Browsing by Author "Reddy, Daya"

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    A virtual element method for hyperelasticity
    (2021) van Huyssteen, Daniel; Reddy, Daya
    This thesis studies the approximation of plane problems of hyperelasticity, using a loworder virtual element method (VEM). The VEM is an extension of the finite element method (FEM). It is characterised by considerable freedom with regard to element geometry, permitting arbitrary polygonal and polyhedral elements in two and three dimensions respectively. Furthermore, the local basis functions are not known explicitly on elements and take the simple form of piecewise-linear Lagrangian functions on element boundaries. All integrations are performed on element edges. The VEM formulation typically involves a consistency term, computed via a projection, and a stabilization term, which must be approximated. Problems concerning isotropic and transversely isotropic hyperelastic material models are considered. Examples of transversely isotropic materials, which are characterised by an axis of symmetry normal to a plane of isotropy, range from simple fibre-reinforced materials to biological tissues. To date, in the context of hyperelasticity, investigation of the performance of VEM has primarily focused on problems involving the isotropic neo-Hookean material model. Furthermore, there has been limited investigation into the behaviour of the VEM in the nearly incompressible and nearly inextensible limits. In this thesis a VEM formulation with a novel approach to the construction of the stabilization term is formulated and implemented for problems involving isotropic and transversely isotropic hyperelastic materials. The governing equations of hyperelasticity are derived and various isotropic and transversely isotropic constitutive models are presented. This is followed by presentation of the virtual element formulation of the hyperelastic problem and a possible approach to its practical implementation. Through a range of numerical examples, the VEM with the proposed stabilization term is found to exhibit robust and accurate behaviour for a variety of mesh types, including those comprising highly non-convex element geometries, and for problems involving severe deformations. Furthermore, the versatility of the proposed VEM formulation is demonstrated through its application to a range of popular isotropic and transversely isotropic material models for a wide variety of material parameters. Through this investigation the VEM is found to exhibit locking-free behaviour in the limiting cases of near-incompressibility and near-inextensibility, both separately and combined.
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    Acceleration waves in constrained thermoelastic materials
    (1989) Bleach, Gordon Phillip; Reddy, Daya
    We study the propagation and growth of acceleration waves in isotropic thermoelastic media subject to a broad class of thermomechanical constraints. The work is based on an existing thermodynamic theory of constrained thermoelastic materials presented by Reddy (1984) for both definite and non- conductors, but we differ by adopting a new definition of a constrained non-conductor and by investigating the consequences of isotropy. The set of constraints considered is not arbitrary but is large enough to include most constraints commonly found in practice. We also extend Reddy's (1984) work by including consideration of sets of constraints for which a set of vectors associated with the constraints is linearly dependent. These vectors play a significant role in the propagation conditions and in the growth equations described below. Propagation conditions (of Fresnel-Hadamard type) are derived for both homothermal and homentropic waves, and solutions for longitudinal and transverse principal waves are discussed. The derivations involve the determination of jumps in the time derivative of constraint multipliers which are required in the solution of the corresponding growth equations, and it is found that these multipliers cannot be separately determined if the set of constraint vectors mentioned above is linearly dependent. This difficulty forces us to restrict the constraint set for which the growth equations for homothermal and homentropic waves can be derived. The growth of plane, cylindrical and spherical waves is considered and solutions are discussed, concentrating on the influence of the constraints on the results.
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    Finite element analysis of flows in secondary settling tanks
    (2002) Kleine, Dorothee; Reddy, Daya; Ekama, George
    Secondary settling tanks (SSTs) form a crucial part of wastewater treatment plants. Besides having to produce the separation of suspended solids and clarified effluent the secondary settling tank is used to concentrate and recycle the settled sludge to the biological reactor. The efficiency of the biological reactor in the waste water treatment system is determined by the efficiency of this final clarifying process. Hydrodynamic models have been developed for simulating secondary settling tanks in order to gain a better understanding of the complex flow patterns in these tanks, and to make design and optimization of the SST internal features possible. These models use mainly the finite volume method. This thesis is concerned with the development and implementation of a finite element approach to the simulation of flows in SSTs. Although it is nowadays also possible to realise an unstructured grid within the FVM, the power of the finite element method (FEM) lies in its higher flexibility in fitting irregular domains and in providing local grid refinement. Generally, unstructured mesh procedures with the FVM require essential, additional orthogonality corrections, which affect the accuracy of the solution, and these corrections increase the computational cost due to the additional computations and increased iteration requirements. Structured mesh discretization may offer significantly shorter computation time. The FEM is therefore convenient for handling arbitrarily shaped domains and adaptation of complex internal features of SSTs, such as inlet and outlet arrangements.
