The conventional finite element models (FEM) of problems in structural mechanics are based on the principles of virtual work and the total potential energy. In these models, the secondary variables, such as the bending moment and shear force, are post-computed and do not yield good accuracy. In addition, in the case of the Timoshenko beam theory, the element with lower-order equal interpolation of the variables suffers from shear locking. In both Euler-Bernoulli and Timoshenko beam theories, the elements based on weak form Galerkin formulation also suffer from membrane locking when applied to geometrically nonlinear problems. In order to alleviate these types of locking, often reduced integration techniques are employed. However, this technique has other disadvantages, such as hour-glass modes or spurious rigid body modes. Hence, it is desirable to develop alternative finite element models that overcome the locking problems. Least-squares finite element models are considered to be better alternatives to the weak form Galerkin finite element models and, therefore, are in this study for investigation. The basic idea behind the least-squares finite element model is to compute the residuals due to the approximation of the variables of each equation being modeled, construct integral statement of the sum of the squares of the residuals (called least-squares functional), and minimize the integral with respect to the unknown parameters (i.e., nodal values) of the approximations. The least-squares formulation helps to retain the generalized displacements and forces (or stress resultants) as independent variables, and also allows the use of equal order interpolation functions for all variables. In this thesis comparison is made between the solution accuracy of finite element models of the Euler-Bernoulli and Timoshenko beam theories based on two different least-square models with the conventional weak form Galerkin finite element models. The developed models were applied to beam problems with different boundary conditions. The solutions obtained by the least-squares finite element models found to be very accurate for generalized displacements and forces when compared with the exact solutions, and they are more accurate in predicting the forces when compared to the conventional finite element models.

The compositions of in-host microbial communities (microbiota) play a significant role in host health, and a better understanding of the microbiota's role in a host's transition from health to disease or vice versa could lead to novel medical treatments. One of the first steps toward this understanding is modeling interaction dynamics of the microbiota, which can be exceedingly challenging given the complexity of the dynamics and difficulties in collecting sufficient data. Methods such as principal differential analysis, dynamic flux estimation, and others have been developed to overcome these challenges for ordinary differential equation models. Despite their advantages, these methods are still vastly underutilized in mathematical biology, and one potential reason for this is their sophisticated implementation. While this work focuses on applying principal differential analysis to microbiota data, we also provide comprehensive details regarding the derivation and numerics of this method. For further validation of the method, we demonstrate the feasibility of principal differential analysis using simulation studies and then apply the method to intestinal and vaginal microbiota data. In working with these data, we capture experimentally confirmed dynamics while also revealing potential new insights into those dynamics. We also explore how we find the forward solution of the model differential equation in the context of principal differential analysis, which amounts to a least-squares finite element method. We provide alternative ideas for how to use the least-squares finite element method to find the forward solution and share the insights we gain from highlighting this piece of the larger parameter estimation problem. / Ph. D.

Composite structures, such as laminated beams, plates and shells, are widely used in the automotive, aerospace and marine industries due to their superior specific strength and tailor-able mechanical properties. Because of their use in a wide range of applications, and their commonplace in the engineering design community, the need to accurately predict their behavior to external stimuli is crucial. We consider in this thesis the application of the least-squares finite element method (LSFEM) to problems of static deformations of laminated and sandwich plates and transient plane stress deformations of sandwich beams. Models are derived to express the governing equations of linear elasticity in terms of layer-wise continuous variables for composite plates and beams, which allow inter-laminar continuity conditions at layer interfaces to be satisfied. When Legendre-Gauss-Lobatto (LGL) basis functions with the LGL nodes taken as integration points are used to approximate the unknown field variables, the methodology yields a system of discrete equations with a symmetric positive definite coefficient matrix. The main goal of this research is to determine the efficacy of the LSFEM in accurately predicting stresses in laminated composites when subjected to both quasi-static and transient surface tractions. Convergence of the numerical algorithms with respect to the LGL basis functions in space and time (when applicable) is also considered and explored. In the transient analysis of sandwich beams, we study the sensitivity of the first failure load to the beam's aspect ratio (AR), facesheet-core thickness ratio (FCTR) and facesheet-core stiffness ratio (FCSR). We then explore how failure of sandwich beams is affected by considering facesheet and core materials with different in-plane and transverse stiffness ratios. Computed results are compared to available analytical solutions, published results and those found by using the commercial FE software ABAQUS where appropriate / Master of Science / Composite materials are formed by combining two or more materials on a macroscopic scale such that they have better engineering properties than either material individually. They are usually in the form of a laminate comprised of numerous plies with each ply having unidirectional fibers. Laminates are used in all sorts of engineering applications, ranging from boat hulls, racing car bodies and storage tanks. Unlike their homogeneous material counterparts, such as metals, laminated composites present structural designers and analysts a number of computational challenges. Chief among these challenges is the satisfaction of the so-called continuity conditions, which require certain quantities to be continuous at the interfaces of the composite's layers. In this thesis, we use a mathematical model, called a state-space model, that allows us to simultaneously solve for these quantities in the composite structure's domain and satisfy the continuity conditions at layer interfaces. To solve the governing equations that are derived from this model, we use a numerical technique called the least-squares method which seeks to minimize the squares of the governing equations and the associated side condition residuals over the computational domain. With this mathematical model and numerical method, we investigate static and dynamic deformations of laminated composites structures. The goal of this thesis is to determine the efficacy of the proposed methodology in predicting stresses in laminated composite structures when subjected to static and transient mechanical loading.

