Browsing by Subject "Finite elements"
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Item Analysis, implementation, and verification of a discontinuous galerkin method for prediction of storm surges and coastal deformation(2011-08) Mirabito, Christopher Michael; Dawson, Clinton N.; Demkowicz, Leszek F.; Gamba, Irene M.; Ghattas, Omar; Kim, WonsuckStorm surge, the pileup of seawater occurring as a result of high surface stresses and strong currents generated by extreme storm events such as hurricanes, is known to cause greater loss of life than these storms' associated winds. For example, inland flooding from the storm surge along the Gulf Coast during Hurricane Katrina killed hundreds of people. Previous storms produced even larger death tolls. Simultaneously, dune, barrier island, and channel erosion taking place during a hurricane leads to the removal of major flow controls, which significantly affects inland inundation. Also, excessive sea bed scouring around pilings can compromise the structural integrity of bridges, levees, piers, and buildings. Modeling these processes requires tightly coupling a bed morphology equation to the shallow water equations (SWE). Discontinuous Galerkin finite element methods (DGFEMs) are a natural choice for modeling this coupled system, given the need to solve these problems on large, complicated, unstructured computational meshes, as well as the desire to implement hp-adaptivity for capturing the dynamic features of the solution. Comprehensive modeling of these processes in the coastal zone presents several challenges and open questions. Most existing hydrodynamic models use a fixed-bed approach; the bottom is not allowed to evolve in response to the fluid motion. With respect to movable-bed models, there is no single, generally accepted mathematical model in use. Numerical challenges include coupling models of processes that exhibit disparate time scales during fair weather, but possibly similar time scales during intense storms. The main goals of this dissertation include implementing a robust, efficient, tightly-coupled morphological model using the local discontinuous Galerkin (LDG) method within the existing Advanced Circulation (ADCIRC) modeling framework, performing systematic code and model verification (using test cases with known solutions, proven convergence rates, or well-documented physical behavior), analyzing the stability and accuracy of the implemented numerical scheme by way of a priori error estimates, and ultimately laying some of the necessary groundwork needed to simultaneously model storm surges and bed morphodynamics during extreme storm events.Item Domain decomposition methods in geomechanics(2012-08) Florez Guzman, Horacio Antonio; Wheeler, Mary F. (Mary Fanett); Delshad, Mojdeh; Mear, Mark; Landis, Chad; Rodriguez, AdolfoHydrocarbon production or injection of fluids in the reservoir can produce changes in the rock stresses and in-situ geomechanics, potentially leading to compaction and subsidence with harmful effects in wells, cap-rock, faults, and the surrounding environment as well. In order to tackle these changes and their impact, accurate simulations are essential. The Mortar Finite Element Method (MFEM) has been demonstrated to be a powerful technique in order to formulate a weak continuity condition at the interface of sub-domains in which different meshes, i.e. non-conforming or hybrid, and / or variational approximations are used. This is particularly suitable when coupling different physics on different domains, such as elasticity and poroelasticity, in the context of coupled flow and geomechanics. In this dissertation, popular Domain Decomposition Methods (DDM) are implemented in order to carry large simulations by taking full advantage of current parallel computer architectures. Different solution schemes can be defined depending upon the way information is exchanged between sub-domain interfaces. Three different schemes, i.e. Dirichlet-Neumann (DN), Neumann-Neumann (NN) and MFEM, are tested and the advantages and disadvantages of each of them are identified. As a first contribution, the MFEM is extended to deal with curve interfaces represented by Non-Uniform Rational B-Splines (NURBS) curves and surfaces. The goal is to have a more robust geometrical representation for mortar spaces, which allows gluing non-conforming interfaces on realistic geometries. The resulting mortar saddle-point problem will be decoupled by means of the DN- and NN-DDM. Additionally, a reservoir geometry reconstruction procedure based on NURBS surfaces is presented as well. The technique builds a robust piecewise continuous geometrical representation that can be exploited by MFEM in order to tackle realistic problems, which is a second contribution. Tensor product meshes are usually propagated from the reservoir in a conforming way into its surroundings, which makes non-matching interfaces highly attractive in this case. In the context of reservoir compaction and subsidence estimation, it is common to deal with serial legacy codes for flow. Indeed, major reservoir simulators such as compositional codes lack parallelism. Another issue is the fact that, generally speaking, flow and mechanics domains are different. To overcome this limitation, a serial-parallel approach is proposed in order to couple serial flow codes with our parallel mechanics code by means of iterative coupling. Concrete results in loosely coupling are presented as a third contribution. As a final contribution, the DN-DDM is applied to couple elasticity and plasticity, which seems very promising in order to speed up computations involving poroplasticity. Several examples