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    Locking-free discontinuous Galerkin methods for problems in elasticity, using linear and multilinear approximations
    (2013) Grieshaber, B J; Reddy, Daya
    With interior penalty discontinuous Galerkin methods well established as locking-free for lowo-rder triangular elements, and thus an effective alternative to the Standard Galerkin method for nearly incompressible materials, substantial numerical evidence in this work shows that this is not the case for quadrilateral elements. Direct comparisons to triangles illustrate the material dependence of three common interior penalty methods for bilinear quadrilaterals, with locking and other manifestations of poor approximations in the near-incompressible regime. To understand this discrepancy with a view to providing a remedy for the problem, an existing convergence analysis for triangles is looked at for possible extension to the case of quadrilaterals. This highlights the need for a suitable interpolant for the error-splitting approach of the proof. To rectify the problem manifesting in the numerical results, a modification to the formulation or elements themselves is necessary, and a preliminary analysis with bilinear elements, assuming the existence of a suitable interpolant with some basic properties, indicates two modifications as potential remedies: edge-term under-integration, and the use of linear rather than multilinear elements.
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    Mathematical simulation of dynamic behaviour of secondary settling tanks
    (1993) Ozinsky, Alison Emslie; Ekama, George A; Reddy, Daya
    The main objective of this thesis was to bring the theoretical and practical aspects of secondary settling tank developments closer together. This was achieved firstly, by evaluating and developing empirical relationships from which the flux theory constants may be derived from simpler sludge settleability measures; and secondly, by developing a computer model for the simulation of dynamic behaviour of full scale secondary settling tanks. The model was initially developed for and tested on laboratory scale data. It was then calibrated with full scale data and used to verify the flux theory by comparing the simulated predictions and the measured results. The simulations demonstrated that a calibrated and verified dynamic settling tank model based on the flux theory and incorporating various refinements such as turbulent diffusivity implicitly encompasses such features of secondary settling tank behaviour such as maximum underflow concentration and sludge storage concentration and capacity. It was concluded that the simulation program is an improvement on previous simulation programs based purely on the steady state flux theory and should be used as a starting point for developing design theories based on the flux theory.
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    The mechanical design aspects of a small diameter vascular prosthesis
    (1999) MacKellar, Iain Campbell; Starke, Greg; Reddy, Daya
    Failure of medium to small diameter vascular grafts is believed to be in part due to the compliance mismatch between the native artery and the implanted graft. Consequently, designers are examining the use of more compliant materials for their manufacture. Ether free polyurethanes are currently amongst the most popular materials for use in biological implants although these materials are inherently too stiff for use in vascular prostheses. These materials can be made more compliant by introducing porosity. Apart from creating a more compliant overall material, under optimal biological conditions, the porosity may lead to cell in growth through the thickness of the graft allowing an endothelial cell layer to form on the inner flow surface. Compliance and cell ingrowth are both important characteristics that determine the successful functioning of the graft. The current work is part of a collaborative venture with the Cardiovascular Research Unit (CVRU) at the University of Cape Town to design and develop a new polyurethane graft. Finite element models are used to facilitate stress analyses and to evaluate the long-term behaviour and compliance of various graft designs made from a bio-inert thermoplastic polyurethane. Material properties of the polyurethane are determined from uniaxial tension tests, simple-shear tests and viscoelastic shear tests. The constitutive equations for a compressible, large strain hyper elastic material model with viscoelasticity are implemented in the finite element code using material constants calculated from the test data. The behaviour of the finite element model is verified by using a single element test and comparing results to the material data. The finite element model is validated for use m more sophisticated problems by comparing axi-symmetric models with in vitro experiments. An artery/graft anastomosis is then analysed by modelling the artery as an incompressible hyperplastic material. Further more complex graft designs are analysed with internal growth channels and spiral reinforcing winds. Viscoelastic effects are also examined. The modelling method is discussed and important results are noted.
