Protection from the potentially damaging effects of shock loading is a common design requirement for diverse mechanical structures ranging from shock accelerometers to spacecraft. High-damping viscoelastic materials are employed in the design of geometrically complex impact absorbent components. Since shock transients have a broadband frequency spectrum, it is imperative to properly model frequency dependence of material parameters. The Anelastic Displacement Fields (ADF) method is employed to develop new axisymmetric and plane stress finite elements that are capable of modeling frequency dependent material behavior of linear viscoelastic materials. The new finite elements are used to model and analyze behavior of viscoelastic structures subjected to shock loads. The development of such ADF-based finite element models offers an attractive analytical tool to aid in the design of shock absorbent mechanical filters. This work will also show that it is possible to determine material properties’ frequency dependence by iteratively fitting ADF model predictions to experimental results. A series of experiments designed to validate the axisymmetric and plane stress finite element models are performed. These experiments involve the propagation of longitudinal waves through elastic and viscoelastic rods, and behavior of elastomeric mechanical filters subjected to shock. Comparison of model predictions to theory and experiments confirm that ADF-based finite element models are capable of capturing phenomena such as geometric dispersion and viscoelastic attenuation of longitudinal waves in rods as well as modeling the behavior of mechanical filters subjected to shock. / Ph. D.

A class of immersed finite element (IFE) spaces is developed for solving elliptic boundary value problems that have interfaces. IFE spaces are finite element approximation spaces which are based upon meshes that can be independent of interfaces in the domain. Three different quadratic IFE spaces and their related biquadratic IFE spaces are introduced here for the purposes of solving both forward and inverse elliptic interface problems in 1D and 2D. These different spaces are constructed by (i) using a hierarchical approach, (ii) imposing extra continuity requirements or (iii) using a local refinement technique. The interpolation properties of each space are tested against appropriate testing functions in 1D and 2D. The IFE spaces are also used to approximate the solution of a forward elliptic interface problem using the Galerkin finite element method and the mixed least squares finite element method. Finally, one appropriate space is selected to solve an inverse interface problem using either an output least squares approach or the least squares with mixed equation error method. / Ph. D.

We study the behavior of numerical stabilization schemes in the context of linear quadratic regulator (LQR) control problems for convection diffusion equations. The motivation for this effort comes from the observation that when linearization is applied to fluid flow control problems the resulting equations have the form of a convection diffusion equation. This effort is focused on the specific problem of computing the feedback functional gains that are the kernels of the feedback operators defined by solutions of operator Riccati equations. We develop a stabilization scheme based on the Galerkin Least Squares (GLS) method and compare this scheme to the standard Galerkin finite element method. We use cubic B-splines in order to keep the higher order terms that occur in GLS formulation. We conduct a careful numerical investigation into the convergence and accuracy of the functional gains computed using stabilization. We also conduct numerical studies of the role that the stabilization parameter plays in this convergence. Overall, we discovered that stabilization produces much better approximations to the functional gains on coarse meshes than the unstabilized method and that adjustments in the stabilization parameter greatly effects the accuracy and convergence rates. We discovered that the optimal stabilization parameter for simulation and steady state analysis is not necessarily optimal for solving the Riccati equation that defines the functional gains. Finally, we suggest that the stabilized GLS method might provide good initial values for iterative schemes on coarse meshes. / Ph. D.

An integrated procedure, combining finite element modeling and fatigue analysis methods, is developed and applied to the fatigue optimization of a notched, induction hardened, steel shaft subjected to combined bending and torsional loading. Finite element analysis is used first to develop unit-load factors for generating stress-time histories, and then, employing thermo-elastic techniques, to determine the residual stresses resulting from induction hardening. These stress fields are combined using elastic superposition, and incorporated in a fatigue analysis procedure to predict failure location and lifetime. Through systematic variation of geometry, processing, and loading parameters, performance surfaces are generated from which optimum case depths for maximizing shaft fatigue performance are determined. General implications of such procedures to the product development process are discussed. / Master of Science

