OpenUCT :: Browsing by Author "Reddy, B Daya"

Browsing by Author "Reddy, B Daya"

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Open Access
A priori error analysis of virtual element method for contact problem
(Springer International Publishing, 2022-04-01) Wang, Fei; Reddy, B Daya
As an extension of the finite element method, the virtual element method (VEM) can handle very general polygonal meshes, making it very suitable for non-matching meshes. In (Wriggers et al. in Comput. Mech. 58:1039–1050, 2016), the lowest-order virtual element method was applied to solve the contact problem of two elastic bodies on non-matching meshes. The numerical experiments showed the robustness and accuracy of the virtual element scheme. In this paper, we establish a priori error estimate of the virtual element method for the contact problem and prove that the lowest-order VEM achieves linear convergence order, which is optimal.
Open Access
Computational modelling of cardiac function and myocardial infarction
(2012) MBewu, James; Reddy, B Daya; Skatulla, Sebastian
Cardiovascular disease is a leading cause of death in South Africa. In particular non-fatal myocardial infarction is a key determinant for future cardiac failure due to adverse remodelling and electrophysiological dysfunction. Computational modelling of the electrophysiology and mechanics of the heart can provide useful insights into the causes of cardiac failure and the efficacy of treatments designed to combat myocardial infarction. A computational model of the healthy and infarcted left ventricle of a rat was developed using the eikonal diffusion equation to simulate the electrophysiology; a continuum mechanical model incorporating a passive mechanical model of Usyk to describe the nonlinear, anisotropic and nearly compressible nature of cardiac tissue; and an active stress model of Guccione to model the contraction of cardiac tissue. Boundary conditions modelling the blood pressure on the heart wall were applied to simulate the cardiac cycle.
Open Access
A computational neuromuscular model of the human upper airway with application to the study of obstructive sleep apnoea
(2014) Pelteret, Jean-Paul; Reddy, B Daya
Numerous challenges are faced in investigations aimed at developing a better understanding of the pathophysiology of obstructive sleep apnoea. The anatomy of the tongue and other upper airway tissues, and the ability to model their behaviour, is central to such investigations. In this thesis, details of the construction and development of a three-dimensional finite element model of soft tissues of the human upper airway, as well as a simplified fluid model of the airway, are provided. The anatomical data was obtained from the Visible Human Project, and its underlying micro-histological data describing tongue musculature were also extracted from the same source and incorporated into the model. An overview of the mathematical models used to describe tissue behaviour, both at a macro- and microscopic level, is given. Hyperelastic constitutive models were used to describe the material behaviour, and material incompressibility was accounted for. An active Hill three-element muscle model was used to represent the muscular tissue of the tongue. The neural stimulus for each muscle group to a priori unknown external forces was determined through the use of a genetic algorithm-based neural control model. The fundamental behaviour of the tongue under gravitational and breathing-induced loading is investigated. The response of the various muscles of the tongue to the complex loading developed during breathing is determined, with a particular focus being placed to that of the genioglossus. It is demonstrated that, when a time-dependent loading is applied to the tongue, the neural model is able to control the position of the tongue and produce a physiologically realistic response for the genioglossus. A comparison is then made to the response determined under quasi-static conditions using the pressure distribution extracted from computational fluid-dynamics results. An analytical model describing the time-dependent response of the components of the tongue musculature most active during oral breathing is developed and validated. It is then modified to simulate the activity of the tongue during sleep and under conditions relating to various possible neural and physiological pathologies. The retroglossal movement of the tongue resulting from the pathologies is quantified and their role in the potential to induce airway collapse is discussed.