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    Numerical simulation of friction welding processes: An arbitrary Lagrangian-Eulerian approach
    (2022) Hamed, Maien Mohamed Osman; Reddy, Daya; McBride, Andrew
    The development and implementation of a finite strain thermo-viscoplasticity solver with thermomechanical friction contact for numerical simulation of friction welding processes are described. A finite strain associative coupled thermoplasticity model is used, which is suited for the large deformations characteristic of friction welding processes, and which resolves the viscoplastic deformations in the thermomechanically affected zone as well as the elastic stresses in the parent material. To prevent the large deformations from causing large distortions and degrading the simulation accuracy, an arbitrary Lagrangian Eulerian (ALE) formulation for coupled finite strain thermoplasticity is developed and incorporated into the solver, in which the motion of the reference configuration is represented incrementally in terms of a reference velocity field. Thus, the deformation from the material configuration is required neither explicitly in terms of a deformation field, nor implicitly in terms of the deformation gradient. The solver is implemented using the deal. II library and programmed for distributed memory parallel computing architectures, which reduces simulation run times and enables simulations with larger meshes than would fit on a single computer. The interprocess communications required in such a distributed memory parallel implementation of the ALE formulation and the thermomechanical friction contact are described and implemented. The axisymmetric solver implementation is validated with benchmark problems and used to simulate a direct drive friction welding process.
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    Numerical solution for subsurface reservoir simulation
    (2017) Etekpo, Kossi; Tambue, Antoine; Reddy, Daya
    Transport problems in porous media constitute an important field of scientific research in modern world, due to their broad applications in area such as petroleum engineering, water resources, pollutants transport and green- house gases sequestration to just mention few. The mathematical models that describe such problems have been developed and form one of the main classes of partial differential equations (PDEs) that scientists encounter in the real-world modeling. Nevertheless, in most of the cases, the exact solutions in the classical sense of those models are not available. The study of numerical approximation of PDEs is therefore an active research area and there is an extensive literature on numerical methods for PDEs. In this work, we review some numerical techniques, more precisely we present finite volume method with two-point flux approximation and mixed finite volume method for spatial discretization of elliptic and parabolic PDEs modeling transport flow in porous media. We then present some standard explicit and implicit methods, Rosenbrock schemes and exponential time stepping schemes for temporal discretization. We finally run some numerical simulations of advection-diffusion-reaction problems in a heterogeneous and an anisotropic porous media.
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    Proper orthogonal decomposition with interpolation-based real-time modelling of the heart
    (2017) Rama, Ritesh Rao; Skatulla, Sebastian; Reddy, Daya
    Several studies have been carried out recently with the aim of achieving cardiac modelling of the whole heart for a full heartbeat. However, within the context of the Galerkin method, those simulations require high computational demand, ranging from 16 - 200 CPUs, and long calculation time, lasting from 1 h - 50 h. To solve this problem, this research proposes to make use of a Reduced Order Method (ROM) called the Proper Orthogonal Decomposition with Interpolation method (PODI) to achieve real-time modelling with an adequate level of solution accuracy. The idea behind this method is to first construct a database of pre-computed full-scale solutions using the Element-free Galerkin method (EFG) and then project a selected subset of these solutions to a low dimensional space. Using the Moving Least Square method (MLS), an interpolation is carried out for the problem-at-hand, before the resulting coefficients are projected back to the original high dimensional solution space. The aim of this project is to tackle real-time modelling of a patient-specific heart for a full heartbeat in different stages, namely: modelling (i) the diastolic filling with variations of material properties, (ii) the isovolumetric contraction (IVC), ejection and isovolumetric relation (IVR) with arbitrary time evolutions, and (iii) variations in heart anatomy. For the diastolic filling, computations are carried out on a bi-ventricle model (BV) to investigate the performance and accuracy for varying the material parameters. The PODI calculations of the LV are completed within 14 s on a normal desktop machine with