royprsa;rspa.2017.00492017-04-26Research articles47322002017004920170049<![CDATA[A scale-entropy diffusion equation to explore scale-dependent fractality]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170054?rss=1
In the last three decades, fractal geometry became a mathematical tool widely used in physics. Nevertheless, it has been observed that real multi-scale phenomena display a departure to fractality that implies an impossibility to define the multi-scale features with an unique fractal dimension, leading to variations in the scale-space. The scale-entropy diffusion equation theorizes the organization of the scale dynamics involving scale-dependent fractals. A study of the theory is possible through the scale-entropy sink term in the equation and corresponds to precise behaviours in scale-space. In the first part of the paper, we study the scale space features when the scale-entropy sink term is modified. The second part is a numerical investigation and analysis of several solutions of the scale-entropy diffusion equation. By a precise measurement of the transition scales tested on truncated deterministic fractals, we developed a new simple method to estimate fractal dimension which appears to be much better than a classical method. Furthermore, we show that deterministic fractals display intrinsic log-periodic oscillations of the fractal dimension. In order to represent this complex behaviour, we introduce a departure to fractal diagram linking scale-space, scale-dependent fractal dimension and scale-entropy sink. Finally, we construct deterministic scale-dependent fractals and verify the results predicted by the scale-entropy diffusion equation.
]]>2017-04-19T00:05:36-07:00info:doi/10.1098/rspa.2017.0054hwp:master-id:royprsa;rspa.2017.00542017-04-19Research articles47322002017005420170054<![CDATA[Large gyres as a shallow-water asymptotic solution of Eulers equation in spherical coordinates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170063?rss=1
Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.
]]>2017-04-12T00:05:32-07:00info:doi/10.1098/rspa.2017.0063hwp:master-id:royprsa;rspa.2017.00632017-04-12Research articles47322002017006320170063<![CDATA[Quantifying the bending of bilayer temperature-sensitive hydrogels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170092?rss=1
Stimuli-responsive hydrogels can serve as manipulators, including grippers, sensors, etc., where structures can undergo significant bending. Here, a finite-deformation theory is developed to quantify the evolution of the curvature of bilayer temperature-sensitive hydrogels when subjected to a temperature change. Analysis of the theory indicates that there is an optimal thickness ratio to acquire the largest curvature in the bilayer and also suggests that the sign or the magnitude of the curvature can be significantly affected by pre-stretches or small pores in the bilayer. This study may provide important guidelines in fabricating temperature-responsive bilayers with desirable mechanical performance.
]]>2017-04-05T00:05:29-07:00info:doi/10.1098/rspa.2017.0092hwp:master-id:royprsa;rspa.2017.00922017-04-05Research articles47322002017009220170092<![CDATA[Uncertainty quantification and optimal decisions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2200/20170115?rss=1
A mathematical model can be analysed to construct policies for action that are close to optimal for the model. If the model is accurate, such policies will be close to optimal when implemented in the real world. In this paper, the different aspects of an ideal workflow are reviewed: modelling, forecasting, evaluating forecasts, data assimilation and constructing control policies for decision-making. The example of the oil industry is used to motivate the discussion, and other examples, such as weather forecasting and precision agriculture, are used to argue that the same mathematical ideas apply in different contexts. Particular emphasis is placed on (i) uncertainty quantification in forecasting and (ii) how decisions are optimized and made robust to uncertainty in models and judgements. This necessitates full use of the relevant data and by balancing costs and benefits into the long term may suggest policies quite different from those relevant to the short term.