Open Access
Computational simulation of bone remodelling post reverse total shoulder arthroplasty
(2017) Liedtke, Helen; McBride, Andrew Trevor; Reddy, B Daya
Bone is a living material. It adapts, in an optimal sense, to loading by changing its density and trabeculae architecture - a process termed remodelling. Implanted orthopaedic devices can significantly alter the loading on the surrounding bone. In addition, these devices rely on bone ingrowth to ensure secure implant fixation. In this project, a computational model that accounts for bone remodelling is developed and used to elucidate the response of bone following a reverse shoulder procedure. The reverse shoulder procedure investigated here is for rotary cuff deficient patients. In this procedure up to 75 % complications are reported in some clinical series. It is therefore necessary, for the design of successful implants, to understand the loading environment to promote bone growth in the correct areas. The physical process of remodelling is modelled using continuum scale, open system thermodynamics whereby the density of bone evolves isotropically in response to the loading it experiences. The fully-nonlinear continuum theory is solved approximately using the finite element method. The finite element library AceGEN forms the basis for the implementation. Several benchmark problems were implemented to validate the code and demonstrate features of the theory. These include several one-dimensional problems, the classical two-dimensional femur benchmark, and a series of three-dimensional examples. The three-dimensional examples include different loading scenarios on a rectangular block, as well as the investigation of the ASTM testing procedure of the glenoid side prosthesis implanted in a polyurethane foam block. The results clearly demonstrate the adaptive behaviour of the bone density in response to the magnitude and duration of the loading. The numerical implementation is also shown to be robust. The remodelling of the scapula post reverse shoulder arthroplasty is then investigated. A statistical shape model of the scapula was obtained from collaborators in the Division of Biomedical Engineering at the University of Cape Town. The finite element model was used to determine the density distribution in the scapula prior to surgery. A virtual surgery was then performed. The resulting geometry provides the input for the pre-processing phase of the post reverse shoulder arthroplasty model. The loading conditions for the reverse shoulder were provided by collaborators in the Division of Biomedical Engineering and the Leon Root Motion Analysis Laboratory at the Hospital for Special Surgery in New York City. The maximal loading condition at 90° abduction is used as the input for the simulation. It was found that the density increases in the vicinity of the screws, where the maximum stresses are concentrated, however, bone resorption is observed directly below and adjacent to the implant. No conclusive statement can be made, however, as only one loading scenario is considered and calibration of the model against experimental results is still outstanding. A unique feature of the code is that the upper and lower bounds of the density do not have to be enforced directly, as done in most bone remodelling theories in the literature. Rather, the bounds of the density are naturally enforced by calibrating the mass flux for the problem at hand. This project lays out the groundwork for a sound remodelling code, which can serve as a predictive tool in the field of orthopaedics.
Open Access
Dense granular flow in rotating drums: a computational investigation of constitutive equations
(2018) Povall, Timothy Mark; Govender, Indresan; McBride, Andrew; Reddy, B Daya
The constitutive laws of dense granular flow are investigated. Simulations of a drum, with periodic boundary conditions, rotating at varying speeds are performed. From the resulting data, kinematic and kinetic fields are extracted and used to investigate the validity of constitutive relations proposed in the literature. Two key constitutive assumptions are (a) isotropy and (b) incompressibility. The rotating drum system is found to be largely isotropic for high rotational speeds. For low rotational speeds, anisotropy is observed in the bottom part of the system, where the particles are flowing upwards. A small degree of compressibility is observed in the downward-flowing layer. The friction coefficient for the granular constitutive relations is also investigated. An empirically-derived friction law has a better fit to the data when compared to other friction laws proposed in the literature. Lastly, two scaling laws are investigated: the scaling between the scaled flow-rate (flux) and the thickness of the downward- flowing layer and the scaling between the dynamic angle of repose of the bed and the flux through the downward- flowing layer. The thickness-flux scaling is measured by interpolating the flux over a number of slices through the flowing layer, this is done in a number of different ways. The size of the measured section through the flowing layer is varied. The orientation of the slices is also varied. Also investigated is whether the total velocity or the tangential velocity produce the same scaling. The size of the section of the flowing layer significantly changes the scaling, this shows that the scaling is not constant throughout the flowing layer. The dynamic angle of repose is determined using two methods, one which is determined unambiguously as the repose angle of the ellipse fitted to the equilibrium surface and the other which is the changing angle of the tangent to the equilibrium surface or free surface. The first repose angle is found to be highly dependent on the flux even in the limit of infinite drum length, which is modelled using axial periodic boundary conditions. The second definition results in two sets of repose angles with complex behaviour that may be due to inertial effects. An instability in the system is observed, this is conjectured to be due to a frictional threshold that is breached as the rotational speed of the drum increases. Algorithms for calculating field variables and features of the charge are presented.
Open Access
Development of a numerical tool for the optimisation of vascular prosthesis towards physiological compliance
(2007) Van der Merwe, Helena; Franz, Thomas; Reddy, B Daya; Zilla, Peter
It has been proposed that if a vascular prosthesis is to more closely approximate the mechanical behaviour of a native vessel, it should similarly feature a multi-component structure. One of the components could be a metal support structure, similar to an endovascular stent. The objective of the project was to develop a numerical tool, using the Finite Element Method (FEM) to aid in the development and optimization of such a metallic support structure. This tool was used to simulate the behaviour of different designs under the simulated in vivo conditions. The numerical results of the predicted mechanical behaviour are then analysed.
