Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2004"

A study of the characteristics of white noise using the empirical mode decomposition method

Abstract: Based on numerical experiments on white noise using the empirical mode decomposition (EMD) method, we find empirically that the EMD is effectively a dyadic filter, the intrinsic mode function (IMF) components are all normally distributed, and the Fourier spectra of the IMF components are all identical and cover the same area on a semi–logarithmic period scale. Expanding from these empirical findings, we further deduce that the product of the energy density of IMF and its corresponding averaged period is a constant, and that the energy–density function is chi–squared distributed. Furthermore, we derive the energy–density spread function of the IMF components. Through these results, we establish a method of assigning statistical significance of information content for IMF components from any noisy data. Southern Oscillation Index data are used to illustrate the methodology developed here.

A new test for chaos in deterministic systems

Abstract: We describe a new test for determining whether a given deterministic dynamical system is chaotic or non-chaotic. In contrast to the usual method of computing the maximal Lyapunov exponent, our method is applied directly to the time-series data and does not require phase-space reconstruction. Moreover, the dimension of the dynamical system and the form of the underlying equations are irrelevant. The input is the time-series data and the output is 0 or 1, depending on whether the dynamics is non-chaotic or chaotic. The test is universally applicable to any deterministic dynamical system, in particular to ordinary and partial differential equations, and to maps. Our diagnostic is the real valued function p(t)=∫φ(xs)cos(θ(s)) ds, where φ is an observable on the underlying dynamics x(t) and θ(t)&=ct+∫φ(xs)cos(θ(s)) ds. The constant c > 0 is fixed arbitrarily. We define the mean-square displacement M(t) for p(t) and set K=limt→∞\logM(t)\logt. Using recent developments in ergodic theory, we argue that, typically, K=0, signifying non-chaotic dynamics, or $K=1$, signifying chaotic dynamics.

Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations

Abstract: In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrodinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

Cubature on Wiener space

Abstract: It is well known that there is a mathematical equivalence between ‘solving’ parabolic partial differential equations (PDEs) and ‘the integration’ of certain functionals on Wiener space. Monte Carlo simulation of stochastic differential equations (SDEs) is a naive approach based on this underlying principle. In finite dimensions, it is well known that cubature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener space. In the process we develop high–order numerical schemes valid for high–dimensional SDEs and semi–elliptic PDEs.

High– and low–cycle fatigue crack initiation using polycrystal plasticity

Abstract: A polycrystal plasticity finite–element model has been developed for nickel–base alloy C263. That is, a representative region of the material, containing about 60 grains, has been modelled using crystal plasticity, taking account of grain morphology and crystallographic orientation. With just a single material property (in addition to standard elastic properties), namely, the critical resolved shear stress, the model is shown to be capable of predicting correctly a wide range of cyclic plasticity behaviour in face–centred cubic nickel alloy C263. A fatigue crack initiation criterion is proposed, based simply on a critical accumulated slip. When this critical slip is achieved within the microstructure, crack initiation is taken to have occurred. The model predicts the development of persistent slip bands within individual grains with a width of ca. 10 μm. The model also predicts that crack initiation can occur preferentially at grain triple points under both low– (LCF) and high–cycle fatigue (HCF). For the case of HCF, this also corresponds to a free surface. The polycrystal plasticity model combined with the fatigue crack initiation criterion are shown to predict correctly the standard Basquin and Goodman correlations in HCF, and the Coffin–Manson correlation in LCF. The model predictions are based on just two material properties: the critical resolved shear stress and the critical accumulated slip. Just one experimental test is required to determine these properties, for a given temperature, which have been obtained for nickel alloy C263. Predictions of life for nickel alloy C263 are then made over a broad range of loading conditions covering both LCF and HCF. Good agreement with experiments is achieved, despite the simplicity of the proposed ‘two–parameter’ model. A simple three–dimensional form of the model has provided an estimate of the fatigue limit for HCF crack initiation in C263.

