Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Finite element modelling of load bearing steel stud walls under real building fires

Fire safety has become an important part in structural design due to the ever increasing loss of properties and lives during fires. Conventionally the fire rating of load bearing wall systems made of Light gauge Steel Frames (LSF) is determined using fire tests based on the standard time-temperature curve given in ISO 834 (ISO, 1999). The standard time-temperature curve given in ISO 834 (ISO, 1999) originated from the application of wood burning furnaces in the early 1900s. However, modern commercial and residential buildings make use of thermoplastic materials, which mean considerably high fuel loads. Hence a detailed fire research study into the performance of LSF walls was undertaken using the developed real fire curves based on Eurocode parametric curves (ECS, 2002) and Barnett’s BFD curves (Barnett, 2002) using both full scale fire tests and numerical studies. It included LSF walls without any insulation, and the recently developed externally insulated composite panel system. This paper presents the details of the numerical studies and the results. It also includes brief details of the development of real building fire curves and experimental studies.

Critical timescales and time intervals for coupled linear processes

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and coworkers rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single–species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalise the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications including the analysis of models describing coupled chemical decay and cell differentiation processes, amongst others.

Residence time investigation in a co-axial dielectric barrier discharge reactor

The residence time distribution (RTD) is a crucial parameter when treating engine exhaust emissions with a Dielectric Barrier Discharge (DBD) reactor. In this paper, the residence time of such a reactor is investigated using a finite element based software: COMSOL Multiphysics 4.3. Non-thermal plasma (NTP) discharge is being introduced as a promising method for pollutant emission reduction. DBD is one of the most advantageous of NTP technologies. In a two cylinder co-axial DBD reactor, tubes are placed between two electrodes and flow passes through the annuals between these barrier tubes. If the mean residence time increases in a DBD reactor, there will be a corresponding increase in reaction time and consequently, the pollutant removal efficiency can increase. However, pollutant formation can occur during increased mean residence time and so the proportion of fluid that may remain for periods significantly longer than the mean residence time is of great importance. In this study, first, the residence time distribution is calculated based on the standard reactor used by the authors for ultrafine particle (10-500 nm) removal. Then, different geometrics and various inlet velocities are considered. Finally, for selected cases, some roughness elements added inside the reactor and the residence time is calculated. These results will form the basis for a COMSOL plasma and CFD module investigation.

Porohyperelastic analysis to explore mechanical properties of chondrocytes using numerical modeling and experiments : a finite element study

There are many continuum mechanical models have been developed such as liquid drop models, solid models, and so on for single living cell biomechanics studies. However, these models do not give a fully approach to exhibit a clear understanding of the behaviour of single living cells such as swelling behaviour, drag effect, etc. Hence, the porohyperelastic (PHE) model which can capture those aspects would be a good candidature to study cells behaviour (e.g. chondrocytes in this study). In this research, an FEM model of single chondrocyte cell will be developed by using this PHE model to simulate Atomic Force Microscopy (AFM) experimental results with the variation of strain rate. This material model will be compared with viscoelastic model to demonstrate the advantages of PHE model. The results have shown that the maximum value of force applied of PHE model is lower at lower strain rates. This is because the mobile fluid does not have enough time to exude in case of very high strain rate and also due to the lower permeability of the membrane than that of the protoplasm of chondrocyte. This behavior is barely observed in viscoelastic model. Thus, PHE model is the better model for cell biomechanics studies.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Finite element modelling of load bearing cold-formed steel wall systems under fire conditions

Light Gauge Steel Framing (LSF) walls are made of cold-formed, thin-walled steel lipped channel studs with plasterboard linings on both sides. However, these thin-walled steel sections heat up quickly and lose their strength under fire conditions despite the protection provided by plasterboards. A new composite wall panel was recently proposed to improve the fire resistance rating of LSF walls, where an insulation layer was used externally between the plasterboards on both sides of the wall frame instead of using it in the cavity. A research study using both fire tests and numerical studies was undertaken to investigate the structural and thermal behaviour of load bearing LSF walls made of both conventional and the new composite panels under standard fire conditions and to determine their fire resistance rating. This paper presents the details of finite element models of LSF wall studs developed to simulate the structural performance of LSF wall panels under standard fire conditions. Finite element analyses were conducted under both steady and transient state conditions using the time-temperature profiles measured during the fire tests. The developed models were validated using the fire test results of 11 LSF wall panels with various plasterboard/insulation configurations and load ratios. They were able to predict the fire resistance rating within five minutes. The use of accurate numerical models allowed the inclusion of various complex structural and thermal effects such as local buckling, thermal bowing and neutral axis shift that occurred in thin-walled steel studs under non-uniform elevated temperature conditions. Finite element analyses also demonstrated the improvements offered by the new composite panel system over the conventional cavity insulated system.

