Abstract
An important class of methodologies for the parallel processing of computational models defined on some discrete geometric data structures (i.e. meshes, grids) is the so calledgeometry decomposition or splitting approach. Compared to the sequential processing of such models, the geometry splitting parallel methodology requires an additional computational phase. It consists of the decomposition of the associated geometric data structure into a number of balancedsubdomains that satisfy a number of conditions that ensure the load balancing and minimum communication requirement of the underlying computations on a parallel hardware platform. It is well known that the implementation of the mesh decomposition phase requires the solution of a computationally intensive problem. For this reason several fast heuristics have been proposed. In this paper we explore a decomposition approach which is part of a parallel adaptive finite element mesh procedure. The proposed integrated approach consists of five steps. It starts with a coarse background mesh that isoptimally decomposed by applying well known heuristics. Then, the initial mesh is refined in each subdomain after linking the new boundaries introduced by its decomposition. Finally, the decomposition of the new refined mesh is improved so that it satisfies the objectives and conditions of the mesh decomposition problem. Extensive experimentation indicates the effectiveness and efficiency of the proposed parallel mesh and decomposition approach.
Similar content being viewed by others
References
Chrisochoides, N.P.; Houstis, E.N.; Rice, J.R. (1994) Mapping algorithms and software environments for data parallel PDE iterative solvers, Journal of Distributed and Parallel Computing, 21, 75–95
Chrisochoides, N.P.; Houstis, E.N.; Houstis, C.E. (1991) Geometry based mapping strategies for PDE computations. Proceedings of the International Conference on Supercomputing, Cologne-Germany, 128–135
Houstis, E.N.; Houstis, C.E.; Rice, J.R.; Weerawarana, S. (1994) PYTHIA: A computationally intelligent paradigm to support smart problem solving environments for PDE based applications, ACM Transactions on Mathematical Software, 1996 (to appear)
Houstis, E.N.; Rice, J.R. (1992) Parallel ELLPACK: A development and problem solving environment for high performance computing machines. Programming Environments for High-level Scientific Problem Solving (Gaffney, P.W.; Houstis, E.N., Editors). North-Holland, Amsterdam, 229–241
Kim, S.; Houstis, E.N.; Rice, J.R. (1994) Parallel stationary iterative methods and their performance. To appear in Proceedings of the INTEL supercomputer users group conference (Marinescu, D.; Frost, R., Editors). San Diego
Farhat, C.; Simon, H.D. (1993) TOP/DOMDEC—a software tool for mesh partitioning and parallel processing. Technical Report, NASA Ames Research Center, RNR-93-011, 1–28
Tragoudas, S. (1994) Min-cut partitioning on underlying tree and graph structures, to appear in IEEE Transactions on Computers
Wu, P.; Houstis, E.N. (1993) Parallel dynamic mesh generation and domain decomposition. Technical Report, Purdue University, Department of Computer Sciences, CSD-TR-93-075, 1–49
Wu, P.; Houstis, E.N. (1993) Parallel electronic prototyping of physical objects. Technical Report, Purdue University, Department of Computer Sciences, CSD-TR-93-026, 1–48
Bykat, A. (1976) Automatic generation of triangular grid: I—Subdivision of a general polygon into convex subregions. II—Triangulation of convex polygons, International Journal for Numerical Methods in Engineering, 10, 1329–1342
Khan, A.I.; Topping, B.H.V. (1991) Parallel adaptive mesh generation, Computing Systems in Engineering, 2(1), 75–101
Lo, S.H. (1991) Automatic mesh generation and adaptation by using contours, International Journal for Numerical Methods in Engineering, 31, 689–707
Sadek, E.A. (1980) A scheme for the automatic generation of triangular finite elements, International Journal for Numerical Methods in Engineering, 15, 1813–1822
Cheng, F.; Jaromczyk, J.W.; Lin, J.; Chang, S.; Lu, J. (1989) A parallel mesh generation algorithm based on the vertex label assignment scheme, International Journal for Numerical Methods in Engineering, 28, 1429–1448
Samet, H. (1984) The quadtree and related hierarchical data structures, Computing Surveys, 16(2), 187–260
Mitchell, W.F. (1987) A comparison of adaptive refinement techniques for elliptic problems. Technical Report, University of Illinois at Urbana-Champaign, Department of Computer Science, UIUCDCS-R-87-1375, 1–14
PATRAN (1992) A Division of PDA Engineering—PATRAN Plus User Manual, Vols 1 and 2, PDA Engineering, PATRAN Division
Deloujie-Rakhshandeh, K. (1990) An approach to the generation of triangular grids possessing few obtuse triangles, International Journal for Numerical Methods in Engineering, 29, 1299–1321