]]>2017-04-26T00:05:45-07:00info:doi/10.1098/rspa.2017.0115hwp:master-id:royprsa;rspa.2017.01152017-04-26Special feature47322002017011520170115<![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[Fluid dynamics of acoustic and hydrodynamic cavitation in hydraulic power systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160345?rss=1
Cavitation is the transition from a liquid to a vapour phase, due to a drop in pressure to the level of the vapour tension of the fluid. Two kinds of cavitation have been reviewed here: acoustic cavitation and hydrodynamic cavitation. As acoustic cavitation in engineering systems is related to the propagation of waves through a region subjected to liquid vaporization, the available expressions of the sound speed are discussed. One of the main effects of hydrodynamic cavitation in the nozzles and orifices of hydraulic power systems is a reduction in flow permeability. Different discharge coefficient formulae are analysed in this paper: the Reynolds number and the cavitation number result to be the key fluid dynamical parameters for liquid and cavitating flows, respectively. The latest advances in the characterization of different cavitation regimes in a nozzle, as the cavitation number reduces, are presented. The physical cause of choked flows is explained, and an analogy between cavitation and supersonic aerodynamic flows is proposed. The main approaches to cavitation modelling in hydraulic power systems are also reviewed: these are divided into homogeneous-mixture and two-phase models. The homogeneous-mixture models are further subdivided into barotropic and baroclinic models. The advantages and disadvantages of an implementation of the complete Rayleigh–Plesset equation are examined.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0345hwp:master-id:royprsa;rspa.2016.03452017-03-15Review articles47321992016034520160345<![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[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[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[A restatement of the natural science evidence concerning catchment-based 'natural flood management in the UK]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160706?rss=1
Flooding is a very costly natural hazard in the UK and is expected to increase further under future climate change scenarios. Flood defences are commonly deployed to protect communities and property from flooding, but in recent years flood management policy has looked towards solutions that seek to mitigate flood risk at flood-prone sites through targeted interventions throughout the catchment, sometimes using techniques which involve working with natural processes. This paper describes a project to provide a succinct summary of the natural science evidence base concerning the effectiveness of catchment-based ‘natural’ flood management in the UK. The evidence summary is designed to be read by an informed but not technically specialist audience. Each evidence statement is placed into one of four categories describing the nature of the underlying information. The evidence summary forms the appendix to this paper and an annotated bibliography is provided in the electronic supplementary material.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0706hwp:master-id:royprsa;rspa.2016.07062017-03-15Review article47321992016070620160706<![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[The cross-over to magnetostrophic convection in planetary dynamo systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160731?rss=1
Global scale magnetostrophic balance, in which Lorentz and Coriolis forces comprise the leading-order force balance, has long been thought to describe the natural state of planetary dynamo systems. This argument arises from consideration of the linear theory of rotating magnetoconvection. Here we test this long-held tenet by directly comparing linear predictions against dynamo modelling results. This comparison shows that dynamo modelling results are not typically in the global magnetostrophic state predicted by linear theory. Then, in order to estimate at what scale (if any) magnetostrophic balance will arise in nonlinear dynamo systems, we carry out a simple scaling analysis of the Elsasser number , yielding an improved estimate of the ratio of Lorentz and Coriolis forces. From this, we deduce that there is a magnetostrophic cross-over length scale, LX(o2/Rmo)D, where _{o} is the linear (or traditional) Elsasser number, Rm_{o} is the system scale magnetic Reynolds number and D is the length scale of the system. On scales well above LX, magnetostrophic convection dynamics should not be possible. Only on scales smaller than LX should it be possible for the convective behaviours to follow the predictions for the magnetostrophic branch of convection. Because LX is significantly smaller than the system scale in most dynamo models, their large-scale flows should be quasi-geostrophic, as is confirmed in many dynamo simulations. Estimating _{o}~=1 and Rm_{o}~=10^{3} in Earth’s core, the cross-over scale is approximately 1/1000 that of the system scale, suggesting that magnetostrophic convection dynamics exists in the core only on small scales below those that can be characterized by geomagnetic observations.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0731hwp:master-id:royprsa;rspa.2016.07312017-03-15Special feature47321992016073120160731<![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[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[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[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[An Euler-Lagrange approach for studying blood flow in an aneurysmal geometry]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160774?rss=1
To numerically study blood flow in an aneurysm, the development of an approach that tracks the moving tissue and accounts for its interaction with the fluid is required. This study presents a mathematical approach that expands fluid mechanics principles, taking into consideration the domain’s motion. The initial fluid equations, derived in Euler form, are expanded to a mixed Euler–Lagrange formulation to study blood flow in the aneurysm during the cardiac cycle. Transport equations are transformed into a moving body-fitted reference frame using generalized curvilinear coordinates. The equations of motion consist of a coupled and nonlinear system of partial differential equations (PDEs). The PDEs are discretized using the finite volume method. Owing to strong coupling and nonlinear terms, a simultaneous solution approach is applied. The results show that velocity is substantially influenced by the pulsating wall. Intensification of polymorphic flow patterns is observed. Increments of Reynolds and Womersley numbers are evident as pulsatility increases. The pressure field reveals areas of a lateral pressure gradient at the aneurysm. As pulsatility increases, the diastolic flow vortex shifts towards the aortic wall, distal to the aneurysmal neck. Wall shear stress is amplified at the shoulders of the moving wall compared with that of the rigid one.
