Browsing by Subject "Linear elasticity"
Now showing 1 - 4 of 4
- Results Per Page
- Sort Options
Item The enriched Galerkin method for linear elasticity and phase field fracture propagation(2015-12) Mital, Prashant; Wheeler, Mary F. (Mary Fanett); Wick, ThomasThis thesis focuses on the application of the discontinuous Galerkin (DG) and enriched Galerkin (EG) methods to the problems of linear elasticity and phase field fracture propagation. The use of traditional and popular continuous Galerkin method (CG) for linear elasticity has posed some challenges. For example, nonphysical stress oscillations often occur in CG solutions for linearly elastic, nearly incompressible materials. Furthermore, CG solutions produce discontinuous stresses at the finite element boundaries which need to be post-processed. Based on the success of the DG methods in solving these challenges, we attempt resolution of the same problems with the yet untested EG method. For phase field fracture propagation, the CG method has been ubiquitously used in the literature. Since the phase field displacement solution is essentially discontinuous across the crack, we hypothesize that the discontinuous DG and EG methods could offer some advantages when applied to the fracture problem. We then perform a comparative analysis of CG, DG and EG applied to the phase field equations to determine if this is indeed the case. We begin by applying a family of DG and EG methods, including Nonsymmetric Interior Penalty Galerkin (NIPG), Symmetric Interior Penalty Galerkin (SIPG), and Incomplete Interior Penalty Galerkin (IIPG) to 2D linear elasticity problems. It is shown that the EG methods are simple and robust for dealing with linear elasticity. They are also shown to converge at the same rates as the corresponding DG methods. A detailed comparison of the performance of NIPG, IIPG, and SIPG is also made. We then propose a novel monolithic scheme with an augmented-Lagrangian method for phase field fracture propagation. We apply CG, DG and EG methods to the scheme and establish convergence in space and time through numerical studies. It is shown that the Newton method used for solving the system of nonlinear equations converges faster for DG and EG than it does for CG.Item Fast algorithms for frequency domain wave propagation(2012-12) Tsuji, Paul Hikaru; Ying, Lexing; Ghattas, Omar N.; Engquist, Bjorn; Fomel, Sergey; Ren, KuiHigh-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time.Item Parallel simulation of coupled flow and geomechanics in porous media(2014-12) Wang, Bin, 1984-; Wheeler, Mary F. (Mary Fanett)In this research we consider developing a reservoir simulator capable of simulating complex coupled poromechanical processes on massively parallel computers. A variety of problems arising from petroleum and environmental engineering inherently necessitate the understanding of interactions between fluid flow and solid mechanics. Examples in petroleum engineering include reservoir compaction, wellbore collapse, sand production, and hydraulic fracturing. In environmental engineering, surface subsidence, carbon sequestration, and waste disposal are also coupled poromechanical processes. These economically and environmentally important problems motivate the active pursuit of robust, efficient, and accurate simulation tools for coupled poromechanical problems. Three coupling approaches are currently employed in the reservoir simulation community to solve the poromechanics system, namely, the fully implicit coupling (FIM), the explicit coupling, and the iterative coupling. The choice of the coupling scheme significantly affects the efficiency of the simulator and the accuracy of the solution. We adopt the fixed-stress iterative coupling scheme to solve the coupled system due to its advantages over the other two. Unlike the explicit coupling, the fixed-stress split has been theoretically proven to converge to the FIM for linear poroelasticity model. In addition, it is more efficient and easier to implement than the FIM. Our computational results indicate that this approach is also valid for multiphase flow. We discretize the quasi-static linear elasticity model for geomechanics in space using the continuous Galerkin (CG) finite element method (FEM) on general hexahedral grids. Fluid flow models are discretized by locally mass conservative schemes, specifically, the mixed finite element method (MFE) for the equation of state compositional flow on Cartesian grids and the multipoint flux mixed finite element method (MFMFE) for the single phase and two-phase flows on general hexahedral grids. While both the MFE and the MFMFE generate cell-centered stencils for pressure, the MFMFE has advantages in handling full tensor permeabilities and general geometry and boundary conditions. The MFMFE also