Significant research is ongoing in the world to meet the needs of social and environmental crisis by harnessing wind and solar energy at high altitudes. One such approach is the use of an inflatable High Altitude Aerial Platform (HAAP). In the presented work, such periodically supported beam-like structures are analyzed using various mathematical models primarily modeling them as an equivalent beam using one-dimensional theories. The Euler-Bernoulli Theory (EBT) has been widely used for high aspect ratio beams, whereas the First Order Shear Deformation Theory (FSDT), or the Timoshenko beam theory, considers transverse shear effects and hence is superior in modeling low aspect ratio beams. First, an Isogeometric Analysis (IGA) is conducted using both FSDT and EBT to predict thermal buckling of periodically supported composite beams. Isogeometric analysis overcomes the limitations of the Gibbs phenomenon at discontinuities for a periodically supported beam using a higher order textit{k}-refinement. Next, an Integral Equation Approach (IEA) is implemented using EBT to obtain natural frequencies and buckling loads of periodically supported non-prismatic beams. Ill-conditioning errors were alleviated using admissible orthogonal Chebychev polynomials to obtain higher modes. We also present the prediction of the onset of flutter instability for metal plate and inflatable wing shaped foam test articles analyzed using finite element analysis (FEA). FEA updating based on modal testing and by conducting a geometrically nonlinear analysis resulted in a good agreement against the experiment tests. Furthermore, a nonlinear co-rotational large displacement/rotation FEA including the effects of the pressure as a follower forces was implemented to predict deformations of an inflatable structures. The developed FEA based tool namely Structural Analysis of Inflatables using FEA (SAIF) was compared with the experiments and available literature. It is concluded that the validity of the developed tool depends on the flexibility of the beam, which further depends upon the length of the beam and the bending rigidity of the beam. Inflatable structures analyzed with materials with high value of the Young's modulus and low to medium slenderness ratio tend to perform better against the experimental data. This is attributed to the presence of wrinkling and/or the Brazier effect (ovalling of the cross section) for flexible beams. The presented work has applications in programmable buckling, uncertainty quantification, and design of futuristic HAAP models to help face the upcoming environmental crises and meet the societal needs. / Doctor of Philosophy / In the future, developed countries are projected to face an increase in renewable energy demands due to environmental crises and increasing societal needs for energy due to urbanization. Wind energy, a renewable source, has received increasing attention. Wind farms require large land space and offshore wind energy harvesting is prone to unstable environments. Crosswind kite power is one of the promising and emerging fields where one can harvest energy from the wind farm inaccessible and apparently endless winds at high altitudes. In this dissertation, we present analysis and experiments on investigating complex structures, such as inflatable high altitude aerial platforms (HAAP) by using various mathematical models, primarily modeling them as an equivalent beam using one-dimensional theories. We investigate the effects of internal pressure on such structures, which unlike many other types of applied loads, follow the direction of the deflections. When supported on multiple supports, these structures are more efficient in terms of increased payload capacity due to a better distribution of loads, despite the increased weight penalty. To name a few, there are direct applications of periodic supports in design of bridges and railway sleepers. To avoid violent vibrations or failure, we also investigate the effect of multiple supports on the so-called natural frequency, vibration frequency under absence of applied loads, and buckling loads, instabilities under compression, of such beam-like structures. The presented work will aid in the design of futuristic HAAP models to help face the upcoming environmental crises and meet the energy demands of society due to urbanization.

