The aviation industry is growing at a steady rate but presently, the industry is highly dependent on fossil fuel. As the world is running out of fossil fuels and the wide-spread acceptance of climate change due to carbon emissions, both the governments and industry are spending a significant amount of resources on research to reduce the weight and hence the fuel consumption of commercial aircraft. A commercial fixed-wing aircraft wing consists of spars which are beams running in span-wise direction, carrying the flight loads and ribs which are panels with holes attached to the spars to preserve the outer airfoil shape of the wing. Kapania et al. at Virginia Tech proposed the concept of reducing the weight of aircraft wing using unconventional design of the internal structure consisting of curvilinear spars and ribs (known as SpaRibs) for enhanced performance. A research code, EBF3GLWingOpt, was developed by the Kapania Group. at Virginia Tech to find the best configuration of SpaRibs in terms of weight saving for given flight conditions. However, this software had a number of limitations and it can only create and analyze limited number of SpaRibs configurations. In this work, the limitations of the EBF3GLWingOpt code has been identified and new algorithms have been developed to make is robust and analyze larger number of SpaRibs configurations. The code also has the capability to create cut-outs in the SpaRibs for passage of fuel pipes and wirings. This new version of the code can be used to find best SpaRibs configuration for multiple objectives such as reduction of weight and increase flutter velocity. The code is developed in Python language and it has parallel computational capabilities. The wing is modeled using commercial FEA software, MSC.PATRAN and analyzed using MSC.NASTRAN which are from within EBF3GLWingOpt. Using this code a significant weight reduction for a transport aircraft wing has been achieved. / PHD

Plastic packaging waste currently forms a significant part of municipal solid waste and as such is causing increasing environmental concerns. Such packaging is largely non-biodegradable and is particularly difficult to recycle or to reuse due largely to its complex compositions. Apart from limited recycling of some easily identifiable packaging wastes that can be separated economically, such as bottles, most packaging waste ends up in landfill sites. In recent years, in an attempt to address this problem in plastic packaging, the development of packaging materials from renewable plant resources has received increasing attention and a wide range of bioplastic materials based on starch are now available. Environmentally these bioplastic materials also reduce reliance on oil resources and have the advantage that they are biodegradable and can be composted upon disposal to reduce the environmental impact. Many food packaging containers are produced by thermoforming processes in which thin sheets are inflated under pressure into moulds to produce the required thin -wall structures. Hitherto these thin sheets have almost exclusively been made of oilbased polymers and it is for these that computational models of thermoforming processes have been developed. Recently, in the context of bioplastics, commercial thermoplastic starch sheet materials have been developed. The behaviour of such materials is influenced both by temperature and, because of the inherent hydrophilic characteristics of the materials, by moisture content. Both of these aspects affect the behaviour of bioplastic sheets during the thermoforming process. This thesis describes experimental work and work on the computational modelling of thermoforming processes for thermoplastic starch sheets using a commercially available material. The experimental work has been carried in order to characterise the deformation behaviour of the material with regard to different temperature, moisture contents and strain rates. Thermoforming of the material was performed and samples produced were used for comparison and verification of the computational modelling of the thermoforming process. In the first attempt to model the thermoforming process, a hyperelastic constitutive equation was established to approximate the material behaviour taking account of the combined effects of temperature and moisture content and a simple ii membrane model with constrained deformation was used to model an axisymmetric case of thermoforming. Simulations with this model showed that moisture content mostly affects the pressure required to push the sheet into the mould while moisture variation during thermoforming has little effect on the final thickness distribution of the product. Considerable discrepancies were found in the thickness distribution between the predictions from the model and the experimental measurements. Further attempts were made to take account of the elasto-plastic behaviour of the material and a more complex three-dimensional FE model was developed using ANSYS/LS-DYNA. Based on the findings in the simpler modelling work, no attempt was made to incorporate the moisture content effect on material behaviour but the material parameters for the elasto-plastic constitutive equation were obtained from high speed tensile tests so that moisture variation during thermoforming could be minimised and neglected. The predictions from this model have led to significant improvements in prediction of the thickness distribution which has become much closer to the experimental measurements in comparison with the hyperelastic model. This work provides some important insights into thermoforming of thermoplastic starch materials: a) Deformation behaviour of such materials depends strongly on the moisture content and the temperature, both of which affect behaviour during thermoforming processes, including the preheating stage; b) moisture variation during the thermoforming process has a significant effect on the pressure required for the deformation. This also leads to variation of moisture content distribution in the final product, which in turn affects the material properties such as ductility or impact strength at different positions in the thermoformed structure; c) thermoforming of thermoplastic starch materials can be simulated more accurately by an elasto-plastic model and the LS-DYNA algorithm in comparison with a hyperelastic membrane model. This work has provided useful information on thermoforming of thermoplastic starch materials with particular reference to the design of thermoforming tools and to the careful control of processing conditions including preheating. It has also laid a solid foundation for future work on how the moisture variation impacts on the formation of defects such as incomplete forming due to material hardening and fracture due to loss of ductility.

