Creation of a validated 3D finite element model of an ovine tibia

Introduction Ovine models are widely used in orthopaedic research. To better understand the impact of orthopaedic procedures computer simulations are necessary. 3D finite element (FE) models of bones allow implant designs to be investigated mechanically, thereby reducing mechanical testing. Hypothesis We present the development and validation of an ovine tibia FE model for use in the analysis of tibia fracture fixation plates. Material & Methods Mechanical testing of the tibia consisted of an offset 3-pt bend test with three repetitions of loading to 350N and return to 50N. Tri-axial stacked strain gauges were applied to the anterior and posterior surfaces of the bone and two rigid bodies – consisting of eight infrared active markers, were attached to the ends of the tibia. Positional measurements were taken with a FARO arm 3D digitiser. The FE model was constructed with both geometry and material properties derived from CT images of the bone. The elasticity-density relationship used for material property determination was validated separately using mechanical testing. This model was then transformed to the same coordinate system as the in vitro mechanical test and loads applied. Results Comparison between the mechanical testing and the FE model showed good correlation in surface strains (difference: anterior 2.3%, posterior 3.2%). Discussion & Conclusion This method of model creation provides a simple method for generating subject specific FE models from CT scans. The use of the CT data set for both the geometry and the material properties ensures a more accurate representation of the specific bone. This is reflected in the similarity of the surface strain results.

Mechanical properties of thin sol-gel films

This thesis presents a study of the mechanical properties of thin films. The main aim was to determine the properties of sol-gel derived coatings. These films are used in a range of different applications and are known to be quite porous. Very little work has been carried out in this area and in order to study the mechanical properties of sol-gel films, some of the work was carried out on magnetron sputtered metal coatings in order to validate the techniques developed in this work. The main part of the work has concentrated on the development of various bending techniques to study the elastic modulus of the thin films, including both a small scale three-point bending, as well as a novel bi-axial bending technique based on a disk resting on three supporting balls. The bending techniques involve a load being applied to the sample being tested and the bending response to this force being recorded. These experiments were carried out using an ultra micro indentation system with very sensitive force and depth recording capabilities. By analysing the result of these forces and deflections using existing theories of elasticity, the elastic modulus may be determined. In addition to the bi-axial bending study, a finite element analysis of the stress distribution in a disk during bending was carried out. The results from the bi-axial bending tests of the magnetron sputtered films was confirmed by ultra micro indentation tests, giving information of the hardness and elastic modulus of the films. It was found that while the three point bending method gave acceptable results for uncoated steel substrates, it was very susceptible to slight deformations of the substrate. Improvements were made by more careful preparation of the substrates in order to avoid deformation. However the technique still failed to give reasonable results for coated specimens. In contrast, biaxial bending gave very reliable results even for very thin films and this technique was also found to be useful for determination of the properties of sol-gel coatings. In addition, an ultra micro indentation study of the hardness and elastic modulus of sol-gel films was conducted. This study included conventionally fired films as well as films ion implanted in a range of doses. The indentation tests showed that for implantation of H⁺ ions at doses exceeding 3x10¹⁶ ions/cm², the mechanical properties closely resembled those of films that were conventionally fired to 450°C.

Dynamics of pattern writing and erasure in photorefractive lithium niobate for data storage applications

