Parallel computing of high‐speed compressible flows using a node‐based finite‐element method

An efficient parallel computing method for high-speed compressible flows is presented. The numerical analysis of flows with shocks requires very fine computational grids and grid generation requires a great deal of time. In the proposed method, all computational procedures, from the mesh generation to the solution of a system of equations, can be performed seamlessly in parallel in terms of nodes. Local finite-element mesh is generated robustly around each node, even for severe boundary shapes such as cracks. The algorithm and the data structure of finite-element calculation are based on nodes, and parallel computing is realized by dividing a system of equations by the row of the global coefficient matrix. The inter-processor communication is minimized by renumbering the nodal identification number using ParMETIS. The numerical scheme for high-speed compressible flows is based on the two-step Taylor–Galerkin method. The proposed method is implemented on distributed memory systems, such as an Alpha PC cluster, and a parallel supercomputer, Hitachi SR8000. The performance of the method is illustrated by the computation of supersonic flows over a forward facing step. The numerical examples show that crisp shocks are effectively computed on multiprocessors at high efficiency. Copyright © 2003 John Wiley & Sons, Ltd.

[1]  Tayfun E. Tezduyar,et al.  Flow simulation and high performance computing , 1996 .

[2]  Tohru Ogawa,et al.  A new algorithm for three-dimensional voronoi tessellation , 1983 .

[3]  Genki Yagawa,et al.  Parallel finite element fluid analysis on an element-by-element basis , 1996 .

[4]  Wing Kam Liu,et al.  Reproducing kernel particle methods for structural dynamics , 1995 .

[5]  Genki Yagawa,et al.  Recent developments of free mesh method , 2000 .

[6]  J. T. Oden,et al.  Adaptive implicit/explicit finite element method for compressible viscous flows , 1992 .

[7]  D. F. Watson.  Computing the n-Dimensional Delaunay Tesselation with Application to Voronoi Polytopes , 1981, Comput. J..

[8]  O. Zienkiewicz,et al.  Shallow water problems: A general explicit formulation , 1986 .

[9]  Genki Yagawa,et al.  Parallel computing for incompressible flow using a Nodal-Based method , 1999 .

[10]  Charbel Farhat,et al.  Transient finite element computations on 65536 processors: The connection machine , 1990 .

[11]  William H. Frey,et al.  An apporach to automatic three‐dimensional finite element mesh generation , 1985 .

[12]  L. Paul Chew,et al.  Constrained Delaunay triangulations , 1987, SCG '87.

[13]  Genki Yagawa,et al.  Quadrilateral approaches for accurate free mesh method , 2000 .

[14]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[15]  Genki Yagawa,et al.  Large-Scale Finite Element Fluid Analysis by Massively Parallel Processors , 1997, Parallel Comput..

[16]  Rainald Löhner,et al.  A parallel advancing front grid generation scheme , 2000, International Journal for Numerical Methods in Engineering.

[17]  Tayfun Tezduyar,et al.  Methods for parallel computation of complex flow problems , 1999, Parallel Comput..

[18]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[19]  Gen-Hua Shi,et al.  Manifold Method of Material Analysis , 1992 .

[20]  Genki Yagawa,et al.  Large-scale parallel flow analysis based on free mesh method: a virtually meshless method , 1999 .

[21]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[22]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[23]  Genki Yagawa,et al.  Node‐by‐node parallel finite elements: a virtually meshless method , 2004 .

[24]  G. Yagawa,et al.  Free mesh method: A new meshless finite element method , 1996 .

[25]  Jonathan Richard Shewchuk,et al.  A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations , 1998, SCG '98.

[26]  Tayfun E. Tezduyar,et al.  Parallel finite element methods for large-scale computation of storm surges and tidal flows , 1997 .

[27]  Tayfun E. Tezduyar,et al.  CFD methods for three-dimensional computation of complex flow problems , 1999 .

[28]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[29]  Rex A. Dwyer.  Higher-dimensional voronoi diagrams in linear expected time , 1991, Discret. Comput. Geom..

[30]  Joseph J Monaghan,et al.  An introduction to SPH , 1987 .

[31]  O. C. Zienkiewicz,et al.  The solution of non‐linear hyperbolic equation systems by the finite element method , 1984 .

[32]  Rainald Löhner,et al.  Parallel unstructured grid generation , 1991 .

[33]  Mark S. Shephard,et al.  Parallel three-dimensional mesh generation , 1994 .

[34]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[35]  M. Shephard,et al.  Geometry-based fully automatic mesh generation and the delaunay triangulation , 1988 .

[36]  D. T. Lee,et al.  Generalized delaunay triangulation for planar graphs , 1986, Discret. Comput. Geom..

[37]  George Karypis,et al.  Multilevel k-way Partitioning Scheme for Irregular Graphs , 1998, J. Parallel Distributed Comput..

[38]  John T. Batina,et al.  A gridless Euler/Navier-Stokes solution algorithm for complex-aircraft applications , 1993 .

[39]  N. Kikuchi,et al.  Homogenization theory and digital imaging: A basis for studying the mechanics and design principles of bone tissue , 1994, Biotechnology and bioengineering.

[40]  O. C. Zienkiewicz,et al.  An adaptive finite element procedure for compressible high speed flows , 1985 .

[41]  Thomas J. R. Hughes,et al.  Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies , 1985 .

[42]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[43]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[44]  S. Koshizuka,et al.  Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid , 1996 .