Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1994. / Vita. Abstract. Includes bibliographical references (leaf 103). Also available via the Internet.

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Problems with time-evolving domains are frequently occurring in computationalfluid dynamics and many other fields of science and engineering.Unfitted methods, where the computational mesh does not conform to thegeometry, are of great interest for handling such problems, since they removethe burden of mesh generation. We work towards the goal of developingan unfitted solver for Navier-Stokes equations on time-evolving domainsby developing and presenting cut finite element (CutFEM) splitting methodsfor solving Navier-Stokes equations. These CutFEM splitting methodsuse Nitsche’s method for incorporating boundary conditions and employpatch-based ghost penalty stabilization of the cut elements to achieve stabilityand optimal order error estimates. Numerical benchmarks are used toverify the methods and implementations. The methods are tested against aproblem with known analytical solution, the Taylor-Green vortex, and alsocompared to the classical Deutsche Forschungsgemeinschaft (DFG) benchmarkproblem with channel flow around a cylinder. For both benchmarks,the methods was shown to be stable when satisfying the parabolic Courant–Friedrichs–Lewy (CFL) condition, and to produce optimal convergencerates.

In the present thesis, a computational framework for the analysis of coupled hydro-fracture flow in deformable porous media using a Finite/Discrete Element Method is presented. In this context, a series of developments have been made in order to provide a more efficient and robust numerical model capable of dealing with oil production and slope stability problems. The mechanical response of the skeleton is highly dependent on its seepage behaviour as pore pressure modifications affect the in situ stress field. The u-p formulation has been employed using an explicit time integration scheme where fully saturated and single-phase partially saturated analysis are incorporated for 2-D and 3-D cases. Owing to their inherent simplicity, low order elements provide an excellent framework in which contact conditions coupled with crack propagation can be dealt with in an effective manner. For linear elements this implies single point integration which, however, can result in spurious zero-energy modes. Therefore, in order to obtain reliable results, a stabilization technique has been devised to eliminate hourglassing. The success of the modelling strategy ultimately depends on the interdependence of different phenomena. The linking between the displacement components, network and pore pressures represents an important role in the efficiency of the overall coupling procedure. Therefore, a master-slave technique is proposed to link seepage and network fields, proving to be particularly attractive from a computational cost point of view. Another important development that has provided substantial savings in CPU times is the use of an explicit-explicit subcycling scheme. Numerical examples have been used to assess the accuracy and efficiency of the proposed framework. Special attention is focused on the investigation of hydraulic fracture propagation in oil production problems and plane failure analysis of the stability of slopes.

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Following a stochastic approach, this thesis presents a numerical framework for elastostatics of random media. Firstly, after a mathematically rigorous investigation of the popular white noise model in an engineering context, the smooth spatial stochastic dependence between material properties is identified as a fundamental feature of practical random media. Based on the recognition of the probabilistic essence of practical random media and driven by engineering simulation requirements, a comprehensive random medium model, namely elementary random media (ERM), is consequently defined and its macro-scale properties including stationarity, smoothness and principles for material measurements are systematically explored. Moreover, an explicit representation scheme, namely the Fourier-Karhunen-Loeve (F-K-L) representation, is developed for the general elastic tensor of ERM by combining the spectral representation theory of wide-sense stationary stochastic fields and the standard dimensionality reduction technology of principal component analysis. Then, based on the concept of ERM and the F-K-L representation for its random elastic tensor, the stochastic partial differential equations regarding elastostatics of random media are formulated and further discretized, in a similar fashion as for the standard finite element method, to obtain a stochastic system of linear algebraic equations. For the solution of the resulting stochastic linear algebraic system, two different numerical techniques, i.e. the joint diagonalization solution strategy and the directed Monte Carlo simulation strategy, are developed. Original contributions include the theoretical analysis of practical random medium modelling, establishment of the ERM model and its F-K-L representation, and development of the numerical solvers for the stochastic linear algebraic system. In particular, for computational challenges arising from the proposed framework, two novel numerical algorithms are developed: (a) a quadrature algorithm for multidimensional oscillatory functions, which reduces the computational cost of the F-K-L representation by up to several orders of magnitude; and (b) a Jacobi-like joint diagonalization solution method for relatively small mesh structures, which can effectively solve the associated stochastic linear algebraic system with a large number of random variables.

