A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

Experimental and computer simulation validation of ultrasound phase interference created by lateral inhomogeneity of transit time in replica bone marrow composite models

Purpose: The measurement of broadband ultrasonic attenuation (BUA) in cancellous bone for the assessment of osteoporosis follows a parabolic-type dependence with bone volume fraction; having minima values corresponding to both entire bone and entire marrow. Langton has recently proposed that the primary BUA mechanism may be significant phase interference due to variations in propagation transit time through the test sample as detected over the phase-sensitive surface of the receive ultrasound transducer. This fundamentally simple concept assumes that the propagation of ultrasound through a complex solid : liquid composite sample such as cancellous bone may be considered by an array of parallel ‘sonic rays’. The transit time of each ray is defined by the proportion of bone and marrow propagated, being a minimum (tmin) solely through bone and a maximum (tmax) solely through marrow. A Transit Time Spectrum (TTS), ranging from tmin to tmax, may be defined describing the proportion of sonic rays having a particular transit time, effectively describing lateral inhomogeneity of transit time over the surface of the receive ultrasound transducer. Phase interference may result from interaction of ‘sonic rays’ of differing transit times. The aim of this study was to test the hypothesis that there is a dependence of phase interference upon the lateral inhomogenity of transit time by comparing experimental measurements and computer simulation predictions of ultrasound propagation through a range of relatively simplistic solid:liquid models exhibiting a range of lateral inhomogeneities. Methods: A range of test models was manufactured using acrylic and water as surrogates for bone and marrow respectively. The models varied in thickness in one dimension normal to the direction of propagation, hence exhibiting a range of transit time lateral inhomogeneities, ranging from minimal (single transit time) to maximal (wedge; ultimately the limiting case where each sonic ray has a unique transit time). For the experimental component of the study, two unfocused 1 MHz ¾” broadband diameter transducers were utilized in transmission mode; ultrasound signals were recorded for each of the models. The computer simulation was performed with Matlab, where the transit time and relative amplitude of each sonic ray was calculated. The transit time for each sonic ray was defined as the sum of transit times through acrylic and water components. The relative amplitude considered the reception area for each sonic ray along with absorption in the acrylic. To replicate phase-sensitive detection, all sonic rays were summed and the output signal plotted in comparison with the experimentally derived output signal. Results: From qualtitative and quantitative comparison of the experimental and computer simulation results, there is an extremely high degree of agreement of 94.2% to 99.0% between the two approaches, supporting the concept that propagation of an ultrasound wave, for the models considered, may be approximated by a parallel sonic ray model where the transit time of each ray is defined by the proportion of ‘bone’ and ‘marrow’. Conclusions: This combined experimental and computer simulation study has successfully demonstrated that lateral inhomogeneity of transit time has significant potential for phase interference to occur if a phase-sensitive ultrasound receive transducer is implemented as in most commercial ultrasound bone analysis devices.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

Deformation with coupled chemical diffusion

The deformation of rocks is commonly intimately associated with metamorphic reactions. This paper is a step towards understanding the behaviour of fully coupled, deforming, chemically reacting systems by considering a simple example of the problem comprising a single layer system with elastic-power law viscous constitutive behaviour where the deformation is controlled by the diffusion of a single chemical component that is produced during a metamorphic reaction. Analysis of the problem using the principles of non-equilibrium thermodynamics allows the energy dissipated by the chemical reaction-diffusion processes to be coupled with the energy dissipated during deformation of the layers. This leads to strain-rate softening behaviour and the resultant development of localised deformation which in turn nucleates buckles in the layer. All such diffusion processes, in leading to Herring-Nabarro, Coble or “pressure solution” behaviour, are capable of producing mechanical weakening through the development of a “chemical viscosity”, with the potential for instability in the deformation. For geologically realistic strain rates these chemical feed-back instabilities occur at the centimetre to micron scales, and so produce structures at these scales, as opposed to thermal feed-back instabilities that become important at the 100–1000 m scales.

