Maximum principle and numerical method for the multi-term time–space Riesz–Caputo fractional differential equations

The maximum principle for the space and time–space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time–space Riesz–Caputo fractional differential equations over an open bounded domain. A maximum principle for the equation is proved. The uniqueness and continuous dependence of the solution are derived. Using a fractional predictor–corrector method combining the L1 and L2 discrete schemes, we present a numerical method for the specified equation. Two examples are given to illustrate the obtained results.

Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Numerical simulation of anomalous infiltration in porous media

Nonlinear time-fractional diffusion equations have been used to describe the liquid infiltration for both subdiffusion and superdiffusion in porous media. In this paper, some problems of anomalous infiltration with a variable-order timefractional derivative in porous media are considered. The time-fractional Boussinesq equation is also considered. Two computationally efficient implicit numerical schemes for the diffusion and wave-diffusion equations are proposed. Numerical examples are provided to show that the numerical methods are computationally efficient.

Experimental and numerical investigation of strain-rate dependent mechanical properties of single living cells

The objective of this project is to investigate the strain-rate dependent mechanical behaviour of single living cells using both experimental and numerical techniques. The results revealed that living cells behave as porohyperlastic materials and that both solid and fluid phases within the cells play important roles in their mechanical responses. The research reported in this thesis provides a better understanding of the mechanisms underlying the cellular responses to external mechanical loadings and of the process of mechanical signal transduction in living cells. It would help us to enhance knowledge of and insight into the role of mechanical forces in supporting tissue regeneration or degeneration.

Numerical investigation of case hardening of plant tissue during dying and its influence on the cellular level shrinkage

Dried plant food materials are one of the major contributors to the global food industry. Widening the fundamental understanding on different mechanisms of food material alterations during drying assists the development of novel dried food products and processing techniques. In this regard, case hardening is an important phenomenon, commonly observed during the drying processes of plant food materials, which significantly influences the product quality and process performance. In this work, a recent meshfree-based numerical model of the authors is further improved and used to simulate the influence of case hardening on shrinkage characteristics of plant tissues during drying. In order to model fluid and wall mechanisms in each cell, Smoothed Particle Hydrodynamics (SPH) and the Discrete Element Method (DEM) are used. The model is fundamentally more capable of simulating large deformation of multiphase materials, when compared with conventional grid-based modelling techniques such as Finite Element Methods (FEM) or Finite Difference Methods (FDM). Case hardening is implemented by maintaining distinct moisture levels in the different cell layers of a given tissue. In order to compare and investigate different factors influencing tissue deformations under case hardening, four different plant tissue varieties (apple, potato, carrot and grape) are studied. The simulation results indicate that the inner cells of any given tissue undergo limited shrinkage and cell wall wrinkling compared to the case hardened outer cell layers of the tissues. When comparing unique deformation characteristics of the different tissues, irrespective of the normalised moisture content, the cell size, cell fluid turgor pressure and cell wall characteristics influence the tissue response to case hardening.

Numerical simulation of railway track supporting system using finite-infinite and thin layer elements under impulsive loads

The present study deals with two dimensional, numerical simulation of railway track supporting system subjected to dynamic excitation force. Under plane strain condition, the coupled finite-infinite elements to represent the near and far field stress distribution and thin layer interface element was employed to model the interfacial behavior between sleepers and ballast. To account for the relative debonding, slipping and crushing that could take place in the contact area between the sleepers and ballast, modified Mohr-Coulomb criterion was adopted. Furthermore an attempt has been made to consider the elasto-plastic material non-linearity of the railway track supporting media by employing different constitutive models to represent steel, concrete and supporting materials. Based on the proposed physical and constitutive modeling a code has been developed for dynamic loads. The applicability of the developed F.E code has been demonstrated by analyzing a real railway supporting structure.