We propose a discontinuous Galerkin finite element method (DGFEM) for fully nonlinear elliptic Hamilton--Jacobi--Bellman (HJB) partial differential equations (PDE) of second order with Cordes coefficients. Our analysis shows that the method is both consistent and stable, with arbitrarily high-order convergence rates for sufficiently regular solutions. Error bounds for solutions with minimal regularity show that the method is generally convergent under suitable choices of meshes and polynomial degrees. The method allows for a broad range of hp-refinement strategies on unstructured meshes with varying element sizes and orders of approximation, thus permitting up to exponential convergence rates, even for nonsmooth solutions. Numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients demonstrate the significant gains in accuracy and computational efficiency over existing methods. We then extend the DGFEM for elliptic HJB equations to a space-time DGFEM for parabolic HJB equations. The resulting method is consistent and unconditionally stable for varying time-steps, and we obtain error bounds for both rough and regular solutions, which show that the method is arbitrarily high-order with optimal convergence rates with respect to the mesh size, time-step size, and temporal polynomial degree, and possibly suboptimal by an order and a half in the spatial polynomial degree. Exponential convergence rates under combined hp- and τq-refinement are obtained in numerical experiments on problems with strongly anisotropic diffusion coefficients and early-time singularities. Finally, we show that the combination of a semismooth Newton method with nonoverlapping domain decomposition preconditioners leads to efficient solvers for the discrete nonlinear problems. The semismooth Newton method has a superlinear convergence rate, and performs very effectively in computations. We analyse the spectral bounds of nonoverlapping domain decomposition preconditioners for a model problem, where we establish sharp bounds that are explicit in both the mesh sizes and polynomial degrees. We then go beyond the model problem and show computationally that these algorithms lead to efficient and competitive solvers in practical applications to fully nonlinear HJB equations.

The main objective of this research work is to apply the discontinuous Galerkin method to a classical partial differential equation to investigate the properties of the numerical solution and compare the numerical solution to the analytical solution by using discontinuous Galerkin method. This scheme is applied to 1-D non-linear conservation equation (Burgers equation) in which the governing differential equation is simplified model of the inviscid Navier-stokes equations. In this work three cases are studied. They are sinusoidal wave profile, initial shock discontinuity and initial linear distribution. A grid and time step refinement is performed. Riemann fluxes at each element interfaces are calculated. This scheme is applied to forward differentiation method (Euler's method) and to second order Runge-kutta method of this work.

