Thesis submitted in fulfilment of the requirements for the degree Magister Technologiae: Mechanical Engineering in the Faculty of Engineering at the CAPE PENINSULA UNIVERSITY OF TECHNOLOGY, 2009 / Finite element analysis modelling of sheet metal stamping is an important step in the design of tooling and process parameters. One of the critical measurements to determine the effectiveness of a numerical model is its capability of accurately predicting failure modes. To be able to make accurate predictions of deformation, tool force, blank design, etc computer simulation is almost necessary. In the automotive industry the tooling design can now be made by computer and analysed with FEA, and the amount of prototypes required for qualifying a design before manufacturing commences is greatly reduced. Tool design is a specialized phase of tool engineering. While there are many diecutting operation, some of which are very complex, they can all be reduce to plain blanking , piercing, lancing, cutting off and parting, notching, shaving and trimming. The cutting action that occurs in the piercing is quite similar to that of the chip formation ahead of a cutting tool. The punch contact the material supported by the die and a pressure builds up occurs, When the elastic limit of the work material is the exceeded the material begins to flow plastically (plastic deformation). It is often impractical to pierce holes while forming, or before forming because they would become distorted in the forming operation. The aim of the research is to develop techniques that would reduce the amount time spent during the tool qualifying stage. By accurately setting a finite element simulation that closely matches the experimental or real-life situation we can great understand the material behaviour and properties before tool designing phase commences. In this analysis, during the piercing process of the drainage hole for a shock absorber seat, there is visible material tearing (on the neck) which as a result the component is rejected. This results in material wastage, and prolonged cycle time since the operation has to be now done separately at a different workstation.

We investigate (i) pull-in instabilities in a microelectromechanical (MEM) beam due to the Coulomb force and in MEM membranes due to the Coulomb and the Casimir forces, and (ii) thermomechanical instability in a heat-conducting thermoelastoviscoplastic solid due to thermal softening overcoming hardening caused by strain- and strain-rate effects. Each of these nonlinear multiphysics problems is analyzed by the meshless local Petrov-Galerkin (MLPG) method. The moving least squares (MLS) approximation is used to generate basis functions for the trial solution, and the basis for test functions is taken to be either the weight functions used in the MLS approximation, or the same as for the trial solution. In this case the method becomes Bubnov-Galerkin. Essential (displacement, temperature, electric potential) boundary conditions are enforced by the method of Lagrange multipliers. For the electromechanical problem, the pull-in voltage and the corresponding deflection are extracted by combining the MLPG method with either the displacement iteration pull-in extraction algorithm or the pseudoarclength continuation method. For the thermomechanical problem, the localization of deformation into narrow regions of intense plastic deformation is delineated. For every problem studied, computed results are found to compare well with those obtained either analytically or by the finite element (FE) method. For the same accuracy, the MLPG method generally requires fewer nodes but more CPU time than the FE method; thus additional computational cost is compensated somewhat by the increased efficiency of the MLPG method. / Ph. D.

There is strong interest in the use of small low-cost highly sensitive magnetic field sensors for applications (such as small memory and biomedical devices) requiring weak field measurements. Among weak-field sensors, the magneto-impedance (MI) sensor has demonstrated an absolute resolution on the order of 10-11 T. The MI effect is a sensitive realignment of a periodic magnetization in response to an external magnetic field within small ferromagnetic structures. However, design of MI sensors has relied primarily on trial and error experimental methods along with decoupled models that separate the micromagnetic and classical electromagnetic equations describing the MI effect. To offer a basis for more cost-effective designs, this thesis research presentation begins with a general formulation describing MI sensors, which relaxes assumptions commonly made leading to decoupling. The coupled set of nonlinear equations is solved numerically using an efficient meshless method in a point collocation formulation. For the problem considered, the chosen method is shown to offer advantages over alternative methods including the finite element method. In the case of time, projection methods are used to stabilize the time discretization algorithm while quasi-Newton methods (nonlinear solver) are shown to be more computationally efficient, as well. Specifically, solutions for two MI sensor element geometries are presented, which were validated against published experimental data. While the examples illustrated here are for MI sensors, the approach presented can also be extended to other weak-field sensors like fluxgate and Hall effect sensors.