obtains accurate fluxes at cell interfaces. These characteristics enable the simulation of more practical problems. For many reservoir simulation applications, for instance, the carbon sequestration simulation, we need to account for thermal effects on the compositional flow phase behavior and the solid structure stress evolution. We explicitly couple the poromechanics equations to a simplified energy conservation equation. A time-split scheme is used to solve heat convection and conduction successively. For the convection equation, a higher order Godunov method is employed to capture the sharp temperature front; for the conduction equation, the MFE is utilized. Simulations of coupled poromechanical or thermoporomechanical processes in field scales with high resolution usually require parallel computing capabilities. The flow models, the geomechanics model, and the thermodynamics model are modularized in the Integrated Parallel Accurate Reservoir Simulator (IPARS) which has been developed at the Center for Subsurface Modeling at the University of Texas at Austin. The IPARS framework handles structured (logically rectangular) grids and was originally designed for element-based data communication, such as the pressure data in the flow models. To parallelize the node-based geomechanics model, we enhance the capabilities of the IPARS framework for node-based data communication. Because the geomechanics linear system is more costly to solve than those of flow and thermodynamics models, the performance of linear solvers for the geomechanics model largely dictates the speed and scalability of the coupled simulator. We use the generalized minimal residual (GMRES) solver with the BoomerAMG preconditioner from the hypre library and the geometric multigrid (GMG) solver from the UG4 software toolbox to solve the geomechanics linear system. Additionally, the multilevel k-way mesh partitioning algorithm from METIS is used to generate high quality mesh partitioning to improve solver performance. Numerical examples of coupled poromechanics and thermoporomechanics simulations are presented to show the capabilities of the coupled simulator in solving practical problems accurately and efficiently. These examples include a real carbon sequestration field case with stress-dependent permeability, a synthetic thermoporoelastic reservoir simulation, poroelasticity simulations on highly distorted hexahedral grids, and parallel scalability tests on a massively parallel computer.Item Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods(2018-06-25) Fuentes, Federico, Ph. D.; Demkowicz, Leszek; Babuska, Ivo M.; Caffarelli, Luis A.; Hughes, Thomas J. R.; Oden, J. Tinsley; Wilder, AletaDiscontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and are crafted to be numerically stable by implicitly computing a near-optimal discrete test space as a function of a discrete trial space. Moreover, they are completely general in the sense that they can be applied to a variety of variational formulations, including non-conventional ones that involve non-symmetric functional settings, such as ultraweak variational formulations. In most cases, these properties have been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like in convection-diffusion problems and other singularly perturbed problems. In this work, other features of DPG methods are systematically exploited and applied to different problems. More specifically, the versatility of DPG methods is elucidated by utilizing the underlying methodology to discretize four distinct variational formulations of the equations of linear elasticity. By taking advantage of interface variables inherent to DPG discretizations, an approach to coupling different variational formulations within the same domain is described and used to solve interesting problems. Moreover, the convenient algebraic structure in DPG methods is harnessed to develop a new family of numerical methods called discrete least-squares (DLS) finite element methods. These involve solving, with improved conditioning properties, a discrete least-squares problem associated with an overdetermined rectangular system of equations, instead of directly solving the usual square systems. Their utility is demonstrated with illustrative examples. Additionally, high-order polygonal DPG (PolyDPG) methods are devised by using the intrinsic discontinuities present in ultraweak formulations. The resulting methods can handle heavily distorted non-convex polygonal elements and discontinuous material properties. A polygonal adaptive strategy was also proposed and compared with standard techniques. Lastly, the natural high-order residual-based a posteriori error estimator ingrained within DPG methods was further applied to problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery.