Vallala, Venkat Pradeep (2013-08). Higher-Order Spectral/HP Finite Element Technology for Structures and Fluid Flows. Doctoral Dissertation.
This study deals with the use of high-order spectral/hp approximation functions in the finite element models of various nonlinear boundary-value and initial-value problems arising in the fields of structural mechanics and flows of viscous incompressible fluids. For many of these classes of problems, the high-order (typically, polynomial order p greater than or equal to 4) spectral/hp finite element technology offers many computational advantages over traditional low-order (i.e., p < 3) finite elements. For instance, higher-order spectral/hp finite 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 fluid flows, when combined with least-squares variational principles, such technology allows us to develop efficient finite element models, that always yield a symmetric positive-definite (SPD) coefficient 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 finite 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 flows. The present study uses spectral/hp approximations in the (1) weak-form Galerkin finite element models of viscoelastic beams, (2) weak-form Galerkin displacement finite element models of shear-deformable elastic shell structures under thermal and mechanical loads, and (3) least-squares formulations for the Navier-Stokes equations governing flows of viscous incompressible fluids. 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 finite element technology.