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Item Metadata only Refined MDP-based branch-and-fix algorithm for the Hamiltonian cycle problem(Informs, 2009) Ejov, V.; Filar, J.; Haythorpe, M.; Nguyen, G.We consider the famous Hamiltonian cycle problem (HCP) embedded in a Markov decision process (MDP). More specifically, we consider the HCP as an optimisation problem over the space of occupation measures induced by the MDP's stationary policies. In recent years, this approach to the HCP has led to a number of alternative formulations and algorithmic approaches. In this paper, we focus on a specific embedding, because of the work of Feinberg. We present a “branch-and-fix” type algorithm that solves the HCP. At each branch of the algorithm, only a linear program needs to be solved and the dimensions of the successive linear programs are shrinking rather than expanding. Because the nodes of the branch-and-fix tree correspond to specially structured 1-randomised policies, we characterise the latter. This characterisation indicates that the total number of such policies is significantly smaller than the subset of all 1-randomised policies. Finally, we present some numerical results.Item Metadata only A coupled damage-plasticity model for concrete based on thermodynamic principles: Part II: non-local regularization and numerical implementation(John Wiley & Sons, 2008) Nguyen, G.; Houlsby, G.AbstractNon‐local regularization is applied to a new coupled damage–plasticity model (Int. J. Numer. Anal. Meth. Geomech. 2007; DOI: 10.1002/nag.627), turning it into a non‐local model. This procedure resolves softening‐related problems encountered in local constitutive models when dealing with softening materials. The parameter identification of the new non‐local coupled damage–plasticity model is addressed, with all parameters being shown to be obtainable from the experimental data on concrete. Because of the appearance of non‐local spatial integrals in the constitutive equations, a new implementation scheme is developed for the integration of the non‐local incremental constitutive equations in nonlinear finite element analysis. The performance of the non‐local model is assessed against a range of two‐dimensional structural tests on concrete, illustrating the stability of the stress update procedure and the lack of mesh dependency of the model. Copyright © 2007 John Wiley & Sons, Ltd.Item Metadata only A coupled damage-plasticity model for concrete based on thermodynamic principles: Part 1: model formulation and parameter identification(John Wiley & Sons, 2008) Nguyen, G.; Houlsby, G.AbstractThe development of a coupled damage‐plasticity constitutive model for concrete is presented. Emphasis is put on thermodynamic admissibility, rigour and consistency both in the formulation of the model, and in the identification of model parameters based on experimental tests. The key feature of the thermodynamic framework used in this study is that all behaviour of the model can be derived from two specified energy potentials, following procedures established beforehand. Based on this framework, a constitutive model featuring full coupling between damage and plasticity in both tension and compression is developed. Tensile and compressive responses of the material are captured using two separate damage criteria, and a yield criterion with a multiple hardening rule. A crucial part of this study is the identification of model parameters, with these all being shown to be identifiable and computable based on standard tests on concrete. Behaviour of the model is assessed against experimental data on concrete. Copyright © 2007 John Wiley & Sons, Ltd.Item Metadata only Development of an approach to constitutive modelling of concrete: isotropic damage coupled with plasticity(Elsevier, 2008) Nguyen, G.; Korsunsky, A.The paper presents an approach to constitutive modelling of concrete using damage mechanics and plasticity theory. The thermodynamic formulation, and parameter identification of a non-local coupled damage-plasticity model are presented in this study. The particular focus is the calibration of model parameters. It is shown that both the local parameters and the parameters governing the non-local interaction can be determined from experimental data reliably and consistently. A novel procedure is developed for parameter identification, using the separation of total dissipation energy into additive parts corresponding to different dissipation mechanisms. The relationship between the local and non-local parameters is also addressed, helping to obtain model responses consistent with the fracture energy of the material. The application of the model and the calibration procedure proposed in this study to the numerical failure analysis of concrete structures is illustrated through a series of real structural tests, showing both the performance of the model and the consistency of the proposed calibration procedure. © 2008 Elsevier Ltd. All rights reserved.Item Metadata only Carbon nanotori as traps for atoms and ions(Elsevier Science BV, 2012) Chan, Y.; Cox, B.; Hill, J.Carbon nanotori surely represent an ideal location to trap both charged and uncharged atoms, since they are open, accessible and possess strong attractive energy. In this paper, we investigate the plausibility of carbon nanotori as atomic traps and we use the continuum approximation together with the Lennard-Jones potential to model the encapsulation of an atom or ion by a nanotorus. The critical geometric factors such as the minor and major radii, i.e. r and R of the nanotorus, for which the maximum interaction between the atom and the nanotorus occurs, are determined. For various atoms, assumed situated along the axis of the torus, the minimum potential energy between the atom and the