Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.

Natural convection in attic-shaped spaces subject to sudden and ramp heating boundary conditions

In this study, a discussion of the fluid dynamics in the attic space is reported, focusing on its transient response to sudden and linear changes of temperature along the two inclined walls. The transient behaviour of an attic space is relevant to our daily life. The instantaneous and non-instantaneous (ramp) heating boundary condition is applied on the sloping walls of the attic space. A theoretical understanding of the transient behaviour of the flow in the enclosure is performed through scaling analysis. A proper identification of the timescales, the velocity and the thickness relevant to the flow that develops inside the cavity makes it possible to predict theoretically the basic flow features that will survive once the thermal flow in the enclosure reaches a steady state. A time scale for the heating-up of the whole cavity together with the heat transfer scales through the inclined walls has also been obtained through scaling analysis. All scales are verified by the numerical simulations.

Natural convection in attics subject to instantaneous and ramp cooling boundary conditions

A fundamental study of the fluid dynamics inside an attic shaped triangular enclosure with cold upper walls and adiabatic horizontal bottom wall is reported in this study. The transient behaviour of the attic fluid which is relevant to our daily life is examined based on a scaling analysis. The transient phenomenon begins with the instantaneous cooling and the cooling with linear decreases of temperature up to some specific time (ramp time) and then maintain constant of the upper sloped walls. It is shown that both inclined walls develop a thermal boundary layer whose thicknesses increase towards steady-state or quasi-steady values. A proper identification of the timescales, the velocity and the thickness relevant to the flow that develops inside the cavity makes it possible to predict theoretically the basic flow features that will survive once the thermal flow in the enclosure reaches a steady state. A time scale for the cooling-down of the whole cavity together with the heat transfer scales through the inclined walls has also been obtained through scaling analysis. All scales are verified by the numerical simulations.

An analytical framework for quantifying aquifer response time scales associated with transient boundary conditions

A major challenge in studying coupled groundwater and surface-water interactions arises from the considerable difference in the response time scales of groundwater and surface-water systems affected by external forcings. Although coupled models representing the interaction of groundwater and surface-water systems have been studied for over a century, most have focused on groundwater quantity or quality issues rather than response time. In this study, we present an analytical framework, based on the concept of mean action time (MAT), to estimate the time scale required for groundwater systems to respond to changes in surface-water conditions. MAT can be used to estimate the transient response time scale by analyzing the governing mathematical model. This framework does not require any form of transient solution (either numerical or analytical) to the governing equation, yet it provides a closed form mathematical relationship for the response time as a function of the aquifer geometry, boundary conditions, and flow parameters. Our analysis indicates that aquifer systems have three fundamental time scales: (i) a time scale that depends on the intrinsic properties of the aquifer; (ii) a time scale that depends on the intrinsic properties of the boundary condition, and; (iii) a time scale that depends on the properties of the entire system. We discuss two practical scenarios where MAT estimates provide useful insights and we test the MAT predictions using new laboratory-scale experimental data sets.

Fluid-structure interaction study on human arterial plaque with patient specific geometry and boundary conditions

Atherosclerotic plaque rupture has been extensively considered as the leading cause of death in western countries. It is believed that high stresses within plaque can be an important factor on triggering the rupture of the plaque. Stress analysis in the coronary and carotid arteries with plaque have been developed by many researchers from 2D to 3-D models, from structure analysis only to the Fluid-Structure Interaction (FSI) models[1].