Optical Tomography is an imaging modality that enhances early diagnosis of disease through use of harmless Near-Infrared rays instead of conventional x-rays. The subsequent images are used to reconstruct the object. However Optical Tomography has not been effectively utilized due to the complicated photon scattering phenomenon and ill-posed nature of the corresponding image reconstruction scheme. The major method for reconstruction of the object is based on an iterative loop that constantly minimizes the difference between the predicted model of photon scattering with acquired images. Currently the most effective method of predicting the photon scattering pattern is the solution of the Radiative Transfer Equation (RTE) using the Finite Elements Method (FEM). However, the conventional FEM uses classical C0 interpolation functions, which have shortcomings in terms of continuity of the solution over the domain as well as proper representation of geometry. Hence higher discretization is necessary to maintain accuracy of gradient-based results which may significantly increase the computational cost in each iteration. This research implements the recently developed Isogeometric Approach (IGA) and particularly IGA-based FEM to address the aforementioned issues. The IGA-based FEM has the potential to enhance adaptivity and reduce the computational cost of discretization schemes. The research in this study applies the IGA method to solve the RTE with the diffusion approximation and studies its behavior in comparison to conventional FEM. The results show comparison of the IGA-based solution with analytical and conventional FEM solutions in terms of accuracy and efficiency. While both methods show high levels of accuracy in reference to the analytical solution, the IGA results clearly excel in accuracy. Furthermore, FE solutions tend to have shorter runtimes in low accuracy results. However, in higher accuracy solutions, where it matters the most, the IGA proves to be considerably faster. / Doctor of Philosophy / CT scans can save lives by allowing medical practitioners observe inside the patient's body without use of invasive surgery. However, they use high energy, potentially harmful x-rays to penetrate the organs. Due to limits of the mathematical algorithm used to reconstruct the 3D figure of the organs from the 2D x-ray images, many such images are required. Thus, a high level of x-ray exposure is necessary, which in periodic use can be harmful. Optical Tomography is a promising alternative which replaces x-rays with harmless Near-infrared (NIR) visible light. However, NIR photons have lower energy and tend to scatter before leaving the organs. Therefore, an additional algorithm is required to predict the distribution of light photons inside the body and their resulting 2D images. This is called the forward problem of Optical Tomography. Only then, like conventional CT scans, can another algorithm, called the inverse solution, reconstruct the 3D image by diminishing the difference between the predicted and registered images. Currently Optical Tomography cannot replace x-ray CT scans for most cases, due to shortcomings in the forward and inverse algorithms to handle real life usages. One obstacle stems from the fact that the forward problem must be solved numerous times for the inverse solution to reach the correct visualization. However, the current numerical method, Finite Element Method (FEM), has limitations in generating accurate solutions fast enough using economically viable computers. This limitation is mostly caused by the FEM's use of a simpler mathematical construct that requires more computations and is limited in accurately modelling the geometry and shape. This research implements the recently developed Isogeometric Analysis (IGA) and particularly IGA-based FEM to address this issue. The IGA-based FEM uses the same mathematical construct that is used to visualize the geometry for complicated applications such as some animations and computer games. They are also less complicated to apply due to much lower need for partitioning the domain. This study applies the IGA method to solve the forward problem of diffuse Optical Tomography and compare the accuracy and speed of IGA solution to the conventional FEM solution. The comparison reveals that while both methods can reach high accuracy, the IGA solutions are relatively more accurate. Also, while low accuracy FEM solutions have shorter runtimes, in solutions with required higher accuracy levels, the IGA proves to be considerably faster.

Contact is one of the most challenging nonlinearities to solve in solid mechanics. In traditional linear finite element analysis, the contact surface is only C^0 continuous, as a result, the normal to the contact surface is not continuous. The normal contact force is directed along the normal in the direction of the contact surface, and therefore, the contact force is discontinuous. This issue is tackled in linear finite element analysis using various surface smoothing techniques, however, a better solution is to use isogeometric analysis where the solution space is spanned by smooth spline basis functions. Unfortunately, spline-based isogeometric contact analysis still has limited applicability to industrial computer aided design (CAD) representations. Building analysis suitable mesh from the industrial CAD representations has been a major bottleneck of the computer aided engineering workflow. One promising alternative field of study, intended to address this challenge, is called the immersed finite element method. In this method, the original CAD domain is immersed in a rectilinear grid called the background mesh. This cuts down the model preparation and the mesh generation time from the original CAD domain, but the method suffers from limited accuracy issues. In this dissertation, the original CAD domain is immersed in an envelope domain which can be of arbitrary topological and geometric complexity and can approximate none, some or all of the features of the original CAD domain. Therefore, the method, called the flex representation method, is much more flexible than the traditional immersed finite element method. Within the framework of the flex representation method, a robust and accurate contact search algorithm is developed, that efficiently computes the collision points between the contacting surfaces in a discrete setting. With this information at hand, a penalty based formulation is derived to enforce the contact constraint weakly for multibody and self-contact problems. In addition, the contact algorithm is used to solve various proof-of-concept academic problems and some real world industrial problems to demonstrate the validity and robustness of the algorithms.