High-fidelity computer simulations of childbirth are time consuming, making them impractical for guiding decision-making during obstetric emergencies. The complex geometry, micro-structure, and large finite deformations undergone by the vagina during childbirth result in material and geometric nonlinearities, complicated boundary conditions, and nonhomogeneities within finite element (FE) simulations. Such nonlinearities pose a significant challenge for numerical solvers, increasing the computational time. Simplifying assumptions can reduce the computational time significantly, but this usually comes at the expense of simulation accuracy. The work herein proposed the use of reduced order modeling (ROM) techniques to create surrogate models that capture experimentally-measured displacement fields of rat vaginal tissue during inflation testing in order to attain both the accuracy of higher-fidelity models and the speed of lower-fidelity simulations. The proper orthogonal decomposition (POD) method was used to extract the significant information from FE simulations generated by varying the luminal pressure and the parameters that introduce the anisotropy in the selected constitutive model. In our first study, a new data-driven (DD) variational multiscale (VMS) ROM framework was extended to obtain the displacement fields of rat vaginal tissue subjected to ramping luminal pressure. For comparison purposes, we also investigated the classical Galerkin ROM (G-ROM). In our numerical study, both the G-ROM and the DD-VMS-ROM decreased the FE computational cost by orders of magnitude without a significant decrease in numerical accuracy. Furthermore, the DD-VMS-ROM improved the G-ROM accuracy at a modest computational overhead. Our numerical investigation showed that ROM had the potential to provide efficient and accurate computational tools to describe vaginal deformations, with the ultimate goal of improving maternal health. Our second study compared two common computational strategies for surrogate modeling, physics-based G-ROM and data-driven machine learning (ML), for decreasing the cost of FE simulations of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. Since there are many methods associated with each modeling approach, to provide a fair and natural comparison, we selected a basic model from each category. From the ROM strategies, we considered a simplified G-ROM that is based on the linearization of the underlying nonlinear FE equations. From the ML strategies, we selected a feed-forward dense neural network (DNN) to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the POD coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical comparisons of G-ROM, ML, and POD-ML took place in the reconstructive regime. The numerical results showed that the G-ROM outperformed the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The POD-ML model improved on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model did not improve on the error performance of the ML. In our last study, we expanded our investigation of ML methods for surrogate modeling by comparing the performance of a DNN similar to what was used previously to that of a convolutional neural network (CNN) using 1-D convolution on the input parameters from FE simulations of active vaginal tearing. The new FE simulations utilized a custom continuum damage model that provided material damage and failure properties to an existing anisotropic hyperelastic constitutive model to replicate experimentally-observed tear propagation behaviors. We employed our DNN and CNN models to create mappings from constitutive model parameters, geometric properties of the propagating tear, and luminal pressure values to either the full FE displacement history or the POD coefficients of the displacement history. The root-mean-square error (RMSE) with respect to the FE displacement history achieved by full order output ML predictions was reproducible with POD-ML using a basis of only dimension l=10. Additionally, an order of magnitude reduction in offline time was observed using POD-ML over full-order ML with minimal difference between DNN and CNN architectures. Differences in online computational costs between ML and POD-ML were found to be negligible, but the DNNs produced predictions slightly faster than the CNNs, though both online times were on the same order of magnitude. While convolution did not significantly aid the regression task at hand, POD-ML was demonstrated to be an efficient and effective approach for surrogate modeling of the FE tear propagation model, approximating the displacement history with RMSE less than 0.1 mm and generating results 7 orders of magnitude faster than the FE model. This set of baseline numerical investigations serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, material properties, and tissue damage mechanics of the human vagina, as well as their changes during labor. / Doctor of Philosophy / Computer simulations of childbirth are extremely time-consuming, making them impractical for guiding decision-making by obstetricians when a patient is entering labor. The complex geometry, material microstructure, and large deformations undergone by the vagina during childbirth result in material and geometric properties that are challenging to mathematically model. Consequently, numerical solver methods (e.g., finite elements) require large amounts of time to simulate childbirth. Simplifying assumptions can reduce computational time, but this simplification usually comes at the expense of simulation accuracy. The work of this dissertation proposes the use of several techniques to reduce model complexity and create accurate approximations and predictions of results from full-order models (FOMs) with profound reductions in computational time. Our first study used reduced order models (ROMs) to extract the significant information from a FOM of the rat vagina subjected to inflation. We compared a basic ROM and an advanced, data-driven ROM. Our second study compared the basic ROM to a basic machine learning (ML) technique for approximating a FOM that simulated inflation of the rat vagina with a pre-imposed tear. A hybrid technique incorporating elements of both ROM and ML to approximate FOM results was also considered. Our final study made use of ML and hybrid techniques using a more advanced neural network (a convolutional neural network). These ML models were used to predict the results of a FOM simulation of vaginal tear propagation. These numerical investigations serve as a starting point for future development of computer simulations using state-of-the-art ROM and ML strategies as well as more realistic models for the mechanics of the human vagina during childbirth.