The present rate of technological advance continues to place significant demands on data storage devices. The sheer amount of digital data being generated each year along with consumer expectations, fuels these demands. At present, most digital data is stored magnetically, in the form of hard disk drives or on magnetic tape. The increase in areal density (AD) of magnetic hard disk drives over the past 50 years has been of the order of 100 million times, and current devices are storing data at ADs of the order of hundreds of gigabits per square inch. However, it has been known for some time that the progress in this form of data storage is approaching fundamental limits. The main limitation relates to the lower size limit that an individual bit can have for stable storage. Various techniques for overcoming these fundamental limits are currently the focus of considerable research effort. Most attempt to improve current data storage methods, or modify these slightly for higher density storage. Alternatively, three dimensional optical data storage is a promising field for the information storage needs of the future, offering very high density, high speed memory. There are two ways in which data may be recorded in a three dimensional optical medium; either bit-by-bit (similar in principle to an optical disc medium such as CD or DVD) or by using pages of bit data. Bit-by-bit techniques for three dimensional storage offer high density but are inherently slow due to the serial nature of data access. Page-based techniques, where a two-dimensional page of data bits is written in one write operation, can offer significantly higher data rates, due to their parallel nature. Holographic Data Storage (HDS) is one such page-oriented optical memory technique. This field of research has been active for several decades, but with few commercial products presently available. Another page-oriented optical memory technique involves recording pages of data as phase masks in a photorefractive medium. A photorefractive material is one by which the refractive index can be modified by light of the appropriate wavelength and intensity, and this property can be used to store information in these materials. In phase mask storage, two dimensional pages of data are recorded into a photorefractive crystal, as refractive index changes in the medium. A low-intensity readout beam propagating through the medium will have its intensity profile modified by these refractive index changes and a CCD camera can be used to monitor the readout beam, and thus read the stored data. The main aim of this research was to investigate data storage using phase masks in the photorefractive crystal, lithium niobate (LiNbO3). Firstly the experimental methods for storing the two dimensional pages of data (a set of vertical stripes of varying lengths) in the medium are presented. The laser beam used for writing, whose intensity profile is modified by an amplitudemask which contains a pattern of the information to be stored, illuminates the lithium niobate crystal and the photorefractive effect causes the patterns to be stored as refractive index changes in the medium. These patterns are read out non-destructively using a low intensity probe beam and a CCD camera. A common complication of information storage in photorefractive crystals is the issue of destructive readout. This is a problem particularly for holographic data storage, where the readout beam should be at the same wavelength as the beam used for writing. Since the charge carriers in the medium are still sensitive to the read light field, the readout beam erases the stored information. A method to avoid this is by using thermal fixing. Here the photorefractive medium is heated to temperatures above 150�C; this process forms an ionic grating in the medium. This ionic grating is insensitive to the readout beam and therefore the information is not erased during readout. A non-contact method for determining temperature change in a lithium niobate crystal is presented in this thesis. The temperature-dependent birefringent properties of the medium cause intensity oscillations to be observed for a beam propagating through the medium during a change in temperature. It is shown that each oscillation corresponds to a particular temperature change, and by counting the number of oscillations observed, the temperature change of the medium can be deduced. The presented technique for measuring temperature change could easily be applied to a situation where thermal fixing of data in a photorefractive medium is required. Furthermore, by using an expanded beam and monitoring the intensity oscillations over a wide region, it is shown that the temperature in various locations of the crystal can be monitored simultaneously. This technique could be used to deduce temperature gradients in the medium. It is shown that the three dimensional nature of the recording medium causes interesting degradation effects to occur when the patterns are written for a longer-than-optimal time. This degradation results in the splitting of the vertical stripes in the data pattern, and for long writing exposure times this process can result in the complete deterioration of the information in the medium. It is shown in that simply by using incoherent illumination, the original pattern can be recovered from the degraded state. The reason for the recovery is that the refractive index changes causing the degradation are of a smaller magnitude since they are induced by the write field components scattered from the written structures. During incoherent erasure, the lower magnitude refractive index changes are neutralised first, allowing the original pattern to be recovered. The degradation process is shown to be reversed during the recovery process, and a simple relationship is found relating the time at which particular features appear during degradation and recovery. A further outcome of this work is that the minimum stripe width of 30 ìm is required for accurate storage and recovery of the information in the medium, any size smaller than this results in incomplete recovery. The degradation and recovery process could be applied to an application in image scrambling or cryptography for optical information storage. A two dimensional numerical model based on the finite-difference beam propagation method (FD-BPM) is presented and used to gain insight into the pattern storage process. The model shows that the degradation of the patterns is due to the complicated path taken by the write beam as it propagates through the crystal, and in particular the scattering of this beam from the induced refractive index structures in the medium. The model indicates that the highest quality pattern storage would be achieved with a thin 0.5 mm medium; however this type of medium would also remove the degradation property of the patterns and the subsequent recovery process. To overcome the simplistic treatment of the refractive index change in the FD-BPM model, a fully three dimensional photorefractive model developed by Devaux is presented. This model shows significant insight into the pattern storage, particularly for the degradation and recovery process, and confirms the theory that the recovery of the degraded patterns is possible since the refractive index changes responsible for the degradation are of a smaller magnitude. Finally, detailed analysis of the pattern formation and degradation dynamics for periodic patterns of various periodicities is presented. It is shown that stripe widths in the write beam of greater than 150 ìm result in the formation of different types of refractive index changes, compared with the stripes of smaller widths. As a result, it is shown that the pattern storage method discussed in this thesis has an upper feature size limit of 150 ìm, for accurate and reliable pattern storage.