Mathematical modelling of controlled drug release from polymer micro-spheres : incorporating the effects of swelling, diffusion and dissolution via moving boundary problems

Controlled drug delivery is a key topic in modern pharmacotherapy, where controlled drug delivery devices are required to prolong the period of release, maintain a constant release rate, or release the drug with a predetermined release profile. In the pharmaceutical industry, the development process of a controlled drug delivery device may be facilitated enormously by the mathematical modelling of drug release mechanisms, directly decreasing the number of necessary experiments. Such mathematical modelling is difficult because several mechanisms are involved during the drug release process. The main drug release mechanisms of a controlled release device are based on the device’s physiochemical properties, and include diffusion, swelling and erosion. In this thesis, four controlled drug delivery models are investigated. These four models selectively involve the solvent penetration into the polymeric device, the swelling of the polymer, the polymer erosion and the drug diffusion out of the device but all share two common key features. The first is that the solvent penetration into the polymer causes the transition of the polymer from a glassy state into a rubbery state. The interface between the two states of the polymer is modelled as a moving boundary and the speed of this interface is governed by a kinetic law. The second feature is that drug diffusion only happens in the rubbery region of the polymer, with a nonlinear diffusion coefficient which is dependent on the concentration of solvent. These models are analysed by using both formal asymptotics and numerical computation, where front-fixing methods and the method of lines with finite difference approximations are used to solve these models numerically. This numerical scheme is conservative, accurate and easily implemented to the moving boundary problems and is thoroughly explained in Section 3.2. From the small time asymptotic analysis in Sections 5.3.1, 6.3.1 and 7.2.1, these models exhibit the non-Fickian behaviour referred to as Case II diffusion, and an initial constant rate of drug release which is appealing to the pharmaceutical industry because this indicates zeroorder release. The numerical results of the models qualitatively confirms the experimental behaviour identified in the literature. The knowledge obtained from investigating these models can help to develop more complex multi-layered drug delivery devices in order to achieve sophisticated drug release profiles. A multi-layer matrix tablet, which consists of a number of polymer layers designed to provide sustainable and constant drug release or bimodal drug release, is also discussed in this research. The moving boundary problem describing the solvent penetration into the polymer also arises in melting and freezing problems which have been modelled as the classical onephase Stefan problem. The classical one-phase Stefan problem has unrealistic singularities existed in the problem at the complete melting time. Hence we investigate the effect of including the kinetic undercooling to the melting problem and this problem is called the one-phase Stefan problem with kinetic undercooling. Interestingly we discover the unrealistic singularities existed in the classical one-phase Stefan problem at the complete melting time are regularised and also find out the small time behaviour of the one-phase Stefan problem with kinetic undercooling is different to the classical one-phase Stefan problem from the small time asymptotic analysis in Section 3.3. In the case of melting very small particles, it is known that surface tension effects are important. The effect of including the surface tension to the melting problem for nanoparticles (no kinetic undercooling) has been investigated in the past, however the one-phase Stefan problem with surface tension exhibits finite-time blow-up. Therefore we investigate the effect of including both the surface tension and kinetic undercooling to the melting problem for nanoparticles and find out the the solution continues to exist until complete melting. The investigation of including kinetic undercooling and surface tension to the melting problems reveals more insight into the regularisations of unphysical singularities in the classical one-phase Stefan problem. This investigation gives a better understanding of melting a particle, and contributes to the current body of knowledge related to melting and freezing due to heat conduction.

Comment on "Local accumulation times for source, diffusion, and degradation models in two and three dimensions" [J. Chem. Phys. 138, 104121 (2013)]

In a recent paper, Gordon, Muratov, and Shvartsman studied a partial differential equation (PDE) model describing radially symmetric diffusion and degradation in two and three dimensions. They paid particular attention to the local accumulation time (LAT), also known in the literature as the mean action time, which is a spatially dependent timescale that can be used to provide an estimate of the time required for the transient solution to effectively reach steady state. They presented exact results for three-dimensional applications and gave approximate results for the two-dimensional analogue. Here we make two generalizations of Gordon, Muratov, and Shvartsman’s work: (i) we present an exact expression for the LAT in any dimension and (ii) we present an exact expression for the variance of the distribution. The variance provides useful information regarding the spread about the mean that is not captured by the LAT. We conclude by describing further extensions of the model that were not considered by Gordon,Muratov, and Shvartsman. We have found that exact expressions for the LAT can also be derived for these important extensions...

