Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2014"

On the spectrum of two different fractional operators

Abstract: In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I. Physical basis

Abstract: To simulate debris-flow behaviour from initiation to deposition, we derive a depth-averaged, two-phase model that combines concepts of critical-state soil mechanics, grain-flow mechanics and fluid ...

Can Bohmian mechanics be made relativistic

Abstract: In relativistic space–time, Bohmian theories can be formulated by introducing a privileged foliation of space–time. The introduction of such a foliation—as extra absolute space–time structure—would seem to imply a clear violation of Lorentz invariance, and thus a conflict with fundamental relativity. Here, we consider the possibility that, instead of positing it as extra structure, the required foliation could be covariantly determined by the wave function. We argue that this allows for the formulation of Bohmian theories that seem to qualify as fundamentally Lorentz invariant. We conclude with some discussion of whether or not they might also qualify as fundamentally relativistic.

Continuity equations and ODE flows with non-smooth velocity*

Abstract: In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

Abstract: We evaluate a new depth-averaged mathematical model that is designed to simulate all stages of debris-flow motion, from initiation to deposition. A companion paper shows how the model9s five governing equations describe simultaneous evolution of flow thickness, solid volume fraction, basal pore-fluid pressure and two components of flow momentum. Each equation contains a source term that represents the influence of state-dependent granular dilatancy. Here, we recapitulate the equations and analyse their eigenstructure to show that they form a hyperbolic system with desirable stability properties. To solve the equations, we use a shock-capturing numerical scheme with adaptive mesh refinement, implemented in an open-source software package we call D-Claw. As tests of D-Claw, we compare model output with results from two sets of large-scale debris-flow experiments. One set focuses on flow initiation from landslides triggered by rising pore-water pressures, and the other focuses on downstream flow dynamics, runout and deposition. D-Claw performs well in predicting evolution of flow speeds, thicknesses and basal pore-fluid pressures measured in each type of experiment. Computational results illustrate the critical role of dilatancy in linking coevolution of the solid volume fraction and pore-fluid pressure, which mediates basal Coulomb friction and thereby regulates debris-flow dynamics.

When does a physical system compute

Abstract: Computing is a high-level process of a physical system. Recent interest in non-standard computing systems, including quantum and biological computers, has brought this physical basis of computing to the forefront. There has been, however, no consensus on how to tell if a given physical system is acting as a computer or not; leading to confusion over novel computational devices, and even claims that every physical event is a computation. In this paper we introduce a formal framework that can be used to determine whether or not a physical system is performing a computation. We demonstrate how the abstract computational level interacts with the physical device level, drawing the comparison with the use of mathematical models to represent physical objects in experimental science. This powerful formulation allows a precise description of the similarities between experiments, computation, simulation, and technology, leading to our central conclusion: physical computing is the use of a physical system to predict the outcome of an abstract evolution. We give conditions that must be satisfied in order for computation to be occurring, and illustrate these with a range of non-standard computing scenarios. The framework also covers broader computing contexts, where there is no obvious human computer user. We define the critical notion of a ‘computational entity’, and show the role this plays in defining when computing is taking place in physical systems.

A haemodynamic predictor of intraluminal thrombus formation in abdominal aortic aneurysms

Abstract: Intraluminal thrombus (ILT) is present in over 75% of all abdominal aortic aneurysms (AAAs) and probably contributes to the complex biomechanics and pathobiology of these lesions. A reliable predictor of thrombus formation in enlarging lesions could thereby aid clinicians in treatment planning. The primary goal of this work was to identify a new phenomenological metric having clinical utility that is motivated by the hypothesis that two basic haemodynamic features must coincide spatially and temporally to promote the formation of a thrombus on an intact endothelium—platelets must be activated within a shear flow and then be presented to a susceptible endothelium. Towards this end, we propose a new thrombus formation potential (TFP) that combines information on the flow-induced shear history experienced by blood-borne particles that come in close proximity to the endothelium with information on both the time-averaged wall shear stress (WSS) and the oscillatory shear index (OSI) that locally affect the endothelial mechanobiology. To illustrate the possible utility of this new metric, we show computational results for 10 carotid arteries from five patients where regions of low WSS and high OSI tend not to be presented with activated platelets (i.e. they have a low TFP), consistent with the thrombo-resistance of the healthy carotid despite its complex haemodynamics. Conversely, we show results for three patients that high TFP co-localizes with regions of observed thin thrombus in AAAs, which contrasts with findings of low TFP for the abdominal aorta of three healthy subjects. We submit that these promising results suggest the need for further consideration of the TFP, or a similar combined metric, as a potentially useful clinical predictor of the possible formation of ILT in AAAs.

The bond-based peridynamic system with Dirichlet-type volume constraint

Abstract: In this paper, the bond-based peridynamic system is analysed as a non-local boundary-value problem with volume constraint. The study extends earlier works in the literature on non-local diffusion and non-local peridynamic models, to include non-positive definite kernels. We prove the well-posedness of both linear and nonlinear variational problems with volume constraints. The analysis is based on some non-local Poincare-type inequalities and the compactness of the associated non-local operators. It also offers careful characterizations of the associated solution spaces, such as compact embedding, separability and completeness. In the limit of vanishing non-locality, the convergence of the peridynamic system to the classical Navier equations of elasticity with Poisson ratio ¼ is demonstrated.

Numerical study of fractional nonlinear Schrödinger equations.

Abstract: Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrodinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrodinger equation.

Buckling of regular, chiral and hierarchical honeycombs under a general macroscopic stress state.

Abstract: An approach to obtain analytical closed-form expressions for the macroscopic ‘buckling strength’ of various two-dimensional cellular structures is presented. The method is based on classical beam-column end-moment behaviour expressed in a matrix form. It is applied to sample honeycombs with square, triangular and hexagonal unit cells to determine their buckling strength under a general macroscopic in-plane stress state. The results were verified using finite-element Eigenvalue analysis.

Chiral skyrmions in the plane

Abstract: Magnets without inversion symmetry are a prime example of a solid-state system featuring topological solitons on the nanoscale, and a promising candidate for novel spintronic applications. Magnetic...

Strain gradient plasticity under non-proportional loading

Abstract: A critical examination is made of two classes of strain gradient plasticity theories currently available for studying micrometre-scale plasticity. One class is characterized by certain stress quant...

Utilization of structural steel in buildings

Abstract: Over one-quarter of steel produced annually is used in the construction of buildings. Making this steel causes carbon dioxide emissions, which climate change experts recommend be reduced by half in the next 37 years. One option to achieve this is to design and build more efficiently, still delivering the same service from buildings but using less steel to do so. To estimate how much steel could be saved from this option, 23 steel-framed building designs are studied, sourced from leading UK engineering firms. The utilization of each beam is found and buildings are analysed to find patterns. The results for over 10 000 beams show that average utilization is below 50% of their capacity. The primary reason for this low value is ‘rationalization’—providing extra material to reduce labour costs. By designing for minimum material rather than minimum cost, steel use in buildings could be drastically reduced, leading to an equivalent reduction in ‘embodied’ carbon emissions.

Corrosion pitting and environmentally assisted small crack growth

Abstract: In many applications, corrosion pits act as precursors to cracking, but qualitative and quantitative prediction of damage evolution has been hampered by lack of insights into the process by which a crack develops from a pit. An overview is given of recent breakthroughs in characterization and understanding of the pit-to-crack transition using advanced three-dimensional imaging techniques such as X-ray computed tomography and focused ion beam machining with scanning electron microscopy. These techniques provided novel insights with respect to the location of crack development from a pit, supported by finite-element analysis. This inspired a new concept for the role of pitting in stress corrosion cracking based on the growing pit inducing local dynamic plastic strain, a critical factor in the development of stress corrosion cracks. Challenges in quantifying the subsequent growth rate of the emerging small cracks are then outlined with the potential drop technique being the most viable. A comparison is made with the growth rate for short cracks (through-thickness crack in fracture mechanics specimen) and long cracks and an electrochemical crack size effect invoked to rationalize the data.

Study of dynamic underwater implosion mechanics using digital image correlation

Abstract: The physical processes associated with the implosion of cylindrical tubes in a hydrostatic underwater environment were investigated using high-speed three-dimensional digital image correlation (3D ...

Micropolar continuum modelling of bi-dimensional tetrachiral lattices.

Abstract: The in-plane behaviour of tetrachiral lattices should be characterized by bi-dimensional orthotropic material owing to the existence of two orthogonal axes of rotational symmetry. Moreover, the constitutive model must also represent the chirality inherent in the lattices. To this end, a bi-dimensional orthotropic chiral micropolar model is developed based on the theory of irreducible orthogonal tensor decomposition. The obtained constitutive tensors display a hierarchy structure depending on the symmetry of the underlying microstructure. Eight additional material constants, in addition to five for the hemitropic case, are introduced to characterize the anisotropy under Z2 invariance. The developed continuum model is then applied to a tetrachiral lattice, and the material constants of the continuum model are analytically derived by a homogenization process. By comparing with numerical simulations for the discrete lattice, it is found that the proposed continuum model can correctly characterize the static and wave properties of the tetrachiral lattice.