Open Access
Development of a patient-specific finite element model of the transcatheter aortic valve implantation (TAVI) procedure
(2016) Shirzadi, Mohammad Mehdi; Reddy, B Daya
Transcatheter Aortic Valve Implantation (TAVI) is a procedure developed for replacing the defective aortic valve of a patient as an alternative to open heart Surgical Aortic Valve Replacement (SAVR). In the TAVI procedure a prosthetic valve, which is assembled on to a stent, is crimped and delivered to the patient's aortic root site through several available percutaneous means. The percutaneous nature of TAVI, which is its core advantage in comparison to other SAVR procedures, can however also be its main disadvantage. This is due to lack of direct access to the calcified leaflets, and hence reliance on the host tissue for the proper positioning and anchorage of the deployed prosthetic valve. Therefore, it is desired to have a preoperative quantitative understanding of patient-specific biomechanical interaction of the stent and the native valve to be able to maximise the chance of success of the procedure. The aim of this study was to develop a patient-specific Finite Element (FE) model of the Transcatheter Aortic Valve Implantation (TAVI) procedure for two patients, using a model of the 23 mm percutaneous prosthetic aortic valve developed by Strait Access Technologies (SAT), for the purpose of its post-operative performance. In this regard, the image processing software ScanIP was used to extract the 3D models of the patient-specific aortic roots and leaflets from the provided Multi-Slice Computer Tomography (MSCT) images of the patients. An anisotropic hyperelastic material model was implemented for the roots and leaflets, using two and one families of collagen fibres for their tissues respectively. The stent is made of a cobalt-chromium alloy and its mechanical response was modelled as an isotropic elastoplastic material, with a linear elastic initial response, followed by plastic behaviour with isotropic hardening. The prosthetic leaflets are made of polymer and were modelled as an isotropic hyperelastic material, using the provided experimental test data. The results for the first patient showed that the stent maintained its structural integrity after deployment, and successfully pushed the native leaflets back to keep the aortic root clear of all impediments. No obstruction of the coronary ostia was observed, and prosthetic leaflets were seen to function normally. The stent radial recoil was calculated to be between 2 to 4.28 % after deployments. Its foreshortening was calculated to be approximately 20%. The stent was observed to move back and forth by approximately 3 mm in the last simulation step in which cardiac cycle pressure were applied to the aortic root and prosthetic leaflets. Also, two openings were observed between the stent and aortic root wall during this simulation step, which indicates the possibility of paravalvular leakage. From the second patient simulation, it was observed that the 23 mm stent was not a good choice for this patient, and will cause severe damage or tissue tearing. The maximum principal stress in the aortic root and valve tissues were observed to follow approximately the defined collagen fibre directions.
Open Access
Direct numerical simulation of bubble-bubble and droplet-droplet interaction using a Surface Thin Film model
(2016) Musehane, Ndivhuwo M; Reddy, B Daya; Oxtoby, Oliver Francis
This dissertation deals with the simulation of dispersed multiphase flow. The particle-particle and particle-fluid interactions in this class of flows play an important role on the hydrodynamics and fluid transport phenomena that govern the overall flow behaviour. Accurate computational modelling of the particle-particle and particle- fluid interactions is thus required to correctly model the flow. The aim of this study is to use a Direct Numerical Simulation approach based on a smoothed Volume Of Fluid method to model particle-particle interactions in a dispersed multiphase flow at a fundamental level, and employing a surface thin film model, to drastically reduce the computational effort required. A multiscale modelling approach is followed with the smoothed Volume Of Fluid simulation on the particle scale and the surface thin film model simulation on the thin- film scale. The resulting governing equations are the Navier-Stokes equations for an incompressible viscous multiphase Newtonian fluid undergoing laminar and isothermal three-dimensional flow, the interface advection equation and the reduced order surface thin film equation. The model equations are discretized using the Finite Volume Method and implemented into the open source software OpenFOAM®. The numerical solution is obtained by solving the resulting non-linear system of equations implicitly on a structured computational grid on parallel processors using a pressure correction algorithm to converge the pressure at each time step. The study is restricted to gas-liquid systems where particles could either be bubbles or droplets; rigid particles are not considered. The model is tested against experimental results from binary collision of hydrocarbon droplets. Good qualitative numerical results are obtained at a practical computational cost.