Smoothed Particle Hydrodynamics (SPH) is a meshless numerical method which has long been put into practice for scientific and engineering applications. It arises as a numerical discretization of convolution-like integral operators that approximate local differential operators. There have been many studies on the SPH with an emphasis on its role as a numerical scheme for partial differential equations while little attention is paid to the underlying continuum nonlocal models that lie intermediate between the two. The main goal of this thesis is to provide mathematical understanding of the SPH-like meshless methods by means of ongoing developments in studies of nonlocal models with a finite range of nonlocal interactions. It is timely for such a work to be initiated with growing interests in the nonlocal models. The thesis touches on numerical, theoretical and modeling aspects of the nonlocal integro-differential equations pertaining to the SPH-like schemes. As illustrative examples of each aspect it presents robust SPH-like schemes for advection-convection equations, discusses the stabilities of nonsymmetric nonlocal gradient operators, and proposes a new formulation of nonlocal Dirichlet-like type boundary conditions.

In recent history, the area of physics-based engineering simulation has seen rapid increases in both computer workstation performance as well as common model complexity, both driven largely in part by advances in memory density and availability of clusters and multi-core processors. While the increase in computation time due to model complexity has been largely offset by the increased performance of modern workstations, the increase in model setup time due to model complexity has continued to rise. As such, the major time requirement for solving an engineering model has transitioned from computation time to problem setup time. This is due to the fact that developing the required mesh for complex geometry can be an extremely complicated and time consuming task. Consequently, new solution techniques which are capable of reducing the required amount of human interaction are desirable. The subject of this thesis is the development of a novel meshless method that promises to eliminate the need for structured meshes, and thus, the need for complicated meshing procedures. Although the savings gain due to eliminating the meshing process would be more than sufficient to warrant further study, the proposed method is also capable of reducing the computation time and memory footprint compared to similar models solved using more traditional finite element, finite difference, finite volume, or boundary element methods. In particular, this thesis will outline the development of an interactive, meshless, physically accurate modeling environment that provides an extensible framework which can be applied to a multitude of governing equations encountered in computational mechanics and thermofluids. Additionally, through the development of tailored preprocessing routines, efficiency and accuracy of the proposed meshless algorithms can be tested in a more realistic and flexible environment. Examples are provided in the areas of elasticity, heat transfer and computational fluid dynamics.

The utility of computational fluid dynamics (CFD) for solving problems of engineering interest has experienced rapid growth due to the improvements in both memory capacity and processing speed of computers. While the capability now exists for the solution of the Navier-Stokes equations about complex and complete aircraft configurations, the bottleneck within the process is the time consuming task of properly generating a mesh that can accurately solve the governing partial differential equations (PDEs). This thesis explored two numerical techniques that attempt to circumvent the difficulty associated with the meshing process by solving a simplified form of the continuity equation within a meshless framework. The continuity equation reduces to the full potential equation by assuming irrotational flow. It is a nonlinear PDE that can describe flows for a wide spectrum of Mach numbers that do not exhibit discontinuities. It may not be an adequate model for the detailed analysis of a complex flowfield since viscous effects are not captured by this equation, but it is an appealing alternative for the aircraft designer because it can provide a quick and simple to implement estimate of the aerodynamic characteristics during the conceptual design phase. The two meshless methods explored in this thesis are the Dual Reciprocity Method (DRM) and the Generalized Finite Difference Method (GFD). The Dual Reciprocity Method was shown to have the capability to solve for the two-dimensional subcritical compressible flow over a Circular Cylinder and the non-lifting flow for a NACA 0012 airfoil. Unfortunately these solutions were obtained with the requirement of a priori knowledge of the solution to tune a parameter necessary for proper convergence of the algorithm. Due to the shortcomings of applying the Dual Reciprocity Method, the Generalized Finite Difference Method was also investigated. The GFD method solves a PDE in differential form and can be thought of as a meshless form of a standard finite difference scheme. This method proved to be an accurate and general technique for solving the previously mentioned cases along with the lifting flow about a NACA 0012 airfoil. It was also demonstrated that the GFD method could be formulated to discretize the full potential equation with second order accuracy. Both solution methods offer their own set of unique advantages and challenges, but it was determined that the GFD Method possessed the flexibility necessary for a meshless technique to become a viable aerodynamic design tool.

