Browsing by Subject "Least-squares"
Now showing 1 - 3 of 3
Results Per Page
Sort Options
Item Higher-Order Spectral/HP Finite Element Technology for Structures and Fluid Flows(2013-06-20) Vallala, Venkat PradeepThis study deals with the use of high-order spectral/hp approximation functions in the ?nite element models of various nonlinear boundary-value and initial-value problems arising in the ?elds of structural mechanics and ?ows of viscous incompressible ?uids. For many of these classes of problems, the high-order (typically, polynomial order p greater than or equal to 4) spectral/hp ?nite element technology o?ers many computational advantages over traditional low-order (i.e., p < 3) ?nite elements. For instance, higher-order spectral/hp ?nite element procedures allow us to develop robust structural elements for beams, plates, and shells in a purely displacement-based setting, which avoid all forms of numerical locking. The higher-order spectral/hp basis functions avoid the interpolation error in the numerical schemes, thereby making them accurate and stable. Furthermore, for ?uid ?ows, when combined with least-squares variational principles, such technology allows us to develop e?cient ?nite element models, that always yield a symmetric positive-de?nite (SPD) coe?cient matrix, and thereby robust direct or iterative solvers can be used. The least-squares formulation avoids ad-hoc stabilization methods employed with traditional low-order weak-form Galerkin formulations. Also, the use of spectral/hp ?nite element technology results in a better conservation of physical quantities (e.g., dilatation, volume, and mass) and stable evolution of variables with time in the case of unsteady ?ows. The present study uses spectral/hp approximations in the (1) weak-form Galerkin ?nite element models of viscoelastic beams, (2) weak-form Galerkin displacement ?nite element models of shear-deformable elastic shell structures under thermal and mechanical loads, and (3) least-squares formulations for the Navier-Stokes equations governing ?ows of viscous incompressible ?uids. Numerical simulations using the developed technology of several non-trivial benchmark problems are presented to illustrate the robustness of the higher-order spectral/hp based ?nite element technology.Item Least squares based finite element formulations and their applications in fluid mechanics(2009-05-15) Prabhakar, VivekIn this research, least-squares based finite element formulations and their applications in fluid mechanics are presented. Least-squares formulations offer several computational and theoretical advantages for Newtonian as well as non-Newtonian fluid flows. Most notably, these formulations circumvent the inf-sup condition of Ladyzhenskaya-Babuska- Brezzi (LBB) such that the choice of approximating space is not subject to any compatibility condition. Also, the resulting coefficient matrix is symmetric and positive-definite. It has been observed that pressure and velocities are not strongly coupled in traditional leastsquares based finite element formulations. Penalty based least-squares formulations that fix the pressure-velocity coupling problem are proposed, implemented in a computational scheme, and evaluated in this study. The continuity equation is treated as a constraint on the velocity field and the constraint is enforced using the penalty method. These penalty based formulations produce accurate results for even low penalty parameters (in the range of 10-50 penalty parameter). A stress based least-squares formulation is also being proposed to couple pressure and velocities. Stress components are introduced as independent variables to make the system first order. The continuity equation is eliminated from the system with suitable modifications. Least-squares formulations are also developed for viscoelastic flows and moving boundary flows. All the formulations developed in this study are tested using several benchmark problems. All of the finite element models developed in this study performed well in all cases. A method to exploit orthogonality of modal bases to avoid numerical integration and have a fast computation is also developed during this study. The entries of the coefficient matrix are calculated analytically. The properties of Jacobi polynomials are used and most of the entries of the coefficient matrix are recast so that they can be evaluated analytically.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.