Flexural behaviour and design of the new built-up LiteSteel beams

LiteSteel Beam (LSB) is a new cold-formed steel beam produced by OneSteel Australian Tube Mills. The new beam is effectively a channel section with two rectangular hollow flanges and a slender web, and is manufactured using a combined cold-forming and electric resistance welding process. OneSteel Australian Tube Mills is promoting the use of LSBs as flexural members in a range of applications, such as floor bearers. When LSBs are used as back to back built-up sections, they are likely to improve their moment capacity and thus extend their applications further. However, the structural behaviour of built-up beams is not well understood. Many steel design codes include guidelines for connecting two channels to form a built-up I-section including the required longitudinal spacing of connections. But these rules were found to be inadequate in some applications. Currently the safe spans of builtup beams are determined based on twice the moment capacity of a single section. Research has shown that these guidelines are conservative. Therefore large scale lateral buckling tests and advanced numerical analyses were undertaken to investigate the flexural behaviour of back to back LSBs connected by fasteners (bolts) at various longitudinal spacings under uniform moment conditions. In this research an experimental investigation was first undertaken to study the flexural behaviour of back to back LSBs including its buckling characteristics. This experimental study included tensile coupon tests, initial geometric imperfection measurements and lateral buckling tests. The initial geometric imperfection measurements taken on several back to back LSB specimens showed that the back to back bolting process is not likely to alter the imperfections, and the measured imperfections are well below the fabrication tolerance limits. Twelve large scale lateral buckling tests were conducted to investigate the behaviour of back to back built-up LSBs with various longitudinal fastener spacings under uniform moment conditions. Tests also included two single LSB specimens. Test results showed that the back to back LSBs gave higher moment capacities in comparison with single LSBs, and the fastener spacing influenced the ultimate moment capacities. As the fastener spacing was reduced the ultimate moment capacities of back to back LSBs increased. Finite element models of back to back LSBs with varying fastener spacings were then developed to conduct a detailed parametric study on the flexural behaviour of back to back built-up LSBs. Two finite element models were developed, namely experimental and ideal finite element models. The models included the complex contact behaviour between LSB web elements and intermittently fastened bolted connections along the web elements. They were validated by comparing their results with experimental results and numerical results obtained from an established buckling analysis program called THIN-WALL. These comparisons showed that the developed models could accurately predict both the elastic lateral distortional buckling moments and the non-linear ultimate moment capacities of back to back LSBs. Therefore the ideal finite element models incorporating ideal simply supported boundary conditions and uniform moment conditions were used in a detailed parametric study on the flexural behaviour of back to back LSB members. In the detailed parametric study, both elastic buckling and nonlinear analyses of back to back LSBs were conducted for 13 LSB sections with varying spans and fastener spacings. Finite element analysis results confirmed that the current design rules in AS/NZS 4600 (SA, 2005) are very conservative while the new design rules developed by Anapayan and Mahendran (2009a) for single LSB members were also found to be conservative. Thus new member capacity design rules were developed for back to back LSB members as a function of non-dimensional member slenderness. New empirical equations were also developed to aid in the calculation of elastic lateral distortional buckling moments of intermittently fastened back to back LSBs. Design guidelines were developed for the maximum fastener spacing of back to back LSBs in order to optimise the use of fasteners. A closer fastener spacing of span/6 was recommended for intermediate spans and some long spans where the influence of fastener spacing was found to be high. In the last phase of this research, a detailed investigation was conducted to investigate the potential use of different types of connections and stiffeners in improving the flexural strength of back to back LSB members. It was found that using transverse web stiffeners was the most cost-effective and simple strengthening method. It is recommended that web stiffeners are used at the supports and every third points within the span, and their thickness is in the range of 3 to 5 mm depending on the size of LSB section. The use of web stiffeners eliminated most of the lateral distortional buckling effects and hence improved the ultimate moment capacities. A suitable design equation was developed to calculate the elastic lateral buckling moments of back to back LSBs with the above recommended web stiffener configuration while the same design rules developed for unstiffened back to back LSBs were recommended to calculate the ultimate moment capacities.

Natural convection boundary-layer adjacent to an inclined flat plate subject to sudden and ramp heating

The natural convection thermal boundary layer adjacent to an inclined flat plate subject to sudden heating and a temperature boundary condition which follows a ramp function up until a specified time and then remains constant is investigated. The development of the flow from start-up to a steady-state has been described based on scaling analyses and verified by numerical simulations. Different flow regimes based on the Rayleigh number are discussed with numerical results for both boundary conditions. For ramp heating, the boundary layer flow depends on the comparison of the time at which the ramp heating is completed and the time at which the boundary layer completes its growth. If the ramp time is long compared with the steady state time, the layer reaches a quasi steady mode in which the growth of the layer is governed solely by the thermal balance between convection and conduction. On the other hand, if the ramp is completed before the layer becomes steady; the subsequent growth is governed by the balance between buoyancy and inertia, as for the case of instantaneous heating.

Experimental investigation of wheel/rail rolling contact at railhead edge

Wheel-rail rolling contact at railhead edge, such as a gap in an insulated rail joint, is a complex problem; there are only limited analytical, numerical and experimental studies available on this problem in the academic literature. This paper describes experimental and numerical investigations of railhead strains in the vicinity of the edge under the contact of a loaded wheel. A full-scale test rig was developed to cyclically apply wheel/rail rolling contact load to the edge zone of the railhead. An image analysis technique was employed to determine the railhead vertical, lateral and shear strain components. The vertical strains determined using the image analysis method have been validated with the strain gauge measurements and used for the calibration of a 3D nonlinear Finite Element Model (FEM) that simulates the wheel/rail contact at the railhead edge and use suitable boundary conditions commensurate to the experimental setup. The FEM was then used to determine other states of strains.