of coupling of elasticity, poroelasticity, and plasticity ranging from near-wellbore applications to field level subsidence computations help to show that the proposed methodology can handle problems of practical interest. In order to facilitate the implementation of complex workflows, an advanced Python wrapper interface that allows programming capabilities have been implemented. The proposed serial-parallel approach seems to be appropriate to handle geomechanical problems involving different meshes for flow and mechanics as well as coupling parallel mechanistic codes with legacy flow simulators.Item Nonconforming finite element methods for fluid-structure interaction(2005-08) Swim, Edward W.; Seshaiyer, Padmanabhan; Allen, Edward J.; Barnard, Roger W.; Manservisi, SandroAccurately simulating the interaction between a fluid and a structure remains a challenging problem in computational mathematics. One difficult aspect of this process is to efficiently couple the geometry of each domain as well as the systems of equations which model the physical properties of each media. The primary objective of this dissertation is to systematically develop and analyze sophisticated computational techniques which employ finite element methods for solving fluid-structure interaction problems that arise in science and engineering applications. First, a one dimensional problem is presented in order to introduce the approximation techniques we will extend to higher dimensions. Additionally, many two-dimensional fluid-structure interactions can be reduced to one dimension under certain assumptions about the geometry of the subdomains along with inflow and outflow boundary conditions. In this context, we will establish consistency and stability properties for our discretization methods. Secondly, we will develop a nonconforming finite element methodology using a three-field formulation to solve a coupled physical system comprised of two adjacent domains, one containing a viscous, incompressible fluid and the other an elastic structure. Our method will employ an arbitrary Lagrangian Eulerian strategy to formulate the governing equations for the fluid coupled with a linear elasticity model describing the deformation of the solid in order to simulate a full unsteady physical phenomenon. This formulation is analogous to the one used in the one-dimensional problem, although more complicated constraints are required due to the geometry of the problem. Again, consistency and stability properties are established for this technique. Finally, computational results which establish consistency and convergence of our numerical methods for the one dimensional problem are presented. These results include verification that our discretization technique is first order in time and that when a nonconforming technique is applied, i.e., when different polynomial degrees of approximation are used in the fluid and structure domains, the solution obtained is no worse than those computed using a conforming method. Additionally, convergence of the method under refinement of the computational grid, known as the h-method, is explored. Thus, our numerical experiments provide confidence in the discretization techniques established in the previous chapters as well as insight on how to appropriately construct finite element code for the two dimensional problem.Item Nonconforming finite element methods for fluid-structure interaction problems(Texas Tech University, 2005-08) Swim, Edward W.; Seshaiyer, Padmanabhan; Allen, Edward J.; Barnard, Roger W.; Manservisi, SandroAccurately simulating the interaction between a fluid and a structure remains a challenging problem in computational mathematics. One difficult aspect of this process is to efficiently couple the geometry of each domain as well as the systems of equations which model the physical properties of each media. The primary objective of this dissertation is to systematically develop and analyze Sophisticated computational techniques which employ finite element methods for solving fluid-structure interaction problems that arise in science and engineering applications. First, a one-dimensional problem is presented in order to introduce the approximation techniques we will extend to higher dimensions. Additionally, many two-dimensional fluid-structure interactions can be reduced to one dimension under certain assumptions about the geometry of the subdomains along with inflow and outflow boundary conditions. In this context, we will establish consistency and stability properties for our discretization methods. Secondly, we will develop a nonconforming finite element methodology using a three-field formulation to solve a coupled physical system comprised of two adjacent domains, one containing a viscous, incompressible fluid and the other an elastic structure. Our method will employ an arbitrary Lagrangian-Eulerian strategy to formulate the governing equations for the fluid coupled with a linear elasticity model describing the deformation of the solid in order to simulate a full unsteady physical phenomenon. This formulation is analogous to the one used in the one-dimensional problem, although more complicated constraints are required due to the geometry of the problem. Again, consistency and stability properties are established for this technique. Finally, computational results which establish consistency and convergence of our numerical methods for the one-dimensional problem are presented. These results include verification that our discretization technique is first order in time and that when a nonconforming technique is applied, i.e., when different polynomial degrees of approximation