Large-amplitude love waves

In the context of the finite elasticity theory, we considera model for compressible solids called ‘compressible neo-Hookeanmaterial’. We show how finite-amplitude inhomogeneousplane wave solutions and finite-amplitude unattenuated solutionscan combine to form a finite-amplitude Love wave. We take alayer of finite thickness overlying a solid half-space, bothmade of different prestressed compressible neo-Hookean materials.We derive an exact solution of the equations of motion and boundaryconditions and also obtain results for the energy density andthe energy flux of the waves. Finally, we investigate the specialcase when the interface between the layer and the substrateis in a principal plane of the prestrain. A numerical exampleis given.     

Mathematical modelling for keyhole surgery simulations: a biomechanical model for spleen tissue

There is growing interest in the potential for computer simulationsto provide good training materials for abdominal endoscopicor ‘keyhole’ surgery. This paper describes a preliminarystudy of the mechanical properties of abdominal (spleen) tissue.We show how experimental force–displacement data can beused to derive a biomechanical model for the tissue as an incompressible,homogeneous, isotropic nonlinear elastic material with an exponentialstress–strain law. We also show how the model can be usedto predict the response of the tissue to a surgical probe. Thisinvolves solving a complicated nonlinear constrained boundary-valueproblem, and there is a good fit between the computed solutionand experimental data.     

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domainswith piecewise smooth boundaries is presented. It is intendedfor problems describing the Stokes flow in the situations whereone has corners or cusps formed by parts of the domain boundaryand, due to the nature of the boundary conditions on these partsof the boundary, these regions have a global effect on the shapeof the whole domain and hence have to be resolved with sufficientaccuracy. The algorithm combines the boundary integral equationmethod for the main part of the flow domain and the finite-elementmethod which is used to resolve the corner/cusp regions. Twoparts of the solution are matched along a numerical ‘internalinterface’ or, as a variant, two interfaces, and theyare determined simultaneously by inverting a combined matrixin the course of iterations. The algorithm is illustrated byconsidering the flow configuration of ‘curtain coating’,a flow where a sheet of liquid impinges onto a moving solidsubstrate, which is particularly sensitive to what happens inthe corner region formed, physically, by the free surface andthe solid boundary. The ‘moving contact line problem’is addressed in the framework of an earlier developed interfaceformation model which treats the dynamic contact angle as partof the solution, as opposed to it being a prescribed functionof the contact line speed, as in the so-called ‘slip models’.     

On the Existence and Stability of Spatially Structured Solutions of the Reaction-Diffusion Equations

The reaction-diffusion equations for the well-known ‘Brusselator’chemical kinetic model are investigated when the model is madeconsistent with the principle of detailed balance. In contrastto the original model, the corrected one does not show solutionswith ‘spatial structure’ i.e. solutions with multipleinternal maxima and multiple internal global minima in bothdependent variables. Sufficient conditions for stability ofthe solutions are given in terms of a Rayleigh quotient forgeneral boundary conditions for nonlinear reaction-diffusionequations in general. For the particular case of the ‘Brusselator’it is shown that conditions for a change of stability are muchmore unlikely to be attained as a result of the restrictionsof detailed balancing. The detailed sufficiency condition forthe stability of any steady-state solution and for no branchingfrom the ‘equilibrium’ branch solution depends onwhether the solution has global extrema inside the region, onits boundary, or both     

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump. In memory of Gene Golub.     

Shape sensitivity of curvilinear cracks on interface to non-linear perturbations

The 2D-model of an anisotropic, non-homogeneous, bonded elastic solid with a crack on the interface is considered. We state the linear problem with the stress-free boundary condition at the crack faces in addition to the transmission condition at the connected part of the interface. The sensitivity of the model to non-linear perturbations of the curvilinear crack along the interface is investigated. We obtain the asymptotic expansion and the corresponding derivatives of the potential energy functional with respect to the crack length via the material derivatives of the solution. This allows us to describe the growth or stationarity, and the local optimality conditions by the Griffith rupture criterion. The integral expression of the energy release rate for the considered problems is obtained, and the Cherepanov-Rice integral is discussed.     

An Exact Riemann Solver for a Fluidized Bed Model

We study a 2 x 2 hyperbolic system of conservation laws withsource term arising in a fluidized bed model. The system issolved numerically and results are presented to demonstratethe occurrence of ‘slugging’ in the full model equations.The numerical procedure is based on operator splitting and Godunov'smethod, for which we derive the exact solution of the Riemannproblem. A second-order improvement due to Davis (1988) mayproduce small oscillations near shocks and these can be reducedif the underlying flux limiter of the Davis method is replacedby the minmod limiter.     