a relative L₂-error norm of 6x10⁻³. These calculations are about 2050 times faster than EFG, with each displacement step generated at a calculation frequency of 1074 Hz. An error sensitivity analysis is consequently carried out to find the most sensitive parameter and optimum dataset to be selected for the PODI calculation. In the second phase of the research, a so-called "time standardisation scheme" is adopted to model a full heartbeat cycle. This is due to the simulation of the IVC, ejection, and IVR phases being carried out using a displacement-driven calculation method which does not use uniform simulation steps across datasets. Generated results are accurate, with the PODI calculations being 2200 faster than EFG. The PODI method is, in the third phase of this work, extended to deal with arbitrary heart meshes by developing a method called "Degrees of freedom standardisation" (DOFS). DOFS consists of using a template mesh over which all dataset result fields are projected. Once the result fields are standardised, they are consequently used for the PODI calculation, before the PODI solution is projected back to the mesh of the problem-at-hand. The first template mesh to be considered is a cube mesh. However, it is found to produce results with high errors and non-physical behaviour. The second template mesh used is a heart template. In this case, a preprocessing step is required where a non-rigid transformation based on the coherent point drift method is used to transform all dataset hearts onto the heart template. The heart template approach generated a PODI solution of higher accuracy at a relatively low computational time. Following these encouraging results, a final investigation is carried out where the PODI method is coupled with a computationally expensive gradient-based optimisation method called the Levenberg- Marquardt (PODI-LVM) method. It is then compared against the full-scale simulation one where the EFG is used with the Levenberg-Marquardt method (EFG-LVM). In this case, the PODI-LVM simulations are 1025 times faster than the EFG-LVM, while its error is less than 1%. It is also observed that since the PODI database is built using EFG simulations, the PODI-LVM behaves similarly to the EFG-LVM one.
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    Shell finite elements, with applications in biomechanics
    (2009) Bartle, Samantha; Reddy, Daya
    This thesis gives a detailed presentation of a formulation for thin shells, and its finite element approximation, with the goal of modelling soft, thin biological tissues. The rigorous but complex theory due to Simo and Fox (1986) is presented in an accessible manner, with detailed derivations where appropriate. The presentation is confined to small strains and linear elasticity, with the constitutive theory extended to take account of transverse isotropy. The finite element formulation is given in such a way as to make various implementational aspects clear. Implementation has been carried out in deal.II, an open source library of finite element code. Substantial detail is given about how the shell formulation was implemented; this includes preprocessing, programming of the solution algorithm, and post-processing of results. The formulation is tested against a series of benchmark problems for flat plates and cylindrical shells, under a variety of loading conditions, and compared with results in the literature. II Two example problems in biomechanics are considered: the problem of arterial clamping, and the modelling of a prosthetic aortic valve. In the case of the clamped artery, the deformed shape for a range of clamp depths compares well with results in the literature obtained using a three-dimensional formulation. The addition of helical fibre families orientated in the same manner as two different arterial layers significantly altered the resulting deformations and agreed qualitatively with those in the literature. Using the geometric and material parameters given in earlier studies of prosthetic aortic valve leaflets, the shell solution algorithm was used to simulate a leaflet with and without transverse isotropy. The deformed leaflet behaved as expected for a diastolic state and showed a significant increase in load carried by the aortic wall with the inclusion of fibres. The work concludes with suggestions for extensions to include, for example, large strains and nonlinear material models.
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    The simulation of single phase, compressible fluid flow in fractured petroleum reservoirs using finite elements
    (2002) Hattingh, Shane Kenneth Francis; Reddy, Daya
    In this thesis, commonly used equations governing the flow of fluids are reviewed, from first principles where appropriate. The assumptions that are made in the process are critically assessed and their limitations are discussed. The equations deal with flow through a porous and permeable medium, a single fracture, a network of fractures, and with the coupling of the fracture network and blocks of matrix material.