Influence of drying conditions on the moisture diffusion during single stage and two stage fluidized bed drying of bovine intestine for pet food

Bovine intestine samples were heat pump fluidized bed dried at atmospheric pressure and at temperatures below and above the material freezing points equipped with a continuous monitoring system. The investigation of the drying characteristics has been conducted in the temperature range -10~25oC and the airflow in the range 1.5~2.5 m/s. Some experiments were conducted as a single temperature drying experiments and others as two stage drying experiments employing two temperatures. An Arrhenius-type equation was used to interpret the influence of the drying air parameters on the effective diffusivity, calculated with the method of slopes in terms of energy activation, and this was found to be sensitivity of the temperature. The effective diffusion coefficient of moisture transfer was determined by Fickian method using uni-dimensional moisture movement in both moisture, removal by evaporation and combined sublimation and evaporation. Correlations expressing the effective moisture diffusivity and drying temperature are reported.

First-principle studies of the formation and diffusion of hydrogen vacancies in magnesium hydride

Ab initio density functional theory (DFT) calculations are performed to study the formation and diffusion of hydrogen vacancies on MgH2(110) surface and in bulk. We find that the formation energies for a single H-vacancy increase slightly from the surface to deep layers. The energies for creating adjacent surface divancacies at two inplane sites and at an inplane and a bridge site are even smaller than that for the formation of a single H-vacancy, a fact that is attributed to the strong vacancy−vacancy interactions. The diffusion of an H-vacancy from an in-plane site to a bridge site on the surface has the smallest activation barrier calculated at 0.15 eV and should be fast at room temperature. The activation barriers computed for H-vacancy diffusion from the surface into sublayers are all less than 0.70 eV, which is much smaller than the activation energy for desorption of hydrogen on the MgH2(110) surface (1.78−2.80 eV/H2). This suggests that surface desorption is more likely than vacancy diffusion to be rate determining, such that finding effective catalyst on the MgH2 surface to facilitate desorption will be very important for improving overall dehydrogenation performance.

Atomic hydrogen diffusion in novel magnesium nanostructures : the impact of incorporated subsurface carbon atoms

Ab initio Density Functional Theory (DFT) calculations are performed to study the diffusion of atomic hydrogen on a Mg(0001) surface and their migration into the subsurface layers. A carbon atom located initially on a Mg(0001) surface can migrate into the sub-surface layer and occupy a fcc site, with charge transfer to the C atom from neighboring Mg atoms. The cluster of postively charged Mg atoms surrounding a sub-surface C is then shown to facilitate the dissociative chemisorption of molecular hydrogen on the Mg(0001) surface, and the surface migration and subsequent diffusion into the subsurface of atomic hydrogen. This helps rationalize the experimentally-observed improvement in absorption kinetics of H2 when graphite or single walled carbon nanotubes (SWCNT) are introduced into the Mg powder during ball milling.

Computer modeling of diffusion in biological tissues

Molecular-level computer simulations of restricted water diffusion can be used to develop models for relating diffusion tensor imaging measurements of anisotropic tissue to microstructural tissue characteristics. The diffusion tensors resulting from these simulations can then be analyzed in terms of their relationship to the structural anisotropy of the model used. As the translational motion of water molecules is essentially random, their dynamics can be effectively simulated using computers. In addition to modeling water dynamics and water-tissue interactions, the simulation software of the present study was developed to automatically generate collagen fiber networks from user-defined parameters. This flexibility provides the opportunity for further investigations of the relationship between the diffusion tensor of water and morphologically different models representing different anisotropic tissues.