In this thesis study, a least-squares finite element method (LSFEM) based conjugate heat transfer solver was developed. In the mentioned solver, fluid flow and heat transfer computations were performed separately. This means that the calculated velocity values in the flow calculation part were exported to the heat transfer part to be used in the convective part of the energy equation. Incompressible Navier-Stokes equations were used in the flow simulations. In conjugate heat transfer computations, it is required to calculate the heat transfer in both flow field and solid region. In this study, conjugate behavior was accomplished in a fully coupled manner, i.e., energy equation for fluid and solid regions was solved simultaneously and no boundary conditions were defined on the fluid-solid interface. To assure that the developed solver works properly, lid driven cavity flow, backward facing step flow and thermally driven cavity flow problems were simulated in three dimensions and the findings compared well with the available data from the literature. Couette flow and thermally driven cavity flow with conjugate heat transfer in two dimensions were modeled to further validate the solver. Finally, a microchannel conjugate heat transfer problem was simulated. In the flow solution part of the microchannel problem, conservation of mass was not achieved. This problem was expected since the LSFEM has problems related to mass conservation especially in high aspect ratio channels. In order to overcome the mentioned problem, weight of continuity equation was increased by multiplying it with a constant. Weighting worked for the microchannel problem and the mass conservation issue was resolved. Obtained results for microchannel heat transfer problem were in good agreement in general with the previous experimental and numerical works. In the first computations with the solver / quadrilateral and triangular elements for two dimensional problems, hexagonal and tetrahedron elements for three dimensional problems were tried. However, since only the quadrilateral and hexagonal elements gave satisfactory results, they were used in all the above mentioned simulations.

Least-squares finite element method (LSFEM) is employed to simulate 2-D and axisymmetric flows governed by the compressible Euler equations. Least-squares formulation brings many advantages over classical Galerkin finite element methods. For non-self-adjoint systems, LSFEM result in symmetric positive-definite matrices which can be solved efficiently by iterative methods. Additionally, with a unified formulation it can work in all flight regimes from subsonic to supersonic. Another advantage is that, the method does not require artificial viscosity since it is naturally diffusive which also appears as a difficulty for sharply resolving high gradients in the flow field such as shock waves. This problem is dealt by employing adaptive mesh refinement (AMR) on triangular meshes. LSFEM with AMR technique is numerically tested with various flow problems and good agreement with the available data in literature is seen.

Au cours de ce travail, un modèle physique et numérique 3D du procédé de soudage à l’arc TIG (Tungsten Inert Gas) a été développé dans l’objectif de prédire, en fonction des paramètres opératoires, les grandeurs utiles au concepteur d’assemblages soudés.Le modèle développé, à l’aide du code de calcul aux éléments finis Cast3M, traite les phénomènes physiques agissant dans la pièce et, plus particulièrement, dans le bain de soudage, l’arc étant traité comme une source. Pour ce faire, les équations non-linéaires de la thermohydraulique couplées à celles de l’électromagnétisme sont résolues en régime stationnaire avec un modèle prenant en compte la surface libre déformable du bain de soudage.Une première étape du développement a porté sur la modélisation des phénomènes électromagnétiques par deux méthodes numériques différentes, à comparer les résultats numériques obtenus avec ceux de la littérature. Ensuite, afin de valider le pouvoir prédictif du modèle, des simulations de différentes configurations de soudage d’intérêt ont été étudiées, en variant la composition chimique du matériau, la vitesse de défilement, la pression d’arc imposée et, plus particulièrement, la position de soudage. Des comparaisons avec des expériences et des modèles numériques de la littérature confirment les bonnes tendances obtenues. Enfin, une approche de la modélisation de l’apport de matière a été abordée et des résultats de cette approche ont été montrés. Notre modèle complet constitue donc une base solide pour le développement de modèles de simulation numérique du soudage (SNS) 3D totalement couplés avec l’arc dans le futur et sera intégré dans le logiciel métier WPROCESS. / In this work, we develop a 3D physical and numerical model of the GTA (gas tungsten arc) welding process in order to predict, for given welding parameters, useful quantities for the designer of welded assembly.The model is developed in the Cast3M finite element software and takes into account the main physical phenomena acting in the workpiece and particularly in the weld pool, subject to source terms modeling the arc part of the welding process. A steady solution of this model is thought for and involves the coupling of the nonlinear thermohydaulics and electromagnetic equations together with the displacement of the deformable free surface of the weld pool.A first step in the development consisted in modeling the electromagnetic phenomena with two different numerical methods, in comparing the numerical results obtained with those of the literature. Then, in order to assess the predictive capability of the model, simulations of various welding configurations are performed : variation in the chemical composition of the material, of the welding speed, of the prescribed arc pressure and of the welding positions, which is a focus of this work, are studied. A good agreement is obtained between the results of our model and other experimental and numerical results of the literature. Eventually, a model accounting for metal filling is proposed and its results are discussed. Thus, our complete model can be seen as a solid foundation towards future totally-coupled 3D welding models including the arc and it will be included in the WPROCESS software dedicated to the numerical simulation of welding.