In der vorliegenden Arbeit wird ein gemischtes Element gegen Locking-Effekte untersucht. Dazu wird ein Fünf-Feld-Hu-Washizu-Funktional (CoFEM-Element) für lineare und quadratische Hexaeder-Elemente unter einer hyperelastischen, isotropen, polykonvexen sowie einer transversal-isotropen Materialformulierung implementiert. Die resultierenden nichtlinearen Gleichungen werden mithilfe eines Mehrebenen-NEWTON-RAPHSON-Verfahren unter Beachtung einer konsistenten Linearisierung gelöst. Als repräsentatives Beispiel der numerischen Untersuchungen dient der einseitig eingespannte Cook-Balken mit einer quadratischen Druckverteilung am Rand. Zur Beurteilung des CoFEM-Elements wird das räumliche Konvergenzverhalten für unterschiedliche Polynomgrade und für verschiedene Netze unter Beachtung der algorithmischen Effizienz untersucht. / This paper presents a mixed finite element formulation of Hu-Washizu type (CoFEM) designed to reduce locking effects with respect to a linear and quadratic approximation in space. We consider a hyperelastic, isotropic, polyconvex material formulation as well as transverse isotropy. The resulting nonlinear algebraic equations are solved with a multilevel NEWTON-RAPHSON method. As a numerical example serves a cook-like cantilever beam with a quadratic distribution of in-plane load on the Neumann boundary. We analyze the spatial convergence with respect to the polynomial degree of the underlying Lagrange polynomials and with respect to the level of mesh refinement in terms of algorithmic efficiency.

Cette thèse s'intéresse à la génération, à l'utilisation et à l'évaluation des maillages hex-dominants, composés d'hexaèdres et de tétraèdres, dans la cadre de la simulation numérique par la méthode des éléments finis. Les éléments finis hexaédriques sont souvent préférés aux éléments tétraédriques car ils offrent un meilleur ratio entre précision et temps de calcul dans un certain nombre de situations. Cependant, si la génération automatique de maillages tétraédriques est aujourd'hui un domaine bien maîtrisé, ce n'est pas le cas de la génération de maillages hexaédriques alignés avec le bord, qui reste un problème largement ouvert. En l'absence de progrès significatifs, les approches actuelles se contentent de maillages hex-dominants afin de tirer parti des performances supérieures des hexaèdres et de la flexibilité géométrique des tétraèdres, qui rend possible le maillage automatique. Dans une première partie, nous développons des algorithmes robustes pour la génération de maillages hex-dominants à partir de champs de directions, notamment pour l'isolement et le remplissage des régions difficiles à mailler (singularités et autres dégénérescences). Dans la seconde partie, nous essayons de déterminer dans quelles situations et dans quelle mesure les maillages hexaédriques, et hex-dominants générés précédemment, sont plus intéressants que les maillages tétraédriques. Ceci implique spécifiquement d'étudier plusieurs manières d'effectuer des simulations par éléments finis avec les maillages hybrides, dont une approche où nous utilisons des contraintes de continuité pour maillages non-conformes. Pour mesurer l'influence du maillage sur l'approximation des solutions, nous proposons une nouvelle méthode d'échantillonnage pour calculer très efficacement des distances globales entre solutions éléments finis définies sur des domaines compliqués / This thesis focuses on generation, usage and evaluation of hex-dominant meshes, which are made of hexaehedra and tetrahedra, in the context of the finite element method. Hexahedron finite elements are often preferred to tetrahedron elements because they offer a better compromise between accuracy and computation time in certain situations. However, if tetrahedral meshing is a well mastered subject, it is not the case of hexahedral meshing. Generating hexahedral meshes with elements aligned to the borders is still an open and difficult problem. Meanwhile, current automated approaches can use hex-dominant meshes in order to take advantage of both hexahedron accuracy and geometrical flexibility of tetrahedra. In the first part, we develop robust algorithms for the generation of hex-dominant meshes with elements aligned with the borders. Specifically, we propose a method to extract and fill the areas where hexahedral meshing is difficult (singularities and degeneracies). In the second part, we try to identify and to quantify the advantages of hexahedral and hex-dominant meshes over tetrehedral ones. This requires to study various ways to apply the finite element method on hybrid meshes, including one in which we propose to use continuity constraints on hexahedral-tetrahedral non-conforming meshes. To measure the impact of meshes on the finite element accuracy, we develop a new sampling method which allows to compute efficiently global distances between finite element solutions defined on complicated 3D domains