]]>2017-03-29T00:05:24-07:00info:doi/10.1098/rspa.2016.0774hwp:master-id:royprsa;rspa.2016.07742017-03-29Research articles47321992016077420160774<![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[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[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[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[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[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[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[Comment on 'Are some people suffering as a result of increasing mass exposure of the public to ultrasound in air?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160828?rss=1
A number of queries regarding the paper ‘Are some people suffering as a result of increasing mass exposure of the public to ultrasound in air?’ (Leighton 2016 Proc. R. Soc. A472, 20150624 (doi:10.1098/rspa.2015.0624)) have been sent in from readers, almost all based around some or all of a small set of questions. These can be grouped into issues of engineering, human factors and timeliness. Those issues (represented by the most typical wording used in queries) and my responses are summarized in this comment.
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0828hwp:master-id:royprsa;rspa.2016.08282017-03-15Comment47321992016082820160828<![CDATA[Mechanics of ultrasound elastography]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160841?rss=1
Ultrasound elastography enables in vivo measurement of the mechanical properties of living soft tissues in a non-destructive and non-invasive manner and has attracted considerable interest for clinical use in recent years. Continuum mechanics plays an essential role in understanding and improving ultrasound-based elastography methods and is the main focus of this review. In particular, the mechanics theories involved in both static and dynamic elastography methods are surveyed. They may help understand the challenges in and opportunities for the practical applications of various ultrasound elastography methods to characterize the linear elastic, viscoelastic, anisotropic elastic and hyperelastic properties of both bulk and thin-walled soft materials, especially the in vivo characterization of biological soft tissues.
]]>2017-03-01T00:05:20-08:00info:doi/10.1098/rspa.2016.0841hwp:master-id:royprsa;rspa.2016.08412017-03-01Review articles47321992016084120160841<![CDATA[Emerging surface characterization techniques for carbon steel corrosion: a critical brief review]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160852?rss=1
Carbon steel is a preferred construction material in many industrial and domestic applications, including oil and gas pipelines, where corrosion mitigation using film-forming corrosion inhibitor formulations is a widely accepted method. This review identifies surface analytical techniques that are considered suitable for analysis of thin films at metallic substrates, but are yet to be applied to analysis of carbon steel surfaces in corrosive media or treated with corrosion inhibitors. The reviewed methods include time of flight-secondary ion mass spectrometry, X-ray absorption spectroscopy methods, particle-induced X-ray emission, Rutherford backscatter spectroscopy, Auger electron spectroscopy, electron probe microanalysis, near-edge X-ray absorption fine structure spectroscopy, X-ray photoemission electron microscopy, low-energy electron diffraction, small-angle neutron scattering and neutron reflectometry, and conversion electron Moessbauer spectrometry. Advantages and limitations of the analytical methods in thin-film surface investigations are discussed. Technical parameters of nominated analytical methods are provided to assist in the selection of suitable methods for analysis of metallic substrates deposited with surface films. The challenges associated with the applications of the emerging analytical methods in corrosion science are also addressed.
]]>2017-03-08T00:05:22-08:00info:doi/10.1098/rspa.2016.0852hwp:master-id:royprsa;rspa.2016.08522017-03-08Review articles47321992016085220160852<![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[Microprocessors: the engines of the digital age]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160893?rss=1
The microprocessor—a computer central processing unit integrated onto a single microchip—has come to dominate computing across all of its scales from the tiniest consumer appliance to the largest supercomputer. This dominance has taken decades to achieve, but an irresistible logic made the ultimate outcome inevitable. The objectives of this Perspective paper are to offer a brief history of the development of the microprocessor and to answer questions such as: where did the microprocessor come from, where is it now, and where might it go in the future?