O Método dos Elementos Finitos é a forma mais difundida de análise estrutural numérica, com aplicações nas mais diversas teorias estruturais. Contudo, no estudo das cascas e alguns outros usos, suas deficiências impulsionaram a pesquisa em outros métodos de resolução de Equações Diferenciais Parciais. O presente trabalho utiliza uma dessas alternativas, o Método de Galerkin Livre de Elementos (Element-Free Galerkin) para estudar as cascas. Inicia com a observação da aproximação usada no método, os Moving Least Squares e os Multiple-Fixed Least Squares. A seguir, estabelece uma formulação que combina a teoria de placas moderadamente espessas de Reissner-Mindlin à teoria da Elasticidade Plana e se utiliza da aproximação estudada para analisar placas e chapas deste tipo. Depois, expõe uma teoria geometricamente exata de cascas inicialmente curvas onde as curvaturas iniciais são impostas como deformações livres de tensão a partir de uma configuração de referência plana. Tal teoria exclui a necessidade de coordenadas curvilíneas e consequentemente da utilização de objetos como os símbolos de Cristoffel, já que todas as integrações e imposições são feitas na configuração plana de referência, em um sistema ortonormal de coordenadas. A imposição das condições essenciais de contorno é feita por forma fraca, resultando em um funcional híbrido de deslocamentos que permite a maleabilidade necessária ao uso dos Moving Least Squares. Esse trabalho se propõe a particularizar tal teoria para o caso de pequenos deslocamentos e deformações (linearidade geométrica), mantendo a consistência das definições de tensões e deformações generalizadas enquanto permite uma imposição da forma fraca resultante, depois de discretizada, por um sistema linear de equações. Por fim, exemplos numéricos são usados para discutir sua eficácia e exatidão. / The Finite Element Method is the most spread numerical analysis tool, applied to a wide range of structural theories. However, for the study of shells and other problems, some of its deficiencies have stimulated research in other methods for solving the derived Partial Differential Equations. The present work uses one of those alternatives, the Element Free Galerkin Method, for the study of shells. It begins with the observation of the approximation used in the method, Moving Least Squares and Multiple-Fixed Least Squares. Then, it establishes a formulation that combines the Reissner-Mindlin moderately thick plate theory with plane elasticity, and uses the proponed approximation to analyze such plates and stabs. Afterwards, it demonstrates a geometrically exact shell theory that accounts for initial curvatures as a stress-free deformation from a flat reference configuration. Such theory precludes the use of curvilinear coordinates and, subsequently, the use of objects such as Cristoffel symbols, as all integrations and impositions are done in the flat reference configuration, in an orthogonal frame. The essential boundary conditions are imposed in a eak statement, rendering a hybrid displacement functional that provides the necessary conditions for the use of Moving Least Squares. This works main objective is the particularization of this theory for the small displacement and strains assumption (geometrical linearity), keeping the consistent definition of generalized stresses and strains, while allowing the imposition of the discretized weak form through a system of linear equations. Lastly, numerical simulations are carried out to assess the methods efficiency and accuracy.