nanotorus is calculated and compared, and shown to be approximately kηε σ2, where η is the uniform atomic density, ε and σ are the Lennard-Jones well depth and the van der Waals radius, respectively, and k is a universal non-dimensional constant with the approximate value -12.42. The results given in this paper might be used for future drug delivery and biosensing design. © 2012 Elsevier B.V. All rights reserved.Item Metadata only Patch dynamics for macroscale modelling in one dimension(Cambridge University Press, 2012) Bunder, J.; Roberts, A.; Engineering Mathematics and Applications Conference (4 Dec 2011 - 7 Dec 2011 : Australia); Faculty of Engineering, Computer & Mathematical SciencesWe discuss efficient macroscale modelling of microscale systems using patch dynamics. This pilot study effectively homogenises microscale varying diffusion in one dimension. The `equation free' approach requires that the microscale model be solved only on small spatial patches. Suitable boundary conditions ensure that these patches are well coupled. By centre manifold theory, an emergent closed model exists on the macroscale. Patch dynamics systematically approximates this macroscale model. The modelling is readily adaptable to higher dimensions and to reaction-diffusion equations.Item Metadata only Averaging approximation to singularly perturbed nonlinear stochastic wave equations(Amer Inst Physics, 2012) Lv, Y.; Roberts, A.; Faculty of Engineering, Computer & Mathematical SciencesAn averaging method is applied to derive effective approximation to a singularly perturbed nonlinear stochastic damped wave equation. Small parameter ν > 0 characterizes the singular perturbation, and νᵅ, 0 ≤ α ≤ 1/2, parametrizes the strength of the noise. Some scaling transformations and the martingale representation theorem yield the effective approximation, a stochastic nonlinear heat equation, for small ν in the sense of distribution.Item Metadata only Average and deviation for slow-fast stochastic partial differential equations(Academic Press Inc, 2012) Wang, W.; Roberts, A.; Faculty of Engineering, Computer & Mathematical SciencesAveraging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the stochastic deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of order O(ε) instead of order O(√ε) attained in previous averaging.Item Metadata only General rolled-up and polyhedral models for carbon nanotubes(Taylor and Francis Inc, 2011) Lee, K.; Cox, B.; Hill, J.In many computational studies of carbon nanotubes, the minimum energy configuration frequently settles on a structure for which the bond lengths are distinct. Here, we extend both the rolled-up and the polyhedral models for SWCNTs to produce general models incorporating either distinct bond lengths and the same bond angle, or distinct bond lengths and distinct bond angles. The CNTs considered here are assumed to be formed by sp2 hybridization but with different bond lengths so that the nanotube structure is assumed to comprise irregular hexagonal patterns. The polyhedral model with distinct bond lengths and distinct bond angles is based on the two postulates that all bonds lying on the same helix are equal in length and that all atoms are equidistant from a common axis of symmetry. The polyhedral model with distinct bond lengths and the same bond angle has the additional postulate that all the adjacent bond angles are equal. We derive exact formulae for the geometric parameters and we present asymptotic expansions for the polyhedral model with distinct bond lengths and distinct bond angles to the first two orders of magnitude. Good agreement is demonstrated for the predictions of the polyhedral model compared with the results obtained from other computational studies.Item Metadata only Unbiased estimation of multi-fractal dimensions of finite data sets(Elsevier, 1996) Roberts, A.; Cronin, A.We present a novel method for determining multi-fractal properties from experimental data. It is based on maximizing the likelihood that the given finite data set comes from a particular set of parameters in a multi-parameter family of well known multi-fractals. By comparing characteristic correlations obtained from the original data with those that occur in artificially generated multi-fractals with the same number of data points, we expect that predicted multi-fractal properties are unbiased by the finiteness of the experimental data.Item Metadata only Design of a two-state shuttle memory device(Tech Science Press, 2010) Lee, K.; Hill, J.In this study, we investigate the mechanics of a metallofullerene shuttle memory device, comprising a metallofullerene which is located inside a closed carbon nanotube. The interaction energy for the system is obtained from the 6-12 Lennard-Jones potential using the continuum approximation, which assumes that a discrete atomic structure can be replaced by an average atomic surface density. This approach shows that the system has two equal minimum energy positions, which are symmetrically located close to the tube extremities, and therefore it gives rise to the possibility of being used as a two-state memory device. On one side the encapsulated metallofullerene represents the zero information state and by applying an external electrical field, the metallofullerene can overcome the energy barrier of the nanotube, and pass from one end of the tube to the other end, where the metallofullerene then represents the one information state. By appropriately selecting different nanotube geometries, the memory device can be designed to have various data transfer rates. In particular, design parameters are presented for the optimization of the data transfer rates and the stabilization of the data storage. The former involves optimization of the nanotube length and the applied electric field, while