]]>2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2016.0893hwp:master-id:royprsa;rspa.2016.08932017-03-15Perspective47321992016089320160893<![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[Resistive-pulse and rectification sensing with glass and carbon nanopipettes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160931?rss=1
Along with more prevalent solid-state nanopores, glass or quartz nanopipettes have found applications in resistive-pulse and rectification sensing. Their advantages include the ease of fabrication, small physical size and needle-like geometry, rendering them useful for local measurements in small spaces and delivery of nanoparticles/biomolecules. Carbon nanopipettes fabricated by depositing a thin carbon layer on the inner wall of a quartz pipette provide additional means for detecting electroactive species and fine-tuning the current rectification properties. In this paper, we discuss the fundamentals of resistive-pulse sensing with nanopipettes and our recent studies of current rectification in carbon pipettes.
]]>2017-03-08T00:05:22-08:00info:doi/10.1098/rspa.2016.0931hwp:master-id:royprsa;rspa.2016.09312017-03-08Special feature47321992016093120160931<![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[Propagation of combustion waves in the shell-core energetic materials with external heat losses]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160937?rss=1
In this paper, the properties and stability of combustion waves propagating in the composite solid energetic material of the shell–core type are numerically investigated within the one-dimensional diffusive-thermal model with heat losses to the surroundings. The flame speed is calculated as a function of the parameters of the model. The boundaries of stability are determined in the space of parameters by solving the linear stability problem and direct integration of the governing non-stationary equations. The results are compared with the characteristics of the combustion waves in pure solid fuel. It is demonstrated that a stable travelling combustion wave solution can exist for the parameters of the model for which the flame front propagation is unstable in pure solid fuel and it can propagate several times faster even in the presence of significant heat losses.
]]>2017-03-29T00:05:24-07:00info:doi/10.1098/rspa.2016.0937hwp:master-id:royprsa;rspa.2016.09372017-03-29Research articles47321992016093720160937<![CDATA[Effect of hydrophobic core on the electrophoresis of a diffuse soft particle]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20160942?rss=1
Electrophoresis of a diffuse soft particle with a charged hydrophobic core is considered under the weak field and low charge density assumptions. The hydrophobic surface of the core is coated with a diffuse polyelectrolyte layer (PEL) in which a gradual transition of the polymer segment distribution from the impenetrable core to the surrounding electrolyte medium is considered. A mathematical model is adopted to analyse the impact of the core hydrophobicity on the diffuse soft particle electrophoresis. The mobility based on the present model for the limiting cases such as bare colloids with hydrophobic core and soft particles with no-slip rigid cores are in good agreement with the existing results. The presence of PEL charges produces the impact of the core hydrophobicity on the soft particle mobility different from the corresponding bare colloid with hydrophobic surface in an electrolyte medium. The impact of the core hydrophobicity is subtle when the hydrodynamic screening length of the PEL is low. Reversal in mobility can be achieved by tuning the core hydrophobicity for an oppositely charged core and PEL.
]]>2017-03-29T00:05:24-07:00info:doi/10.1098/rspa.2016.0942hwp:master-id:royprsa;rspa.2016.09422017-03-29Research articles47321992016094220160942<![CDATA[Introduction for perspectives in geophysical and astrophysical fluids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20170078?rss=1
2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2017.0078hwp:master-id:royprsa;rspa.2017.00782017-03-15Special feature47321992017007820170078<![CDATA[Obituary: Prof. Anjan Kundu]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2199/20170107?rss=1
2017-03-15T00:05:21-07:00info:doi/10.1098/rspa.2017.0107hwp:master-id:royprsa;rspa.2017.01072017-03-15Editorial47321992017010720170107<![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[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[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[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[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[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[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[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[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[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[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[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[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[Challenges to deployment of twenty-first century nuclear reactor systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2198/20160815?rss=1
The science and engineering of materials have always been fundamental to the success of nuclear power to date. They are also the key to the successful deployment and operation of a new generation of nuclear reactor systems and their associated fuel cycles. This article reflects on some of the historical issues, the challenges still prevalent today and the requirement for significant ongoing materials R&D and discusses the potential role of small modular reactors.
]]>2017-02-01T01:05:46-08:00info:doi/10.1098/rspa.2016.0815hwp:master-id:royprsa;rspa.2016.08152017-02-01Perspective47321982016081520160815<![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[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