A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters

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• 作者：Joachim Berdal Haga (1) jobh@simula.no
  Harald Osnes (2)
  Hans Petter Langtangen (3)
• 关键词：Parallel computing &#8211 ; Block preconditioning &#8211 ; Poroelasticity &#8211 ; Low ; permeable media &#8211 ; Finite elements
• 刊名：Computational Geosciences
• 出版年：2012
• 出版时间：June 2012
• 年：2012
• 卷：16
• 期：3
• 页码：723-734
• 全文大小：522.4 KB
• 参考文献：1. Adams, M.: Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics. Int. J. Numer. Methods Eng. 55, 519–534 (2002). doi:
  2. Adams, M.F., Bayraktar, H.H., Keaveny, T.M., Papadopoulos, P.: Ultrascalable implicit finite element analyses in solid mechanics with over a half a billion degrees of freedom. In: Proceedings of the ACM/IEEE Conference on Supercomputing (SC2004), IEEE Computer Society, p. 34 (2004)
  3. The Simula computer cluster bigblue. http://simula.no/research/sc/cbc/events/2008/081105-slides/bigblue-intro.pdf
  4. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941). doi:
  5. Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adaptive smoothed aggregation (αSA). SIAM J. Sci. Comput. 25(6), 1896–1920 (2004). doi:
  6. Catalyurek, U.V., Boman, E.G., Devine, K.D., Bozdag, D., Heaphy, R.T., Riesen, L.A.: Hypergraph-based dynamic load balancing for adaptive scientific computations. In: Proc. of 21st International Parallel and Distributed Processing Symposium (IPDPS’07). IEEE (2007)
  7. Chow, E., Falgout, R.D., Hu, J.J., Tuminaro, R., Yang, U.M.: A survey of parallelization techniques for multigrid solvers. In: Heroux M.A., Raghavan P., Simon H.D. (eds.) Parallel Processing for Scientific Computing, pp. 179–202. SIAM, Philadelphia (2006)
  8. Library for numerical solution of PDEs from inuTech GmbH. http://www.diffpack.com/
  9. Doi, S., Washio, T.: Ordering strategies and related techniques to overcome the trade-off between parallelism and convergence in incomplete factorizations. Parallel Comput. 25, 1995–2014 (1999). doi:
  10. Elman, H.C., Howle, V.E., Shadid, J.N., Shuttleworth, R., Tuminaro, R.S.: A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations. J. Comput. Phys. 227(3), 1790–1808 (2007). doi:
  11. Elman, H.C., Howle, V.E., Shadid, J.N., Tuminaro, R.S.: A parallel block multi-level preconditioner for the 3D incompressible Navier–Stokes equations. J. Comput. Phys. 187(2), 504–523 (2003). doi:
  12. Fletcher, R.: Conjugate gradient methods for indefinite systems. In: Watson G. (ed.) Numerical Analysis, Lecture Notes in Mathematics, vol. 506, pp. 73–89. Springer, Berlin (1976). doi:10.1007/BFb0080116
  13. Gee, M.W., Siefert, C.M., Hu, J.J., Tuminaro, R.S., Sala, M.G.: ML 5.0 smoothed aggregation user’s guide. Tech. Rep. SAND2006-2649, Sandia National Laboratories (2006). http://software.sandia.gov/trilinos/packages/ml/
  14. George, A., Ng, E.: On the complexity of sparse QR and LU factorization of finite-element matrices. SIAM J. Sci. Stat. Comput. 9, 849 (1988). doi:
  15. Hackbush, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, Berlin (1995)
  16. Haga, J.B., Langtangen, H.P., Nielsen, B.F., Osnes, H.: On the performance of an algebraic multigrid preconditioner for the pressure equation with highly discontinuous media. In: Skallerud B., Andersson H.I. (eds.) Proceedings of MekIT’09, pp. 191–204. NTNU, Tapir (2009). http://simula.no/research/sc/publications/Simula.SC.568
  17. Haga, J.B., Langtangen, H.P., Osnes, H.: On the causes of pressure oscillations in low-permeable and low-compressible porous media (2010). Int. J. Numer. Anal. Methods Geomech. (http://simula.no/publications/Simula.simula.18)