Phase manipulation and the harmonic components of ringing forces on a surface-piercing column

Abstract: A general phase-based harmonic separation method for the hydrodynamic loading on a fixed structure in water waves of moderate steepness is proposed. An existing method demonstrated in the experimental study described by Zang et al. (Zang et al. 2010 In Proc. Third Int. Conf. on Appl. of Phys. Modelling to Port and Coastal Protection. pp. 1–7.) achieves the separation of a total diffraction force into odd and even harmonics by controlling the phase of incident focused waves. Underlying this method is the assumption that the hydrodynamic force in focused waves possesses a Stokes-like structure. Under the same assumption, it is shown here how the harmonic separation method can be generalized, so that the first four sum harmonics can be separated by phase control and linear combinations of the resultant time-histories. The effectiveness of the method is demonstrated by comparisons of the Fourier transforms of the combined time-histories containing the harmonics of interest. The local wave elevations around the focus time are also visualized for the first three harmonics in order to reveal the local dynamics driving components within the wave force time-history.

Inertial manifolds and finite-dimensional reduction for dissipative PDEs ∗

Abstract: This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mane projection theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.

Organic synthesis: the art and science of replicating the molecules of living nature and creating others like them in the laboratory

Abstract: Synthetic organic chemists have the power to replicate some of the most intriguing molecules of living nature in the laboratory and apply their developed synthetic strategies and technologies to construct variations of them. Such molecules facilitate biology and medicine, as they often find uses as biological tools and drug candidates for clinical development. In addition, by employing sophisticated catalytic reactions and appropriately designed synthetic processes, they can synthesize not only the molecules of nature and their analogues, but also myriad other organic molecules for potential applications in many areas of science, technology and everyday life. After a short historical introduction, this article focuses on recent advances in the field of organic synthesis with demonstrative examples of total synthesis of complex bioactive molecules, natural or designed, from the author’s laboratories, and their impact on chemistry, biology and medicine.

Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents.

Abstract: Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension, with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose two dimensions, exactly solvable nonlinear Schrodinger (NLS) equation derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed two-dimensional equation exhibits modulation instability and frequency correction induced by the nonlinear effect, with a directional preference, all of which can be determined through precise analytic result. The two-dimensional NLS equation allows also an exact lump soliton which can model a full-grown surface rogue wave with adjustable height and modular inclination. The lump soliton under the influence of an ocean current appears and disappears preceded by a hole state, with its dynamics controlled by the current term. These desirable properties make our exact model promising for describing ocean rogue waves.

Superfluid light in bulk nonlinear media.

Abstract: We review how the paraxial approximation naturally leads to a hydrodynamic description of light propagation in a bulk Kerr nonlinear medium in terms of a wave equation analogous to the Gross–Pitaevskii equation for the order parameter of a superfluid. The main features of the many-body collective dynamics of the fluid of light in this propagating geometry are discussed: generation and observation of Bogoliubov sound waves in the fluid of light is first described. Experimentally accessible manifestations of superfluidity are then highlighted. Perspectives in view of realizing analogue models of gravity are finally given.

Auxetic two-dimensional lattices with Poisson's ratio arbitrarily close to 1

Abstract: In this paper, we propose a class of lattice structures with macroscopic Poisson's ratio arbitrarily close to the stability limit 1. We tested experimentally the effective Poisson's ratio of the mi...

Boundedness and stabilization in a multi-dimensional chemotaxis—haptotaxis model

Abstract: This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given bywhere χ, ξ and μ are positive parameters and Ω ⊂ ℝn (n ≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for any μ > χ and any sufficiently smooth initial data (u0, w0) satisfying u0 ≥ 0 and w0 > 0, the associated initial–boundary-value problem possesses a unique global smooth solution that is uniformly bounded. Moreover, we analyse the stability and attractivity properties of the non-trivial homogeneous equilibrium (u, v, w) ≡ (1,1, 0) and establish a quantitative result relating the domain of attraction of this steady state to the size of μ. In particular, this will imply that whenever u0 > 0 and 0 < w0 < 1 in there exists a positive constant μ* depending only on χ, ξ, Ω, u0 and w0 such that for any μ < μ* the above global solution (u, v, w) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.

Stiffest elastic networks

Abstract: The rigidity of a network of elastic beams is closely related to its microstructure. We show both numerically and theoretically that there is a class of isotropic networks, which are stiffer than a...