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    Some theoretical aspects of fibre suspension flows
    (1999) Diatezua, Jacquie Kiangebeni; Reddy, Daya
    This thesis is concerned with properties of equations governing fibre suspensions. Of particular interest is the extent to which solutions, and their properties, depend on the type of closure used. For this purpose two closure rules are investigated: the linear and the quadratic closures. We show that the equations are consistent with the second law of thermodynamics, or dissipation inequality, when the quadratic closure is used. When the linear closure is used, a sufficient condition for consistency is that the particle number Np satisfies Np ≤ 35/2. Likewise, flows are found to be monotonically stable for the quadratic closure, and for the linear closure with Np ≤ 35/2. The second part of the thesis is concerned with one-dimensional problems, and their solution by finite element. The hyperbolic nature of the evolution equation for the orientation tensor necessitates a modification of the standard Galerkin-based approach. We investigate the conditions under which convergence is obtained, for unidirectional flows, with the use of the Streamline Upwind (SU) method, and the Streamline upwind Petrov/Galerkin (SUPG) method.
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    Three-field mixed finite element approximations for problems in elasticity
    (2013) Chama, Abdoulkadri; Reddy, Daya
    This thesis is concerned with three-field mixed methods for elasticity (often referred to as Hu-Washizu formulations) in which the variables are, for small-strain problems, the displacement, stress and strain. For problems in nonlinear elasticity the corresponding variables are the displacement, first Piola-Kirchhoff stress, and deformation gradient. Of particular interest is the design and analysis of mixed formulations that are uniformly stable in the incompressible limit. The first part of the thesis deals with problems in linear elasticity. Lamichhane, Reddy and Wohlmuth (Numer. Math., 104 (2006)) have shown that the conditions for stability and uniform convergence include an ellipticity condition and, secondly, a condition that the displacement together with a discrete pressure, suitably defined, constitute a stable Stokes pair. The latter condition implies that the inf-sup condition for the three-field formulation is satisfied. In the thesis, families of new stable mixed elements are generated by the following approach. First, a stable Stokes pair is chosen. Then, the space of discrete stresses is defined such that the associated discrete pressure corresponds to that of the Stokes pressure. The space of strains is defined such that it forms a superset of the space of stresses. The final task is that of showing that the spaces chosen in this way satisfy the discrete ellipticity condition. A number of new families of mixed elements are designed and analyzed in this way, and numerical examples in two and three space dimensions are presented to illustrate the theory. The second part of the thesis comprises a short chapter in which the displacement-dilatation- pressure formulation of Taylor (Int. J. Numer. Meth. Engng, 47 (2000)) is shown to be a special case of the general three-field formulation, and is then shown to be uniformly convergent. The final part of the thesis is concerned with the extension of the earlier approach to problems of nonlinear elasticity. The problem considered is the incremental or linearized version, of the kind that forms part of a Newton-Raphson process in numerical implementations, with the unknown variables being the increments in displacement, first Piola-Kirchhoff stress, and deformation gradient. In the discrete formulation the elasticity tensor (that is, the second derivative of the strain energy with respect to deformation gradient) is approximated by its mean value on each element. Conditions are established for the resulting incremental formulation to be stable and uniformly convergent, assuming that the continuous problem is stable. The analysis is illustrated through selected numerical examples.
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    Time dependent finite element simulations of a generalized Oldroyd-B fluid
    (2014-08-15) Donev,Ivajlo Georgiev; Reddy, Daya
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    A variational approach to local optimality in control theory
    (2001) Brown, Bruce J L; Reddy, Daya; Yavin, Prof
    A new approach to control theory is investigated in this thesis. The approach is based on a locally specified state space model of the control dynamics; together with a goal function, which defines a generalized distance from each state position to the desired equilibrium point or trajectory. A feedback control function is sought, which will result in a system response which approximates the gradient descent trajectories of the specified goal function. The approximation is chosen so that the resulting trajectories satisfy a certain local optimality criterion, involving the averaged second derivative of the goal function along the trajectories.
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    Well-posedness and long-time dynamics of β-plane ageostrophic flows
    (2004) Tladi, Maleafisha Stephen; Reddy, Daya
    Includes bibliographical references.