Perforated Muzzle Brake
Rensselaer Polytechnic Institute

Parallel Adaptive Finite Element Computation

Perforated Muzzle Brake

Joseph E. Flaherty, Raymond M. Loy, Can Ozturan, Mark S. Shephard, James D. Teresco, Louis H. Ziantz

Scientific Computation Research Center (SCOREC) and
Department of Computer Science,
Rensselaer Polytechnic Institute, Troy, NY

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.


Several tools that facilitate parallel adaptive FEM computation have been developed at the Rensselaer Polytechnic Institute. The SCOREC Mesh Database (MDB) provides a hierarchical representation of a finite element mesh along with operators to query and update the mesh data structure. The SCOREC Finite Octree Automatic Mesh Generator uses a geometric (CAD) model of the domain to generate an initial mesh. Mesh enrichment procedures perform parallel h-refinement using error indicator information and enrichment thresholds. The Parallel Mesh Database (PMDB), built on top of MDB, provides operators to create and manipulate distributed meshes. PMDB holds MDB meshes on each processor that are subsets of the complete one. Interprocessor boundary entity lists and interprocessor links provide fast query and update operations on mesh structures. PMDB provides arbitrary multiple mesh migration, and routines to analyze partition quality. MPI is used for interprocessor communication.

Partitioning and Rebalancing

Initial mesh partitioning uses either Inertial Recursive Bisection (IRB), which repeatedly bisects the mesh in a direction orthogonal to the principal axis of inertia of the domain, or Octree Partitioning (OCT), which uses the octree structure underlying the mesh to achieve a balanced load. Rebalancing must be parallel since the mesh is distributed. Available dynamic load balancing procedures include Iterative Tree Balancing (ITB), which migrates entities between partitions by optimizing neighborhood load-request trees, Parallel Sort Inertial Recursive Bisection (PSIRB), which performs IRB with a parallel sort of the data, and OCT.

Sample Results

As an example, consider the three-dimensional unsteady compressible (EULER) flow in a cylinder containing a cylindrical ``vent hole.'' This problem simulates a (simplified) perforated muzzle brake for a cannon. Using symmetry, the flow need only be solved in one half of the domain bounded by a plane through the vent. The initial mesh contained 80,791 tetrahedral elements. The larger cylinder initially contains helium gas moving at Mach 1.23 while the vent is quiet. A hypothesized diaphragm between the two cylinders is ruptured to begin the flow.

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. Mon Feb 3 23:01:05 EST 1997