Enhancing the Hodgkin-Huxley Equations: Simulations Based on the First Publication in the Biophysical Journal.

The experiments in the Cole and Moore article in the first issue of the Biophysical Journal provided the first independent experimental confirmation of the Hodgkin-Huxley (HH) equations. A log-log plot of the K current versus time showed that raising the HH variable n to the sixth power provided the best fit to the data. Subsequent simulations using n(6) and setting the resting potential at the in vivo value simplifies the HH equations by eliminating the leakage term. Our article also reported that the K current in response to a depolarizing step to ENa was delayed if the step was preceded by a hyperpolarization. While the interpretation of this phenomenon in the article was flawed, subsequent simulations show that the effect completely arises from the original HH equations.

An extension of Purcell's vector method with applications to panel element equations

The classical Purcell's vector method, for the construction of solutions to dense systems of linear equations is extended to a flexible orthogonalisation procedure. Some properties are revealed of the orthogonalisation procedure in relation to the classical Gauss-Jordan elimination with or without pivoting. Additional properties that are not shared by the classical Gauss-Jordan elimination are exploited. Further properties related to distributed computing are discussed with applications to panel element equations in subsonic compressible aerodynamics. Using an orthogonalisation procedure within panel methods enables a functional decomposition of the sequential panel methods and leads to a two-level parallelism.

Fire field modelling using the SMARTFIRE automated dynamic solution control environment

SMARTFIRE is a fire field model based on an open architecture integrated CFD code and knowledge-based system. It makes use of the expert system to assist the user in setting up the problem specification and new computational techniques such as Group Solvers to reduce the computational effort involved in solving the equations. This paper concentrates on recent research into the use of artificial intelligence techniques to assist in dynamic solution control of fire scenarios being simulated using fire field modelling techniques. This is designed to improve the convergence capabilities of the software while further decreasing the computational overheads. The technique automatically controls solver relaxations using an integrated production rule engine with a blackboard to monitor and implement the required control changes during solution processing. Initial results for a two-dimensional fire simulation are presented that demonstrate the potential for considerable savings in simulation run-times when compared with control sets from various sources. Furthermore, the results demonstrate enhanced solution reliability due to obtaining acceptable convergence within each time step unlike some of the comparison simulations.

Discretisation procedures for multi-physics phenomena

Procedures are described for solving the equations governing a multi-physics process. Finite volume techniques are used to discretise, using the same unstructured mesh, the equations of fluid flow, heat transfer with solidification, and solid deformation. These discretised equations are then solved in an integrated manner. The computational mechanics environment, PHYSICA, which facilitates the building of multi-physics models, is described. Comparisons between model predictions and experimental data are presented for the casting of metal components.

Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method

A semi-Lagrangian finite volume scheme for solving viscoelastic flow problems is presented. A staggered grid arrangement is used in which the dependent variables are located at different mesh points in the computational domain. The convection terms in the momentum and constitutive equations are treated using a semi-Lagrangian approach in which particles on a regular grid are traced backwards over a single time-step. The method is applied to the 4 : 1 planar contraction problem for an Oldroyd B fluid for both creeping and inertial flow conditions. The development of vortex behaviour with increasing values of We is analyzed.

A semi-Lagrangian finite volume method for Newtonian contraction flows

A new finite volume method for solving the incompressible Navier--Stokes equations is presented. The main features of this method are the location of the velocity components and pressure on different staggered grids and a semi-Lagrangian method for the treatment of convection. An interpolation procedure based on area-weighting is used for the convection part of the computation. The method is applied to flow through a constricted channel, and results are obtained for Reynolds numbers, based on half the flow rate, up to 1000. The behavior of the vortex in the salient corner is investigated qualitatively and quantitatively, and excellent agreement is found with the numerical results of Dennis and Smith [Proc. Roy. Soc. London A, 372 (1980), pp. 393-414] and the asymptotic theory of Smith [J. Fluid Mech., 90 (1979), pp. 725-754].

On the coupling of Navier–Stokes and linearised Euler equations for aeroacoustic simulation

Aerodynamic generation of sound is governed by the Navier–Stokes equations while acoustic propagation in a non-uniform medium is effectively described by the linearised Euler equations. Different numerical schemes are required for the efficient solution of these two sets of equations, and therefore, coupling techniques become an essential issue. Two types of one-way coupling between the flow solver and the acoustic solver are discussed: (a) for aerodynamic sound generated at solid surfaces, and (b) in the free stream. Test results indicate how the coupling achieves the necessary accuracy so that Computational Fluid Dynamics codes can be used in aeroacoustic simulations.

An improved result of exponential stability of stochastic semilinear evolution equations

Sufficient conditions for the exponential stability of a class ofnonlinear, non-autonomous stochastic differential equations in infinitedimensions are studied. The analysis consists of introducing a suitableapproximating solution systems and using a limiting argument to pass onstability of strong solutions to mild ones. As a consequence, the classicalcriteriaof stability in A. Ichikawa [8] are improved and extended to cover a class ofnon-autonomous stochastic evolution equations.Two examples are investigated to illustrate our theory.

