The objective of this project is to investigate the complex fracture of ice and understand its role within larger ice sheet simulations and global climate change. At the present time, ice fracture is not explicitly considered within ice sheet models due in part to large computational costs associated with the accurate modeling of this complex phenomena. However, fracture not only plays an extremely important role in regional behavior but also influences ice dynamics over much larger zones in ways that are currently not well understood. Dramatic illustrations of fracture-induced phenomena most notably include the recent collapse of ice shelves in Antarctica (e.g. partial collapse of the Wilkins shelf in March of 2008 and the diminishing extent of the Larsen B shelf from 1998 to 2002). Other fracture examples include ice calving (fracture of icebergs) which is presently approximated in simplistic ways within ice sheet models, and the draining of supraglacial lakes through a complex network of cracks, a so called ice sheet plumbing system, that is believed to cause accelerated ice sheet flows due essentially to lubrication of the contact surface with the ground. These dramatic changes are emblematic of the ongoing change in the Earth's polar regions and highlight the important role of fracturing ice. To model ice fracture, a simulation capability will be designed centered around extended finite elements and solved by specialized multigrid methods on parallel computers. In addition, appropriate dynamic load balancing techniques will be employed to ensure an approximate equal amount of work for each processor.
In this report we summarize research into new parallel algebraic multigrid (AMG) methods. We first provide a introduction to parallel AMG. We then discuss our research in parallel AMG algorithms for very large scale platforms. We detail significant improvements in the AMG setup phase to a matrix-matrix multiplication kernel. We present a smoothed aggregation AMG algorithm with fewer communication synchronization points, and discuss its links to domain decomposition methods. Finally, we discuss a multigrid smoothing technique that utilizes two message passing layers for use on multicore processors.
In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code.
We report on algebraic multilevel preconditioners for the parallel solution of linear systems arising from a Newton procedure applied to the finite-element (FE) discretization of the incompressible Navier-Stokes equations. We focus on the issue of how to coarsen FE operators produced from high aspect ratio elements.