are used in the fluid and structure domains, the solution obtained is no worse than those computed using a conforming method. Additionally, convergence of the method under refinement of the computational grid, known as the h-method, is explored. Thus, our numerical experiments provide confidence in the discretization techniques established in the previous chapters as well as insight on how to appropriately construct finite element code for the two-dimensional problem.Item On the crushing of honeycomb under axial compression(2010-12) Wilbert, Adrien; Kyriakides, S.; Ravi-Chandar, KrishnaswamyThis thesis presents a comprehensive study of the compressive response of hexagonal honeycomb panels from the initial elastic regime to a fully crushed state. Expanded aluminum alloy honeycomb panels with a cell size of 0.375 in (9.53 mm), a relative density of 0.026, and a height of 0.625 in (15.9 mm) are laterally compressed quasi statically between rigid platens under displacement control. The cells buckle elastically and collapse at a higher stress due to inelastic action. Deformation then first localizes at mid-height and the cells crush by progressive formation of folds; associated with each fold family is a stress undulation. The response densifies when the whole panel height is consumed by folds. The buckling, collapse, and crushing events are simulated numerically using finite element models involving periodic domains of a single or several characteristic cells. The models idealize the microstructure as hexagonal, with double walls in one direction. The nonlinear behavior is initiated by elastic buckling while inelastic collapse that leads to the localization observed in the experiments occurs at a significantly higher load. The collapse stress is found to be mildly sensitive to various problem imperfections. For the particular honeycomb studied, the collapse stress is 67% higher than the buckling stress. It was also shown that all aspects of the compressive behavior can be reproduced numerically using periodic domains with a fine mesh capable of capturing the complexity of the folds. The calculated buckling stress is reduced when considering periodic square domains as the compatibility of the buckles between neighboring cells tends to make the structure more compliant. The mode consisting of three half waves is observed in every simulation but its amplitude is seen to be accented at the center of the domains. The calculated crushing response is shown to better resemble measured ones when a 4x4 cell domain is used, which is smoother and reproduces decays in the amplitude of load peaks. However, the average crushing stress can be captured with engineering accuracy even from a single cell domain.Item On the effect of Lüders bands on the bending of steel tubes(2011-12) Hallai, Julian de Freitas; Kyriakides, S.; Engelhardt, Michael D.; Landis, Chad M.; Liechti, Kenneth M.; Ravi-Chandar, KrishnaswaIn several practical applications, hot-finished steel pipe that exhibits Lüders bands is bent to strains of 2-3%. Lüders banding is a material instability that leads to inhomogeneous plastic deformation in the range of 1-4%. This work investigates the influence of Lüders banding on the inelastic response and stability of tubes under rotation controlled pure bending. It starts with the results of an experimental study involving tubes of several diameter-to-thickness ratios in the range of 33.2 to 14.7 and Lüders strains of 1.8% to 2.7%. In all cases, the initial elastic regime terminates at a local moment maximum and the local nucleation of narrow angled Lüders bands of higher strain on the tension and compression sides of the tube. As the rotation continues, the bands multiply and spread axially causing the affected zone to bend to a higher curvature while the rest of the tube is still at the curvature corresponding to the initial moment maximum. With further rotation of the ends, the higher curvature zone(s) gradually spreads while the moment remains essentially unchanged. For relatively low D/t tubes and/or short Lüders strains, the whole tube eventually is deformed to the higher curvature entering the usual hardening regime. Subsequently it continues to deform uniformly until the usual limit moment instability is reached. For high D/t tubes and/or materials with longer Lüders strains, the propagation of the larger curvature is interrupted by collapse when a critical length is Lüders deformed leaving behind part of the structure essentially undeformed. The higher the D/t and/or the longer the Lüders strain is, the shorter the critical length. This class of problems is analyzed using 3D finite elements while the material is modeled as an elastic-plastic solid with an “up-down-up” response over the extent of the Lüders strain, followed by hardening. The analysis reproduces the main features of the mechanical behavior provided the unstable part of the response is suitably calibrated. The uniform curvature elastic regime terminates with the nucleation of localized banded deformation. The bands appear in pockets on the most deformed sites of the tube and propagate into the hitherto intact part of the structure while the moment remains essentially unchanged. The Lüders-deformed section has a higher curvature, ovalizes more than the rest of the tube, and develops wrinkles with a characteristic wavelength. For every tube D/t there exists a threshold of Lüders strain separating the two types of behavior. This bounding value of Lüders strain was studied parametrically.Item On the hydraulic bulge testing of thin sheets(2013-12) Mersch, John Philip; Kyriakides, S.The bulge test is a commonly used experiment to establish