Holistic projection of initial conditions onto a finite difference approximation

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (Archilla, 1998). However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretisation of \pde{}s such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky equation (MacKenzie, 2000). In this paper the geometric view of a centre manifold is used to provide correct initial conditions for numerical discretisations (Roberts, 1997). The derived projection of initial conditions follows from the physical processes expressed in the PDEs and so is appropriately conservative. This rational approach increases the accuracy of forecasts made with finite difference models.Comment: 8 pages, LaTe

Efficient design of tall tapered feeders

Toowoomba Foundry is seeking help with the design of tall tapered feeders, which supply extra molten iron to poured castings as they cool. There is a problem with the reliability of the feeders, particularly for thicker castings of the newer spheroidal graphite irons. An effective feeder remains molten until the casting has set. The setting of the melt in the feeder is delayed by making it large enough to retain its heat for longer than the casting, and by placing it close to the thermal centre of the casting. So a larger feeder is more reliable. But any metal remaining in the feeder is recycled, and has an associated energy cost. If a feeder is too small, it will set too soon, and the casting will have unwanted holes in it that may require the entire casting to be recycled. Thus there is a tension between making the feeder smaller so as to minimise recycled metal, and having the feeder large enough that the casting is good. Existing design methods use purely conductive models of heat transport. We investigate the relevance of convection in the cooling feeder, and set up a boundary-layer model of flow driven by density differences. We find that convection is a significant factor in the design of a feeder, effectively maintaining constant temperature across it. The height of the feeder is important mainly in providing the driving force for this flow

The inertial dynamics of thin film flow of non-Newtonian fluids

Consider the flow of a thin layer of non-Newtonian fluid over a solid surface. I model the case of a viscosity that depends nonlinearly on the shear-rate; power law fluids are an important example, but the analysis here is for general nonlinear dependence. The modelling allows for large changes in film thickness provided the changes occur over a large enough lateral length scale. Modifying the surface boundary condition for tangential stress forms an accessible base for the analysis where flow with constant shear is a neutral critical mode, in addition to a mode representing conservation of fluid. Perturbatively removing the modification then constructs a model for the coupled dynamics of the fluid depth and the lateral momentum. For example, the results model the dynamics of gravity currents of non-Newtonian fluids even when the flow is not very slow

Initial Conditions for Models of Dynamical Systems

The long-time behaviour of many dynamical systems may be effectively predicted by a low-dimensional model that describes the evolution of a reduced set of variables. We consider the question of how to equip such a low-dimensional model with appropriate initial conditions, so that it faithfully reproduces the long-term behaviour of the original high-dimensional dynamical system. Our method involves putting the dynamical system into normal form, which not only generates the low-dimensional model, but also provides the correct initial conditions for the model. We illustrate the method with several examples. Keywords: normal form, isochrons, initialisation, centre manifoldComment: 24 pages in standard LaTeX, 66K, no figure

Thermomechanical Behavior of Thermoelectric SMA Actuators

In this paper we analyse numerically thermomechanical behaviour of a sandwich-type actuator where a shape memory alloy layer is located between two oppositely doped semiconductors. The mathematical model for this analysis is based on a coupled system of partial differential equations with constitutive equations taken in the Falk form. The system is solved using an efficient differential-algebraic solver and computational results describing thermomechanical fields in such devices are presented

Springs–neaps cycles in daily total seabed light: Daylength-induced changes

AbstractIn shallow, tidal seas, daily total seabed light is determined largely by the interaction of the solar elevation cycle, the tidal cycle in water depth, and any temporal variability in turbidity. Since tidal range, times of low water, and often turbidity vary in regular ways over the springs–neaps cycle, daily total seabed light exhibits cycles of the same periodicity. Corresponding cycles are likely to be induced in the daily total primary production of benthic algae and plants, particularly those light-limited specimens occupying the lower reaches of a sub-tidal population. Consequently, this effect is an important control on the growth patterns, depth distribution and survival of, for example, macroalgal forests and seagrass meadows.Seasonal changes in daylength exert an important additional control on these cycles, as they alter the fraction of the tidal and turbidity cycles occurring within daylight hours. Bowers et al. (1997) modelled this phenomenon numerically and predicted that for a site with low water at about midday and midnight at neaps tides, 6am and 6pm at springs, daily total seabed light peaks at neaps in winter, but the ‘sense’ of the cycle ‘switches’ so that it peaks at springs in summer — the longer daylength permits the morning and evening low water springs to contribute substantially to the daily total. Observations for such a site in North Wales (UK), presented in this paper, show that no such ‘switch’ occurs, and neaps tides host the largest daily totals throughout the year. The predicted ‘switch’ is not observed because turbidity increases generally at spring tides, and specifically at low water springs, both of which were not accounted for in the model. Observations at a second site in Brittany (France), diametrically opposite in terms of the times of low water at neaps and at springs, indicate a peak at springs throughout the year.Analytical tools are developed to calculate the percentage of daily total sea surface irradiance reaching the bed at a site of interest on any given day, and to determine the sense of any springs–neaps cycle thereof for a given season. The conditions required for a ‘switch’ are explored graphically, resulting in the identification of criteria (and a useful parameter) for predicting their occurrence. Consequences for the growth patterns, depth limits and long-term survival of benthic algae and plants are discussed

Novel expression of Haemonchus contortus vaccine candidate aminopeptidase H11 using the free-living nematode Caenorhabditis elegans

With the problem of parasitic nematode drug resistance increasing, vaccine development offers an alternative sustainable control approach. For some parasitic nematodes, native extracts enriched for specific proteins are highly protective. However, recombinant forms of these proteins have failed to replicate this protection. This is thought to be due to differences in glycosylation and/or conformation between native and recombinant proteins. We have exploited the free-living nematode Caenorhabditis elegans to examine its suitability as an alternative system for recombinant expression of parasitic nematode vaccine candidates. We focussed on Haemonchus contortus aminopeptidase H11 glycoprotein, which is enriched in a gut membrane fraction capable of inducing significant protection against this important ovine gastrointestinal nematode. We show that H. contortus H11 expressed in C. elegans is enzymatically active and MALDI mass spectrometry identifies similar di- and tri-fucosylated structures to those on native H11, with fucose at the 3- and/or 6-positions of the proximal GlcNAc. Some glycan structural differences were observed, such as lack of LDNF. Serum antibody to native H11 binds to C. elegans recombinant H11 and most of the antibody to rH11 or native H11 is directed to glycan moieties. Despite these similarities, no reduction in worm burden or faecal egg count was observed following immunisation of sheep with C. elegans-expressed recombinant H11 protein. The findings suggest that the di- and tri-fucosylated N-glycans expressed on rH11 do not contribute to the protective effect of H11 and that additional components present in native H11-enriched extract are likely required for enhancing the antibody response necessary for protection

Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling

Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale system at a desired macroscale system level. In this scheme, the given microscale model is computed in small patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate simulations, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve translational invariance, rotational invariance, and self-adjoint symmetry, thus guaranteeing that conservation laws associated with these symmetries are preserved in the macroscale simulation. Spectral and algebraic analyses of the proposed scheme in both one and two dimensions reveal mechanisms for further improving the accuracy of the simulations. Consistency of the patch scheme's macroscale dynamics with the original microscale model is proved. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation in a wide range of multiscale scenarios of interest to scientists and engineers