Application of fast parallel and sequential tree codes to computing three-dimensional flows with the vortex element and boundary element methods

Abstract

A fast parallel oct-tree code originally developed for three-dimensional N-body gravitational simulations was modified into (1) a fast N-vortex code for viscous and inviscid vortex flow computations using the regularized vortex particle method (VEM), and (2) a fast N-panel code for solving boundary integral equations in potential flow aerodynamics using the boundary element method (BEM). The core of the fast tree code remains essentially unchanged between the different application codes: gravitation, VEM, BEM, etc. Only the modules that actually encode the physical model are changed. Particular attention is given to controlling the error introduced by the use of multipole expansions to represent the field produced by groups of elements, i.e., the tree code error. In particular, the acceptable error bound for use of any multipole expansion approximation is a run-time parameter. Program outputs include statistics on the errors for the field evaluation at all element locations. Problems in VEM and BEM involving N in the range 10 4 to over 10 6 are computed on parallel supercomputers. Problems with N in the range 10 3 to 10 5 are computed on workstations. Performance results are presented, together with sample computational results. For the VEM method, a high order particle redistribution scheme has been incorporated, in an efficient way, into the parallel tree code. It is applied, if necessary, to ensure that the core overlapping condition remains satisfied in long time computations. In addition, two different relaxation schemes have also been incorporated and partially tested. Such schemes are applied, if necessary, to ensure that the particle representation of the vorticity field remains a good representation of the true divergence free vorticity field in long time computations.