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On the application of the Wiener-Hopf technique to problems in dynamic elasticity
No full textMany problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener–Hopf functional equations defined in a strip in a complex transform plane. Apart from a few special cases, the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener–Hopf kernels are of matrix form. The key step in the solution of a Wiener–Hopf equation, which is to decompose the kernel into a product of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from special matrices which yield commutative factorizations, no procedure has yet been devised to factorize exactly general matrix kernels.
This paper shall demonstrate, by way of example, that the Wiener–Hopf approximant matrix (WHAM) procedure for obtaining approximate factors of matrix kernels (recently introduced by the author in [SIAM J. Appl. Math. 57 (2) (1997) 541]) is applicable to the class of matrix kernels found in elasticity, and in particular to problems in QNDE. First, as a motivating example, the kernel arising in the model of diffraction of skew incident elastic waves on a semi-infinite crack in an isotropic elastic space is studied. This was first examined in a seminal work by Achenbach and Gautesen [J. Acoust. Soc. Am. 61 (2) (1977) 413] and here three methods are offered for deriving distinct non-commutative factorizations of the kernel. Second, the WHAM method is employed to factorize the matrix kernel arising in the problem of radiation into an elastic half-space with mixed boundary conditions on its face. Third, brief mention is made of kernel factorization related to the problems of flexural wave diffraction by a crack in a thin (Mindlin) plate, and body wave scattering by an interfacial crack
On transient oscillations of plates in moving fluids
No full textIn recent years, various groups of researchers have looked at the two-dimensional motions of an undamped infinite thin elastic plate lying under a uniformly moving incompressible inviscid fluid. The plate is driven, usually by a single frequency time-harmonic line-source switched on at a finite time. The system’s behaviour is interesting as it can be shown to be absolutely unstable for flow velocities above a critical value, and below this the long-time solution is convectively unstable (downstream of the source) for a sufficiently low forcing frequency. These results do not appear particularly plausible from a physical point of view, and there is some question regarding the realisation of long-time steady behaviour, and so this article attempts to examine ways in which the model problem can be improved. In particular, the effects of introducing plate thickness and fluid compressibility to the model are studied. This is carried out by comparing the morphology of the original and modified solutions in the complex wavenumber space. It is found that, in the limit of small fluid-to-plate density ratio, the two problems exhibit qualitatively identical behaviour. However, the addition of structural damping is shown herein to lead to a very different solution – the initial boundary value problem is absolutely unstable at all flow velocities. Various other modifications to the original model, including finiteness of the plate, three-dimensional effects and nonlinearity, are discussed and their impact on the long-time response of the system is assessed
The Ziegler and Zariski spectra of some domestic string algebras
No full textIt was a conjecture of the second author that the Cantor–Bendixson rank of the Ziegler spectrum of a finite-dimensional algebra is either less than or equal to 2 or is undefined. Here we refute this conjecture by describing the Ziegler spectra of some domestic string algebras where arbitrary finite values greater than 2 are obtained. We give a complete description of the Ziegler and Gabriel–Zariski spectra of the simplest of these algebras. The conjecture has been independently refuted by Schröer who, extending his work (1997) on these algebras, computed their Krull–Gabriel dimension
Local times of fractional Brownian sheet
No full textLet B0H = {B0H(t), t Î \mathbb R+N } be a real-valued fractional Brownian sheet. Consider the (N, d) Gaussian random field BH defined by
BH(t) = (BH1(t), ¼, BHd(t)) (t Î \mathbb R+N),
where BH1, ¼, BHd are independent copies of B0H. In this paper, the existence and joint continuity of the local times of BH are established
Continua of states in boundary-layer flows
Get PDFWe consider a class of three-dimensional boundary-layer flows, which may be viewed as an extension of the Falkner–Skan similarity form, to include a cross-flow velocity component, about a plane of symmetry. In general, this provides a range of three- dimensional boundary-layer solutions, parameterized by a Falkner–Skan similarity parameter, n, together with a further parameter, [Psi][infty infinity], which is associated with a cross-flow velocity component in the external flow. In this work two particular cases are of special interest: for n = 0 the similarity equations possess a family of solutions related to the Blasius boundary layer; for n = 1 the similarity solution provides an exact reduction of the Navier–Stokes equations corresponding to the flow near a saddle point of attachment. It is known from the work of Davey (1961) that in this latter class of flow, a continuum of solutions can be found. The continuum arises (in general) because it is possible to find states with an algebraic, rather than exponential, behaviour in the far field. In this work we provide a detailed overview of the continuum states, and show that a discrete infinity of ‘exponential modes’ are smoothly embedded within the ‘algebraic modes’ of the continuum. At a critical value of the cross-flow, these exponential modes appear as a cascade of eigensolutions to the far-field equations, which arise in a manner analogous to the energy eigenstates found in quantum mechanical problems described by the Schrödinger equation.
