Logo PTI
Polish Information Processing Society
Logo FedCSIS

Annals of Computer Science and Information Systems, Volume 8

Proceedings of the 2016 Federated Conference on Computer Science and Information Systems

Block Subspace Projection PCG Method for Solution of Natural Vibration Problem in Structural Analysis


DOI: http://dx.doi.org/10.15439/2016F88

Citation: Proceedings of the 2016 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, Vol. 8, pages 669672 ()

Full text

Abstract. The block subspace projection preconditioned conjugate gradient method for analysis of natural vibration frequencies and modes applying to large problems of structural mechanics is proposed. It is oriented at the usage in finite element analysis software operated on multi-core desktop computers with restricted amount of core memory as an alternative approach to widespread block Lanczos method and subspace iteration method. We focused our attention on achievement of high computational stability and parallelization of proposed algorithm. The solution of real-life large problems confirms the reliability of proposed approach.


  1. S. Yu. Fialko, “Natural vibrations of complex bodies,” Int. Applied Mechanics, vol. 40, no. 1, pp. 83 – 90, 2004, http://dx.doi.org/10.1023/B:INAM.0000023814.13805.34.
  2. S. Fialko, “Aggregation Multilevel Iterative Solver for Analysis of Large-Scale Finite Element Problems of Structural Mechanics: Linear Statics and Natural Vibrations”, in PPAM 2001, R. Wyrzykowski et al. (Eds.), LNCS 2328, Springer-Verlag Berlin Heidelberg, 2002, pp. 663–670 http://dx.doi.org/10.1007/1-4020-5370-3_41.
  3. S. Yu. Fialko, E. Z. Kriksunov and V. S. Karpilovskyy, “A block Lanczos method with spectral transformations for natural vibrations and seismic analysis of large structures in SCAD software,” in Proc. CMM-2003 – Computer Methods in Mechanics, Gliwice, Poland, 2003, pp. 129—130.
  4. S. Yu. Fialko, “Iterative methods for solving large-scale problems of structural mechanics using multi-core computers,” Archieves of Civil and Mechanical Engineering, vol. 14, pp. 190 – 203, 2014, http://dx.doi.org/10.1016/j.acme.2013.05.009.
  5. S. Yu. Fialko, F. Żegleń, “Block Preconditioned Conjugate Gradient Method for Extraction of Natural Vibration Frequencies in Structural Analysis”, Proceedings of the FedCSIS. Łódż, 2015. IEEE Xplore Digital Library, pp. 655 – 662. http://dx.doi.org/10.15439/2015F87.
  6. S. Yu. Fialko, “PARFES: A method for solving finite element linear equations on multi-core computers,” Advances in Engineering software, vol. 40, no. 12, pp. 1256-1265, 2010, http://dx.doi.org/10.1016/j.advengsoft.2010.09.002.
  7. S. Yu. Fialko, “Parallel direct solver for solving systems of linear equations resulting from finite element method on multi-core desktops and workstations”, Computers and Mathematics with Applications 70, pp. 2968–2987, 2015. http://dx.doi.org/10.1016/j.camwa.2015.10.009
  8. C. K. Gan, P. D. Haynes and M. C. Payne, “Preconditioned conjugate gradient method for sparse generalized eigenvalue problem in electronic structure calculations,” Computer Physics Communications, vol 134, nr. 1, pp. 33 – 40, 2001, http://dx.doi.org/10.1016/S0010-4655(00)00188-0.
  9. V. Hernbadez, J. E. Roman, A. Tomas and V. Vidal, “A survey a software for sparse eigenvalue problems,” Universitat Politecnica De Valencia, SLEPs technical report STR-6, 2009.
  10. A. V. Knyazev and K. Neymayr, “Efficient solution of symmetric eigenvalue problem using multigrid preconditioners in the locally optimal block conjugate gradient method,” Electronic Transactions on Numerical Analysis, vol. 15, pp. 38 – 55, 2003.
  11. A. V. Knyazev, M. E. Argentati, I. Lashuk, E.E. Ovtchinnikov, “Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in HYPRE and PETSC”. URL: http://arxiv.org/pdf/0705.2626.pdf.
  12. Y. Saad, Numerical methods for large eigenvalue problems, Revised edition, Classics in applied mathematics. SIAM, 2011, http://dx.doi.org/10.1137/1.9781611970739.
  13. Intel Math Kernel Library Reference Manual. https://software.intel.com/ru-ru/node/521001