Novel channel tracking and equalization methods in MU-MIMO-OFDM systems

The methodology undertaken, the channel model and the system model created for developing a novel adaptive equalization method and a novel channel tracking method for uplink of MU-MIMO-OFDM systems is presented in this paper. The results show that the channel tracking method works with 97% accuracy, while the training-based initial channel estimation method shows poor performance in estimating the actual channel comparatively.

How to increase the accuracy of analysis and reduce the computational time in ANSYS in the case of deformation study of orthopedic bone plates

Currently, finite element analyses are usually done by means of commercial software tools. Accuracy of analysis and computational time are two important factors in efficiency of these tools. This paper studies the effective parameters in computational time and accuracy of finite element analyses performed by ANSYS and provides the guidelines for the users of this software whenever they us this software for study on deformation of orthopedic bone plates or study on similar cases. It is not a fundamental scientific study and only shares the findings of the authors about structural analysis by means of ANSYS workbench. It gives an idea to the readers about improving the performance of the software and avoiding the traps. The solutions provided in this paper are not the only possible solutions of the problems and in similar cases there are other solutions which are not given in this paper. The parameters of solution method, material model, geometric model, mesh configuration, number of the analysis steps, program controlled parameters and computer settings are discussed through thoroughly in this paper.

Real - time prediction of rainfall induced instability of residual soil slopes associated with deep cracks