Submitted by PEDRO ROGERIO CLETO null (pedro.constant@gmail.com) on 2016-06-28T20:02:04Z No. of bitstreams: 1 Dissertacao_PedroCleto_VF.pdf: 6736443 bytes, checksum: adef1b42d29662c6340d24f74ffa54ec (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-06-30T17:45:25Z (GMT) No. of bitstreams: 1 cleto_pr_me_bauru.pdf: 6736443 bytes, checksum: adef1b42d29662c6340d24f74ffa54ec (MD5) / Made available in DSpace on 2016-06-30T17:45:25Z (GMT). No. of bitstreams: 1 cleto_pr_me_bauru.pdf: 6736443 bytes, checksum: adef1b42d29662c6340d24f74ffa54ec (MD5) Previous issue date: 2016-05-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A técnica de fraturamento hidráulico é amplamente utilizada na indústria petrolífera para aumentar a permeabilidade da rocha-reservatório numa região próxima ao poço e permitir a extração, e consequente produção, de hidrocarbonetos armazenados em seus poros. Primeiramente a rocha é perfurada criando-se um poço e então injeta-se fluido a uma pressão suficientemente alta para fraturar a rocha. A injeção contínua de fluido permite que as fraturas se propaguem pelo reservatório, formando assim canais de alta permeabilidade. A modelagem e simulação computacional de fraturamento hidráulico são complexas em função da física envolvida no processo. O presente trabalho objetiva o estudo da formação e propagação de fraturas induzidas hidraulicamente em meios rochosos de baixa permeabilidade e também tem o propósito de verificar se a metodologia adotada é capaz de reproduzir características apresentadas num processo de fraturamento hidráulico, como a pressão necessária para causar a ruptura da rocha. Para tal, apresenta-se a técnica de fragmentação da malha utilizando elementos finitos de elevada razão de aspecto (ou elementos de interface) para representar a fratura, aos quais são atribuídas relações constitutivas baseadas na mecânica do dano. Além disso, os elementos de interface também possuem um acoplamento hidromecânico capaz de representar o canal de alta permeabilidade devido à ocorrência da fratura. Os resultados obtidos mostraram que os elementos de interface associados à técnica de fragmentação da malha foram capazes de representar tanto a formação quanto a propagação das fraturas induzidas hidraulicamente. Os resultados também mostraram que as curvas de pressão obtidas corresponderam àquelas idealizadas teoricamente. / The hydraulic fracturing technique is widely used to increase the permeability of reservoirs in the vicinity of the well and to allow the extraction and subsequent production of hydrocarbons trapped in its pores. Firstly, the rock is drilled, creating a well and then a fluid is injected at a sufficiently high pressure to fracture the rock. The continuous fluid injection allows the fractures to propagate through the reservoir, thereby forming some high permeability paths. The computer modeling and simulation of hydraulic fracturing are complex due to the physics involved in the process. This work aims to study the formation and propagation of hydraulically induced fractures in rocky media with low permeability and also aims to verify if the adopted methodology is able to reproduce the characteristics presented in a hydraulic fracturing process, as for instance, the required pressure to cause the breakdown of the rock. For this purpose, it is presented the mesh fragmentation technique using finite elements with a high aspect ratio (or interface elements) to represent the fracture, which are assigned constitutive relations based on damage mechanics. Besides, the interface elements also have a hydromechanical coupling which is able to represent the high permeability path due to the fracture. The results showed that the interface elements associated with the mesh fragmentation technique were able to represent both the formation and the propagation of hydraulically induced fractures. The results also showed that the obtained pressure curves corresponded to those theoretically idealized.