In this dissertation, I provide a robust, efficient inverse mapping algorithm for use in immersed simulation methods, specifically in the Flex Representation Method. I also explore a structural theory that unifies the theories of solids, shells, beams, and rigid bodies. As part of this, I preform a preliminary exploration of applying the Flex Representation Method to shells. Finally, I explore why higher order elements suffer from small critical time steps in explicit dynamics. I then propose a simple method of remedying this issue by exploiting the properties of U-splines.

This research endeavor develops Isogeometric approach for analysis of composite structures and take advantage of higher order continuity, smoothness and variation diminishing property of Nurbs basis for stress analysis of composite and sandwich beams and plates. This research also computes stress concentration factor in a composite plate with a hole. Isogeometric nonlinear/linear finite element code is developed for static and dynamic analysis of laminated composite plates. Nurbs linear, quadratic, higher-order and k-refined elements are constructed using various refinement procedures and validated with numerical testing. Nurbs post-processor for in-plane and interlaminar stress calculation in laminated composite and sandwich plates is developed. Nurbs post-processor is found to be superior than regular finite element and in good agreement with the literature. Nurbs Isgoemetric analysis is used for stress analysis of laminated composite plate with open-hole. Stress concentration factor is computed along the hole edge and good agreement is obtained with the literature. Nurbs Isogeometric finite element code for free-vibration and linear dynamics analysis of laminated composite plates also obtain good agreement with the literature. Main highlights of the research are newly developed 9 control point linear Nurbs element, k-refined and higher-order Nurbs elements in isogeometric framework. Nurbs elements remove shear-locking and hourglass problems in thin plates in context of first-order shear deformation theory without the additional step and compute better stresses than Lagrange finite element and higher order shear deformation theory for comparatively thick plates i.e. a/h = 4. Also, Nurbs Isogeometric analysis perform well for vibration and dynamic problems and for straight and curved edge problems. / Ph. D.

A análise isogeométrica (AIG) de estruturas consiste em construir a geometria exata ou aproximada de um modelo computacional a partir de funções criadas por meio de tecnologias de Computer Aided Design (CAD), tais como B-Splines, NURBS (Non-Uniform Rational BSplines) e T-splines, e aplicar o conceito de análise isoparamétrica, ou seja, representar o espaço de solução para as variáveis independentes em termos das mesmas funções que representam a geometria. O presente trabalho visa o estudo da análise isogeométrica aplicada a vigas planas, com a utilização de B-Splines e NURBS para aproximação de deslocamentos. São desenvolvidos modelos isogeométricos de vigas planas baseados nas hipóteses de Bernoulli- Euler e Timoshenko, e alguns exemplos de aplicação são realizados a fim de comparar os resultados numéricos com soluções analíticas, mostrando boa concordância. Uma questão pertinente à AIG corresponde à imposição de vínculos em pontos do domínio em que as funções básicas não sejam interpolatórias ou os vínculos desejados não forem diretamente relacionados aos graus de liberdade do elemento, que é o caso do elemento de viga de Bernoulli-Euler, já que as rotações geralmente não são tidas como graus de liberdade mas há a necessidade de se prescrever condições de contorno/conexão nas mesmas para descrever problemas físicos. Essa questão é tratada no presente trabalho através dos Métodos de Lagrange e de penalidade. São realizados exemplos de aplicação construídos com elementos de viga de Bernoulli-Euler utilizando os métodos de Lagrange e de penalidade na imposição de vínculos e na conexão entre pontos de regiões de domínio. / Isogeometric analysis (IGA) consists on building the geometry of the computational model with functions created by Computer Aided Design (CAD) technologies, such as B-Splines, NURBS (Non-Uniform Rational B-Splines) and T-Splines. Then, isoparametric concept is employed, that is, the solution space is represented by means of the same functions used to describe the geometry. The aim of the present contribution is the study of isogeometric analysis applied to 2D beams with interpolation via B-splines and NURBS. Two-dimensional isogeometric beam formulations based on Bernoulli-Euler and Timoshenko assumptions are presented. Some examples of application are given and results are compared to analytical solutions, showing good agreement. An important issue about IGA corresponds to the imposition of constraints at points of domain in which the shape functions are not interpolatory, or the desired constraints are not directly related to the degrees of freedoms. This may occur for Bernoulli-Euler beams since rotations are not usually defined as degrees of freedom, but they need to be assessed for prescription of some boundary/connection conditions. This is done in present contribution by employing both Lagrange and penalty methods. Some examples of structures composed by 2D isogeometric Bernoulli-Euler beam elements are solved by using Lagrange and Penalty methods to impose constraints and to make the connection between domain regions.