This dissertation introduces novel methods for solving highly challenging model- ing and control problems, motivated by advanced aerospace systems. Adaptable, ro- bust and computationally effcient, multi-resolution approximation algorithms based on Radial Basis Function Network and Global-Local Orthogonal Mapping approaches are developed to address various problems associated with the design of large scale dynamical systems. The main feature of the Radial Basis Function Network approach is the unique direction dependent scaling and rotation of the radial basis function via a novel Directed Connectivity Graph approach. The learning of shaping and rota- tion parameters for the Radial Basis Functions led to a broadly useful approximation approach that leads to global approximations capable of good local approximation for many moderate dimensioned applications. However, even with these refinements, many applications with many high frequency local input/output variations and a high dimensional input space remain a challenge and motivate us to investigate an entirely new approach. The Global-Local Orthogonal Mapping method is based upon a novel averaging process that allows construction of a piecewise continuous global family of local least-squares approximations, while retaining the freedom to vary in a general way the resolution (e.g., degrees of freedom) of the local approximations. These approximation methodologies are compatible with a wide variety of disciplines such as continuous function approximation, dynamic system modeling, nonlinear sig-nal processing and time series prediction. Further, related methods are developed for the modeling of dynamical systems nominally described by nonlinear differential equations and to solve for static and dynamic response of Distributed Parameter Sys- tems in an effcient manner. Finally, a hierarchical control allocation algorithm is presented to solve the control allocation problem for highly over-actuated systems that might arise with the development of embedded systems. The control allocation algorithm makes use of the concept of distribution functions to keep in check the "curse of dimensionality". The studies in the dissertation focus on demonstrating, through analysis, simulation, and design, the applicability and feasibility of these ap- proximation algorithms to a variety of examples. The results from these studies are of direct utility in addressing the "curse of dimensionality" and frequent redundancy of neural network approximation.

O σκοπός της παρούσας διδακτορικής διατριβής είναι διττός και, εμπεριέχει δραστηριότητα τόσο στο κομμάτι της εφαρμοσμένης όσο και της βασικής έρευνας. Πιο συγκεκριμένα, περιλαμβάνει την εφαρμογή σύγχρονων υπολογιστικών μεθόδων (Μέθοδος Πεπερασμένων Στοιχείων, Μέθοδος Πεπερασμένων Όγκων) στη μελέτη της ροής του αίματος, καθώς και την ανάπτυξη σύγχρονων υπολογιστικών μεθοδολογιών που δε στηρίζονται στη χρήση πλέγματος. Ταυτόχρονα μελετάται η αποτελεσματική εφαρμογή των μεθόδων της Υπολογιστικής Ρευστομηχανικής στην ιατρική πρακτική (Simulation Treatment Planning), για την αποτελεσματική πρόληψη, διάγνωση και θεραπευτική αντιμετώπιση των νόσων του καρδιακού και περιφερικού αγγειακού συστήματος. Η διαδικασία υλοποίησης των προσομοιώσεων έχουν σκοπό την υποβοήθηση του θεράποντα ιατρού στη λήψη ιατρικής απόφασης σχετικά με τη θεραπευτική αγωγή. Παράλληλα, στο κομμάτι της βασικής έρευνας αναπτύσσονται σύγχρονες υπολογιστικές μέθοδοι οι οποίες πρόκειται να καλύψουν τις αδυναμίες που παρουσιάζουν οι διαδεδομένες υπολογιστικές μέθοδοι. Η υλοποίησή τους βρίσκει ανταπόκριση και πεδίο εφαρμογής σε διάφορους τομείς της Επιστήμης και της Μηχανικής. Έτσι, θα παρουσιαστούν αποτελέσματα μόνο στο τομέα της Υπολογιστικής Ρευστομηχανικής. Τα κριτήρια αυτά θα στηριχθούν σε ποσοτικές συνδυαστικές αναλύσεις, οι οποίες ενσωματώνουν το state-of-the-art της μορφολογικής απεικόνισης των αγγείων στο state-of-the-art των μεθοδολογιών της υπολογιστικής ρευστομηχανικής, οι οποίες είναι άμεσα σχετιζόμενες με την εκτίμηση των αιμοδυναμικών παραγόντων και τάσεων που εφαρμόζονται σε πάσχουσες περιοχές του αγγειακού συστήματος, όπως στενώσεις, θρόμβοι και ανευρύσματα. / The aim of present doctoral thesis is double fold and, includes research activity both in the fields of applied and basic research. More precisely, it includes the application of modern numerical methods (Finite Element Method, Finite Volume Method) for the study of blood flow, as well as the development of modern numerical methodologies, which do not rely on the use of a computational mesh, that is the so-called meshless or Meshfree methods. Furthermore, the effective application of sophisticated numerical methods in the medical practice (Simulation Treatment Planning) has been studied, since there is a great necessity for effective prevention, diagnosis and therapeutic confrontation of illnesses the cardiac and vascular system. The simulation conducted they aim to assist the doctor in the decision-making. At the same time, regarding of area of the basic research, sophisticated numerical methods were developed and applied to various applications of science and engineering. More precisely, results will be presented for Computational Fluid Dynamics problems.