On two-phase sediment transport: sheet flow of massive particles

Abstract: A model is presented for concentrated sediment transport that is driven by strong, fully developed turbulent shear flows over a mobile bed. Balance equations for the average mass, momentum and energy for the two phases are phrased in terms of concentration–weighted (Favre averaged) velocities. Closures for the correlations between fluctuations in concentration and particle velocities are based on those for collisional grain flow. This is appropriate for particles that are so massive that their fall velocity exceeds the friction velocity of the turbulent fluid flow. Particular attention is given to the slow flow in the region of high concentration above the stationary bed. A failure criterion is introduced to determine the location of the stationary bed. The proposed model is solved numerically with a finite–difference algorithm in both steady and unsteady conditions. The predictions of sediment concentration and velocity are tested against experimental measurements that involve massive particles. The model is further employed to study several global features of sheet flow such as the total sediment transport rate in steady and unsteady conditions.

A two-layer approach to wave modelling

Abstract: A set of model equations for water–wave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers. Within each layer, an independent velocity profile is derived. With two separate velocity profiles, matched at the interface of the two layers, the resulting set of equations has three free parameters, allowing for an optimization with known analytical properties of water waves. The optimized model equations show good linear wave characteristics up to kh ≈ 6 , while the second–order nonlinear behaviour is captured to kh ≈ 6 as well. A numerical algorithm for solving the model equations is developed and tested against one– and two–horizontal–dimension cases. Agreement with laboratory data is excellent.

Peeling from a biomimetically patterned thin elastic film

Abstract: Inspired by the observation that many naturally occurring adhesives arise as texturedthin films, we consider the displacement-controlled peeling of a flexible plate from an incision-patterned thin adhesive elastic layer. We find that crack initiation from an incision on the film occurs at a load much higher than that required to propagate it on a smooth adhesive surface; multiple incisions thus cause the crack to propagate intermittently. Microscopically, this mode of crack initiation and propagation in geometrically confined thin adhesive films is related to the nucleation of cavitation bubbles behind the incision which must grow and coalesce before a viable crack propagates. Our theoretical analysis allows us to rationalize these experimental observations qualitatively and quantitatively and suggests a simple design criterion for increasing the interfacial fracture toughness of adhesive films.

Mechanisms of deep penetration of soft solids, with application to the injection and wounding of skin

Abstract: Micromechanical models are developed for the deep penetration of a soft solid by a flatbottomed and by a sharptipped cylindrical punch. The soft solid is taken to represent mammalian skin and silic...

Numerical methods for strong solutions of stochastic differential equations: an overview

Abstract: This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Substructural organization, dislocation plasticity and harmonic generation in cyclically stressed wavy slip metals

Abstract: Organized substructural arrangements of dislocations formed in wavy slip, face–centred–cubic metals during cyclic stress–induced fatigue are shown analytically to engender a substantial nonlinearity in the microelastic–plastic deformation resulting from an impressed stress perturbation. The non–Hookean stress–strain relationship is quantified by a material nonlinearity parameter βthat for a given fatigue state is highly sensitive to the volume fractions of veins and persistent slip bands (PSBs), PSB internal stresses, dislocation multipole configurations, dislocation loop lengths, dipole heights and the densities of secondary dislocations in the substructures. The effects on β of vacancy, microcrack and macrocrack formation are also addressed. The connection between β and acoustic harmonic generation is obtained. The model is applied to calculations of β for fatigued polycrystalline nickel as a function of per cent life to fracture. For cyclic stress–controlled loading at 241 MPa, the model predicts a monotonic increase in β of ca. 360% over the fatigue life. For strain–controlled loading at a total strain of 1.75 × 10 −3 , a monotonic increase in β of ca. 375% over the fatigue life is predicted.