O presente trabalho tem por objetivo desenvolver uma formulação numérica baseada em Análise Isogeométrica para o estudo de escoamentos incompressíveis isotérmicos de fluidos newtonianos. Com o emprego desta metodologia, os procedimentos de pré-processamento e análise são unificados, melhorando as condições de continuidade das funções de base empregadas tanto na discretização espacial do problema como na aproximação das variáveis do sistema de equações. O sistema de equações fundamentais do escoamento é formado pelas equações de Navier-Stokes e pela equação de conservação de massa, descrita segundo a hipótese de pseudo-compressibilidade, além de uma equação constitutiva para fluidos viscosos de acordo com a hipótese de Stokes. Para problemas com escoamentos turbulentos emprega-se a Simulação de Grandes Escalas - LES (Large Eddy Simulation), na qual o modelo clássico de Smagorinsky é utilizado para a representação das escalas inferiores à resolução da malha. O esquema explícito de dois passos de Taylor-Galerkin é aplicado no contexto da Análise Isogeométrica para a discretização das equações governantes, sendo que a discretização espacial é realizada empregando-se funções NURBS (Non Uniform Rational Basis B-Splines). Essas funções base apresentam vantagens em relação às tradicionais funções utilizadas no MEF (Método dos Elementos Finitos), principalmente no que diz respeito à facilidade de obtenção de continuidade superior a C0 entre os elementos e representação precisa das geometrias. Propõe-se também o desenvolvimento de ferramentas de pré e pós-processamento baseadas na estrutura de dados da Análise Isogeométrica para a geração de malhas e visualização de resultados. Alguns problemas clássicos da Dinâmica dos Fluidos Computacional são analisados para a validação da metodologia apresentada. Os resultados apresentados demonstram boa aproximação da formulação em relação a dados de referência, além de maior versatilidade quanto à discretização espacial dos problemas em comparação com as tradicionais formulações baseadas em elementos finitos. / This work aims to develop a numerical formulation based on Isogeometric Analysis for the study of incompressible flows of Newtonian fluids under isothermal conditions. By using this methodology, pre-processing and analysis procedures are unified, improving the conditions of continuity of the basis functions utilized in the approximations of the equation variables and spatial discretization of the problem. The system of fundamental equations of the fluid flow is constituted by the Navier-Stokes equations and the mass conservation equation, which is described according to the pseudo-compressibility hypothesis. In addition, a constitutive equation for viscous fluids according to Stokes' hypothesis is also provided. Turbulent flows are analyzed using LES (Large Eddy Simulation), where the Smagorinsky’s model is adopted for sub-grid scales. The explicit two-step Taylor-Galerkin method is applied into the context of Isogeometric Analysis for the discretization of the flow equations, where spatial discretization is carried out taking into account Non Uniform Rational Basis B-Splines (NURBS) basis functions. These basis functions have advantages over traditional functions employed in the FEM (Finite Element Method). Particularly, it is easier to obtain continuity order higher than C0 between adjacent elements and geometry representation is more accurate. Pre and post-processing tools for mesh generation and results visualization are also proposed considering the data structure inherent to Isogeometric Analysis. Some classic problems of Computational Fluid Dynamics are analyzed in order to validate the proposed methodology. Results obtained here show that the present formulation has good approximation when compared with predictions obtained by reference authors. Moreover, Isogeometric Analysis is more versatile than traditional finite element formulations when spatial discretization procedures are considered.