Open Access
The effect of viscoelastic deformation in pipe cracks on leakage response to variations in pressure
(2014) Ssozi, Eva Nantongo; Van Zyl, J E; Reddy, B Daya
Water is an important and increasingly scarce resource in the world today. Unfortunately, a lot of water is lost through leakage since most distribution systems are deteriorating. Therefore research in leakage management is necessary in order to improve the utilization of water resources. Leakage may be reduced by managing the water pressure in water distribution systems. One of the important factors affecting the pressure-leakage relationship is pipe material behaviour (Van Zyl & Clayton 2007; Greyvenstein & Van Zyl 2007). The pressure – leakage relationship has been described by several relationships such as the Torricelli equation, the Fixed and Variable Area Discharge (FAVAD) concept and the conventional equation. Pipe material behaviour affects leakage parameters in the pressure-leakage relationship such as the leak area and the leak exponent (Cassa et al. 2010). For this project, the pressure-leakage relationships in High Density Polyethylene (HDPE) and Polyvinylchloride (PVC) pipes are investigated. HDPE and PVC are polymeric materials and therefore exhibit a viscoelastic response to applied stress and applied strain. Viscoelastic responses include creep, relaxation, hysteresis and time dependency. When these pipes experience stresses due to water pressures, failure and fracture may occur as leaks. The viscoelastic properties of these materials therefore affect how the leaks respond to pressure change. The effect of viscoelastic deformations in leaks was investigated using Finite Element Analysis (FEA) software, Abaqus. Round holes and longitudinal cracks were represented as individual leaks in HDPE and PVC pipes in Abaqus. Pressure was applied to each pipe model for different time periods, and the deformed leak areas were obtained. Further analysis was carried out to determine the effects of pressure on leak parameters such as the gradient of the leak area-pressure relationship, leak exponent and the leakage number. The analysis shows that viscoelastic deformations have an effect on the pressure-leakage relationship. A linear relationship exists between the leak area and pressure for all time periods investigated and therefore gradients could be obtained. Deformed leak areas, gradients and leak exponents all increased with time and therefore confirmed that the time dependency of viscoelastic materials affected the pressure-leakage relationship. The leakage exponents for both materials were found to vary between 0.5 and 1.5 for both HDPE and PVC. HDPE also exhibited higher leak exponents, gradients and larger deformed leak areas than PVC for the same leaks. It was also found that leakage in viscoelastic materials may be analysed using the leakage number, developed for elastically deforming materials by Van Zyl & Cassa (2013).
Open Access
Efficient and robust partitioned solution schemes for fluid-structure interactions
(2015) Bogaers, Alfred Edward Jules; Reddy, B Daya; Kok, Schalk; Franz, Thomas
In this thesis, the development of a strongly coupled, partitioned fluid-structure interactions (FSI) solver is outlined. Well established methods are analysed and new methods are proposed to provide robust, accurate and efficient FSI solutions. All the methods introduced and analysed are primarily geared towards the solution of incompressible, transient FSI problems, which facilitate the use of black-box sub-domain field solvers. In the first part of the thesis, radial basis function (RBF) interpolation is introduced for interface information transfer. RBF interpolation requires no grid connectivity information, and therefore presents an elegant means by which to transfer information across a non-matching and non-conforming interface to couple finite element to finite volume based discretisation schemes. The transfer scheme is analysed, with particular emphasis on a comparison between consistent and conservative formulations. The primary aim is to demonstrate that the widely used conservative formulation is a zero order method. Furthermore, while the consistent formulation is not provably conservative, it yields errors well within acceptable levels and converges within the limit of mesh refinement. A newly developed multi-vector update quasi-Newton (MVQN) method for implicit coupling of black-box partitioned solvers is proposed. The new coupling scheme, under certain conditions, can be demonstrated to provide near Newton-like convergence behaviour. The superior convergence properties and robust nature of the MVQN method are shown in comparison to other well-known quasi-Newton coupling schemes, including the least squares reduced order modelling (IBQN-LS) scheme, the classical rank-1 update Broyden's method, and fixed point iterations with dynamic relaxation. Partitioned, incompressible FSI, based on Dirichlet-Neumann domain decomposition solution schemes, cannot be applied to problems where the fluid domain is fully enclosed. A simple example often provided in the literature is that of balloon inflation with a prescribed inflow velocity. In this context, artificial compressibility (AC) will be shown to be a useful method to relax the incompressibility constraint, by including a source term within the fluid continuity equation. The attractiveness of AC stems from the fact that this source term can readily be added to almost any fluid field solver, including most commercial solvers. AC/FSI is however limited in the range of problems it can effectively be applied to. To this end, the combination of the newly developed MVQN method with AC/FSI is proposed. In so doing, the AC modified fluid field solver can continue to be treated as a black-box solver, while the overall robustness and performance are significantly improved. The study concludes with a demonstration of the modularity offered by partitioned FSI solvers. The analysis of the coupled environment is extended to include steady state FSI, FSI with free surfaces and an FSI problem with solid-body contact.