Numerical modeling of railway track supporting system using finite–infinite and thin layer elements

The present contribution deals with the numerical modelling of railway track-supporting systems-using coupled finite-infinite elements-to represent the near and distant field stress distribution, and also employing a thin layer interface element to account for the interfacial behaviour between sleepers and ballast. To simulate the relative debonding, slipping and crushing at the contact area between sleepers and ballast, a modified Mohr-Coulomb criterion was adopted. Further more an attempt was made to consider the elasto plastic materials’ non-linearity of the railway track supporting media by employing different constitutive models to represent steel, concrete and other supporting materials. It is seen that during an incremental-iterative mode of load application, the yielding initially started from the edge of the sleepers and then flowed vertically downwards and spread towards the centre of the railway supporting system.

Numerical investigation of motion and deformation of a single red blood cell in a stenosed capillary

It is generally assumed that influence of the red blood cells (RBCs) is predominant in blood rheology. The healthy RBCs are highly deformable and can thus easily squeeze through the smallest capillaries having internal diameter less than their characteristic size. On the other hand, RBCs infected by malaria or other diseases are stiffer and so less deformable. Thus it is harder for them to flow through the smallest capillaries. Therefore, it is very important to critically and realistically investigate the mechanical behavior of both healthy and infected RBCs which is a current gap in knowledge. The motion and the steady state deformed shape of the RBCs depend on many factors, such as the geometrical parameters of the capillary through which blood flows, the membrane bending stiffness and the mean velocity of the blood flow. In this study, motion and deformation of a single two-dimensional RBC in a stenosed capillary is explored by using smoothed particle hydrodynamics (SPH) method. An elastic spring network is used to model the RBC membrane, while the RBC's inside fluid and outside fluid are treated as SPH particles. The effect of RBC's membrane stiffness (kb), inlet pressure (P) and geometrical parameters of the capillary on the motion and deformation of the RBC is studied. The deformation index, RBC's mean velocity and the cell membrane energy are analyzed when the cell passes through the stenosed capillary. The simulation results demonstrate that the kb, P and the geometrical parameters of the capillary have a significant impact on the RBCs' motion and deformation in the stenosed section.

Numerical treatment of a two-dimensional variable-order fractional nonlinear reaction-diffusion model

A two-dimensional variable-order fractional nonlinear reaction-diffusion model is considered. A second-order spatial accurate semi-implicit alternating direction method for a two-dimensional variable-order fractional nonlinear reaction-diffusion model is proposed. Stability and convergence of the semi-implicit alternating direct method are established. Finally, some numerical examples are given to support our theoretical analysis. These numerical techniques can be used to simulate a two-dimensional variable order fractional FitzHugh-Nagumo model in a rectangular domain. This type of model can be used to describe how electrical currents flow through the heart, controlling its contractions, and are used to ascertain the effects of certain drugs designed to treat arrhythmia.

Numerical modeling of thin steel roof battens under wind uplift loads

Extreme wind events such as tropical cyclones, tornadoes and storms are more likely to impact the Australian coastal regions due to possible climate changes. Such events can be extremely destructive to building structures, in particular, low-rise buildings with lightweight roofing systems that are commonly made of thin steel roofing sheets and battens. Large wind uplift loads that act on the roofs during high wind events often cause premature roof connection failures. Recent wind damage investigations have shown that roof failures have mostly occurred at the batten to rafter or truss screw connections. In most of these cases, the screw fastener heads pulled through the bottom flanges of thin steel roof battens. This roof connection failure is very critical as both roofing sheets and battens will be lost during the high wind events. Hence, a research study was conducted to investigate this critical pull-through failure using both experimental and numerical methods. This paper presents the details of numerical modeling and the results.

Numerical computation of an Evans function for travelling waves

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller–Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c≥sqrt(2δ) in the F-KPP equation.

