Gabriel J. Lord

Cellular buckling in long structures

A long structural system with an unstable (subcritical)post-buckling response that subsequently restabilizes typically deformsin a cellular manner, with localized buckles first forming and thenlocking up in sequence. As buckling continues over a growing number ofcells, the response can be described by a set of lengthening homoclinicconnections from the fundamental equilibrium state to itself. In thelimit, this leads to a heteroclinic connection from the fundamentalunbuckled state to a post-buckled state that is periodic. Under suchprogressive displacement the load tends to oscillate between twodistinct values.The paper is both a review and a pointer tofuture research. The response is described via a typical system, asimple but ubiquitous model of a strut on a foundation which includesinitially-destabilizing and finally-restabilizing nonlinear terms. Anumber of different structural forms, including the axially-compressedcylindrical shell, a typical sandwich structure, a model of geologicalfolding and a simple link model are shown to display such behaviour. Amathematical variational argument is outlined for determining the globalminimum postbuckling state under controlled end displacement (rigidloading). Finally, the paper stresses the practical significance of aMaxwell-load instability criterion for such systems. This criterion,defined under dead loading to be where the pre-buckled and post-buckledstate have the same energy, is shown to have significance in the presentsetting under rigid loading also. Specifically, the Maxwell load isargued to be the limit of minimum energy localized solutions asend-shortening tends to infinity.

Waves and bumps in neuronal networks with axo-dendritic synaptic interactions

We consider a firing rate model of a neuronal network continuum that incorporates axo-dendritic synaptic processing and the finite conduction velocities of action potentials. The model equation is an integral one defined on a spatially extended domain. Apart from a spatial integral mixing the network connectivity function with space-dependent delays, arising from non-instantaneous axonal communication, the integral model also includes a temporal integration over some appropriately identified distributed delay kernel. These distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs. They are interpreted as Green’s functions of some linear dierential operator. Exploiting this Green’s function description we discuss formal reductions of this non-local system to equivalent partial dierential equation (PDE) models. We distinguish between those spatial connectivity functions that give rise to local PDE models and those that give rise to PDE models with delayed non-local terms. For cases in which local PDEs are derived, we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions. We also calculate exact solutions, parameterized by axonal conduction velocity, for the Heaviside firing rate function (the sigmoidal firing rate function in the limit of infinte gain). The inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses, which we study analytically and numerically in terms of a global homoclinic orbit. Finally, we consider the impact of dendritic interactions on waves and on static spatially localized solutions. Exact analysis for

An Introduction to Computational Stochastic PDEs

This book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of the art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB codes included (and downloadable) allows readers to perform computations themselves and solve the test problems discussed. Practical examples are drawn from finance, mathematical biology, neuroscience, fluid flow modeling and materials science.

Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells

Abstract A structural system with an unstable post-buckling response that subsequently restabilizes has the potential to exhibit homoclinic connections from the fundamental equilibrium state to itself over a range of loads, and heteroclinic connections between fundamental and periodic equilibrium states over a different (smaller) range of loads. It is argued that such equilibrium configurations are important in the interpretation of observed behaviour, and govern the minimum possible post-buckling loads. To illustrate this, the classical problem of a long thin axially-compressed cylindrical shell is revisited from three different perspectives: asymptotic conjecture, analogy with nonlinear dynamics, and numerical continuation analysis of a partial spectral decomposition of the underlying equilibrium equations. The nonlinear dynamics analogy demonstrates that the structure of the heteroclinic connections is more complicated than that indicated by the asymptotics: this is confirmed by the numerics. However, when the asymptotic portrayal is compared to the numerics, it turns out to be surprisingly accurate in its Maxwell-load prediction of the practically-significant first minimum to appear in the post-buckling regime.

Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem

Abstract This paper concerns homoclinic solutions to periodic orbits in a fourth-order Hamiltonian system arising from a reduction of the classical water-wave problem in the presence of surface tension. These solutions correspond to travelling solitary waves which converge to non-decaying ripples at infinity. An analytical result of Amick and Toland (1992), showing the existence of such homoclinic orbits to small-amplitude periodic orbits in a singular limit, is extended numerically. Also, a related result by Amick and McLeod (1991), showing the non-existence of homoclinic solutions to zero, is motivated geometrically. A general boundary-value method is constructed for continuation of homoclinic orbits to periodic orbits in Hamiltonian and reversible systems. Numerical results are presented using the path-following software auto , showing that the Amick-Toland solutions persist well away from the singular limit and for large-amplitude periodic orbits. Special account is taken of the phase shift between the two periodic solutions in the asymptotic limits. Furthermore, new multi-modal homoclinic solutions to periodic orbits are shown to exist under a transversality hypothesis, which is verified a posteriori by explicit computation. Continuation of these new solutions reveals limit points with respect to the singular parameter.

The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds

We present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions. We discretise in space by a Galerkin method and in time by using a stochastic exponential integrator. We show that for spatially regular (smooth) noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces (and the noise becomes rougher).

Computation of localized post buckling in long axially compressed cylindrical shells

Buckling is investigated of a long thin cylindrical shell under longitudinal compression as modelled by the von Kármán–Donnell equations. Evidence is reviewed for the buckling being localized to some portion of the axial length. In accordance with this observed behaviour the equations are first approximated circumferentially by a Galerkin procedure, whereupon cross–symmetric homoclinic solutions of the resulting system of ordinary differential equations are sought in the axial direction. Results are compared with experimental and other numerical data. Excellent agreement with experiments is achieved with fewer approximating modes than other methods.

Do true elevation gravity–capillary solitary waves exist? A numerical investigation

This paper extends the numerical results of Hunter & Vanden-Broeck (1983) and Vanden-Broeck (1991) which were concerned with studies of solitary waves on the surface of fluids of finite depth under the action of gravity and surface tension. The aim of this paper is to answer the question of whether small-amplitude elevation solitary waves exist. Several analytical results have proved that bifurcating from Froude number F = 1, for Bond number τ between 0 and 1/3, there are families of ‘generalized’ solitary waves with periodic tails whose minimum amplitude is an exponentially small function of F−1. An open problem (which, for τ sufficiently close to 1/3, was recently proved by Sun 1999 to be false) is whether this amplitude can ever be zero, which would give a truly localized solitary wave. The problem is first addressed in terms of model equations taking the form of generalized fifth-order KdV equations, where it is demonstrated that if such a zero-tail-amplitude solution occurs, it does so along codimension-one lines in the parameter plane. Moreover, along solution paths of generalized solitary waves a topological distinction is found between cases where the tail does vanish and those where it does not. This motivates a new set of numerical results for the full problem, formulated using a boundary integral method, namely to probe the size of the tail amplitude as τ varies for fixed F > 1. The strong conclusion from the numerical results is that true solitary waves of elevation do not exist for the steady gravity–capillary water wave problem, at least for 9/50 < τ < 1=3. This finding confirms and explains previous asymptotic results by Yang & Akylas.

Postprocessing for Stochastic Parabolic Partial Differential Equations

We investigate the strong approximation of stochastic parabolic partial differential equations with additive noise. We introduce postprocessing in the context of a standard Galerkin approximation, although other spatial discretizations are possible. In time, we follow [G. J. Lord and J. Rougemont, IMA J. Numer. Anal., 24 (2004), pp. 587-604] and use an exponential integrator. We prove strong error estimates and discuss the best number of postprocessing terms to take. Numerically, we evaluate the efficiency of the methods and observe rates of convergence. Some experiments with the implicit Euler-Maruyama method are described.

Cylinder Buckling: The Mountain Pass as an Organizing Center

We revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder, we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this estimate is the existence of a mountain pass point for the system. We prove the existence on bounded domains of such solutions for almost all loads and then numerically compute example mountain pass solutions. Numerically the mountain pass solution with lowest energy has the form of a single dimple. We interpret these results and validate the lower bound against some experimental results available in the literature.