  18. Haga, J.B., Osnes, H., Langtangen, H.P.: Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media. Int. J. Numer. Anal. Methods Geomech. (2010). http://simula.no/research/sc/publications/Simula.SC.660
  19. Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S.: An overview of the Trilinos project. ACM Trans. Math. Softw. 31(3), 397–423 (2005). doi:
  20. The NOTUR computer cluster hexagon. http://www.notur.no/hardware/hexagon
  21. Joubert, W., Cullum, J.: Scalable algebraic multigrid on 3500 processors. Electron. Trans. Numer. Anal. 23, 105–128 (2006)
  22. Karypis, G., Schloegel, K., Kumar, V.: ParMETIS parallel graph partitioning and sparse matrix ordering library, version 3.1. University of Minnesota, Minneapolis (2003). http://glaros.dtc.umn.edu/gkhome/metis/parmetis/overview
  23. Langtangen, H.P.: Computational Partial Differential Equations: Numerical Methods and Diffpack Programming, 2nd edn. Springer, Berlin (2003)
  24. Petroleum systems modelling software from Schlumberger Aachen Technology Center. http://www.petromod.com/
  25. Phoon, K.K., Toh, K.C., Chan, S.H., Lee, F.H.: An efficient diagonal preconditioner for finite element solution of Biot’s consolidation equations. Int. J. Numer. Methods Eng. 55, 377–400 (2002). doi:
  26. Thakur, R., Gropp, W.: Improving the performance of collective operations in MPICH. In: Recent Adv. Parallel Virtual Mach. Message Passing Interface, pp. 257–267 (2003)
  27. Toh, K.C., Phoon, K.K., Chan, S.H.: Block preconditioners for symmetric indefinite linear systems. Int. J. Numer. Methods Eng. 60, 1361–1381 (2004). doi:
  28. Tuminaro, R.S., Tong, C.: Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines. In: Proceedings of the 2000 ACM/IEEE conference on Supercomputing. IEEE Computer Society (2000). doi:10.1109/SC.2000.10008
  29. Yang, U.M.: Parallel algebraic multigrid methods—high performance preconditioners. In: Bruaset A.M., Tveito A. (eds.) Numerical Solution of Partial Differential Equations on Parallel Computers, pp. 209–236. Springer, Berlin (2006). doi:
• 作者单位：1. Computational Geosciences, Simula Research Laboratory, PO Box 134, 1325 Lysaker, Norway2. Department of Mathematics, University of Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway3. Center for Biomedical Computing, Simula Research Laboratory, PO Box 134, 1325 Lysaker, Norway
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Modeling and IndustrialMathematics
  Geotechnical Engineering
  Hydrogeology
  Soil Science and Conservation
  
• 出版者：Springer Netherlands
• ISSN：1573-1499

文摘

Large-scale simulations of coupled flow in deformable porous media require iterative methods for solving the systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in realistic geological applications, such as basin evolution at regional scales. The success of iterative methods for such problems depends strongly on finding effective preconditioners with good parallel scaling properties, which is the topic of the present paper. We present a parallel preconditioner for Biot’s equations of coupled elasticity and fluid flow in porous media. The preconditioner is based on an approximation of the exact inverse of the two-by-two block system arising from a finite element discretisation. The approximation relies on a highly scalable approximation of the global Schur complement of the coefficient matrix, combined with generally available state-of-the-art multilevel preconditioners for the individual blocks. This preconditioner is shown to be robust on problems with highly heterogeneous material parameters. We investigate the weak and strong parallel scaling of this preconditioner on up to 512 processors and demonstrate its ability on a realistic basin-scale problem in poroelasticity with over eight million tetrahedral elements.