Joseph E. Flaherty, Raymond M. Loy, Can
Ozturan, Mark S. Shephard,
James D. Teresco, Louis H. Ziantz
Scientific Computation Research Center (SCOREC) and
Adaptive finite element methods (FEMs) for partial differential equations (PDEs) provide greater efficiency, reliability, and robustness than classical methods by automatically refining or coarsening portions of the space-time domain (h-refinement), varying the method order (p-refinement), and/or moving the mesh to follow evolving phenomena (r-refinement). As such, extensive computational effort is confined to regions where solution resolution is inadequate. Estimates of discretization errors are used to control solution accuracy and provide a measure of confidence in the results.
In order to solve large problems in reasonable times, adaptive methods must execute efficiently on parallel computers. A good initial partitioning of the space-time domain is not sufficient to assure high performance throughout the computation. Adaptive enrichment causes load imbalances that necessitate a dynamic redistribution of data.
Time steps are either accepted or rejected based on whether or not elemental error indicators exceed a prescribed tolerance. Rejected time steps are repeated subsequent to adaptive h-refinement and rebalancing. Coarsening is essential to keep mesh sizes manageable as areas of interest move through the domain. Upon h-refinement, the solution is interpolated to the new mesh and the integration continues with, possibly, a new time step. Accepted steps continue with, possibly, a new global time step.
The accompanying picture, obtained using 16 processors of an IBM SP-2, displays the Mach number with velocity vectors and mesh partitioning using PSIRB. Flow features compare favorably with experimental and numerical results of Nagamatsu, Carofano, et al. (US Army ARDEC, ARCCB-TR-87016, 1987). Animations of the solution density and solution mach number are available. There are also animations of solution mach number with velocity vectors, a large format and smaller format.
We have demonstrated a capability to perform large-scale parallel adaptive FEM computation. We have used these tools to examine the Euler flow in intersecting cylinders simulating a perforated muzzle brake. The tools and techniques referenced here are generic and are being used to solve problems in many areas including materials processing, crystal growth, and biomechanics.
email@example.com- Mon Feb 3 23:01:05 EST 1997