Numerical optimization of soft-mold aided co-curing process of advanced grid-stiffened composite structures

The co-curing process for advanced grid-stiffened (AGS) composite structure is a promising manufacturing process, which could reduce the manufacturing cost, augment the advantages and improve the performance of AGS composite structure. An improved method named soft-mold aided co-curing process which replaces the expansion molds by a whole rubber mold is adopted in this paper. This co-curing process is capable to co-cure a typical AGS composite structure with the manufacturer’s recommended cure cycle (MRCC). Numerical models are developed to evaluate the variation of temperature and the degree of cure in AGS composite structure during the soft-mold aided co-curing process. The simulation results were validated by experimental results obtained from embedded temperature sensors. Based on the validated modeling framework, the cycle of cure can be optimized by reducing more than half the time of MRCC while obtaining a reliable degree of cure. The shape and size effects of AGS composite structure on the distribution of temperature and degree of cure are also investigated to provide insights for the optimization of soft-mold aided co-curing process.

Numerical investigation of aerosol particle transport and deposition in realistic lung airway

Different human activities like combustion of fossil fuels, biomass burning, industrial and agricultural activities, emit a large amount of particulates into the atmosphere. As a consequence, the air we inhale contains significant amount of suspended particles, including organic and inorganic solids and liquids, as well as various microorganism, which are solely responsible for a number of pulmonary diseases. Developing a numerical model for transport and deposition of foreign particles in realistic lung geometry is very challenging due to the complex geometrical structure of the human lung. In this study, we have numerically investigated the airborne particle transport and its deposition in human lung surface. In order to obtain the appropriate results of particle transport and deposition in human lung, we have generated realistic lung geometry from the CT scan obtained from a local hospital. For a more accurate approach, we have also created a mucus layer inside the geometry, adjacent to the lung surface and added all apposite mucus layer properties to the wall surface. The Lagrangian particle tracking technique is employed by using ANSYS FLUENT solver to simulate the steady-state inspiratory flow. Various injection techniques have been introduced to release the foreign particles through the inlet of the geometry. In order to investigate the effects of particle size on deposition, numerical calculations are carried out for different sizes of particles ranging from 1 micron to 10 micron. The numerical results show that particle deposition pattern is completely dependent on its initial position and in case of realistic geometry; most of the particles are deposited on the rough wall surface of the lung geometry instead of carinal region.

Numerical modelling and design of unlined cold-formed steel wall frames

Cold-formed steel wall frame systems using lipped or unlipped C-sections and gypsum plasterboard lining are commonly utilised in the construction of both the load bearing and non-load bearing walls in the residential, commercial and industrial buildings. However, the structural behaviour of unlined and lined stud wall frames is not well understood and adequate design rules are not available. A detailed research program was therefore undertaken to investigate the behaviour of stud wall frame systems. As the first step in this research, the problem relating to the degree of end fixity of stud was investigated. The studs are usually connected to the top and bottom tracks and the degree of end fixity provided by these tracks is not adequately addressed by the design codes. A finite element model of unlined frames was therefore developed, and validated using full scale experimental results. It was then used in a detailed parametric study to develop appropriate design rules for unlined wall frames. This study has shown that by using appropriate effective length factors, the ultimate load and failure modes of the unlined studs can be accurately predicted using the provisions of Australian or American cold-formed steel structures design codes. This paper presents the details of the finite element analyses, the results and recommended design rules for unlined wall frames.

A preconditioned numerical solver for stiff nonlinear reaction-diffusion equations with fractional Laplacians that avoids dense matrices

The numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the spatial nonlocality of the fractional differential operators. The dense coefficient matrices that arise from spatial discretisation of these operators mean that even one-dimensional problems can be difficult to solve using standard methods on grids comprising thousands of nodes or more. In this work we address this issue of efficiency for one-dimensional, nonlinear space-fractional reaction–diffusion equations with fractional Laplacian operators. We apply variable-order, variable-stepsize backward differentiation formulas in a Jacobian-free Newton–Krylov framework to advance the solution in time. A key advantage of this approach is the elimination of any requirement to form the dense matrix representation of the fractional Laplacian operator. We show how a banded approximation to this matrix, which can be formed and factorised efficiently, can be used as part of an effective preconditioner that accelerates convergence of the Krylov subspace iterative solver. Our approach also captures the full contribution from the nonlinear reaction term in the preconditioner, which is crucial for problems that exhibit stiff reactions. Numerical examples are presented to illustrate the overall effectiveness of the solver.