Numerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation. In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions. Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.

Within this thesis, thin walled shell structures are discussed with modern element formulationsin the context of the Isogeometric Analysis (IGA). IGA was designed to achieve a directinterface from CAD to analysis. According to the concept of IGA, Non-Uniform RationalB-Splines (NURBS) are used as shape functions in the design and the analysis. Dependingon the polynomial order, NURBS can come along with a high order continuity. Therefore,the curvature of a shell surface can be described directly by the shape function derivativeswhich is not possible within the classical Finite Element Analysis (FEA) using linear meshes.This description of the curvature gives rise to the application of the Kirchho-Love shellformulation, which describes the curvature stiness with the dierentiation of the spatialdegrees of freedom. Based upon this, the formulation can be enhanced with further kinematicalexpressions as the shear dierence vector, which leads to a 5-parameter Reissner-Mindlinformulation. This kinematic formulation is intrinsically free from transverse shear lockingdue to the split into Kirchho-Love and additional shear contributions. The formulation canbe further extended to a 7-parameter three-dimensional shell element, which considers volumetriceects in the thickness direction. Two additional parameters are engaged to describethe related thickness changes under load and to enable the use of three-dimensional materiallaws. In general, three-dimensional shell elements suer from curvature thickness and Poisson'sthickness locking. However, these locking phenomena are intrinsically avoided by thehierarchic application of the shear dierence vector and the 7th parameter respectively. The3-parameter Kirchho-Love, the 5-parameter Reissner-Mindlin and the 7-parameter 3D shellelement build a hierarchic family of model-adaptive shells.This hierarchic family of shell elements is presented and discussed in the scope of this thesis.The concept and the properties of the single elements are elaborated and the dierences arediscussed. Geometrically linear and non-linear benchmark examples are simulated. Convergencestudies are performed and the results are validated against analytical solutionsand solutions from literature, taking into account deections and internal forces. Furthermore,the dierent locking phenomena which occur in analyses with shell formulations areexamined. Several test cases are designed to ensure a validated implementation of the hierarchicshell elements. The element formulations and further pre- and postprocessing featuresare implemented and validated within the open-source software environment Kratos Multi-physics.

<div>Many Engineering structures, including electronic component assemblies, are inherently multi-scale in nature. These structures often experience complex local nonlinear behavior such as plasticity, damage or fracture. These local behaviors eventually lead to the failure at the macro length scale. Connecting the behavior across the length scales to develop an understanding of the failure mechanism is important for developing reliable products.</div><div><br></div><div>To solve multi-scale problems in which the critical region is much smaller than the entire structure, an iterative solution approach based on domain decomposition techniques is proposed. Two independent models are constructed to model the global and local substructures respectively. The unbalanced force at the interface is iteratively reduced to ensure force equilibrium of the overall structure in the final solution. The approach is non-intrusive since only nodal values on the interface are transferred between the global and local models. Solution acceleration using SR1 and BFGS updates is also demonstrated. Equally importantly, the two updates are applied in a non-intrusive manner, meaning that the technique is implemented without needing access to the codes using which the sub-domains are analyzed. Code- and mesh-agnostic solutions for problems with local nonlinear material behavior or local crack growth are demonstrated. Analysis in which the global and local models are solved using two different commercial codes is also demonstrated.</div><div><br></div><div>Engineering analysis using numerical models are helpful in providing insight into the connection between the structure, loading history, behavior and failure. Specifically, Isogeometric analysis (IGA) is advantageous for engineering problems with evolving geometry compared to the traditional finite element method (FEM). IGA carries out analysis by building behavioral approximations isoparametrically on the geometrical model (commonly NURBS) and is thus a promising approach to integrating Computer-Aided Design (CAD) with Computer-Aided Engineering (CAE).</div><div><br></div><div>In enriched isogeometric Analysis (EIGA), the solution is enriched with known behavior on lower dimensional geometrical features such as crack tips or interfaces. In the present research, enriched field approximation techniques are developed for the application of boundary conditions, coupling patches with non-matching discretizations and for modeling singular stresses in the structure.</div><div><br></div><div>The first problem solution discussed is to apply Dirichlet and Neumann boundary conditions on boundary representation (B-rep) CAD models immersed in an underlying domain of regular grid points. The boundary conditions are applied on the degrees of freedom of the lower dimensional B-rep part directly. The solution approach for the immersed analysis uses signed algebraic level sets constructed from the B-rep surfaces to blend the enriched</div><div>field with the underlying field. The algebraic level sets provide a surrogate for distance, are non-iteratively (or algebraically) computed and allow implicit Boolean compositions.</div><div><br></div><div>The methodology is also applied to couple solution approximations of decomposed patches by smoothly blending incompatible geometries to an arbitrary degree of smoothness. A parametrically described frame or interface is introduced to “stitch” the adjacent patches. A hierarchical blending procedure is then developed to stitch multiple unstructured patches including those with T-junctions or extraordinary vertices.</div><div><br></div><div>Finally, using the EIGA technique, a computational method for analyzing general multimaterial sharp corners that enables accurate estimations of the generalized stress intensity factors is proposed. Explicitly modeled geometries of material junctions, crack tips and deboned interfaces are isogeometrically and hierarchically enriched to construct approximations with the known local behavior. specifically, a vertex enrichment is used to approximate the asymptotic field near the re-entrant corner or crack tip, Heaviside function is used to approximate the discontinuous crack face and the parametric smooth stitching technique is used to approximate the behavior across material interface. The developed method allows direct extraction of generalized stress intensity factors without needing a posteriori evaluation of path independent integrals for decisions on crack propagation. The numerical implementation is validated through analysis of a bi-material corner, interface crack and growth of an inclined crack in a homogeneous solid. The developed procedure demonstrates rapid convergence to the solution stress intensity factors with relatively fewer degrees of freedom, even with uniformly coarse discretizations.</div>

Non-uniform rational B-splines (NURBS) methodology is presented, on which the isogeometric analysis is based. T-splines are also introduced as a surface design methodology, which are a generalization of NURBS and permit local refinement. Isogeometric analysis using NURBS and T-splines are applied separately to a structural mechanics problem. The results are compared with the closed-form solution. The desirable performance of isogeometric analysis using T-splines on engineering analysis is demonstrated.

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.