Le calcul de la durée de vie des Composites tissés à Matrice Céramique (CMC) nécessite de déterminer l’évolution de la densité de fissures dans le matériau(pouvant atteindre 10 mm-1). Afin de les représenter finement on se propose de travailler à l’échelle mésoscopique. Les méthodes de type Embedded Finite Element (EFEM) nous ont paru être les plus adaptées au problème. Elles permettent une représentation discrète des fissures sans introduire de degrés de liberté additionnels.Notre choix s’est porté sur une EFEM s’affranchissant d’itérations élémentaires et appelée Augmented Finite Element Method (AFEM). Une variante d’AFEM, palliant des lacunes de la méthode originale, a été développée. Nous avons démontré que,sous certaines conditions, AFEM et la méthode des éléments finis classique (FEM) étaient équivalentes. Nous avons ensuite comparé la précision d’AFEM et de FEM pour représenter des discontinuités fortes et faibles. Les travaux de thèse se concluent par des exemples d’application de la méthode aux CMC. / Computing the lifetime of woven Ceramic Matrix Composites (CMC) requires evaluating the crack density in the material (which can reach 10 mm-1). Numerical simulations at the mesoscopic scale are needed to precisely estimate it. Embedded Finite Element Methods (EFEM) seem to be the most appropriate to do so. They allow for a discrete representation of cracks with no additional degrees of freedom.We chose to work with an EFEM free from local iterations named the Augmented Finite Element Method (AFEM). Improvements over the original AFEM have been proposed. We also demonstrated that, under one hypothesis, the AFEM and the classical Finite Element Method (FEM) are fully equivalent. We then compare the accuracy of the AFEM and the classical FEM to represent weak and strong discontinuities. Finally, some examples of application of AFEM to CMC are given.

In der vorliegenden Arbeit wird ein gemischtes Element gegen Locking-Effekte untersucht. Dazu wird ein Fünf-Feld-Hu-Washizu-Funktional (CoFEM-Element) für lineare und quadratische Hexaeder-Elemente unter einer hyperelastischen, isotropen, polykonvexen sowie einer transversal-isotropen Materialformulierung implementiert. Die resultierenden nichtlinearen Gleichungen werden mithilfe eines Mehrebenen-NEWTON-RAPHSON-Verfahren unter Beachtung einer konsistenten Linearisierung gelöst. Als repräsentatives Beispiel der numerischen Untersuchungen dient der einseitig eingespannte Cook-Balken mit einer quadratischen Druckverteilung am Rand. Zur Beurteilung des CoFEM-Elements wird das räumliche Konvergenzverhalten für unterschiedliche Polynomgrade und für verschiedene Netze unter Beachtung der algorithmischen Effizienz untersucht. / This paper presents a mixed finite element formulation of Hu-Washizu type (CoFEM) designed to reduce locking effects with respect to a linear and quadratic approximation in space. We consider a hyperelastic, isotropic, polyconvex material formulation as well as transverse isotropy. The resulting nonlinear algebraic equations are solved with a multilevel NEWTON-RAPHSON method. As a numerical example serves a cook-like cantilever beam with a quadratic distribution of in-plane load on the Neumann boundary. We analyze the spatial convergence with respect to the polynomial degree of the underlying Lagrange polynomials and with respect to the level of mesh refinement in terms of algorithmic efficiency.