A century of variation in the dependence of Greenland iceberg calving on ice sheet surface mass balance and regional climate change

Abstract: Iceberg calving is a major component of the total mass balance of the Greenland ice sheet (GrIS). A century-long record of Greenland icebergs comes from the International Ice Patrol's record of icebergs (I48N) passing latitude 48° N, off Newfoundland. I48N exhibits strong interannual variability, with a significant increase in amplitude over recent decades. In this study, we show, through a combination of nonlinear system identification and coupled ocean–iceberg modelling, that I48N's variability is predominantly caused by fluctuation in GrIS calving discharge rather than open ocean iceberg melting. We also demonstrate that the episodic variation in iceberg discharge is strongly linked to a nonlinear combination of recent changes in the surface mass balance (SMB) of the GrIS and regional atmospheric and oceanic climate variability, on the scale of the previous 1–3 years, with the dominant causal mechanism shifting between glaciological (SMB) and climatic (ocean temperature) over time. We suggest that this is a change in whether glacial run-off or under-ice melting is dominant, respectively. We also suggest that GrIS calving discharge is episodic on at least a regional scale and has recently been increasing significantly, largely as a result of west Greenland sources.

On nonlinear viscoelastic deformations: a reappraisal of Fung's quasi-linear viscoelastic model.

Abstract: This paper offers a reappraisal of Fung's model for quasi-linear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models. Results from the new model are contrasted with those in the literature for the case of uniaxial elongation of a bar: for an imposed stretch of an incompressible bar and for an imposed load. In the latter case, a numerical solution to a Volterra integral equation is required to obtain the results. This is achieved by a high-order discretization scheme. Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan–Murphy and Gent.

Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

Abstract: Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima–Zwanzig–Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection–reaction problems.

Subglacial hydrology and the formation of ice streams

Abstract: Antarctic ice streams are associated with pressurized subglacial meltwater but the role this water plays in the dynamics of the streams is not known. To address this, we present a model of subglacial water flow below ice sheets, and particularly below ice streams. The base-level flow is fed by subglacial melting and is presumed to take the form of a rough-bedded film, in which the ice is supported by larger clasts, but there is a millimetric water film which submerges the smaller particles. A model for the film is given by two coupled partial differential equations, representing mass conservation of water and ice closure. We assume that there is no sediment transport and solve for water film depth and effective pressure. This is coupled to a vertically integrated, higher order model for ice-sheet dynamics. If there is a sufficiently small amount of meltwater produced (e.g. if ice flux is low), the distributed film and ice sheet are stable, whereas for larger amounts of melt the ice–water system can become unstable, and ice streams form spontaneously as a consequence. We show that this can be explained in terms of a multi-valued sliding law, which arises from a simplified, one-dimensional analysis of the coupled model.

Three-dimensionally deformable, highly stretchable, permeable, durable and washable fabric circuit boards

Abstract: This paper reports fabric circuit boards (FCBs), a new type of circuit boards, that are three-dimensionally deformable, highly stretchable, durable and washable ideally for wearable electronic applications. Fabricated by using computerized knitting technologies at ambient dry conditions, the resultant knitted FCBs exhibit outstanding electrical stability with less than 1% relative resistance change up to 300% strain in unidirectional tensile test or 150% membrane strain in three-dimensional ball punch test, extraordinary fatigue life of more than 1 000 000 loading cycles at 20% maximum strain, and satisfactory washing capability up to 30 times. To the best of our knowledge, the performance of new FCBs has far exceeded those of previously reported metal-coated elastomeric films or other organic materials in terms of changes in electrical resistance, stretchability, fatigue life and washing capability as well as permeability. Theoretical analysis and numerical simulation illustrate that the structural conversion of knitted fabrics is attributed to the effective mitigation of strain in the conductive metal fibres, hence the outstanding mechanical and electrical properties. Those distinctive features make the FCBs particularly suitable for next-to-skin electronic devices. This paper has further demonstrated the application potential of the knitted FCBs in smart protective apparel for in situ measurement during ballistic impact.

Energy-based analysis of biochemical cycles using bond graphs.

Abstract: Thermodynamic aspects of chemical reactions have a long history in the physical chemistry literature. In particular, biochemical cycles require a source of energy to function. However, although fundamental, the role of chemical potential and Gibb's free energy in the analysis of biochemical systems is often overlooked leading to models which are physically impossible. The bond graph approach was developed for modelling engineering systems, where energy generation, storage and transmission are fundamental. The method focuses on how power flows between components and how energy is stored, transmitted or dissipated within components. Based on the early ideas of network thermodynamics, we have applied this approach to biochemical systems to generate models which automatically obey the laws of thermodynamics. We illustrate the method with examples of biochemical cycles. We have found that thermodynamically compliant models of simple biochemical cycles can easily be developed using this approach. In particular, both stoichiometric information and simulation models can be developed directly from the bond graph. Furthermore, model reduction and approximation while retaining structural and thermodynamic properties is facilitated. Because the bond graph approach is also modular and scaleable, we believe that it provides a secure foundation for building thermodynamically compliant models of large biochemical networks.