O projeto de transdutores piezelétricos envolve modelagens matemáticas e verificações experimentais, que são necessárias para validá-los. Através das análises dos transdutores verificam-se as influências das condições de contorno experimentais e as dificuldades para modelá-las. No entanto, na literatura atual este ponto não é discutido de forma abrangente. Além disso, as propriedades elétricas, piezelétricas e mecânicas dos materiais devem ser conhecidas com precisão para que o modelo represente um transdutor físico. Estas propriedades são fornecidas pelos fabricantes em valores aproximados, porém algumas delas, como por exemplo, o amortecimento mecânico, não são fornecidas, o que dificulta a modelagem. Por estes motivos, este trabalho trata do desenvolvimento de técnicas de verificações experimentais de transdutores piezelétricos, visando esclarecer efeitos relacionados às variações nas condições de contorno devido à fixação de pequenos espelhos e a fixação do próprio transdutor num suporte, e propor formas de incorporar esses efeitos nos modelos matemáticos. Além disso, propõe-se uma metodologia para a determinação do coeficiente de amortecimento das cerâmicas piezelétricas. São apresentadas as técnicas ópticas utilizadas na medição dinâmica e estática de deslocamentos de transdutores piezelétricos, os problemas das influências mecânicas, como vibrações ambientes, nas análises ópticas e as respectivas soluções adotadas. As análises numéricas foram conduzidas com o software de elementos finitos ANSYS e os resultados comparados com as verificações experimentais. Os materiais analisados são piezocerâmicas PZT-5A, piezoatuadores flextensionais, projetados pelo método de otimização topológica e um piezoatuador bilaminar. Nos resultados apresentados são mostrados os desvios e os comentários sobre as possíveis causas. / Piezoelectric transducer design is done by using numerical method and experimental analysis, which are needed to validate it. Transducers analysis allow us to verify the influence of experimental boundary conditions and the difficulties to model them. However, in the current literature this point is not discussed in a comprehensive way. In addiction, electrical, piezoelectric and mechanical material properties must be known with accuracy, to represent a real transducer. Approximately values of these properties are provided by manufacturers, however other property values are not provided, such as for example, mechanical damping, which causes difficulties for modelling. Therefore, this work is about the development of experimental techniques to analyze piezoelectric transducers, aiming to understand the effects related to boundary conditions changes due to stick of small mirrors and the way the transducer is held, and suggest of way to incorporate these effects in mathematical model. In addiction, suggests a methodology to determine the piezoceramics damping coefficient. It is described optical techniques used for displacement measurements of piezoelectric transducer in dynamic and static operation, problems related to mechanical influences in optical analysis, such as environment vibration, and respective applied solutions. Numerical analysis are conducted by using finite element software, ANSYS, and the results are compared with experimental verification. The materials analyzed are piezoceramic PZT-5A, flextensional piezoactuators, designed by using topology optimization and a bimorph pizoactuator. In the present results is shown the deviation and comments about possible cause.

Submitted by Maria Cristina (library@lncc.br) on 2015-11-25T16:47:47Z No. of bitstreams: 1 Tese (Iury_Igreja).pdf: 8846320 bytes, checksum: fd47a9c83cd8a2b8393741418c9a3962 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2015-11-25T16:48:02Z (GMT) No. of bitstreams: 1 Tese (Iury_Igreja).pdf: 8846320 bytes, checksum: fd47a9c83cd8a2b8393741418c9a3962 (MD5) / Made available in DSpace on 2015-11-25T16:48:12Z (GMT). No. of bitstreams: 1 Tese (Iury_Igreja).pdf: 8846320 bytes, checksum: fd47a9c83cd8a2b8393741418c9a3962 (MD5) Previous issue date: 2015-06-22 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / In this thesis we propose stabilized hybrid finite elemento formulations to the Stokes and Darcy problems that preserve main properties of the Galerkin discontinuous methods associated with these problems. Moreover, combining these formulations we propose a stabilized hybrid finite element method for the Stokes-Darcy coupled problem. Lagrange multipliers are introduced to weakly impose continuity at the element interfaces giving rise to a global system involving only degrees-of-freedom associated with the multipliers. Knowing the multipliers, the quantities of interest are obtained from local problems that are solved at the element level. Different choices for the multipliers are evaluated for the Stokes and Darcy problems separately and for the Stokes-Darcy coupled problem. With proper choice of multipliers, Beavers-Joseph and Beavers-Joseph-Saffman interface conditions are naturally imposed. Through stabilization techniques hybrid finite element methods are generated with great flexibility in the choice of approximation spaces allowing, for example, the same order of approximations for all variables (velocity, pressure and multiplier). To validate the formulations of the Stokes, Darcy and Stokes-Darcy coupled problems several numerical experiments are performed to illustrate the flexibility and robustness of the proposed formulations and show optimal convergence rates. / Nesta tese propomos formula ções de elementos finitos híbridas estabilizadas para os problemas de Stokes e de Darcy que preservam as principais propriedades dos métodos de Galerkin descontínuo associados a estes problemas. Além disso, combinando estas formulações propomos um método de elementos finitos híbrido estabilizado para o problema de Stokes-Darcy acoplado. Multiplicadores de Lagrange são introduzidos para impor fracamente a continuidade nas interfaces dos elementos dando origem há um sistema global que envolve apenas graus de liberdade associados aos multiplicadores. Conhecidos os multiplicadores, as variáveis de interesse são obtidas através dos problemas locais que são resolvidos em nível de elemento. Diferentes escolhas para os multiplicadores são avaliadas para os problemas de Stokes e de Darcy isoladamente e para o problema acoplado Stokes-Darcy. Com uma adequada escolha dos multiplicadores, as condi ções de interface de Beavers-Joseph e Beavers-Joseph-Saffman são impostas naturalmente. Através de técnicas de estabiliza ção são gerados métodos de elementos finitos híbridos com grande flexibilidade na escolha dos espa os de aproxima ção. Permitindo-se, por exemplo, o uso de aproxima ções de mesma ordem para todas as variáveis (velocidade, pressão e multiplicador). Para validar as formula ções do problema de Stokes, de Darcy e de Stokes-Darcy acoplado são realizados vários experimentos numéricos que ilustram a flexibilidade e robustez das formula ›es propostas e mostram taxas ótimas de convergê ncia.