On thermomechanical restrictions of continua

Abstract: The central idea proposed here is that, in entropy–producing processes, a specific choice from among a competing class of constitutive functions can be made so that the state variables evolve in a way that maximizes the rate of entropy production. When attention is restricted to quadratic forms for the rate of entropy production, the assumption leads to results that are fully in keeping with linear phenomenological relations that satisfy the Onsager relations. In other words, the usual linear evolution laws such as Fourier's law of heat conduction, Fick's law, Darcy's law, Newton's law of viscosity, etc., all corroborate this assumption. We clarify the difference between the maximum rate of entropy production criterion that characterizes choices among constitutive relations and the minimum entropy production theorem due to Onsager (1931) that characterizes steady states for special choices of the rate of entropy production . We then show that for other forms of entropy production that are not quadratic for which the Onsager relations and related theorems cannot be applied, we can use the procedure described here to obtain nonlinear laws. We demonstrate by means of an example that even yield–type phenomena can be accommodated within this framework, while they cannot within the framework of Onsager. We also discuss issues concerning constraints, especially in thermoelasticity within the context of our ideas.

Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion

Abstract: This paper considers the nonlinear stability oftravelling wavefronts of a time-delayed diffusive Nicholson blowflies equation. We prove that, under a weighted L2 norm, ifa solution is sufficiently close to a travelling wave front initially, it converges exponentially to the wavefront as t → ∞. The rate ofconvergence is also estimated.

Coiling of viscous jets

Abstract: A stream of viscous fluid falling from a sufficient height onto a surface forms a series of regular coils. I use a numerical model for a deformable fluid thread to predict the coiling frequency as a function of the thread's radius, the flow rate, the fall height, and the fluid viscosity. Three distinct modes of coiling can occur: viscous (e.g. toothpaste), gravitational (honey falling from a moderate height) and inertial (honey falling from a great height). When inertia is significant, three states of steady coiling with different frequencies can exist over a range of fall heights. The numerically predicted coiling frequencies agree well with experimental measurements in the inertial coiling regime.

The Hilbert spectrum via wavelet projections

Abstract: Non-stationary signals are increasingly analysed in the time-frequency domain to determine the variation of frequency components with time. It was recently proposed in this journal that such signals could be analysed by projections onto the time-frequency plane giving a set of monocomponent signals. These could then be converted to ‘analytic’ signals using the Hilbert transform and their instantaneous frequency calculated, which when weighted by the energy yields the ‘Hilbert energy spectrum’ for that projection. Agglomeration over projections yields the complete Hilbert spectrum. We show that superior results can be obtained using waveletbased projections. The maximal-overlap (undecimated/stationary/translation invariant) discrete wavelet transform and wavelet packet transforms are used, with

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

Abstract: Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

A theory of stress softening of elastomers based on finite chain extensibility

Abstract: In this paper we develop a theory to describe the Mullins effect in rubber-like solids, based on the notion of limiting chain extensibility associated with the Gent model of rubber elasticity. We relate the theory to the mechanisms of network alteration and to the pseudo-elasticity theory of the Mullins effect. The inherently anisotropic nature of the Mullins effect is also discussed.

Kinetics of the bainite transformation

Abstract: A theory is developed for the evolution of bainite as a function of time, temperature, chemical composition and austenite grain size. The model takes into account the details of the mechanism of transformation, including the fact that nucleation begins at the austenite grain surfaces, and that the growth of a sheaf occurs by the repeated nucleation of small platelets. Predictions made using the model are shown to compare well against published isothermal and continuous cooling transformation data.

Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution

Abstract: Analytic approximations for percolation points in two–dimensional and three–dimensional particulate arrays have been reported for only a very few, simple particle geometries. Here, an analytical approach is presented to determine the percolative properties (i.e. statistical cluster properties) of permeable ellipsoids of revolution. We generalize a series expansion technique, previously used by other authors to study arrays of spheres and cubes. Our analytic solutions are compared with Monte Carlo simulation results, and show good agreement at low particle aspect ratio. At higher aspect ratios, the analytic approximation becomes even more computationally intensive than direct simulation of a number of realizations. Additional simulation results, and simplified, closed–form bounding expressions for percolation thresholds are also presented. Limitations and applications of the asymptotic expressions are discussed in the context of materials design and design of sensor arrays.