O Método dos Elementos Finitos é a forma mais difundida de análise estrutural numérica, com aplicações nas mais diversas teorias estruturais. Contudo, no estudo das cascas e alguns outros usos, suas deficiências impulsionaram a pesquisa em outros métodos de resolução de Equações Diferenciais Parciais. O presente trabalho utiliza uma dessas alternativas, o Método de Galerkin Livre de Elementos (Element-Free Galerkin) para estudar as cascas. Inicia com a observação da aproximação usada no método, os Moving Least Squares e os Multiple-Fixed Least Squares. A seguir, estabelece uma formulação que combina a teoria de placas moderadamente espessas de Reissner-Mindlin à teoria da Elasticidade Plana e se utiliza da aproximação estudada para analisar placas e chapas deste tipo. Depois, expõe uma teoria geometricamente exata de cascas inicialmente curvas onde as curvaturas iniciais são impostas como deformações livres de tensão a partir de uma configuração de referência plana. Tal teoria exclui a necessidade de coordenadas curvilíneas e consequentemente da utilização de objetos como os símbolos de Cristoffel, já que todas as integrações e imposições são feitas na configuração plana de referência, em um sistema ortonormal de coordenadas. A imposição das condições essenciais de contorno é feita por forma fraca, resultando em um funcional híbrido de deslocamentos que permite a maleabilidade necessária ao uso dos Moving Least Squares. Esse trabalho se propõe a particularizar tal teoria para o caso de pequenos deslocamentos e deformações (linearidade geométrica), mantendo a consistência das definições de tensões e deformações generalizadas enquanto permite uma imposição da forma fraca resultante, depois de discretizada, por um sistema linear de equações. Por fim, exemplos numéricos são usados para discutir sua eficácia e exatidão. / The Finite Element Method is the most spread numerical analysis tool, applied to a wide range of structural theories. However, for the study of shells and other problems, some of its deficiencies have stimulated research in other methods for solving the derived Partial Differential Equations. The present work uses one of those alternatives, the Element Free Galerkin Method, for the study of shells. It begins with the observation of the approximation used in the method, Moving Least Squares and Multiple-Fixed Least Squares. Then, it establishes a formulation that combines the Reissner-Mindlin moderately thick plate theory with plane elasticity, and uses the proponed approximation to analyze such plates and stabs. Afterwards, it demonstrates a geometrically exact shell theory that accounts for initial curvatures as a stress-free deformation from a flat reference configuration. Such theory precludes the use of curvilinear coordinates and, subsequently, the use of objects such as Cristoffel symbols, as all integrations and impositions are done in the flat reference configuration, in an orthogonal frame. The essential boundary conditions are imposed in a eak statement, rendering a hybrid displacement functional that provides the necessary conditions for the use of Moving Least Squares. This works main objective is the particularization of this theory for the small displacement and strains assumption (geometrical linearity), keeping the consistent definition of generalized stresses and strains, while allowing the imposition of the discretized weak form through a system of linear equations. Lastly, numerical simulations are carried out to assess the methods efficiency and accuracy.