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.

Computational modelling of metal extrusion and forging processes

The computational modelling of extrusion and forging processes is now well established. There are two main approaches: Lagrangian and Eulerian. The first has considerable complexities associated with remeshing, especially when the code is parallelised. The second approach means that the mould has to be assumed to be entirely rigid and this may not be the case. In this paper, a novel approach is described which utilises finite volume methods on unstructured meshes. This approach involves the solution of free surface non-Newtonian fluid flow equations in an Eulerian context to track the behaviour of the workpiece and its extrusion/forging, and the solution of the solid mechanics equations in the Lagrangian context to predict the deformation/stress behaviour of the die. Test cases for modelling extrusion and forging problems using this approach will be presented.

A mixed Eulerian-Lagrangian approach for metal extrusion

In this paper a mixed Eulerian-Lagrangian approach for the modelling metal extrusion processes is presented. The approach involves the solution of non-Newtonian fluid flow equations in an Eulerian context, using a free-surface algorithm to track the behaviour of the workpiece and its extrusion. The solid mechanics equations associated with the tools are solved in Lagangrian context. Thermal interactions between the workpiece are modelled and a fluid-structure interaction technique is employed to model the effect of the fluid traction load imposed by the workpiece on the tools. Two extrusion test cases are investigated and the results obtained show the potential of the model with regard to representing the physics of the process and the simulation time.

Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval

In this note, we consider the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval on the machine under the non-resumable scenario. Together with a recent 2-approximation algorithm designed by Kacem [I. Kacem, Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers & Industrial Engineering 54 (2008) 401–410], this paper is the first successful attempt to develop a constant ratio approximation algorithm for this problem. We present two approaches to designing such an algorithm. Our best algorithm guarantees a worst-case performance ratio of 2+ε. © 2008 Elsevier B.V. All rights reserved.

Multiscale numerical modelling of crack propagation in two-dimensional metal plate

A novel multiscale model of brittle crack propagation in an Ag plate with macroscopic dimensions has been developed. The model represents crack propagation as stochastic drift-diffusion motion of the crack tip atom through the material, and couples the dynamics across three different length scales. It integrates the nanomechanics of bond rupture at the crack tip with the displacement and stress field equations of continuum based fracture theories. The finite element method is employed to obtain the continuum based displacement and stress fields over the macroscopic plate, and these are then used to drive the crack tip forward at the atomic level using the molecular dynamics simulation method based on many-body interatomic potentials. The linkage from the nanoscopic scale back to the macroscopic scale is established via the Ito stochastic calculus, the stochastic differential equation of which advances the tip to a new position on the macroscopic scale using the crack velocity and diffusion constant obtained on the nanoscale. Well known crack characteristics, such as the roughening transitions of the crack surfaces, crack velocity oscillations, as well as the macroscopic crack trajectories, are obtained.

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.

Determining surface tension using DC magnetic levitation

The values of material physical properties are vital for the successful use of numerical simulations for electromagnetic processing of materials. The surface tension of materials can be determined from the experimental measurement of the surface oscillation frequency of liquid droplets. In order for this technique to be used, a positioning field is required that results in a modification to the oscillation frequency. A number of previous analytical models have been developed that mainly focus on electrically conducting droplets positioned using an A.C. electromagnetic field, but due to the turbulent flow resulting from the high electromagnetic fields required to balance gravity, reliable measurements have largely been limited to microgravity. In this work axisymmetric analytical and numerical models are developed, which allow the surface tension of a diamagnetic droplet positioned in a high DC magnetic field to be determined from the surface oscillations. In the case of D.C. levitation there is no internal electric currents with resulting Joule heating, Marangoni flow and other effects that introduce additional physics that complicates the measurement process. The analytical solution uses the linearised Navier-Stokes equations in the inviscid case. The body force from a DC field is potential, in contrast to the AC case, and it can be derived from Maxwell equations giving a solution for the magnetic field in the form of a series expansion of Legendre polynomials. The first few terms in this expansion represent a constant and gradient magnetic field valid close to the origin, which can be used to position the droplet. Initially the mathematical model is verified in microgravity conditions using a numerical model developed to solve the transient electromagnetics, fluid flow and thermodynamic equations. In the numerical model (as in experiment) the magnetic field is obtained using electrical current carrying coils, which provides the confinement force for a liquid droplet. The model incorporates free surface deformation to accurately model the oscillations that result from the interaction between the droplet and the non-uniform external magnetic field. A comparison is made between the analytical perturbation theory and the numerical pseudo spectral approximation solutions for small amplitude oscillations.