Minimal surfaces and multifunctionality

Abstract: Triply periodic minimal surfaces are objects of great interest to physical scientists, biologists and mathematicians. It has recently been shown that triply periodic two-phase bicontinuous composites with interfaces that are the Schwartz primitive (P) and diamond (D) minimal surfaces are not only geometrically extremal but extremal for simultaneous transport of heat and electricity. More importantly, here we further establish the multifunctionality of such two-phase systems by showing that they are also extremal when a competition is set up between the effective bulk modulus and the electrical (or thermal) conductivity of the composite. The implications of our findings for materials science and biology, which provides the ultimate multifunctional materials, are discussed.

Stability of steady states and existence of travelling waves in a vector-disease model

Abstract: In this paper, a host-vector model is considered for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modelled in the partial integro-differential equation by a diffusion term. For the general model, we first study the stability of the steady states using the contracting-convex-sets technique. When the spatial variable is one dimensional and the delay kernel assumes some special form, we establish the existence of travelling wave solutions by using the linear chain trick and the geometric singular perturbation method.

An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion

Abstract: Fractional Brownian motion (FBM) with Hurst parameter index between 0 and 1 is a stochastic process originally introduced by Kolmogorov in a study of turbulence. Many other applications have subsequently been suggested. In order to obtain good mathematical models based on FBM, it is necessary to have a stochastic calculus for such processes. The purpose of this paper is to give an introduction to this newly developed theory of stochastic integration for FBM based on white-noise theory and (Malliavin–type) differentiation.

Periodic solutions of higher-dimensional discrete systems

Abstract: Consider the second-order discrete system where f ∈ C (R × Rm, Rm), f(t + M, Z) = f(t, Z) for any (t, Z) ∈ R × Rm and M is a positive integer. By making use of critical-point theory, the existence of M-periodic solutions of (*) is obtained.

Derivation of the Born rule from operational assumptions

Abstract: The Born rule is derived from operational assumptions, together with assumptions of quantum mechanics that concern only the deterministic development of the state. Unlike Gleason's theorem, the argument applies even if probabilities are defined for only a single resolution of the identity, so it applies to a variety of foundational approaches to quantum mechanics. It also provides a probability rule for state spaces that are not Hilbert spaces.

New additive results for the g-Drazin inverse

Abstract: The paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

Detailed analyses of grain–scale plastic deformation in columnar polycrystalline aluminium using orientation image mapping and crystal plasticity models

Abstract: Deformation studies at grain level have been performed in order to model how individual crystals in a polycrystalline material deform. The experiment was carried out by plane–strain compression of a high–purity polycrystalline aluminium with columnar grain structure with near 〈100〉 fibre texture parallel to the constrained direction in the channel die. This structure was chosen to allow a fully three–dimensional characterization of the grain structure. The grain orientations were mapped by orientation image microscopy, as the directionally solidified material was deformed in steps of 10% to a total height reduction of 40%. The grains were found either to show nearly uniform rotations or to split into two types of deformation bands, either with repeating orientation fields or with non–repeating orientation fields. The Taylor model and the finite–element method (FEM) were, as usual, quite successful in predicting the average deformation texture, but the Taylor model failed totally to predict the rotation of individual grains. The FEM was more successful in predicting the individual grain rotations but did not, as in a previous study, predict the morphology of the deformation bands. The significant discovery, made here, was that it appeared possible to model the local deformation at a grain scale, from the observed individual deviations of the grain rotations from those predicted if each grain underwent just the plane–strain conditions imposed on the sample. Plastic work rates were computed allowing four shears (two shears in each of the two contact planes) that are compatible with the channel–die geometry. It was found that in all the ‘hard’ grains (those with high Taylor factors), the additional shears (in type and magnitude) that minimized the plastic energy dissipation rate were the same shears that were needed to match the observed grain rotations. Adjacent Taylor ‘soft’ grains were found to have been subjected to the additional shears imposed by their neighbouring hard grains. This was true even when these shears raised the plastic work of the soft grains. This effect was most marked when the soft grains were small in size. These additional shears found by this plastic work analysis were consistent with the observed additional shear seen in the overall shape change of the sample. The grains forming non–repeating orientation fields had low initial Taylor factors and were surrounded by high–Taylor–factor grains, usually of larger size, but which had adopted somewhat different extra shears. The grains showing repeating orientation fields were found to have an orientation, near ‘cube’, (001) 〈100〉, which was initially unstable, leading to a break–up into different orientation fields when deformed. These differing deformation bands in the cube grains followed different strain paths, which also minimized their plastic work.

