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.
terescoj@cs.rpi.edu- Mon Feb 3 23:01:05 EST 1997