Open Access
Electromagnetic field solutions via the finite element method
(1986) Watkins, L R; Reddy, B Daya
This thesis examines the application of the finite element method to the solution of two important equations which govern electromagnetic fields, namely, the Poisson equation and the Helmholtz equation. These equations. together with appropriate boundary conditions, describe boundary value and eigenvalue problems respectively. Attention will be restricted to boundary value and eigenvalue problems on domains in R² in which more than one dielectric medium may be present. Weak or variational statements of these problems are derived, comprising the governing partial differential equation and appropriate boundary conditions. The Galerkin method is used to pose the variational statement of the problem in a finite-dimensional subspace and the finite element method employed to generate an appropriate set of basis functions which spans this subspace. A linear matrix or linear matrix eigenvalue problem results and suitable techniques for their numerical solution are presented. Various corrputat ional aspects of the finite element method and the solution techniques are discussed. Finite element programs capable of running on a microcomputer are developed and are used to analyse a number of electromagnetic field problems. The results of these analyses are compared, where possible, with analytic solutions, and elsewhere with results obtained by other methods.
Open Access
Estimates for the rate of convergence of finite element approximations of the solution of a time-dependent variational inequality
(1993) Schroeder, Gregory C; Reddy, B Daya
The main aim of this thesis is to analyse two types of general finite element approximations to the solution of a time-dependent variational inequality. The two types of approximations considered are the following: 1. Semi-discrete approximations, in which only the spatial domain is discretised by finite elements; 2. fully discrete approximations, in which the spatial domain is again discretised by finite elements and, in addition, the time domain is discretised and the time-derivatives appearing in the variational inequality are approximated by backward differences. Estimates of the error inherent in the above two types of approximations, in suitable Sobolev norms, are obtained; in particular, these estimates express the rate of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and, where appropriate, in terms of the time step size k. In addition, the above analysis is preceded by related results concerning the existence and uniqueness of the solution to the variational inequality and is followed by an application in elastoplasticity theory.
Open Access
Existence and stability of solutions to the equations of fibre suspension flows
(1999) Munganga, Justin Manango Wazute; Reddy, B Daya
A popular approach to formulating the initial-boundary value problem for fibre suspension flows is that in which fibre orientation is accounted for in an averaged sense, through the introduction of a second-order orientation tensor A. This variable, together with the velocity and pressure, then constitutes the set of unknown variables for the problem. The governing equations are balance of linear momentum, the incompressibility condition, an evolution equation for A, and a constitutive equation for the stress. The evolution equation contains a fourth-order orientation tensor A, and it is necessary to approximate A as a function of A, through a closure relation. The purpose of this these is to examine the well-posedness of the equations governing fibre fibre suspension ﬂows, for various closure relations. It has previously been shown by GP Galdi and BD Reddy that, for the linear closure, the problem is wellposed provided that the particle number, a material constant, is less than a critical value. The work by Galdi and Reddy made of a model in which rotary diffusivity is a function of the ﬂow. This thesis re-examines these issues in two different ways. First, the second law of thermodynamics is used to establish the constraints that the constitutive equations have to satisfy in order to be compatible with this law. This investigation is carried out for a variety of closure rules. The second contribution of the thesis concerns the existence and uniqueness of solutions to the governing equations, for the linear and quadratic closures; for a model in which the rotary diffusivity is treated as a constant, local and global existence of solutions are established, for sufficiently small data, and in the case of the linear closure, for admissible values of the particle number. The existence theory uses a Schauder fixed point approach.