Market power analysis in electricity markets using supply function equilibrium model

This paper presents a surveillance method based on the gametheory which is used by the ISO to find whether a power supplierin an electricity market has market power. The paper uses thesupply function equilibrium model to analyse the generationsuppliers’ bidding behaviour and models the ISO's marketpower monitoring problem as a bi-level multi-objective problem.The outer sub-problem is a multi-objective problem which maximizessuppliers’ payoffs, while the inner one is the ISO's marketclearing problem based on the locational marginal pricing mechanism.A discrete method is adopted to find ‘good enough’solutions, in a continuous bidding strategy space, which arethe intersection of all suppliers’ optimal response spacesaccording to Nash equilibrium. The paper utilizes the IEEE 118-bussystem to illustrate the application of the proposed methodwith three suppliers as price setters in the energy market andthe other generators as price takers. The numerical resultsshow that the transmission congestion may enhance the suppliers’ability to exercise market power. Likewise, suppliers’gaming behaviour could relieve the transmission congestion.It is shown that applying price caps is an efficient way ofmitigating market power.     

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.     

Some new aspects of the transient analysis of discrete-parameter Markov models with an application to the evaluation of repair events in power transmission

In Markov reliability modelling, a partitioned state space isused to describe the behaviour of a system each state of whichis associated with the system either being functional or underrepair. Such a system alternates between working and repairperiods indefinitely. Recent research results on the distributionof the sequences of the lengths of working and repair periodsafford the reliability analyst a set of system characteristicswhich can be used in addition to the traditional ones (reliability,point availability, etc.) to describe the system‘s transientbehaviour. In this paper, we present a concise derivation ofclosed-form expressions for the probability mass function andthe factorial moments of the total cumulative ’time‘spent in a subset of the state space by an irreducible or absorbingdiscrete-parameter Markov chain during the first n time instances.This result is then applied to analyse the sequence of repairevents categorized as ’minor‘ and ’major‘of a Markov model of a power transmission system. The numericalimplementation using the Macintosh version of MatLab is alsodiscussed.     

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Variational ‘self-consistent’ estimates for nonlinearproblems are formulated, building on a variational formulationpreviously developed by the authors. The formulation employsa linear ‘comparison medium’ for whose propertiessome ‘self-consistent’ choice is made. In contrastto linear problems, three possible self-consistent choices presentthemselves. The results that they give are analysed for twoparticular systems (a nonlinear dielectric and a nonlinear lossycomposite) for which bounds are already available. Estimatesbased on self-consistent embedding of a single inclusion ina homogeneous matrix composed of ‘comparison material’are also developed.     

Contrary Statements About Mathematics

The preceding paper ‘Strong statements of analysis’by A. R. D. Mathias defends a so-called full-blooded set theorywithout full detail [3]. He again objects to a weak set theorywhich he calls ‘Mac’, in which the usual Zermelo–Fraenkelseparation scheme is required only for formulas with suitably‘restricted’ quantifiers. I had proposed that suchseparation is adequate for all standard uses of set theory inmathematics. But Mathias has not produced any counter examplesof actual mathematics which requires the use of a stronger separation.     

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

Planar elasticity problems are considered for thin domains fixedalong a small part of the end region boundary. The analysisinvolves two small parameters: the normalized thickness of thebody and the normalized length of the fixed part of the boundary.The aim of the paper is to derive an asymptotic approximationof the solution to a boundary-value problem in such a domainand, in particular, analyze the ‘effective boundary conditions’,which occur for the leading-order terms of the asymptotics.We include applications for problems of both anti-plane shearand plane strain elasticity.     

Spatial discretization of partial differential equations with integrals

We consider the problem of constructing spatial finite-differenceapproximations on an arbitrary fixed grid which preserve anynumber of integrals of the partial differential equation andpreserve some of its symmetries. A basis for the space of suchfinite-difference operators is constructed; most cases of interestinvolve a single such basis element. (The ‘Arakawa’Jacobian is such an element, as are discretizations satisfying‘summation by parts’ identities.) We show how thegrid, its symmetries, and the differential operator interactto affect the complexity of the finite difference.     