Moment decay rates of solutions of stochastic differential equations

The objective of this paper is to investigate the p-ίh moment asymptotic stability decay rates for certain finite-dimensional Itό stochastic differential equations. Motivated by some practical examples, the point of our analysis is a special consideration of general decay speeds, which contain as a special case the usual exponential or polynomial type one, to meet various situations. Sufficient conditions for stochastic differential equations (with variable delays or not) are obtained to ensure their asymptotic properties. Several examples are studied to illustrate our theory.

Dynamic fluid–structure interaction using finite volume unstructured mesh procedures

A three-dimensional finite volume, unstructured mesh (FV-UM) method for dynamic fluid–structure interaction (DFSI) is described. Fluid structure interaction, as applied to flexible structures, has wide application in diverse areas such as flutter in aircraft, wind response of buildings, flows in elastic pipes and blood vessels. It involves the coupling of fluid flow and structural mechanics, two fields that are conventionally modelled using two dissimilar methods, thus a single comprehensive computational model of both phenomena is a considerable challenge. Until recently work in this area focused on one phenomenon and represented the behaviour of the other more simply. More recently, strategies for solving the full coupling between the fluid and solid mechanics behaviour have been developed. A key contribution has been made by Farhat et al. [Int. J. Numer. Meth. Fluids 21 (1995) 807] employing FV-UM methods for solving the Euler flow equations and a conventional finite element method for the elastic solid mechanics and the spring based mesh procedure of Batina [AIAA paper 0115, 1989] for mesh movement. In this paper, we describe an approach which broadly exploits the three field strategy described by Farhat for fluid flow, structural dynamics and mesh movement but, in the context of DFSI, contains a number of novel features: • a single mesh covering the entire domain, • a Navier–Stokes flow, • a single FV-UM discretisation approach for both the flow and solid mechanics procedures, • an implicit predictor–corrector version of the Newmark algorithm, • a single code embedding the whole strategy.

Modelling air quality in street canyons: a review

High pollution levels have been often observed in urban street canyons due to the increased traffic emissions and reduced natural ventilation. Microscale dispersion models with different levels of complexity may be used to assess urban air qualityand support decision-making for pollution control strategies and traffic planning. Mathematical models calculate pollutant concentrations by solving either analytically a simplified set of parametric equations or numerically a set of differential equations that describe in detail wind flow and pollutant dispersion. Street canyon models, which might also include simplified photochemistry and particle deposition–resuspension algorithms, are often nested within larger-scale urban dispersion codes. Reduced-scale physical models in wind tunnels may also be used for investigating atmospheric processes within urban canyons and validating mathematical models. A range of monitoring techniques is used to measure pollutant concentrations in urban streets. Point measurement methods (continuous monitoring, passive and active pre-concentration sampling, grab sampling) are available for gaseous pollutants. A number of sampling techniques (mainlybased on filtration and impaction) can be used to obtain mass concentration, size distribution and chemical composition of particles. A combination of different sampling/monitoring techniques is often adopted in experimental studies. Relativelysimple mathematical models have usually been used in association with field measurements to obtain and interpret time series of pollutant concentrations at a limited number of receptor locations in street canyons. On the other hand, advanced numerical codes have often been applied in combination with wind tunnel and/or field data to simulate small-scale dispersion within the urban canopy.

On a flexible elimination algorithm with applications to panel element equations

A flexible elimination algorithm is presented and is applied to the solution of dense systems of linear equations. Properties of the algorithm are exploited in relation to panel element methods for potential flow and subsonic compressible flow. Further properties in relation to distributed computing are also discussed.

Details of an integrated approach to three-dimensional dynamic fluid structure interaction

Fluid structure interaction, as applied to flexible structures, has wide application in diverse areas such as flutter in aircraft, wind response of buildings, flows in elastic pipes and blood vessels. Numerical modelling of dynamic fluid-structure interaction (DFSI) involves the coupling of fluid flow and structural mechanics, two fields that are conventionally modelled using two dissimilar methods, thus a single comprehensive computational model of both phenomena is a considerable challenge and until recently work in this area focused on one phenomenon and represented the behaviour of the other more simply. A single, finite volume unstructured mesh (FV-UM) spatial discretisation method has been employed on a single mesh for the entire domain. The Navier Stokes equations for fluid flow are solved using a SIMPLE type procedure and the Newmark b algorithm is employed for solving the dynamic equilibrium equations for linear elastic solid mechanics and mesh movement is achieved using a spring based mesh procedure for dynamic mesh movement. In the paper we describe a number of additional computation issues for the efficient and accurate modelling of three-dimensional, dynamic fluid-structure interaction problems.