Two methods for overcoming locking phenomena in isogeometric beams are presented. The first method blends the rotation of a Timoshenko beam with the rotation of a Bernoulli beam to produce realistic displacements in straight beams. The second method uses mixed variational principles, specifically the Hu-Washizu Principle, to produce realistic displacements as well as realistic strains without post-processing.

Neste trabalho buscou-se consolidar aspectos referentes à otimização de problemas envolvidos na mecânica dos meios contínuos, envolvendo diferentes áreas do conhecimento, tais como: otimização matemática, diferenciação automática, análise estrutural, análise aerodinâmica, parametrização de curvas, superfícies e sólidos do tipo B-spline racionais não-uniformes (NURBS, acrônimo do inglês), análise IsoGeométrica (IGA, acrônimo do inglês) e análise por Elementos Finitos (FEA, acrônimo do inglês). Como objetivo final busca-se otimizar formas de cascas estruturais e formas de corpos aerodinâmicos imersos em escoamentos compressíveis. No que concerne à análise estrutural, esta é realizada via análise IsoGeométrica utilizando elementos sólidos para modelar cascas. Uma cinemática co-rotacional abrangente e precisa baseada na exata decomposição polar é desenvolvida, para lidar com problemas estáticos e dinâmicos altamente não lineares. Na análise estática foram implementados o método de Newton-Raphson e controle de deslocamentos generalizado, para problemas dinâmicos foram implementados o método -generalizado (G) e o método energia momento generalizado (GEMM+). A análise aerodinâmica é realizada via análise por Elementos Finitos para modelar escoamentos compressíveis viscosos e não viscosos em regimes transônicos e supersônicos. Um esquema característico baseado na separação da equação de momento (CBS, acrônimo do inglês) é utilizado para obter uma adequada integração temporal. No que concerne à otimização matemática, é utilizado um método baseado em gradientes, conhecido por programação quadrática sequencial (SQP, acrônimo do inglês), onde a avaliação as derivadas de Fréchet são levadas a cabo via diferenciação automática (AD, acrônimo do inglês). No que concerne aos resultados finais é realizada a otimização estrutural de forma de cascas modeladas como sólidos são apresentados, evidenciando um desempenho ótimo com respeito à energia de deformação interna. Os resultados de otimização aerodinâmica bidimensionais apresentam perfis aerodinâmicos ótimos com respeito à relação arrasto/sustentação para uma ampla gama de número de Mach, enquanto um resultado tridimensional é apresentado evidenciando a robustez e eficiência da implementação proposta. Pretendese estabelecer com este trabalho as bases para pesquisas em problemas de otimização aeroelástica. / Consolidation of the link among optimization problems in continuum mechanics, involving different fields, such as mathematical optimization, automatic differentiation, structural analysis, aerodynamic analysis, curves, surfaces and solids parameterization using Non Uniform Rational B-spline (NURBS), IsoGeometric Analysis (IGA), Finite Element Analysis (FEA) is looked for. Structural shape optimization of shell structures and aerodynamic shape optimization of immersed bodies in compressible flows are the main goals of this work. Concerning structural analysis, the so-called IsoGeometric analysis is employed. An accurate and comprehensive corotational kinematic based on the exact polar decomposition is developed in order to study highly nonlinear static and dynamic problems. Static analysis is carried out with Newton-Raphson and Generalized Displacement Control Method, while dynamic analysis is carried out with Generalized- (G) and Generalized Energy-Momentum Method (GEMM+). Aerodynamic analysis is carried out via Finite Element Analysis (FEA) in order to solve compressible flows in transonic and supersonic regimes. A Characteristic Based Split (CBS) method is employed to obtain an accurate time integration, which is based on the splitting of the momentum equation. Concerning mathematical optimization, the so-called Sequential Quadratic Programming (SQP) is employed, which is a gradient-based method, where the Fréchet derivatives are evaluated using Automatic Differentiation (AD). Final results consisting in structural optimization shown an optimal behaviour with respect to internal strain energy. While, results concerning aerodynamic bi-dimensional shape optimization exhibit a optimal behaviour with respect drag/lift ratio, for a large range of Mach number, and a simple result for tri-dimensional case is presented in order to show the efficiency and robustness of the implementation. Bases for future research in aeroelastic optimization problems are established in this work.