Quadratisation de Certaines Structures de Poisson

On sait associer à certaines structures de Poisson surRⁿ, de 1-jet nul en 0, des actions de R² sur Rⁿ, donnéespar le ‘rotationnel’ de leur partie quadratiqueet un autre champ de vecteurs. Lorsque ces actions sont ‘nonrésonantes’ et ‘hyperboliques’, onmontre que ces structures sont ‘quadratisables’,en ce sens qu'il existe des coordonnées dans lesquelles,elles sont quadratiques. Dans le cas de la dimension 3, nosrésultats mènent à la ‘non-dégénérescence’générique des structures de Poisson quadratiquesà rotationnels inversibles. We can associate with some Poisson structures defined on Rⁿwith a zero 1-jet at zero, actions from R² on Rⁿ, given by the‘curl’ of their quadratic part and another vectorfield. Assuming that those actions are ‘hyperbolics’and without ‘resonances’, we give a normal formfor those structures. On R³, we prove that every quadratic Poissonstructure with invertible curl, is generically ‘non degenerate’.     

Homogenization of very rough interfaces for the micropolar elasticity theory

In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar isotropic micropolar elastic solids is investigated. The interface is assumed to oscillate between two parallel straight lines. The main aim is to derive homogenized equations in explicit form. These equations are obtained by the homogenization method along with the matrix formalism of the theory of micropolar elasticity. Since obtained homogenized equations are totally explicit, they are a powerful tool for solving various practical problems. As an example, the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type is investigated. The closed-form formulas for the reflection and transmission coefficients have been derived. Based on these formulas, some numerical examples are carried out to show the dependence of the reflection and transmission coefficients on the incident angle and the geometry parameter of the interface.     

Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials

Hubbard trees are invariant trees connecting the points of thecritical orbits of post-critically finite polynomials. Douadyand Hubbard showed in the Orsay Notes that they encode all combinatorialproperties of the Julia sets. For quadratic polynomials, onecan describe the dynamics as a subshift on two symbols, anditinerary of the critical value is called the kneading sequence.Whereas every (pre)periodic sequence is realized by an abstractHubbard tree (see the authors’ preprint from 2007), notevery such tree is realized by a quadratic polynomial. In thispaper, we give an Admissibility Condition that describes preciselywhich sequences correspond to quadratic polynomials. We identifythe occurrence of the so-called ‘evil branch points’as the sole obstruction to being realizable. We also show howto derive the properties of periodic (branch) points in thetree (their periods, relative positions, number of arms andwhether they are evil or not) from the kneading sequence.     

Investigation of the prefracture zone at the tip of a mode I crack reaching a nonsmooth interface of an elastic media

We perform calculation of the initial prefracture zone at the tip of a mode I crack that reaches a nonsmooth interface of two dissimilar elastic media at its corner point by the Wiener–Hopf method. The zone is modeled by a line of normal displacement fracture on a crack continuation. Expressions for the length of the prefracture zone and the potential energy accumulated in it are obtained. Their numerical values are compared with the corresponding values for the prefracture zone in a bonding material on the interface of the media, on the basis of which we draw a conclusion on the possible direction of development of the zone.     

压电材料中两平行不相等界面裂纹的动态特性研究

利用Schmidt方法,研究了压电材料中两个平行不相等的可导通界面裂纹对简谐反平面剪切波的散射问题．利用Fourier变换,使问题的求解转换为对两对以裂纹面张开位移为未知变量的对偶积分方程的求解．数值计算结果表明,动态应力强度因子及电位移强度因子受裂纹的几何参数、入射波频率的影响．在特殊情况下,与已有结果进行了比较分析．同时,电位移强度因子远小于不可导通电边界条件下相应问题的结果．     

Systems with Time Delay in the Calculus of Variations: The Method of Steps

In a recent article (see [8]), we derived necessary and sufficientconditions for minima for the fixed-endpoint problem in thecalculus of variations involving a constant delay in the phasecoordinates. These conditions are expressed, explicitly, interms of the first and second variations. The vanishing of thefirst variation is characterized in terms of an extended Euler'sequation, just as for delay-free problems, but the characterizationof the conditions on the second variation remained unsolved.In this paper we convert, through the ‘method of steps’,the delay problem into one without delay. Although this problemwill not have fixed-endpoint constraints, it allows us to introduce,in a natural way, the concept of ‘conjugate sequence’;this solves the main difficulty for delay problems, namely,to give initial conditions for existence and uniqueness of solutionsof the Hamiltonian system (which is a difference–differentialsystem with both advanced and retarded arguments). The conditionson the second variation are then characterized by an extra conditionwhich is based exclusively on a solution of a given matrix Riccatiequation.