the latter involves the nanotube radius and the choice of metallofullerene.Item Metadata only Polyhedral models and geometric structures for nanotubes(Cambridge University Press, 2011) Lee, Richard K. F.; School of Mathematical Sciences : Applied MathematicsItem Metadata only Modal analysis of a small ship sea keeping trial(Australian Mathematical Society, 2005) Metcalfe, A.; Maurits, L.; Svenson, T.; Thach, R.; Hearn, G.Data from sea keeping trials of a Scottish trawler are analyzed. The trawler sailed an octagonal course, each leg took over 20 minutes and data recorded twice a second. The natural frequencies of vibration for each of the six rigid body modes are estimated from the heave, surge, sway, pitch, yaw and roll time series. The time series are investigated for evidence of non-linearity. A time domain model is fitted to a roll time series, and second order amplitude response functions are then obtained using autoregressive estimators.Item Metadata only Choose inter-element coupling to preserve self-adjoint dynamics in multiscale modelling and computation(Elsevier Science BV, 2010) Roberts, A.Consider the macroscale modelling of microscale spatio-temporal dynamics. Here we develop an approach to ensure coarse scale discrete models preserve important self-adjoint properties of the microscale dynamics. The first part explores the discrete modelling of microscale continuum dynamics in multiple spatial dimensions. The second part addresses how dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids); for simplicity we address only one-dimensional lattices. Such mapping of discrete lattice dynamics may be iterated to empower future research to explore scale dependent emergent phenomena. The support of the dynamical systems theory of centre manifolds ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain, and for all time. The accuracy of the modelling is limited by the asymptotic resolution of subgrid scale processes. As given examples demonstrate, the approach developed here ensures the preservation of important symmetries of the microscale dynamics.Item Metadata only A corrected quadrature formula and applications(Australian Mathematical Society, 2004) Ujevic, N.; Roberts, A.A straightforward three-point quadrature formula of closed type is derived that improves on Simpson’s rule. Just using the additional information of the integrand’s derivative at the two endpoints we show the error is sixth order in grid spacing. Various error bounds for the quadrature formula are obtained to quantify more precisely the errors. Applications in numerical integration are given. With these error bounds, which are generally better than the usual Peano bounds, the composite formulas can be applied to integrands with lower order derivatives.Item Metadata only Managing river flow in arid regions with matrix analytic methods(Elsevier Science BV, 2010) Fisher, A.; Green, D.; Metcalfe, A.Matrix analytic methods (MAM) support non-geometric transitions between sets of states, termed levels, by introducing states, known as phases, within levels. The phases can correspond to an observed variable or they can be hidden. MAM models have been extensively used in telecommunications, and efficient algorithms for evaluation of performance continue to be developed. Following a review of MAM in discrete time, the daily flows of the ephemeral Cooper Creek, in South Australia, are modelled. Hidden phases are used in the distribution of the duration of the dry spells, whereas observed phases, are used within non-zero flow levels. Seasonal and non-seasonal models are compared. A second application is managing the level of a reservoir in Queensland, using MAM within stochastic dynamic programming. Inflows to the reservoir are seasonal and influenced by the value of the Southern Oscillation Index (SOI), which is used to define phases. The benefit of including SOI in the decision process is shown.Item Metadata only Low-dimensional modelling of a generalized Burgers equation(Research India Publications, 2007) Li, Z.; Roberts, A.Burgers equation is one of the simplest nonlinear partial differential equations—it combines the basic processes of diffusion and nonlinear steepening. In some applications it is appropriate for the diffusion coefficient to be a time-dependent function. Using a Wayne's transformation and centre manifold theory, we derive lmode and 2-mode centre manifold models of the generalised Burgers equations for bounded smooth time dependent coefficients. These modellings give some interesting extensions to existing results such as the similarity solutions using the similarity method.Item Metadata only Computer algebra compares the stochastic superslow manifold of an averaged SPDE with that of the original slow-fast SPDE(2010) Roberts, A.The computer algebra routines documented here empower you to reproduce and check many of the details described by an article on large deviations for slow-fast stochastic systems [Wang et al., 2010]. We consider a `small' spatial domain with two coupled concentration fields, one governed by a `slow' reaction-diffusion equation and one governed by a stochastic `fast' linear equation. In the regime of a stochastic bifurcation, we derive two superslow models of the dynamics: the first is of the averaged model of the slow dynamics derived via large deviation principles; and the second is of the original fast-slow dynamics. Comparing the two superslow models validates the averaging in the large deviation principle in this parameter regimeItem Metadata only The solution of a free boundary problem related to environmental management systems(Marcel Dekker Inc, 2007) Elliott, R.; Filinkov, A.Item Metadata only Elementary Calculus of Financial Mathematics(Society for Industrial and Applied Mathematics, 2008) Roberts, A.