The mechanical heterogeneity of the hard callus influences local tissue strains during bone healing : a finite element study based on sheep experiments

During secondary fracture healing, various tissue types including new bone are formed. The local mechanical strains play an important role in tissue proliferation and differentiation. To further our mechanobiological understanding of fracture healing, a precise assessment of local strains is mandatory. Until now, static analyses using Finite Elements (FE) have assumed homogenous material properties. With the recent quantification of both the spatial tissue patterns (Vetter et al., 2010) and the development of elastic modulus of newly formed bone during healing (Manjubala et al., 2009), it is now possible to incorporate this heterogeneity. Therefore, the aim of this study is to investigate the effect of this heterogeneity on the strain patterns at six successive healing stages. The input data of the present work stemmed from a comprehensive cross-sectional study of sheep with a tibial osteotomy (Epari et al., 2006). In our FE model, each element containing bone was described by a bulk elastic modulus, which depended on both the local area fraction and the local elastic modulus of the bone material. The obtained strains were compared with the results of hypothetical FE models assuming homogeneous material properties. The differences in the spatial distributions of the strains between the heterogeneous and homogeneous FE models were interpreted using a current mechanobiological theory (Isakson et al., 2006). This interpretation showed that considering the heterogeneity of the hard callus is most important at the intermediate stages of healing, when cartilage transforms to bone via endochondral ossification.

Mechanical properties of cold-formed steels at elevated temperatures

Mechanical properties have an important role in the fire safety design of cold-formed steel structures due to the rapid reduction in mechanical properties such as yield strength and elastic modulus under fire conditions and associated reduction to the load carrying capacities. Hence there is a need to fully understand the deterioration characteristics of yield strength and elastic modulus of cold-formed steels at elevated temperatures. Although past research has produced useful experimental data on the mechanical properties of cold-formed steels at elevated temperatures, such data do not yet cover different cold-formed steel grades and thicknesses. Therefore, an experimental study was undertaken to investigate the elevated temperature mechanical properties of two low and high strength steels with two thicknesses that are commonly used in Australia. Tensile coupon tests were undertaken using a steady state test method for temperatures in the range 20–700 °C. Test results were compared with the currently available reduction factors for yield strength and elastic modulus, and stress–strain curves, based on which further improvements were made. For this purpose, test results of many other cold-formed steels were also used based on other similar studies undertaken at the Queensland University of Technology. Improved equations were developed to predict the yield strength and elastic modulus reduction factors and stress–strain curves of a range of cold-formed steel grades and thicknesses used in Australia. This paper presents the results of this experimental study, comparisons with the results of past research and steel design standards, and the new predictive equations.

A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

We present a mass-conservative vertex-centred finite volume method for efficiently solving the mixed form of Richards’ equation in heterogeneous porous media. The spatial discretisation is particularly well-suited to heterogeneous media because it produces consistent flux approximations at quadrature points where material properties are continuous. Combined with the method of lines, the spatial discretisation gives a set of differential algebraic equations amenable to solution using higher-order implicit solvers. We investigate the solution of the mixed form using a Jacobian-free inexact Newton solver, which requires the solution of an extra variable for each node in the mesh compared to the pressure-head form. By exploiting the structure of the Jacobian for the mixed form, the size of the preconditioner is reduced to that for the pressure-head form, and there is minimal computational overhead for solving the mixed form. The proposed formulation is tested on two challenging test problems. The solutions from the new formulation offer conservation of mass at least one order of magnitude more accurate than a pressure head formulation, and the higher-order temporal integration significantly improves both the mass balance and computational efficiency of the solution.