A study of the diffusion characteristics of normal, delipidized and relipidized articular cartilage using magnetic resonance imaging

This paper assesses the capacity to provide semipermeability of the synthetic layer of surface-active phospholipids created to replace the depleted surface amorphous layer of articular cartilage. The surfaces of articular cartilage specimens in normal, delipidized, and relipidized conditions following incubation in dipalmitoyl-phosphatidylcholine and palmitoyl-oleoyl-phosphatidylcholine components of the joint lipid mixture were characterized nanoscopically with the atomic force microscope and also imaged as deuterium oxide (D2O) diffused transiently through these surfaces in a magnetic resonance imaging enclosure. The MR images were then used to determine the apparent diffusion coefficients in a purpose-built MATLAB®-based algorithm. Our results revealed that all surfaces were permeable to D2O, but that there was a significant difference in the semipermeability of the surfaces under the different conditions, relative to the apparent diffusion coefficients. Based on the results and observations, it can be concluded that the synthetic lipid that is deposited to replace the depleted SAL of articular cartilage is capable of inducing some level of semipermeability.

Numerical simulation of anomalous diffusion with application to medical imaging

The first objective of this project is to develop new efficient numerical methods and supporting error and convergence analysis for solving fractional partial differential equations to study anomalous diffusion in biological tissue such as the human brain. The second objective is to develop a new efficient fractional differential-based approach for texture enhancement in image processing. The results of the thesis highlight that the fractional order analysis captured important features of nuclear magnetic resonance (NMR) relaxation and can be used to improve the quality of medical imaging.

Metrological performances of a diffusion charger particle counter for personal monitoring

Airborne particles have been shown to be associated with a wide range of adverse health effects, which has led to a recent increase in medical research to gain better insight into their health effects. However, accurate evaluation of the exposure-dose-response relationship is highly dependent on the ability to track actual exposure levels of people to airborne particles. This is quite a complex task, particularly in relation to submicrometer and ultrafine particles, which can vary quite significantly in terms of particle surface area and number concentrations. Therefore, suitable monitors that can be worn for measuring personal exposure to these particles are needed. This paper presents an evaluation of the metrological performance of six diffusion charger sensors, NanoTracer (Philips Aerasense) monitors, when measuring particle number and surface area concentrations, as well as particle number distribution mean when compared to reference instruments. Tests in the laboratory (by generating monodisperse and polydisperse aerosols) and in the field (using natural ambient particles) were designed to evaluate the response of these devices under both steady-state and dynamics conditions. Results showed that the NanoTracers performed well when measuring steady state aerosols, however they strongly underestimated actual concentrations during dynamic response testing. The field experiments also showed that, when the majority of the particles were smaller than 20 nm, which occurs during particle formation events in the atmosphere, the NanoTracer underestimated number concentration quite significantly. Even though the NanoTracer can be used for personal monitoring of exposure to ultrafine particles, it also has limitations which need to be considered in order to provide meaningful results.

Lateral impact response and parametric studies of axially loaded square concrete filled steel tube columns

This paper presents a numerical study on the response of axially loaded slender square concrete filled steel tube (CFST) columns under low velocity lateral impact loading. A finite element analysis (FEA) model was developed using the explicit dynamic nonlinear finite element code LS -DYNA in which the strain rate effects of both steel and concrete, contact between steel tube and concrete and confinement effect provided by the steel tube for the concrete were considered. The model also benefited from a relatively recent feature of LS-DYNA for applying a pre-loading in the explicit solver. The developed numerical model was verified for its accuracy and adequacy by comparing the results with experimental results available in the literature. The verified model was then employed to conduct a parametric study to investigate the influence of axial load level, impact location, support conditions, and slenderness ratio on the response of the CFST columns. A good agreement between the numerical and experimental results was achieved. The model could reasonably predict the impact load-deflection history and deformed shape of the column at the end of the impact event. The results of the parametric study showed that whilst impact location, axial load level and slenderness ratio can have a significant effect on the peak impact force, residual lateral deflection and maximum lateral deflection, the influence of support fixity is minimal. With an increase of axial load to up to a certain level, the peak force increases; however, a further increase in the axial load causes a decrease in the peak force. Both residual lateral deflection and maximum lateral deflection increase as axial load level increases. Shifting the impact location towards the supports increases the peak force and reduces both residual and maximum lateral deflections. A rise in slenderness ratio decreases the peak force and increases the residual and maximum lateral deflections.

A simplified approach for calculating the moments of action for linear reaction-diffusion equations

The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time under-approximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.