Open Access
Finite element method using vector finite elements applied to eddy current problems
(2011) Adams, Leila; Reddy, B Daya; Wilkinson, Andrew John
Vector fields found in electromagnetics are fundamentally different to vector fields found in other research areas such as structural mechanics. Electromagnetic vector fields possess different physical behaviour patterns and different properties in comparison to the other vector fields and therein lies the necessity of the development of a finite element which would be able to cater for these differences . The vector finite element was then developed and used within the finite element method specifically for the approximation of electromagnetic problems. This dissertation investigates the partial differential equation that governs eddy current behaviour. A finite element algorithm is coded and used to solve this partial differential equation and produce vector field simulations for fundamental eddy current problems.
Open Access
A finite strain theory of elastoplasticity and its application to wave propagation
(1993) Gültop, Tekin; Reddy, B Daya
A constitutive theory of finite strain plasticity is developed by using the methods of convex analysis. The theory abstracts and extends the classical assumptions of a convex region of admissible stresses, and the normality law. The overall effects of plastic behaviour are contained in the theory through the presence of one or more internal variables. The thermodynamic restrictions of the second law together with the use of results of convex analysis lead in a natural way to the evolution equation or flow law. Non-smooth yield surfaces are included in the theory; nevertheless, the form of this theory makes a study of propagation of singular surfaces awkward. With a view to carrying out such a study, an alternative means of treating non-smooth convex yield surfaces is developed. This alternative theory is essentially a synthesis of the theory of Sewell, and that presented earlier in the thesis. The theory of singular surfaces is reviewed in the context of finite strain elastoplasticity, and necessary conditions for the propagation of acceleration waves are derived. A comparison of elastic and plastic wave speeds is made, and inequalities similar to those of Mandel for the small-strain case are derived. The propagation conditions for principal waves in both longitudinal and transverse directions, and the corresponding wave speeds, are found and compared for solids obeying a neo-Hookean elastic law, and with either the von Mises or Tresca yield criteria.
Open Access
Fluid-structure interaction modelling of a patient-specific arteriovenous access fistula
(2016) Guess, Winston; Reddy, B Daya; McBride, Andrew Trevor
This research forms part of an interdisciplinary project that aims to improve the detailed understanding of the haemodynamics and vascular mechanics in arteriovenous shunts that are required for haemodialysis treatments. A combination of new PCMRA imaging and computational modelling of in vivo blood flow aims to determine the haemodynamic conditions that may lead to the high failure rate of vascular access in these circumstances. This thesis focuses on developing a patient-specific fluid-structure interaction (FSI) model of a PC-MRA imaged arteriovenous fistula. The numerical FSI model is developed and simulated within the commercial multiphysics simulation package ANSYS® Academic Research, Release 16. The blood flow is modelled as a Newtonian fluid with the finite-volume method solver ANSYS® Fluent®. A pulsatile mass-flow boundary condition is applied at the artery inlet and a three-element Windkessel model at the artery and vein outlets. ANSYS® Mechanical™, a finite element method solver, is used to model the nonlinear behaviour of the vessel walls. The artery and vein walls are assumed to follow a third-order Yeoh model, and are differentiated by thickness and by material strength characteristics. The staggered FSI model is configured and executed in ANSYS® Workbench™, forming a semi-implicit coupling of the blood flow and vessel wall models. This work shows the effectiveness of combining a number of stabilisation techniques to simultaneously overcome the added-mass effect and optimise the efficiency of the overall model. The PC-MRA data, fluid model, and FSI model show almost identical flow features in the fistula; this applies in particular to a flow recirculation region in the vein that could potentially lead to fistula failure.
Open Access
Formulation and implementation of conforming finite element approximations to static and eigenvalue problems for thin elastic shells
(1987) Eve, Robin Andrew; Reddy, B Daya
In deriving asymptotic error estimates for a conforming finite element analyses of static thin elastic shell problems, the French mathematician Ciarlet (1976) proposed an approach to the formulation of such problems. The formulation he uses is based on classical shell theory making use of Kirchhoff-Koiter assumptions. The shell problem is posed in two-dimensional space to which the real problem, in three-dimensional space, is related by a mapping of the domain of the problem to the shell mid-surface. The finite element approximation is formulated in terms of the covariant components of the shell mid-surface displacement field. In this study, Ciarlet's formulation is extended to include the eigenvalue problem for the shell. In addition to this, the aim of the study is to obtain some indication of how well this approach might be expected to work in practice. The conforming finite element approximation of both the static and eigenvalue problems are implemented. Particular attention is paid to allowing generality of the shell surface geometry through the use of an approximate mapping. The use of different integration rules, in-plane displacement component interpolation schemes and approximate geometry schemes are investigated. Results are presented for shells of different geometries for both static and eigenvalue analyses; these are compared with independently obtained results.