The early warning based on real-time prediction of rain-induced instability of natural residual slopes helps to minimise human casualties due to such slope failures. Slope instability prediction is complicated, as it is influenced by many factors, including soil properties, soil behaviour, slope geometry, and the location and size of deep cracks in the slope. These deep cracks can facilitate rainwater infiltration into the deep soil layers and reduce the unsaturated shear strength of residual soil. Subsequently, it can form a slip surface, triggering a landslide even in partially saturated soil slopes. Although past research has shown the effects of surface-cracks on soil stability, research examining the influence of deep-cracks on soil stability is very limited. This study aimed to develop methodologies for predicting the real-time rain-induced instability of natural residual soil slopes with deep cracks. The results can be used to warn against potential rain-induced slope failures. The literature review conducted on rain induced slope instability of unsaturated residual soil associated with soil crack, reveals that only limited studies have been done in the following areas related to this topic: - Methods for detecting deep cracks in residual soil slopes. - Practical application of unsaturated soil theory in slope stability analysis. - Mechanistic methods for real-time prediction of rain induced residual soil slope instability in critical slopes with deep cracks. Two natural residual soil slopes at Jombok Village, Ngantang City, Indonesia, which are located near a residential area, were investigated to obtain the parameters required for the stability analysis of the slope. A survey first identified all related field geometrical information including slope, roads, rivers, buildings, and boundaries of the slope. Second, the electrical resistivity tomography (ERT) method was used on the slope to identify the location and geometrical characteristics of deep cracks. The two ERT array models employed in this research are: Dipole-dipole and Azimuthal. Next, bore-hole tests were conducted at different locations in the slope to identify soil layers and to collect undisturbed soil samples for laboratory measurement of the soil parameters required for the stability analysis. At the same bore hole locations, Standard Penetration Test (SPT) was undertaken. Undisturbed soil samples taken from the bore-holes were tested in a laboratory to determine the variation of the following soil properties with the depth: - Classification and physical properties such as grain size distribution, atterberg limits, water content, dry density and specific gravity. - Saturated and unsaturated shear strength properties using direct shear apparatus. - Soil water characteristic curves (SWCC) using filter paper method. - Saturated hydraulic conductivity. The following three methods were used to detect and simulate the location and orientation of cracks in the investigated slope: (1) The electrical resistivity distribution of sub-soil obtained from ERT. (2) The profile of classification and physical properties of the soil, based on laboratory testing of soil samples collected from bore-holes and visual observations of the cracks on the slope surface. (3) The results of stress distribution obtained from 2D dynamic analysis of the slope using QUAKE/W software, together with the laboratory measured soil parameters and earthquake records of the area. It was assumed that the deep crack in the slope under investigation was generated by earthquakes. A good agreement was obtained when comparing the location and the orientation of the cracks detected by Method-1 and Method-2. However, the simulated cracks in Method-3 were not in good agreement with the output of Method-1 and Method-2. This may have been due to the material properties used and the assumptions made, for the analysis. From Method-1 and Method-2, it can be concluded that the ERT method can be used to detect the location and orientation of a crack in a soil slope, when the ERT is conducted in very dry or very wet soil conditions. In this study, the cracks detected by the ERT were used for stability analysis of the slope. The stability of the slope was determined using the factor of safety (FOS) of a critical slip surface obtained by SLOPE/W using the limit equilibrium method. Pore-water pressure values for the stability analysis were obtained by coupling the transient seepage analysis of the slope using finite element based software, called SEEP/W. A parametric study conducted on the stability of an investigated slope revealed that the existence of deep cracks and their location in the soil slope are critical for its stability. The following two steps are proposed to predict the rain-induced instability of a residual soil slope with cracks. (a) Step-1: The transient stability analysis of the slope is conducted from the date of the investigation (initial conditions are based on the investigation) to the preferred date (current date), using measured rainfall data. Then, the stability analyses are continued for the next 12 months using the predicted annual rainfall that will be based on the previous five years rainfall data for the area. (b) Step-2: The stability of the slope is calculated in real-time using real-time measured rainfall. In this calculation, rainfall is predicted for the next hour or 24 hours and the stability of the slope is calculated one hour or 24 hours in advance using real time rainfall data. If Step-1 analysis shows critical stability for the forthcoming year, it is recommended that Step-2 be used for more accurate warning against the future failure of the slope. In this research, the results of the application of the Step-1 on an investigated slope (Slope-1) showed that its stability was not approaching a critical value for year 2012 (until 31st December 2012) and therefore, the application of Step-2 was not necessary for the year 2012. A case study (Slope-2) was used to verify the applicability of the complete proposed predictive method. A landslide event at Slope-2 occurred on 31st October 2010. The transient seepage and stability analyses of the slope using data obtained from field tests such as Bore-hole, SPT, ERT and Laboratory tests, were conducted on 12th June 2010 following the Step-1 and found that the slope in critical condition on that current date. It was then showing that the application of the Step-2 could have predicted this failure by giving sufficient warning time.

Quantifying alterations in cell migration : tracking fluorescently-tagged migrating cells by FACs and live-imaging

Cell migration is fundamental to many different physiological processes including embryonic development, inflammation and wound healing. Given the range and importance cell migration plays a number of assays have been developed to measure different aspects of cell migration. Here we describe two different methods to analyze cell migration. The first method analyzes the migration of fluorescently tagged cells using Boyden chambers and FACs and the second looks at migration properties using time-lapse microscopy.

Channel tracking in SDMA-based multi-user MIMO-OFDM communications systems

Capacity of current and future high data rate wireless communications depend significantly on how well changes in the wireless channel are predicted and tracked. Generally, this can be estimated by transmitting known symbols. However, this increases overheads if the channel varies over time. Given today’s bandwidth demand and the increased necessity for mobile wireless devices, the contributions of this research are very significant. This study has developed a novel and efficient channel tracking algorithm that can recursively update the channel estimation for wireless broadband communications reducing overheads, therefore increasing the speed of wireless communication systems.