O projeto de transdutores piezelétricos envolve modelagens matemáticas e verificações experimentais, que são necessárias para validá-los. Através das análises dos transdutores verificam-se as influências das condições de contorno experimentais e as dificuldades para modelá-las. No entanto, na literatura atual este ponto não é discutido de forma abrangente. Além disso, as propriedades elétricas, piezelétricas e mecânicas dos materiais devem ser conhecidas com precisão para que o modelo represente um transdutor físico. Estas propriedades são fornecidas pelos fabricantes em valores aproximados, porém algumas delas, como por exemplo, o amortecimento mecânico, não são fornecidas, o que dificulta a modelagem. Por estes motivos, este trabalho trata do desenvolvimento de técnicas de verificações experimentais de transdutores piezelétricos, visando esclarecer efeitos relacionados às variações nas condições de contorno devido à fixação de pequenos espelhos e a fixação do próprio transdutor num suporte, e propor formas de incorporar esses efeitos nos modelos matemáticos. Além disso, propõe-se uma metodologia para a determinação do coeficiente de amortecimento das cerâmicas piezelétricas. São apresentadas as técnicas ópticas utilizadas na medição dinâmica e estática de deslocamentos de transdutores piezelétricos, os problemas das influências mecânicas, como vibrações ambientes, nas análises ópticas e as respectivas soluções adotadas. As análises numéricas foram conduzidas com o software de elementos finitos ANSYS e os resultados comparados com as verificações experimentais. Os materiais analisados são piezocerâmicas PZT-5A, piezoatuadores flextensionais, projetados pelo método de otimização topológica e um piezoatuador bilaminar. Nos resultados apresentados são mostrados os desvios e os comentários sobre as possíveis causas. / Piezoelectric transducer design is done by using numerical method and experimental analysis, which are needed to validate it. Transducers analysis allow us to verify the influence of experimental boundary conditions and the difficulties to model them. However, in the current literature this point is not discussed in a comprehensive way. In addiction, electrical, piezoelectric and mechanical material properties must be known with accuracy, to represent a real transducer. Approximately values of these properties are provided by manufacturers, however other property values are not provided, such as for example, mechanical damping, which causes difficulties for modelling. Therefore, this work is about the development of experimental techniques to analyze piezoelectric transducers, aiming to understand the effects related to boundary conditions changes due to stick of small mirrors and the way the transducer is held, and suggest of way to incorporate these effects in mathematical model. In addiction, suggests a methodology to determine the piezoceramics damping coefficient. It is described optical techniques used for displacement measurements of piezoelectric transducer in dynamic and static operation, problems related to mechanical influences in optical analysis, such as environment vibration, and respective applied solutions. Numerical analysis are conducted by using finite element software, ANSYS, and the results are compared with experimental verification. The materials analyzed are piezoceramic PZT-5A, flextensional piezoactuators, designed by using topology optimization and a bimorph pizoactuator. In the present results is shown the deviation and comments about possible cause.