the material stress-strain response at the highest possible strain levels. It consists of a metal sheet placed in a die with a circular opening. It is clamped in place and inflated with hydraulic pressure. In this thesis, a bulge testing apparatus was designed, fabricated, calibrated and used to measure the stress-strain response of an aluminum sheet metal and establish its onset of failure. The custom design incorporates a draw-bead for clamping the plate. A closed loop controlled servohydraulic pressurization system consisting of a pressure booster is used to pressurize the specimens. Deformations of the bulge are monitored with a 3D digital image correlation (DIC) system. Bulging experiments on 0.040 in thick Al-2024-T3 sheets were successfully performed. The 3D nature of the DIC enables simultaneous estimates of local strains as well as the local radius of curvature. The successful performance of the tests required careful design of the draw-bead clamping arrangement. Experiments on four plates are presented, three of which burst in the test section as expected. Finite deformation isotropic plasticity was used to extract the true equivalent stress-strain responses from each specimen. The bulge test results correlated well with the uniaxial results as they tended to fall between tensile test results in the rolling and transverse directions. The bulge tests results extended the stress-strain response to strain levels of the order of 40%, as opposed to failure strains of the order of 10% for the tensile tests. Three-dimensional shell and solid models were used to investigate the onset of localization that precedes failure. In both models, the calculated pressure-deformation responses were found to be in reasonable agreement with the measured ones. The solid element model was shown to better capture the localization and its evolution. The corresponding pressure maximum was shown to be imperfection sensitive.Item Space-time discontinuous Petrov-Galerkin finite elements for transient fluid mechanics(2016-05) Ellis, Truman Everett; Demkowicz, Leszek; Moser, Robert D; Hughes, Thomas J.R; Dawson, Clint N; Bui, TanInitial mesh design for computational fluid dynamics can be a time-consuming and expensive process. The stability properties and nonlinear convergence of most numerical methods rely on a minimum level of mesh resolution. This means that unless the initial computational mesh is fine enough, convergence can not be guaranteed. Any meshes below this minimum resolution level are termed to be in the ``pre-asymptotic regime.'' This condition implies that meshes need to in some way anticipate the solution before it is known. On top of the minimum requirement that the surface meshes must adequately represent the geometry of the problem under consideration, resolution requirements on the volume mesh make the CFD practitioner's job significantly more time consuming. In contrast to most other numerical methods, the discontinuous Petrov-Galerkin finite element method retains exceptional stability on extremely coarse meshes. DPG is also inherently very adaptive. It is possible to compute the residual error without knowledge of the exact solution, which can be used to robustly drive adaptivity. This results in a very automated technology, as the user can initialize a computation on the coarsest mesh which adequately represents the geometry then step back and let the program solve and adapt iteratively until it resolves the solution features. A common complaint of minimum residual methods by computational fluid dynamics practitioners is that they are not locally conservative. In this thesis, this concern is addressed by developing a locally conservative DPG formulation by augmenting the system with Lagrange multipliers. The resulting DPG formulation is then proved to be robust and shown to produce superior numerical results over standard DPG on a selection of test problems. Adaptive convergence to steady incompressible and compressible Navier-Stokes solutions was explored in Jesse Chan's and Nathan Roberts' dissertations. Space-time offers a natural extension to transient problems as it preserves the stability and adaptivity properties of DPG in the time dimension. Space-time also offers more extensive parallelization capability than problems treated with traditional time stepping as it allows multigrid concurrently in both space and time. A proof of concept space-time DPG formulation is developed for transient convection-diffusion. The robust test norms derived for steady convection-diffusion are extended to the space-time case and proofs of robustness are provided. Numerical results verify the robust behavior and near $L^2$ optimality of the resulting solutions. The space-time formulation for convection-diffusion is then extended to transient incompressible and compressible Navier-Stokes by analogy. Several numerical experiments are performed, but a mathematical analysis is not attempted for these nonlinear problems. Several side topics are explored such as a study of the compressible Navier-Stokes equations under various variable transformations and the development of consistent test norms through the concept of physical entropy.Item Spectral/hp Finite Element Models for Fluids and Structures(2012-07-16) Payette, GregoryWe consider the application of high-order spectral/hp finite element technology to the numerical solution of boundary-value problems arising in the fields of fluid and solid mechanics. For many problems in these areas, high-order finite element procedures offer many theoretical and practical computational advantages over the low-order finite element technologies that have come to