Queensland’s high risk foot database : tracking the length and width of Queensland’s foot ulcers

Background Foot ulcers are a leading cause of avoidable hospital admissions and lower extremity amputations. However, large clinical studies describing foot ulcer presentations in the ambulatory setting are limited. The aim of this descriptive observational paper is to report the characteristics of ambulatory foot ulcer patients managed across 13 of 17 Queensland Health & Hospital Services. Methods Data on all foot ulcer patients registered with a Queensland High Risk Foot Form (QHRFF) was collected at their first consult in 2012. Data is automatically extracted from each QHRFF into a Queensland high risk foot database. Descriptive statistics display age, sex, ulcer types and co-morbidities. Statewide clinical indicators of foot ulcer management are also reported. Results Overall, 2,034 people presented with a foot ulcer in 2012. Mean age was 63(±14) years and 67.8% were male. Co-morbidities included 85% had diabetes, 49.7% hypertension, 39.2% dyslipidaemia, 25.6% cardiovascular disease, 13.7% kidney disease and 12.2% smoking. Foot ulcer types included 51.6% neuropathic, 17.8% neuro-ischaemic, 7.2% ischaemic, 6.6% post-surgical and 16.8% other; whilst 31% were infected. Clinical indicator results revealed 98% had their wound categorised, 51% received non-removable offloading, median ulcer healing time was 6-weeks and 37% had ulcer recurrence. Conclusion This paper details the largest foot ulcer database reported in Australia. People presenting with foot ulcers appear predominantly older, male with several co-morbidities. Encouragingly it appears most patients are receiving best practice care. These results may be a factor in the significant reduction of Queensland diabetes foot-related hospitalisations and amputations recently reported.

Local stress-strain distribution and load transfer across cartilage matrix at micro-scale using combined microscopy-based finite element method

Articular cartilage is the load-bearing tissue that consists of proteoglycan macromolecules entrapped between collagen fibrils in a three-dimensional architecture. To date, the drudgery of searching for mathematical models to represent the biomechanics of such a system continues without providing a fitting description of its functional response to load at micro-scale level. We believe that the major complication arose when cartilage was first envisaged as a multiphasic model with distinguishable components and that quantifying those and searching for the laws that govern their interaction is inadequate. To the thesis of this paper, cartilage as a bulk is as much continuum as is the response of its components to the external stimuli. For this reason, we framed the fundamental question as to what would be the mechano-structural functionality of such a system in the total absence of one of its key constituents-proteoglycans. To answer this, hydrated normal and proteoglycan depleted samples were tested under confined compression while finite element models were reproduced, for the first time, based on the structural microarchitecture of the cross-sectional profile of the matrices. These micro-porous in silico models served as virtual transducers to produce an internal noninvasive probing mechanism beyond experimental capabilities to render the matrices micromechanics and several others properties like permeability, orientation etc. The results demonstrated that load transfer was closely related to the microarchitecture of the hyperelastic models that represent solid skeleton stress and fluid response based on the state of the collagen network with and without the swollen proteoglycans. In other words, the stress gradient during deformation was a function of the structural pattern of the network and acted in concert with the position-dependent compositional state of the matrix. This reveals that the interaction between indistinguishable components in real cartilage is superimposed by its microarchitectural state which directly influences macromechanical behavior.

The velocity synchronous discrete Fourier transform for order tracking in the field of rotating machinery

Diagnostics of rotating machinery has developed significantly in the last decades, and industrial applications are spreading in different sectors. Most applications are characterized by varying velocities of the shaft and in many cases transients are the most critical to monitor. In these variable speed conditions, fault symptoms are clearer in the angular/order domains than in the common time/frequency ones. In the past, this issue was often solved by synchronously sampling data by means of phase locked circuits governing the acquisition; however, thanks to the spread of cheap and powerful microprocessors, this procedure is nowadays rarer; sampling is usually performed at constant time intervals, and the conversion to the order domain is made by means of digital signal processing techniques. In the last decades different algorithms have been proposed for the extraction of an order spectrum from a signal sampled asynchronously with respect to the shaft rotational velocity; many of them (the so called computed order tracking family) use interpolation techniques to resample the signal at constant angular increments, followed by a common discrete Fourier transform to shift from the angular domain to the order domain. A less exploited family of techniques shifts directly from the time domain to the order spectrum, by means of modified Fourier transforms. This paper proposes a new transform, named velocity synchronous discrete Fourier transform, which takes advantage of the instantaneous velocity to improve the quality of its result, reaching performances that can challenge the computed order tracking.