Die Least-Squares Finite-Elemente-Methoden (LSFEMn) basieren auf der Minimierung des Least-Squares-Funktionals, das aus quadrierten Normen der Residuen eines Systems von partiellen Differentialgleichungen erster Ordnung besteht. Dieses Funktional liefert einen a posteriori Fehlerschätzer und ermöglicht die adaptive Verfeinerung des zugrundeliegenden Netzes. Aus zwei Gründen versagen die gängigen Methoden zum Beweis optimaler Konvergenzraten, wie sie in Carstensen, Feischl, Page und Praetorius (Comp. Math. Appl., 67(6), 2014) zusammengefasst werden. Erstens scheinen fehlende Vorfaktoren proportional zur Netzweite den Beweis einer schrittweisen Reduktion der Least-Squares-Schätzerterme zu verhindern. Zweitens kontrolliert das Least-Squares-Funktional den Fehler der Fluss- beziehungsweise Spannungsvariablen in der H(div)-Norm, wodurch ein Datenapproximationsfehler der rechten Seite f auftritt. Diese Schwierigkeiten führten zu einem zweifachen Paradigmenwechsel in der Konvergenzanalyse adaptiver LSFEMn in Carstensen und Park (SIAM J. Numer. Anal., 53(1), 2015) für das 2D-Poisson-Modellproblem mit Diskretisierung niedrigster Ordnung und homogenen Dirichlet-Randdaten. Ein neuartiger expliziter residuenbasierter Fehlerschätzer ermöglicht den Beweis der Reduktionseigenschaft. Durch separiertes Markieren im adaptiven Algorithmus wird zudem der Datenapproximationsfehler reduziert. Die vorliegende Arbeit verallgemeinert diese Techniken auf die drei linearen Modellprobleme das Poisson-Problem, die Stokes-Gleichungen und das lineare Elastizitätsproblem. Die Axiome der Adaptivität mit separiertem Markieren nach Carstensen und Rabus (SIAM J. Numer. Anal., 55(6), 2017) werden in drei Raumdimensionen nachgewiesen. Die Analysis umfasst Diskretisierungen mit beliebigem Polynomgrad sowie inhomogene Dirichlet- und Neumann-Randbedingungen. Abschließend bestätigen numerische Experimente mit dem h-adaptiven Algorithmus die theoretisch bewiesenen optimalen Konvergenzraten. / The least-squares finite element methods (LSFEMs) base on the minimisation of the least-squares functional consisting of the squared norms of the residuals of first-order systems of partial differential equations. This functional provides a reliable and efficient built-in a posteriori error estimator and allows for adaptive mesh-refinement. The established convergence analysis with rates for adaptive algorithms, as summarised in the axiomatic framework by Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl., 67(6), 2014), fails for two reasons. First, the least-squares estimator lacks prefactors in terms of the mesh-size, what seemingly prevents a reduction under mesh-refinement. Second, the first-order divergence LSFEMs measure the flux or stress errors in the H(div) norm and, thus, involve a data resolution error of the right-hand side f. These difficulties led to a twofold paradigm shift in the convergence analysis with rates for adaptive LSFEMs in Carstensen and Park (SIAM J. Numer. Anal., 53(1), 2015) for the lowest-order discretisation of the 2D Poisson model problem with homogeneous Dirichlet boundary conditions. Accordingly, some novel explicit residual-based a posteriori error estimator accomplishes the reduction property. Furthermore, a separate marking strategy in the adaptive algorithm ensures the sufficient data resolution. This thesis presents the generalisation of these techniques to three linear model problems, namely, the Poisson problem, the Stokes equations, and the linear elasticity problem. It verifies the axioms of adaptivity with separate marking by Carstensen and Rabus (SIAM J. Numer. Anal., 55(6), 2017) in three spatial dimensions. The analysis covers discretisations with arbitrary polynomial degree and inhomogeneous Dirichlet and Neumann boundary conditions. Numerical experiments confirm the theoretically proven optimal convergence rates of the h-adaptive algorithm.