Modeling the diffusion of solid copper into liquid solder alloys

During the soldering process, the copper atoms diffuse into liquid solders. The diffusion process determines integrity and the reworking possibility of a solder joint. In order to capture the diffusion scenarios of solid copper into liquid Sn–Pb and Sn–Cu solders, a computer modeling has been performed for 10 s. An analytical model has also been proposed for calculating the diffusion coefficient of copper into liquid solders. It is found that the diffusion coefficient for Sn–Pb solder is 2.74 × 10− 10 m2/s and for Sn–Cu solder is 6.44 × 10−9 m2/s. The modeling results reveal that the diffusion coefficient is one of the major factors that govern the rate at which solid Cu dissolve in the molten solder. The predicted dissolved amounts of copper into solders have been validated with the help of scanning electron microscopic analysis.

Simultaneous monitoring of drug and solvent diffusion across a model membrane using ATR-FTIR spectroscopy

Attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy has been used to simultaneously follow the diffusion of model drugs and solvent across polydimethylsiloxane (silicone) membrane. Three model drugs, cyanophenol (CNP), methyl nicotinate (MN) and butyl paraben (BP) were selected to cover a range of lipophilicities. Isostearyl isostearate (ISIS) was chosen as the solvent because its large molecular weight should facilitate observation of whether the drug molecules are able to diffuse through the membrane independently of the solvent. The diffusion of the three drugs and the solvent was successfully described by a Fickian model. The effects of parameters such as the absorption wavelength used to follow diffusion on the calculated diffusion coefficient were investigated. Absorption wavelength which affects the depth of penetration of the infrared radiation into the membrane did not significantly affect the calculated diffusion coefficient over the wavelength range tested. Each of the model drugs was observed to diffuse independently of the solvent across the membrane. The diffusion of a CNP-ISIS hydrogen bonded complex across the membrane was also monitored. The relative diffusion rates of the solute and solvent across the membrane can largely be accounted for by the molecular size of the permeant.

ATR-FTIR spectroscopy and spectroscopic imaging of solvent and permeant diffusion across model membranes

The uptake and diffusion of solvents across polymer membranes is important in controlled drug delivery, effects on drug uptake into, for example, infusion bags and containers, as well as transport across protective clothing. Attenuated Total Reflectance Fourier Transform Infrared (ATR-FTIR) spectroscopy has been used to monitor the effects of different solvents on the diffusion of a model compound, 4-cyanophenol (CNP) across silicone membrane and on the equilibrium concentration of CNP obtained in the membrane following diffusion. ATR-FTIR spectroscopic imaging of membrane diffusion was used to gain an understanding of when the boundary conditions applied to Fick's second law, used to model the diffusion of permeants across the silicone membrane do not hold. The imaging experiments indicated that when the solvent was not taken up appreciably into the membrane, the presence of discrete solvent pools between the ATR crystal and the silicone membrane can affect the diffusion profile of the permeant. This effect is more significant if the permeant has a high solubility in the solvent. In contrast, solvents that are taken up into the membrane to a greater extent, or those where the solubility of the permeant in the vehicle is relatively low, were found to show a good fit to the diffusion model. As such these systems allow the ATR-FTIR spectroscopic approach to give mechanistic insight into how the particular solvents enhance permeation. The solubility of CNP in the solvent and the uptake of the solvent into the membrane were found to be important influences on the equilibrium concentration of the permeant obtained in the membrane following diffusion. In general, solvents which were taken up to a significant extent into the membrane and which caused the membrane to swell increased the diffusion coefficient of the permeant in the membrane though other factors such as solvent viscosity may also be important.