Propagation through nonlinear time-dependent bubble clouds and the estimation of bubble populations from measured acoustic characteristics

Abstract: For several decades the propagation characteristics of acoustic pulses (attenuation and sound speed) have been inverted in attempts to measure the size distributions of gas bubbles in liquids. While this has biomedical and industrial applications, most notably it has been attempted in the ocean for defence and environmental purposes, where the bubbles are predominantly generated by breaking waves. Such inversions have required assumptions, and the state–of–the–art technique still assumes that the bubbles undergo linear, steady–state monochromatic pulsations in the free field, without interacting. The measurements always violate, to a greater or lesser extent, these assumptions. The errors incurred by the use of such assumptions have been difficult to quantify, but are expected to be most severe underneath breakers in the surf zone, where the void fraction is greatest. Very few measurements have been made in this important region of the ocean. This paper provides a method by which attenuation can be predicted through clouds of bubbles which need not be homogeneous, nor restricted to linear steady–state monochromatic pulsations. To allow inversion of measured surf zone attenuations to estimate bubble populations with current computational facilities, this model is simplified such that the bubble cloud is assumed to be homogeneous and the bubbles oscillating in steady state (although still nonlinearly). The uses of the new methods for assessing the errors introduced in using state–of–the–art inversions are discussed, as are their implications for oceanographic and industrial nonlinear bubble counters, for biomedical contrast agents, and for sonar target detection in the surf zone.

The multiple-soliton solution of the Camassa-Holm equation

Abstract: This paper refines Johnso's implementation of Constantin's method for solving the Camassa–Holm equation for a multiple–soliton solution. An analytical formula for the q(y) and an explicit relation between x and y are found. An algorithm of solving for u(y) is presented. How to introduce time variable t into the solution is also clearly explained.

Simulation of nonlinear interactions between waves and floating bodies through a finite-element-based numerical tank

Abstract: Fully nonlinear water–wave interactions with a floating structure are investigated through a numerical towing tank. A wave maker is installed on one end of the tank while a numerical beach based on a combination of damping zone and Sommerfeld condition is adopted on the other side of the tank. A floating body is placed at a location in the tank, where it will be set into motion by the waves generated by the wave maker. The simulation is based on the velocity potential theory together with the finite–element method. The mesh used follows the deformation of the free surface and the body motion. Its structure is adjusted and the distribution of elements is completely rearranged when the motion is big to avoid an over–distorted grid. Auxiliary functions are introduced to decouple the nonlinear mutual dependence between the hydrodynamic force and the body motion. Extensive numerical results are provided for vertical circular cylinders and a simplified floating production, storage and offloading, for which meshes are obtained through an efficient scheme based on a two–dimensional tri–tree method.

On the boundary-layer equations for power-law fluids

Abstract: We reconsider the problem of the boundary–layer flow of a non-Newtonian fluid whose constitutive law is given by the classical Ostwald–de Waele power–law model. The boundary–layer equations are solved in similarity form. The resulting similarity solutions for shear–thickening fluids are shown to have a finite–width crisis resulting in the prediction of a finite–width boundary layer. A secondary viscous adjustment layer is required in order to smooth out the solution and to ensure correct matching with the far–field boundary conditions. In the case of shear–thinning fluids, the similarity forms have solutions whose decay into the far field is strongly algebraic. Smooth matching between these inner algebraically decaying solutions and an outer uniform flow is achieved via the introduction of a viscous diffusion layer.