Open Access
Formulation, analysis and solution algorithms for a model of gradient plasticity within a discontinuous Galerkin framework
(2008) McBride, Andrew Trevor; Reddy, B Daya
An investigation of a model of gradient plasticity in which the classical von Mises yield function is augmented by a term involving the Laplacian of the equivalent plastic strain is presented. The theory is developed within the framework of non-smooth convex analysis by exploiting the equivalence between the primal and dual expressions of the plastic deformation evolution relations. The nonlocal plastic evolution relations for the case of gradient plasticity are approximated using a discontinuous Galerkin finite element formulation. Both the small- and finite-strain theories are investigated. Considerable attention is focused on developing a firm mathematical foundation for the model of gradient plasticity restricted to the infinitesimal-strain regime. The key contributions arising from the analysis of the classical plasticity problem and the model of gradient plasticity include demonstrating the consistency of the variational formulation, and analyses of both the continuous-in-time and fully-discrete approximations; the error estimates obtained correspond to those for the conventional Galerkin approximations of the classical problem. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution for both hardening and softening problems. It is well known that classical finite element method simulations of softening problems are pathologically dependent on the discretisation.
Open Access
The general continuum model for structured populations, with two case studies in plant ecology
(1994) Laurie, Henri De Guise; Reddy, B Daya; Cowling, Richard M
The broad aim of this thesis is to investigate the formulation and usefulness of a very general model for plant population dynamics. In chapter 1, the goal of generality is discussed, particularly in the light of the lack of interaction between field and experimental population studies on the one hand and theoretical population dynamics on the other hand. A distinction is ma.de between descriptive and axiomatic theories, and it is suggested that they serve different purposes. The advantages of a. rigorous framework are pointed out and the basic elements of the continuum approach are introduced. In chapter 2, the model is proposed, the existence and uniqueness of solutions to its equations is proved, and an algorithm for numerically -approximating transient solutions is discussed. The question of generality is addressed in two places, and it is argued that the basic framework presented here is in principle adequate to model the processes of plant population dynamics in full detail, though the existence proof cannot to accommodate all possible models. In particular, models with time lags are excluded. Further limitations of the existence proof ill terms of constitutive relations are pointed out. In consequence, the theory here presented does not fully exploit the possibilities for generality inherent in the basic equations. In chapter 3, the question of what data would allow identification of factors determining somatic growth and mortality is investigated computationally. It is shown that using only the average size is insufficient. A class of models which includes all possible combinations of three types of size dependence in somatic growth and mortality is formulated. Qualitative parameter estimation for the various models yields size distributions that can be classified into the following biologically meaningful groups: group (i) has no models that use dependence on relative size; group (ii) has all the models in which somatic growth depends on relative size group (iii) has the models where only mortality depends on relative size. Thus it appears that size distribution may be used to distinguish various forms of size dependence in somatic growth and mortality. In chapter 4, a lottery model criterion for coexistence of plants with disjoint generations is developed, which is shown to require relative density dependence. Computer simulations aiming to initiate the use of exploratory calculations in studies of coexisting serotinous proteoids in fynbos indicate that the aspect of plant population dynamics most sensitive to density dependence is seed production, then somatic growth, while mortality is least sensitive to density dependence.
Open Access
The impact of thermophysical properties on nanofluid-based solar collector performance
(2016) Gakingo, Godfrey Kabungo; Reddy, B Daya; Macdevette, Michelle
Nanofluids are a novel class of heat transfer fluids in which nanoparticles are dispersed in traditional heat transfer fluids. They offer enhanced thermophysical, rheological and radiative properties. These enhancements have resulted in recent research being centred on the application of nanofluids to various systems. An example of such systems is the solar volumetric flow receiver in which great efficiency improvements have been reported. To explain this efficiency increase, researchers have evaluated the impact of enhanced radiative properties of nanofluids while largely neglecting that of enhanced thermophysical properties. This study looks at the impact of enhanced thermophysical properties on the performance of nanofluid-based solar volumetric receivers. Particular focus is drawn to the impact of temperature dependent conductivity and volumetric specific heat capacity. Copper oxide - water nanofluid is employed as its temperature dependent properties have been characterised. [Please note: this thesis file has been deferred until June 2016]