Computation of unsteady incompressible flows with the stabilized finite element methods: Space-time formulations, iterative strategies and massively parallel implementations

We discuss the stabilized finite element computation of unsteady incompressible flows, with emphasis on the space-time formulations, iterative solution techniques and implementations on the massively parallel architectures such as the Connection Machines. The stabilization technique employed in this paper is the Galerkin/least-squares (GLS) method. The Deformable-Spatial-Domain/Stabilized-Space-Time (DSD/SST) formulation was developed for computation of unsteady viscous incompressible flows which involve moving boundaries and interfaces. In this approach, the stabilized finite element formulations of the governing equations are written over the space-time domain of the problem, and therefore the deformation of the spatial domain with respect to time is taken into account automatically. This approach gives us the capability to solve a large class of problems with free surfaces, moving interfaces, and fluid-structure and fluid-particle interactions. In the DSD/SST approach the frequency of remeshing is minimized to minimize the projection errors involved in remeshing and also to increase the parallelization potential of the computations. We present a new mesh moving scheme that minimizes the need for remeshing; in this scheme the motion of the mesh is governed by the modified equations of linear homogeneous elasticity. The implicit equation systems arising from the finite element discretizations are solved iteratively by using the GMRES search technique with the clustered element-by-element, diagonal and nodal-block-diagonal preconditioners. Formulations with diagonal and nodal-block-diagonal preconditioners have been implemented on the Connection Machines CM-200 and CM-5. We also describe a new mixed preconditioning method we developed recently, and discuss the extension of this method to totally unstructured meshes. This mixed preconditioning method is similar, in philosophy, to multi-grid methods, but does not need any intermediate grid levels, and therefore is applicable to unstructured meshes and is simple to implement. The application problems considered include various free-surface flows and simple fluid-structure interaction problems such as vortex-induced oscillations of a cylinder and flow past a pitching airfoil.