Radiation effect on free convection laminar flow along a vertical flat plate with streamwise sinusoidal surface temperature

The effect of thermal radiation on a steady two-dimensional natural convection laminar flow of viscous incompressible optically thick fluid along a vertical flat plate with streamwise sinusoidal surface temperature has been investigated in this study. Using the appropriate variables; the basic governing equations are transformed to convenient form and then solved numerically employing two efficient methods, namely, Implicit finite difference method (IFD) together with Keller box scheme and Straight forward finite difference (SFFD) method. Effects of the variation of the physical parameters, for example, conduction-radiation parameter (Planck number), surface temperature parameter, and the amplitude of the surface temperature, are shown on the skin friction and heat transfer rate quantitatively are shown numerically. Velocity and temperature profiles as well as streamlines and isotherms are also presented and discussed for the variation of conduction-radiation parameter. It is found that both skin-friction and rate of heat transfer are enhanced considerably by increasing the values of conduction radiation parameter, Rd.

The effect of temperature dependent viscosity on MHD natural convection flow from an isothermal sphere

Laminar magnetohydrodynamic (MHD) natural convection flow from an isothermal sphere immersed in a fluid with viscosity proportional to linear function of temperature has been studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations are reduced to convenient form which are solved numerically by two very efficient methods, namely, (i) Implicit finite difference method together with Keller box scheme and (ii) Direct numerical scheme. Numerical results are presented by velocity and temperature distribution, streamlines and isotherms of the fluid as well as heat transfer characteristics, namely the local skin-friction coefficients and the local heat transfer rate for a wide range of magnetohydrodynamic paramagnet and viscosity-variation parameter.