O presente trabalho tem por objetivo desenvolver uma formulação numérica baseada em Análise Isogeométrica para o estudo de escoamentos incompressíveis isotérmicos de fluidos newtonianos. Com o emprego desta metodologia, os procedimentos de pré-processamento e análise são unificados, melhorando as condições de continuidade das funções de base empregadas tanto na discretização espacial do problema como na aproximação das variáveis do sistema de equações. O sistema de equações fundamentais do escoamento é formado pelas equações de Navier-Stokes e pela equação de conservação de massa, descrita segundo a hipótese de pseudo-compressibilidade, além de uma equação constitutiva para fluidos viscosos de acordo com a hipótese de Stokes. Para problemas com escoamentos turbulentos emprega-se a Simulação de Grandes Escalas - LES (Large Eddy Simulation), na qual o modelo clássico de Smagorinsky é utilizado para a representação das escalas inferiores à resolução da malha. O esquema explícito de dois passos de Taylor-Galerkin é aplicado no contexto da Análise Isogeométrica para a discretização das equações governantes, sendo que a discretização espacial é realizada empregando-se funções NURBS (Non Uniform Rational Basis B-Splines). Essas funções base apresentam vantagens em relação às tradicionais funções utilizadas no MEF (Método dos Elementos Finitos), principalmente no que diz respeito à facilidade de obtenção de continuidade superior a C0 entre os elementos e representação precisa das geometrias. Propõe-se também o desenvolvimento de ferramentas de pré e pós-processamento baseadas na estrutura de dados da Análise Isogeométrica para a geração de malhas e visualização de resultados. Alguns problemas clássicos da Dinâmica dos Fluidos Computacional são analisados para a validação da metodologia apresentada. Os resultados apresentados demonstram boa aproximação da formulação em relação a dados de referência, além de maior versatilidade quanto à discretização espacial dos problemas em comparação com as tradicionais formulações baseadas em elementos finitos. / This work aims to develop a numerical formulation based on Isogeometric Analysis for the study of incompressible flows of Newtonian fluids under isothermal conditions. By using this methodology, pre-processing and analysis procedures are unified, improving the conditions of continuity of the basis functions utilized in the approximations of the equation variables and spatial discretization of the problem. The system of fundamental equations of the fluid flow is constituted by the Navier-Stokes equations and the mass conservation equation, which is described according to the pseudo-compressibility hypothesis. In addition, a constitutive equation for viscous fluids according to Stokes' hypothesis is also provided. Turbulent flows are analyzed using LES (Large Eddy Simulation), where the Smagorinsky’s model is adopted for sub-grid scales. The explicit two-step Taylor-Galerkin method is applied into the context of Isogeometric Analysis for the discretization of the flow equations, where spatial discretization is carried out taking into account Non Uniform Rational Basis B-Splines (NURBS) basis functions. These basis functions have advantages over traditional functions employed in the FEM (Finite Element Method). Particularly, it is easier to obtain continuity order higher than C0 between adjacent elements and geometry representation is more accurate. Pre and post-processing tools for mesh generation and results visualization are also proposed considering the data structure inherent to Isogeometric Analysis. Some classic problems of Computational Fluid Dynamics are analyzed in order to validate the proposed methodology. Results obtained here show that the present formulation has good approximation when compared with predictions obtained by reference authors. Moreover, Isogeometric Analysis is more versatile than traditional finite element formulations when spatial discretization procedures are considered.