Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

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.

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.

Development of lightweight concrete with high resistance to water and chloride-ion penetration

This paper presents an experimental study to evaluate the influence of coarse lightweight aggregate (LWA), fine LWA and the quality of the paste matrix on water absorption and permeability, and resistance to chloride-ion penetration in concrete. The results indicate that incorporation of pre-soaked coarse LWA in concrete increases water sorptivity and permeability slightly compared to normal weight concrete (NWC) of similar water-to-cementitious materials ratio (w/cm). Furthermore, resistance of the sand lightweight concrete (LWC) to water permeability and chloride-ion penetration decreases with an increase in porosity of the coarse LWA. The use of fine LWA including a crushed fraction <1.18 mm reduced resistance of the all-LWC to water and chloride-ion penetration compared with the sand-LWC which has the same coarse LWA. Overall, the quality of the paste matrix was dominant in controlling the transport properties of the concrete, regardless of porosity of the aggregates used. With low w/cm and silica fume, low unit weight LWC (_1300 kg/m3) was produced with a higher resistance to water and chloride-ion penetration compared with NWC and LWC of higher unit weights.

Estimation Of concrete resistance To chloride-ion penetration based On water accessible porosity and unit weight Of concrete

To ensure better concrete quality and long-term durability, there has been an increasing focus in recent years on the development of test methods for quality control of concrete. This paper presents a study to evaluate the effect of water accessible porosity and oven-dry unit weight on the resistance of concrete to chloride-ion penetration. Based on the experimental results and regression analyses, empirical relationships of the charge passed (ASTM C 1202) and chloride migration coefficient (NT Build 492) versus the water accessible porosity and oven dry unit weight of the concrete are established. Using basic physical properties of water accessible porosity and oven dry unit weight which can be easily determined, total charge passed and migration coefficient of the concrete can be estimated for quality control and for estimating durability of concrete.

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...

Water and chloride ion penetration resistance of highstrength ultra lightweight cement composite

Durability is a significant issue to focus on for newly developed structural lightweight cement composite (ULCC). This paper presents an experimental study to evaluate the resistance of ULCC to water and chloride ion penetration. Chloride penetrability and sorptivity were evaluated for ULCC (unit weight about 1450 kg/m3) and compared with those of a normal weight concrete (NWC), a lightweight aggregate concrete (LWC), and an ultra lightweight composite with proprietary cementitious binder (DB) (unit weight about 1450 kg/m3) at similar compressive strength of about 60 MPa. Rapid chloride penetrability test, rapid migration test, water absorption (sorptivity) test, and water permeability test were conducted on these mixtures. Results indicate that ULCC and DB had comparable performance. Compared with control LWC and NWC at similar strength level, the ULCC and DB mixtures had higher resistance to chloride ion penetration, lower water absorption and virtually impermeable to water penetration.

Development of lightweight aggregate concrete with high resistance to water and chloride-ion penetration

This paper presents an experimental study to evaluate the effect of coarse and fine LWA in concrete on its water absorption and permeability, and resistance to chloride-ion penetration. In additions, LWC with lower unit weight of about 1300 kg/m3 but high resistance to water and chloride-ion penetration was developed and evaluated. The results indicate that the incorporation of coarse LWA in concrete increases water sorptivity and permeability slightly compared to NWC of similar w/c. The resistance of the sand-LWC to chloride-ion penetration depends on porosity of the coarse LWA. Fine LWA has more influence on the transport proper-ties of concrete than coarse LWA. Use of lightweight crushed sand <1.18 mm reduced the resistance of the LWC to water and chloride-ion penetration to some extent. With low w/cm and silica fume, low unit weight LWC (~1300 kg/m3) was produced with higher resistance to water and chloride ion penetration compared with concretes of higher unit weights.

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.