Properties of lignin and poly(hydroxybutyrate) blends

The Queensland University of Technology (QUT) allows the presentation of a thesis for the Degree of Doctor of Philosophy in the format of published or submitted papers, where such papers have been published, accepted or submitted during the period of candidature. This thesis is composed of Seven published/submitted papers and one poster presentation, of which five have been published and the other two are under review. This project is financially supported by the QUTPRA Grant. The twenty-first century started with the resurrection of lignocellulosic biomass as a potential substitute for petrochemicals. Petrochemicals, which enjoyed the sustainable economic growth during the past century, have begun to reach or have reached their peak. The world energy situation is complicated by political uncertainty and by the environmental impact associated with petrochemical import and usage. In particular, greenhouse gasses and toxic emissions produced by petrochemicals have been implicated as a significant cause of climate changes. Lignocellulosic biomass (e.g. sugarcane biomass and bagasse), which potentially enjoys a more abundant, widely distributed, and cost-effective resource base, can play an indispensible role in the paradigm transition from fossil-based to carbohydrate-based economy. Poly(3-hydroxybutyrate), PHB has attracted much commercial interest as a plastic and biodegradable material because some its physical properties are similar to those of polypropylene (PP), even though the two polymers have quite different chemical structures. PHB exhibits a high degree of crystallinity, has a high melting point of approximately 180°C, and most importantly, unlike PP, PHB is rapidly biodegradable. Two major factors which currently inhibit the widespread use of PHB are its high cost and poor mechanical properties. The production costs of PHB are significantly higher than for plastics produced from petrochemical resources (e.g. PP costs $US1 kg-1, whereas PHB costs $US8 kg-1), and its stiff and brittle nature makes processing difficult and impedes its ability to handle high impact. Lignin, together with cellulose and hemicellulose, are the three main components of every lignocellulosic biomass. It is a natural polymer occurring in the plant cell wall. Lignin, after cellulose, is the most abundant polymer in nature. It is extracted mainly as a by-product in the pulp and paper industry. Although, traditionally lignin is burnt in industry for energy, it has a lot of value-add properties. Lignin, which to date has not been exploited, is an amorphous polymer with hydrophobic behaviour. These make it a good candidate for blending with PHB and technically, blending can be a viable solution for price and reduction and enhance production properties. Theoretically, lignin and PHB affect the physiochemical properties of each other when they become miscible in a composite. A comprehensive study on structural, thermal, rheological and environmental properties of lignin/PHB blends together with neat lignin and PHB is the targeted scope of this thesis. An introduction to this research, including a description of the research problem, a literature review and an account of the research progress linking the research papers is presented in Chapter 1. In this research, lignin was obtained from bagasse through extraction with sodium hydroxide. A novel two-step pH precipitation procedure was used to recover soda lignin with the purity of 96.3 wt% from the black liquor (i.e. the spent sodium hydroxide solution). The precipitation process is presented in Chapter 2. A sequential solvent extraction process was used to fractionate the soda lignin into three fractions. These fractions, together with the soda lignin, were characterised to determine elemental composition, purity, carbohydrate content, molecular weight, and functional group content. The thermal properties of the lignins were also determined. The results are presented and discussed in Chapter 2. On the basis of the type and quantity of functional groups, attempts were made to identify potential applications for each of the individual lignins. As an addendum to the general section on the development of composite materials of lignin, which includes Chapters 1 and 2, studies on the kinetics of bagasse thermal degradation are presented in Appendix 1. The work showed that distinct stages of mass losses depend on residual sucrose. As the development of value-added products from lignin will improve the economics of cellulosic ethanol, a review on lignin applications, which included lignin/PHB composites, is presented in Appendix 2. Chapters 3, 4 and 5 are dedicated to investigations of the properties of soda lignin/PHB composites. Chapter 3 reports on the thermal stability and miscibility of the blends. Although the addition of soda lignin shifts the onset of PHB decomposition to lower temperatures, the lignin/PHB blends are thermally more stable over a wider temperature range. The results from the thermal study also indicated that blends containing up to 40 wt% soda lignin were miscible. The Tg data for these blends fitted nicely to the Gordon-Taylor and Kwei models. Fourier transform infrared spectroscopy (FT-IR) evaluation showed that the miscibility of the blends was because of specific hydrogen bonding (and similar interactions) between reactive phenolic hydroxyl groups of lignin and the carbonyl group of PHB. The thermophysical and rheological properties of soda lignin/PHB blends are presented in Chapter 4. In this chapter, the kinetics of thermal degradation of the blends is studied using thermogravimetric analysis (TGA). This preliminary investigation is limited to the processing temperature of blend manufacturing. Of significance in the study, is the drop in the apparent energy of activation, Ea from 112 kJmol-1 for pure PHB to half that value for blends. This means that the addition of lignin to PHB reduces the thermal stability of PHB, and that the comparative reduced weight loss observed in the TGA data is associated with the slower rate of lignin degradation in the composite. The Tg of PHB, as well as its melting temperature, melting enthalpy, crystallinity and melting point decrease with increase in lignin content. Results from the rheological investigation showed that at low lignin content (.30 wt%), lignin acts as a plasticiser for PHB, while at high lignin content it acts as a filler. Chapter 5 is dedicated to the environmental study of soda lignin/PHB blends. The biodegradability of lignin/PHB blends is compared to that of PHB using the standard soil burial test. To obtain acceptable biodegradation data, samples were buried for 12 months under controlled conditions. Gravimetric analysis, TGA, optical microscopy, scanning electron microscopy (SEM), differential scanning calorimetry (DSC), FT-IR, and X-ray photoelectron spectroscopy (XPS) were used in the study. The results clearly demonstrated that lignin retards the biodegradation of PHB, and that the miscible blends were more resistant to degradation compared to the immiscible blends. To obtain an understanding between the structure of lignin and the properties of the blends, a methanol-soluble lignin, which contains 3× less phenolic hydroxyl group that its parent soda lignin used in preparing blends for the work reported in Chapters 3 and 4, was blended with PHB and the properties of the blends investigated. The results are reported in Chapter 6. At up to 40 wt% methanolsoluble lignin, the experimental data fitted the Gordon-Taylor and Kwei models, similar to the results obtained soda lignin-based blends. However, the values obtained for the interactive parameters for the methanol-soluble lignin blends were slightly lower than the blends obtained with soda lignin indicating weaker association between methanol-soluble lignin and PHB. FT-IR data confirmed that hydrogen bonding is the main interactive force between the reactive functional groups of lignin and the carbonyl group of PHB. In summary, the structural differences existing between the two lignins did not manifest itself in the properties of their blends.

Prediction of structural integrity, robustness and service life using advanced finite element methods

Corrosion is a common phenomenon and critical aspects of steel structural application. It affects the daily design, inspection and maintenance in structural engineering, especially for the heavy and complex industrial applications, where the steel structures are subjected to hash corrosive environments in combination of high working stress condition and often in open field and/or under high temperature production environments. In the paper, it presents the actual engineering application of advanced finite element methods in the predication of the structural integrity and robustness at a designed service life for the furnaces of alumina production, which was operated in the high temperature, corrosive environments and rotating with high working stress condition.