Baase, S. (1988) Graphs and Digraphs, Computer Algorithms: Introduction to Design and Analysis, Addison-Wesley, Reading, MA, Ch. 4, 145–207
Farhat, C. (1988) A simple and efficient automatic FEM domain decomposer, Computers & Structures, 28(5), 579–602
Farhat, C.; Lesoinne, M. (1993) Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics, International Journal for Numerical Methods in Engineering, 36, 745–764
George, A; Liu, J.W.H. (1979) An implementation of a pseudoperipheral node finder, ACM Transactions on Mathematical Software, 5(3), 284–295
Gibbs, N.E.; Poole, J.W.G.; Stockmeyer, P.K. (1976) An algorithm for reducing the bandwidth and profile of a sparse matrix, SIAM Journal of Numerical Analysis, 13(2), 236–250
Malone, J.G. (1988) Automatic mesh decomposition and concurrent finite element analysis for hypercube multiprocessor computers, Computer Methods in Applied Mechanics and Engineering, 70, 27–58
Pissanetsky, S. (1984) Ordering for Gauss elimination: symmetry matrices, Sparse Matrix Technology, Academic Press, Orlando, Ch. 4, 94–158
Chan, W.M.; George, A. (1980) A linear time implementation of the reverse Cuthill-McKee algorithm, BIT, 20, 8–14
George, A.; Liu, J.W.H. (1978) Algorithms for matrix partitioning and the numerical solution of finite element systems, SIAM Journal of Numerical Analysis, 15(2), 297–327
Barnard, S.T.; Simon, H.D. (1993) A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems, Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, 711–718
Fiedler, M. (1975) A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory, Czechoslovak Mathematical Journal, 25(100), 619–633
Fiedler, M. (1975) Eigenvectors of acyclic matrices, Czechoslovak Mathematical Journal, 25(100), 607–618
Fiedler, M. (1973) Algebraic connectivity of graphs, Czechoslovak Mathematical Journal, 23(98), 298–305
Hendrickson, B.; Leland, R. (1993) An improved spectral load balancing method, Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, 953–961
Simon, H.D. (1991) Partitioning of unstructured problems for parallel processing, Computing Systems in Engineering, 2(2/3), 135–148
Venkatakrishnan, V.; Simon, H.D. (1992) A MIMD implementation of a parallel Euler solver for unstructured grids, The Journal of Supercomputing, 6, 117–137
Loriot, M.; Fezoui, L. (1988) Mesh-splitting preprocessor. Unpublished manuscripts
Williams, R. (1992) Parallel meshes for complex geometry, PML, 302–312
Lohner, R.; Camberos, J.; Merriam, M. (1992) Parallel unstructured grid generation, Computer Methods in Applied Mechanics and Engineering, 95, 343–357
Kernighan, B.W.; Lin, S. (1970) An efficient heuristic procedure for partitioning graphs, The Bell System Technical Journal, 49, 291–307
Kirkpatrick, S.; Gelatt, J.C.D.; Vecchi, M.P. (1983) Optimization by simulated annealing, Science, 220(4598), 671–680
Johnson, D.S.; Aragon, C.R.; McGeoch, L.A.; Schevon, C. (1989) Optimization by simulated annealing: An experimental evaluation; Part I, Graph partitioning, Operations Research, 37(6), 865–892
Saab, Y.G.; Rao, V.B. (1991) Combinatorial optimization by stochastic evolution, IEEE Transactions on Computer-Aided Design, 10(4), 525–535
Hertz, A.; Werra, D. (1987) Using Tabu search techniques for graph coloring, Computing,39, 345–351
Glover, F.; McMillan, C. (1985/6) Interactive decision software and computer graphics for architectural and space planning, Annals of Operations Research, 5, 557–573
Savage, J.E.; Wloka, M.G. (1991) Parallelism in graph-partitioning, Journal of Parallel and Distributed Computing, 13, 257–272
Geist, G.A.; Heath, M.T.; Peyton, B.W.; Worley, P.H. (1992) A users' guide to PICL, a portable instrumented communication library. Technical Report, Oak Ridge National Laboratory, Oak Ridge, TN, ORNL/TM-11616, 1–22
Geist, G.A.; Heath, M.T.; Peyton, B.W.; Worley, P.H. (1990) PICL: a portable instrumented communication library, C reference manual. Technical Report, Oak Ridge National Laboratory, Oak Ridge, TN, ORNL/TM-11130, 1–146
Heath, M.T.; Finger, J.E. (1993) ParaGraph: A tool for visualizing performance of parallel programs. Oak Ridge National Laboratory, Oak Ridge, RN, 1–50
Heath, M.T. (1993) Recent developments and case studies in performance visualization using ParaGraph. Performance Measurement and Visualization of Parallel Systems (Haring, G.; Kotsis, G., Editors). Elsevier Science Publishers, Amsterdam, 175–200
Heath, M.T.; Finger, J.E. (1991) Visualizing the performance of parallel programs, IEEE Software, 8(5), 29–39
Al-Nastra, M.; Nguyen, D.T. (1991) An algorithm for domain decomposition in finite element analysis, Computers & Structures, 39(3/4), 277–289
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, P., Houstis, E.N. Parallel adaptive mesh generation and decomposition. Engineering with Computers 12, 155–167 (1996). https://doi.org/10.1007/BF01198731
Issue Date:
DOI: https://doi.org/10.1007/BF01198731