dominate much of the academic research and commercial software of the last several decades. Most notably, we may avoid various forms of locking which, without suitable stabilization, often plague low-order least-squares finite element models of incompressible viscous fluids as well as weak-form Galerkin finite element models of elastic and inelastic structures. The research documented in this dissertation includes applications of spectral/hp finite element technology to an analysis of the roles played by the linearization and minimization operators in least-squares finite element models of nonlinear boundary value problems, a novel least-squares finite element model of the incompressible Navier-Stokes equations with improved local mass conservation, weak-form Galerkin finite element models of viscoelastic beams and a high-order seven parameter continuum shell element for the numerical simulation of the fully geometrically nonlinear mechanical response of isotropic, laminated composite and functionally graded elastic shell structures. In addition, we also present a simple and efficient sparse global finite element coefficient matrix assembly operator that may be readily parallelized for use on shared memory systems. We demonstrate, through the numerical simulation of carefully chosen benchmark problems, that the finite element formulations proposed in this study are efficient, reliable and insensitive to all forms of numerical locking and element geometric distortions.Item Three transdimensional factors for the conversion of 2D acoustic rough surface scattering model results for comparison with 3D scattering(2013-12) Tran, Bryant Minh; Wilson, Preston S.; Isakson, Marcia J.Rough surface scattering is a problem of interest in underwater acoustic remote sensing applications. To model this problem, a fully three-dimensional (3D) finite element model has been developed, but it requires an abundance of time and computational resources. Two-dimensional (2D) models that are much easier to compute are often employed though they don’t natively represent the physical environment. Three quantities have been developed that, when applied, allow 2D rough surface scattering models to be used to predict 3D scattering. The first factor, referred to as the spreading factor, adopted from the work of Sumedh Joshi [1], accounts for geometrical differences between equivalent 2D and 3D model environments. A second factor, referred to as the perturbative factor, is developed through the use of small perturbation theory. This factor is well-suited to account for differences in the scattered field between a 2D model and scattering from an isotropically rough 2D surface in 3D. Lastly, a third composite factor, referred to as the combined factor, of the previous two is developed by taking their minimum. This work deals only with scattering within the plane of the incident wave perpendicular to the scatterer. The applicability of these factors are tested by comparing a 2D scattering model with a fully three-dimensional Monte Carlo finite element method model for a variety of von Karman and Gaussian power spectra. The combined factor shows promise towards a robust method to adequately characterize isotropic 3D rough surfaces using 2D numerical simulations.Item Three-Dimensional Modeling of Shape Memory Polymers Considering Finite Deformations and Heat Transfer(2012-10-16) Volk, Brent Louis 1985-Shape memory polymers (SMPs) are a relatively new class of active materials that can store a temporary shape and return to the original configuration upon application of a stimulus such as temperature. This shape changing ability has led to increased interest in their use for biomedical and aerospace applications. A major challenge, however, in the advancement of these applications is the ability to accurately predict the material behavior for complex geometries and boundary conditions. This work addresses this challenge by developing an experimentally calibrated and validated constitutive model that is implemented as a user material subroutine in Abaqus ? a commercially available finite element software package. The model is formulated in terms of finite deformations and assumes the SMP behaves as a thermoelastic material, for which the response is modeled using a compressible neo-Hookean constitutive equation. An internal state variable, the glassy volume fraction, is introduced to account for the phase transformation and associated stored deformation upon cooling from the rubbery phase to the glassy phase and subsequently recovered upon heating. The numerical implementation is performed such that a system of equations is solved using a Newton-Raphson method to find the updated stress in the material. The conductive heat transfer is incorporated through solving Fourier's law simultaneously with the constitutive equations. To calibrate and validate the model parameters, thermomechanical experiments are performed on an amorphous, thermosetting polyurethane shape memory polymer. Strains of 10-25% are applied and both free recovery (zero load) and constrained displacement recovery boundary conditions are considered for each value of applied strain. Using the uniaxial experimental data, the model is then calibrated and compared to the 1-D experimental results. The validated finite element analysis tool is then used to model biomedical devices, including cardiovascular tubes and thrombectomy devices, fabricated from shape memory polymers. The effects of heat transfer and complex thermal boundary conditions are evaluated using coupled thermal-displacement analysis, for which the thermal material properties were experimentally calibrated.