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A Parallel Hierarchical-Element Method for Contour Dynamics Simulations

R.M. Schoemaker

Introduction

The HEM is a modified contour dynamics (CD) method in the sense that it accelerates highly complex 2D inviscid incompressible vortex flows for 1 processor systems [1]. The HEM has been parallelised on its turn for even faster computation, using OpenMP for up to 16 processors on an Origin 3800 shared-memory archictecture. CD is based on the fact that the evolution of a patch of uniform vorticity is fully determined by the evolution of its bounding contour [2].

A continuous vorticity distribution (red) is approximated by nested patches of uniform vorticity (blue). The contours are discretised by nodes and linear elements. The increasing complexity of the evolving contours demand node redistribution.

The HEM makes use of a hierarchical tree of levels with boxes and is based on the Fast Multipole Method, i.e. sources (vorticity in this case) are approximated by a computationally simpler description. The size of of the collected sources depend on the distance to the evaluation point.

Example of the velocity contributions from the far field to a certain node in the blue box. Faraway vorticity is approximated by large rings. These contributions are passed on to a dotted ring which lies above the blue box in the hierarchical tree and so on for each level. The light orange areas indicate regions for the optimal ring size at each level. The little dark orange area just around the blue box contain conventional interactions due to accuracy requirements.

Parallelisation Strategy

The HEM has been parallelised along the boxes one level at a time in the hierarchical tree. A fixed domain decomposition has been compared to a dynamical domain scheduling policy. Domain decomposition implies assigning processors to certain areas of a boxed level. In the static scheduling policy of OpenMP, processors have to wait for each other. The dynamic scheduling permits `grazing' of the level, implying a load balancing effect for non-uniform node distributions.

Numerical Experiments and Discussion

In the load-imbalanced case (left) the domain is subdivided into P=2 fixed areas. In the load-balanced case (right) two processors 'graze' the domain with one box per processor and if a processor has finished one box it will resume on the next available box and will not wait for the other processors ($P = 2 is arbitrary). In the load-imbalanced case (left) the domain is subdivided into P=2 fixed areas. In the load-balanced case (right) two processors 'graze' the domain with one box per processor and if a processor has finished one box it will resume on the next available box and will not wait for the other processors (P = 2 is arbitrary)

References

1. P.W.C. Vosbeek, H.J.H. Clercx, and R.M.M. Mattheij, Acceleration of ContourDynamics Simulations with a Hierarchical-Element Method, J. Comp. Phys., 161, 287--311 (2000)
2. D.G. Dritschel, Contour Dynamics and Contour Surgery: Numerical algorithms for extendedhigh-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows, Comp. Phys. Rep., 10, 77 (1989)