In this research, the finite element method (FEM) was used to solve the nonlinear, incompressible, transient, three dimensional Navier-Stokes equations in their non-conservative form. Linear tetrahedron elements were employed with the elegant, equal order interpolation for both pressure and velocity. The characteristic based split scheme was formulated in a fully implicit manner to circumvent the time step restrictions of the classical explicit formulations. The monolithic (single step, fully coupled solution procedure for pressures and velocity) form of the CBS scheme was also derived and its suitability was positively demonstrated. Casting the CBS scheme in a monolithic framework, results in the generation of a pressure stabilization term in the mass conservation equation, thereby circumventing the LBB restriction by the elimination of the zero pressure block. An account of all the steps involved in discretizing the Navier-Stokes equations (both in split and monolithic frameworks) was presented in meticulous detail, which included the derivation of the convective and pressure stabilization terms, linearization of the non-linear terms and the consequent derivation of the highly efficient analytical jacobian matrix, along with the temporal and spatial discretizations of the corresponding terms. The monolithic and the split version of the CBS scheme were integrated into a parallel, scalable and extensible Fortran90 software called IFENs. The development of IFENs started during the course of this research and all of its components have been designed and implemented by the author of this thesis. Multi processor parallelism was achieved using the Intel implementation of the most widely used/preferred, Message Passing Interface (MPI) standard. The parallel support needed for the use of a variety of parallel, linear, iterative solvers belonging to the Krylov subspace family (e.g. GMRES and its variants, CG, BiCG, BiCG- stab, etc.), parallel non linear solvers belonging to the Newton-Krylov family (line search newton, trust region newton, nonlinear GMRES, etc.) and parallel preconditioners (incomplete LU, Additive Shwarz Method - ASM, algebraic multigrid, etc.), was provided by the incorporation of PETSc into IFENs. PETSc is a state of the art, non-trivial toolkit, which represents a collection of several parallel libraries useful in high performance scientific computing. Keeping in mind the specific requirements of IFENs, a custom mesh partitioner was implemented. It operated on meshes that were renumbered using bandwidth reducing algorithms like Revere Cuthill Mckee. The possibility of using established domain decomposition libraries like ParMETIS was explored and demonstrated to be counter productive for the demands of this research. After the preliminary testing and validation of the procedures adopted before and during the execution of IFENs, large, high definition domains representative of human arteries (specifically, carotid bifurcations, found in the neck) were considered and the complete incompressible set of Navier-Stokes equations were solved for pressure and velocity fields. During the tenure of this research more than 1000 recorded parallel test cases were executed to test various components of IFENs, as well as various simulations representative of a wide variety of problems. IFENs can easily handle meshes with tens of millions of elements. The largest mesh used for the purpose of this research contained 14.58 million tetrahedrons and 2.489 million nodes, which on average required just 7 minutes per timestep, while executing the classical split framework of the CBS scheme. Results from the simulation of 9 carotid meshes, representative of 4 carotid geometries were presented and found to be in good agreement with the available ultrasound data. The flow fields were analysed and post processed using different techniques for each case. The haemodynamic wall parameters like time averaged wall shear stress and oscillatory shear index were calculated and mapped onto the corresponding boundary nodes. The region in the carotid bifurcation susceptible to the deposition of plaques and consequent stenosis were pointed out and other anomalies were highlighted.

M.Ing. (Civil Engineering) / Bridge design/analysis in South Africa is seldom done from the dynamic point of view, and then only .in most exceptional cases. The topic of dynamic behavior of bridges is fairly complex and not always well understood by the average design engineer. Until the advent of 3-D finite element packages, dynamic design was usually done by assuming behaviour of simplebeams and/orplatesandequating these to the real life situation. Thesevaluescan be found in standard designbooks and tables drawnup as a result of empirical studies and analyses done in the past The author felt that there was a need for a better understanding of the dynamic behavior of bridges and to develop skills in finite element modelling and analysis. He therefore undertook to investigate and study the dynamic behavior of the Bothasfontein interchange bridge. An empirical evaluation of the composite deck was developed using Finite Element Analysis, and the results compared with the actual behavior of the bridge. Good correlation between the model and the measured values was obtained.

M.Ing. (Mechanical Engineering) / Please refer to full text to view abstract

M.Ing. (Mechanical Engineering) / This dissertation evaluates the suitability of the finite element method as a tool for the design and analysis of elastomeric materials in general and flexible shaft couplings in particular. The theoretical background covers numerous aspects that are essential to the comprehension of the functioning of elastomeric materials and the difficulties inherent to the numerical modelling of such materials. These aspects include the properties of rubber, the functioning and selection of flexible couplings and some details regarding linear -, non-linear - 'and dynamic finite element analysis. The problems investigated for the purposes of this study may be divided into three categories: • The capabilities of the finite element method to compare different variations of a flexible coupling design parametrically is investigated. • Uni-axial tensile - and compressive material tests are numerically simulated to assess the ability of the finite element method to predict the response of materials subjected to large-scale nonlinear deformation. The numerical results are also verified by means of physical material tests. • Based on the modelling methods that were optimized in the first two categories, a numerical model of a flexible coupling in start up mode is developed. The accuracy of predictions is evaluated by comparison with physically measured results.