Exponential integrators and a dual-scale model for wood drying

For the timber industry, the ability to simulate the drying of wood is invaluable for manufacturing high quality wood products. Mathematically, however, modelling the drying of a wet porous material, such as wood, is a diffcult task due to its heterogeneous and anisotropic nature, and the complex geometry of the underlying pore structure. The well{ developed macroscopic modelling approach involves writing down classical conservation equations at a length scale where physical quantities (e.g., porosity) can be interpreted as averaged values over a small volume (typically containing hundreds or thousands of pores). This averaging procedure produces balance equations that resemble those of a continuum with the exception that effective coeffcients appear in their deffnitions. Exponential integrators are numerical schemes for initial value problems involving a system of ordinary differential equations. These methods differ from popular Newton{Krylov implicit methods (i.e., those based on the backward differentiation formulae (BDF)) in that they do not require the solution of a system of nonlinear equations at each time step but rather they require computation of matrix{vector products involving the exponential of the Jacobian matrix. Although originally appearing in the 1960s, exponential integrators have recently experienced a resurgence in interest due to a greater undertaking of research in Krylov subspace methods for matrix function approximation. One of the simplest examples of an exponential integrator is the exponential Euler method (EEM), which requires, at each time step, approximation of φ(A)b, where φ(z) = (ez - 1)/z, A E Rnxn and b E Rn. For drying in porous media, the most comprehensive macroscopic formulation is TransPore [Perre and Turner, Chem. Eng. J., 86: 117-131, 2002], which features three coupled, nonlinear partial differential equations. The focus of the first part of this thesis is the use of the exponential Euler method (EEM) for performing the time integration of the macroscopic set of equations featured in TransPore. In particular, a new variable{ stepsize algorithm for EEM is presented within a Krylov subspace framework, which allows control of the error during the integration process. The performance of the new algorithm highlights the great potential of exponential integrators not only for drying applications but across all disciplines of transport phenomena. For example, when applied to well{ known benchmark problems involving single{phase liquid ow in heterogeneous soils, the proposed algorithm requires half the number of function evaluations than that required for an equivalent (sophisticated) Newton{Krylov BDF implementation. Furthermore for all drying configurations tested, the new algorithm always produces, in less computational time, a solution of higher accuracy than the existing backward Euler module featured in TransPore. Some new results relating to Krylov subspace approximation of '(A)b are also developed in this thesis. Most notably, an alternative derivation of the approximation error estimate of Hochbruck, Lubich and Selhofer [SIAM J. Sci. Comput., 19(5): 1552{1574, 1998] is provided, which reveals why it performs well in the error control procedure. Two of the main drawbacks of the macroscopic approach outlined above include the effective coefficients must be supplied to the model, and it fails for some drying configurations, where typical dual{scale mechanisms occur. In the second part of this thesis, a new dual{scale approach for simulating wood drying is proposed that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of softwood at low temperatures and is valid in the so{called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapour in the pores. Coupling between scales is achieved by imposing the macroscopic gradient on the microscopic field using suitably defined periodic boundary conditions, which allows the macroscopic ux to be defined as an average of the microscopic ux over the unit cell. This formulation provides a first step for moving from the macroscopic formulation featured in TransPore to a comprehensive dual{scale formulation capable of addressing any drying configuration. Simulation results reported for a sample of spruce highlight the potential and flexibility of the new dual{scale approach. In particular, for a given unit cell configuration it is not necessary to supply the effective coefficients prior to each simulation.

Two novel numerical methods for solving the two-dimensional fractional Fitzhugh-Nagumo-monodomain model

Fractional mathematical models represent a new approach to modelling complex spatial problems in which there is heterogeneity at many spatial and temporal scales. In this paper, a two-dimensional fractional Fitzhugh-Nagumo-monodomain model with zero Dirichlet boundary conditions is considered. The model consists of a coupled space fractional diffusion equation (SFDE) and an ordinary differential equation. For the SFDE, we first consider the numerical solution of the Riesz fractional nonlinear reaction-diffusion model and compare it to the solution of a fractional in space nonlinear reaction-diffusion model. We present two novel numerical methods for the two-dimensional fractional Fitzhugh-Nagumo-monodomain model using the shifted Grunwald-Letnikov method and the matrix transform method, respectively. Finally, some numerical examples are given to exhibit the consistency of our computational solution methodologies. The numerical results demonstrate the effectiveness of the methods.

An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

Similarity solutions for flow and heat transfer of non-Newtonian fluid over a stretching surface

Similarity solutions are carried out for flow of power law non-Newtonian fluid film on unsteady stretching surface subjected to constant heat flux. Free convection heat transfer induces thermal boundary layer within a semi-infinite layer of Boussinesq fluid. The nonlinear coupled partial differential equations (PDE) governing the flow and the boundary conditions are converted to a system of ordinary differential equations (ODE) using two-parameter groups. This technique reduces the number of independent variables by two, and finally the obtained ordinary differential equations are solved numerically for the temperature and velocity using the shooting method. The thermal and velocity boundary layers are studied by the means of Prandtl number and non-Newtonian power index plotted in curves.

Generalized azimuthal shear deformations in compressible isotropic elastic materials

In this article we study the azimuthal shear deformations in a compressible Isotropic elastic material. This class of deformations involves an azimuthal displacement as a function of the radial and axial coordinates. The equilibrium equations are formulated in terms of the Cauchy-Green strain tensors, which form an overdetermined system of partial differential equations for which solutions do not exist in general. By means of a Legendre transformation, necessary and sufficient conditions for the material to support this deformation are obtained explicitly, in the sense that every solution to the azimuthal equilibrium equation will satisfy the remaining two equations. Additionally, we show how these conditions are sufficient to support all currently known deformations that locally reduce to simple shear. These conditions are then expressed both in terms of the invariants of the Cauchy-Green strain and stretch tensors. Several classes of strain energy functions for which this deformation can be supported are studied. For certain boundary conditions, exact solutions to the equilibrium equations are obtained. © 2005 Society for Industrial and Applied Mathematics.

A numerical method for the fractional Fitzhugh–Nagumo monodomain model

A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach.