Higher-Order Spectral/HP Finite Element Technology for Structures and Fluid Flows
This 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.