Philosophiae Doctor - PhD / Numerical approximations of multiscale problems of important applications in ecology are investigated. One of the class of models considered in this work are singularly perturbed (slow-fast) predator-prey systems which are characterized by the presence of a very small positive parameter representing the separation of time-scales between the fast and slow dynamics. Solution of such problems involve multiple scale phenomenon characterized by repeated switching of slow and fast motions, referred to as relaxationoscillations, which are typically challenging to approximate numerically. Granted with a priori knowledge, various time-stepping methods are developed within the framework of partitioning the full problem into fast and slow components, and then numerically treating each component differently according to their time-scales. Nonlinearities that arise as a result of the application of the implicit parts of such schemes are treated by using iterative algorithms, which are known for their superlinear convergence, such as the Jacobian-Free Newton-Krylov (JFNK) and the Anderson’s Acceleration (AA) fixed point methods.

La simulation numérique fait partie intégrante du processus d'analyse. Que l'on veuille concevoir le profil d'un véhicule, ou chercher à prévoir le résultat d'un forage pétrolier, la simulation numérique est devenue un outil complémentaire à la théorie et aux expérimentations. Cet outildoit produire des résultats précis en un minimum de temps. Pour cela, nous avons à disposition des méthodes numériques précises, et des machines de calcul aux performances importantes. Cet outil doit être générique sur les maillages, l'ordre de la solution, les méthodes numériques, et doitmaintenir ses performances sur les machines de calculs modernes avec une hiérarchie complexes d'unité de calculs. Nous présentons dans cette thèse le background mathématiques de deux classes de schémas numériques, les méthodes aux éléments finis continus et discontinus. Puis nous présentons les enjeux de la conception d'une plateforme en prenant en compte l'ensemble de ces contraintes. Ensuite nous nous intéressons au sous-problème de l'assemblage au dessus d'un support d'exécution. L'opération d'assemblage se retrouve en algèbre linéaire dans les méthodes multi-frontales ou dans les applications de simulations assemblant un système linéaire. Puis, nous concluons en dressant un bilan sur la plateforme AeroSol et donnons des pistes d'évolution possibles. / Numerical simulation is nowadays an essential part of engineering analysis, be it to design anew plane, or to detect underground oil reservoirs. Numerical simulations have indeed become an important complement to theoretical and experimental investigation, allowing one to reduce the cost of engineering design processes. In order to achieve a high level of precision, one need to increase the resolution of his computational domain. So to keep getting results in reasonable time, one shall nd a way to speed-up computations. To do this, we use high performance computing, HPC, to exploit the complex architecture of modern supercomputers. Under these two constraints, and some other like the genericity of finite elements, or the mesh dimension, we developed a new platform AeroSol. In this thesis, we present the mathematical background, and the two types of schemes that are implemented in the platform, the continuous finite elements method, and the discontinuous one. Then, we present the design choices made in the platform,then, we study a sub-problem, the assembly operation, which can be found in linear algebra multi-frontal methods.

In this thesis we develop and analyze the performance ofadaptive finite element methods for multiphysics problems. Inparticular, we propose a methodology for deriving computable errorestimates when solving unidirectionally coupled multiphysics problemsusing segregated finite element solvers.  The error estimates are of a posteriori type and are derived using the standard frameworkof dual weighted residual estimates.  A main feature of themethodology is its capability of automatically estimating thepropagation of error between the involved solvers with respect to anoverall computational goal. The a posteriori estimates are used todrive local mesh refinement, which concentrates the computationalpower to where it is most needed.  We have applied and numericallystudied the methodology to several common multiphysics problems usingvarious types of finite elements in both two and three spatialdimensions. Multiphysics problems often involve convection-diffusion equations for whichstandard finite elements can be unstable. For such equations we formulatea robust discontinuous Galerkin method of optimal order with piecewiseconstant approximation. Sharp a priori and a posteriori error estimatesare proved and verified numerically. Fractional step methods are popular for simulating incompressiblefluid flow. However, since they are not genuine Galerkin methods, butrather based on operator splitting, they do not fit into the standardframework for a posteriori error analysis. We formally derive an aposteriori error estimate for a prototype fractional step method byseparating the error in a functional describing the computational goalinto a finite element discretization residual, a time steppingresidual, and an algebraic residual.