The presence of a discrete infinity of exponential modes is shown to be a generic property of the similarity equations derived for a general n. Furthermore, we show that there may also exist non-uniqueness of the continuum; that is, more than one continuum of states can exist, that are isolated for fixed n and [Psi][infty infinity], but which are connected through an unfolded transcritical bifurcation at a critical value of the cross-flow parameter, [Psi][infty infinity].
The multiplicity of states raises the question of solution selection, which is addressed using two stability analyses that assume the same basic symmetry properties as the base flow. In one case we consider a steady, algebraic form in the ‘streamwise’ direction, whilst in the other a temporal form is assumed. In both cases it is possible to extend the analysis to consider a continuous spectrum of disturbances that decay algebraically in the wall-normal direction. We note some obvious parallels that exist between such stability analyses and the approach to the continua of states described earlier in the paper.
We also discuss the appearance of analogous non-unique states to the Falkner–Skan equation in the presence of an adverse pressure gradient (i.e. n < 0) in an appendix
Differential algebraic equations with after-effect
No full textIn this paper, we are concerned with the solution of delay differential algebraic equations. These are differential algebraic equations with after-effect, or constrained delay differential equations. The general semi-explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments. Even simply stated problems of this type can give rise to difficult analytical and numerical problems. The more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations. Our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems, and to indicate problems that require further research
Configurations of 2n - 2 quadrics in Rn with 3 2n - 1 common tangent lines
No full textWe construct 2n-2 smooth quadrics in Rn whose equations have the same degree 2 homogeneous parts such that these quadrics have 3· 2n-1 isolated common real tangent lines. Special cases of the construction give examples of 2n-2 spheres with affinely dependent centres such that all but one of the radii are equal, and of 2n-2 quadrics which are translated images of each other
Short-term growth in children with growth disorders
No full textObjective : We have previously demonstrated that normal prepubertal growth over 1 year is composed of growth spurts lasting an average of 8 weeks, separated by periods of very slow growth or stasis. We have now analysed short-term growth patterns in eight children with different growth disorders: Turner syndrome ( n = 2), intrauterine growth retardation (IUGR, n = 1) and growth hormone (GH) deficiency (GHD, n = 5).
Methodology : Height was measured daily in the morning by parents over 4-12 months. Regression and time series analysis were used to characterize short-term growth. In two boys (GHD and IUGR) their normal twin brother was measured in parallel.
Results : All height velocity curves, based on regression analysis, showed a biphasic pattern, characterized by growth spurts of varying amplitudes and periods of very slow growth or growth stasis. When compared to growth curves in normal children, the principal qualitative differences in GHD and Turner syndrome were increased stasis time and reduced growth spurt amplitude. In IUGR reduced amplitude and length of growth spurts were seen, but the time spent in stasis was similar to normal children. Two naive patients with GHD increased the amplitude of their growth spurts by a mean 0.013 cm day -1 on GH treatment, with the mean length of their growth spurts increasing by 10 days. Their time spent in stasis decreased from 19% to 6% on GH. In two subjects with GHD the growth pattern during maintenance GH treatment was similar to that seen in normal children. Using time series analysis significant periodicities in height measurements were seen in the majority of children with growth disorders, which disappeared in patients with GHD in the catch-up phase after commencing GH therapy.
Conclusions : (1) The growth spurts and stases seen in normal children are also observed in those with growth disorders, (2) different growth disorders have variable effects on the spurt-stasis model of childhood growth, (3) catch-up growth on GH in children with GHD was achieved by increasing the amplitude of the growth spurts and reducing the time spent in stasis
Interval Analysis in Matlab
Get PDFThe introduction of fast and efficient software for interval arithmetic, such as the MATLAB toolbox INTLAB, has resulted in the increased popularity of the use of interval analysis. We give an introduction to interval arithmetic and explain how it is implemented in the toolbox INTLAB. A tutorial is provided for those who wish to learn how to use INTLAB. We then focus on the interval versions of some important problems in numerical analysis. A variety of techniques for solving interval linear systems of equations are discussed, and these are then tested to compare timings and accuracy. We consider univariate and multivariate interval nonlinear systems and describe algorithms that enclose all the roots. Finally, we give an application of interval analysis. Interval arithmetic is used to take account of rounding errors in the computation of Viswanath's constant, the rate at which a random Fibonacci sequence increases