Effects of corrugation frequency and aspect ratio on natural convection within an enclosure having sinusoidal corrugation over a heated top surface

Natural convection of a two-dimensional laminar steady-state incompressible fluid flow in a modified rectangular enclosure with sinusoidal corrugated top surface has been investigated numerically. The present study has been carried out for different corrugation frequencies on the top surface as well as aspect ratios of the enclosure in order to observe the change in hydrodynamic and thermal behavior with constant corrugation amplitude. A constant flux heat source is flush mounted on the top sinusoidal wall, modeling a wavy sheet shaded room exposed to sunlight. The flat bottom surface is considered as adiabatic, while the both vertical side walls are maintained at the constant ambient temperature. The fluid considered inside the enclosure is air having Prandtl number of 0.71. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. The results in terms of isotherms, streamlines and average Nusselt numbers are obtained for the Rayleigh number ranging from 10^3 to 10^6 with constant physical properties for the fluid medium considered. It is found that the convective phenomena are greatly influenced by the presence of the corrugation and variation of aspect ratios.

Volatile properties of particles emitted by compressed natural gas and diesel buses during steady state and transient driving modes

Volatile properties of particle emissions from four compressed natural gas (CNG) and four diesel buses were investigated under steady state and transient driving modes on a chassis dynamometer. The exhaust was diluted utilising a full-flow continuous volume sampling system and passed through a thermodenuder at controlled temperature. Particle number concentration and size distribution were measured with a condensation particle counter and a scanning mobility particle sizer, respectively. We show that, while almost all the particles emitted by the CNG buses were in the nanoparticle size range, at least 85% and 98% were removed at 100ºC and 250ºC, respectively. Closer analysis of the volatility of particles emitted during transient cycles showed that volatilisation began at around 40°C with the majority occurring by 80°C. Particles produced during hard acceleration from rest exhibited lower volatility than that produced during other times of the cycle. Based on our results and the observation of ash deposits on the walls of the tailpipes, we suggest that these non-volatile particles were composed mostly of ash from lubricating oil. Heating the diesel bus emissions to 100ºC removed ultrafine particle numbers by 69% to 82% when a nucleation mode was present and just 18% when it was not.

Low temperature CO sensitive nanostructured WO3 thin films doped with Fe

Nanostructured tungsten oxide thin film based gas sensors have been developed by thermal evaporation method to detect CO at low operating temperatures. The influence of Fe-doping and annealing heat treatment on microstructural and gas sensing properties of these films have been investigated. Fe was incorporated in WO3 film by co-evaporation and annealing was performed at 400oC for 2 hours in air. AFM analysis revealed a grain size of about 10-15 nm in all the films. GIXRD analysis showed that as-deposited films are amorphous and annealing at 400oC improved the crystallinity. Raman and XRD analysis indicated that Fe is incorporated in the WO3 matrix as a substitutional impurity, resulting in shorter O-W-O bonds and lattice cell parameters. Doping with Fe contributed significantly towards CO sensing performance of WO3 thin films. A good response to various concentrations (10-1000 ppm) of CO has been achieved with 400oC annealed Fe-doped WO3 film at a low operating temperature of 150oC.

MHD natural convection flow from an isothermal horizontal circular cylinder under consideration of temperature dependent viscosity

Magnetohydrodynamic (MHD) natural convection laminar flow from an iso-thermal horizontal circular cylinder immersed in a fluid with viscosity proportional to a linear function of temperature will be discussed with numerical simulations. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equa-tions are reduced to convenient form, which are solved numerically by two very efficient methods, namely, (i) Implicit finite difference method together with Keller box scheme and (ii) Direct numerical scheme. Numerical results are presented by velocity and temperature distributions of the fluid as well as heat transfer characteristics, namely the shearing stress and the local heat transfer rate in terms of the local skin-friction coefficient and the local Nusselt number for a wide range of magnetohydrodynamic parameter, viscosity-variation parameter and viscous dissipation parameter. MHD flow in this geometry with temperature dependent viscosity is absent in the literature. The results obtained from the numerical simulations have been veri-fied by two methodologies.