Accelerating Computation of a Reduced Order Model of a Structural System Resulting from Craig–Bampton Reduction Using GPU Programming

  • Piotr Gorecki General Electric Company Polska sp. z o.o., Poland
  • Miłosz Kalinowski Lukasiewicz Research Network - Institute of Aviation, Engineering Design Center, Poland
  • Łukasz Jeziorek Lukasiewicz Research Network - Institute of Aviation, Engineering Design Center, Poland
  • Jakub Broniszewski General Electric Company Polska sp. z o.o., Poland
  • Tomasz Koziara Lukasiewicz Research Network - Institute of Aviation, Engineering Design Center, Poland


The Craig–Bampton (CB) method is a well-known substructuring technique that reduces the size of a finite element model (FEM) using a set of vibration modes. For large FEA models, the reduction process could be computationally expensive since it requires algebra operations on FEM mode shapes and FEM system sparse matrices. In this paper, we investigate the potential of usage of GPU parallel processing to speed up solving the system of linear equations that results from the CB reduction process made for a model of cyclic structures. A Python based high-level approach, employing the CuPy, GinkGo and STRUMPACK libraries on the GPU, is compared with an optimized Fortran code. In side-to-side comparisons, employing the same inputs, the Python-GPU code is run on a single GPU device and the Fortran code is run on a multi-core compute node. The CB reduction process was split into several parts, each dealing with different kind of algebraic formulation of the problem. Performance comparisons were focused on the sparse system linear solver, since it turned out to be the most time-consuming part. The results suggest that the current GPU-based linear sparse solvers do not surpass the state-of-the-art CPU-based MKL PARDISO solver (at least up to 1M DOFs).


GPU, CPU, reduced order model, structural model, CuPy,


1. K. Lu et al., A review of model order reduction methods for large-scale structure systems, Shock and Vibration, 2021: 6631180, 2021, doi: 10.1155/2021/6631180.
2. R.R. Craig, Coupling of substructures for dynamic analyses: an overview, [in:] Proceeding 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Atlanta, USA, AIAA-2000-1573, 2000, doi: 10.2514/6.2000-1573.
3. R.R. Craig, A.J. Kurdila, Fundamentals of Structural Dynamics, Wiley, New York, 2006.
4. D. de Klerk, D.J. Rixen, S.N. Voormeeren, General framework for dynamic substructuring: history, review, and classification of techniques, AIAA Journal, 46(5): 1169–1181, 2008, doi: 10.2514/1.33274.
5. R. Okuta, Y. Unno, D. Nishino, S. Hido, C. Loomis, CuPy: A NumPy-Compatible Library for NVIDIA GPU Calculations, [in:] Proceedings of Workshop on Machine Learning Systems (LearningSys) in The Thirty-first Annual Conference on Neural Information Processing Systems (NIPS), Long Beach, CA, USA, 2017,
6. P. Ghysels, R. Synk, High performance sparse multifrontal solvers on modern GPUs, Parallel Computing, 110: 102897, 2022, doi: 10.1016/j.parco.2022.102897,
7. A. Abdelfattah, P. Ghysels, W. Boukaram, S. Tomov, X. Li, J. Dongarra, Addressing irregular patterns of matrix computations on GPUs and their impact on applications powered by sparse direct solvers, [in:] SC22: International Conference for High Performance Computing, Networking, Storage and Analysis, Dallas, TX, USA, pp. 1–14, 2022, doi: 10.1109/SC41404.2022.00031.
8. T. Cojean, Y.H. Tsai, H. Anzt, Ginkgo – A math library designed for platform portability, Parallel Computing, 111: 102902, 2022, doi: 10.1016/j.parco.2022.102902,
9. H. Anzt et al., Ginkgo: A modern linear operator algebra framework for high performance computing, ACM Transactions on Mathematical Software, 48(1): 2, pp. 1–33, 2022, doi: 10.1145/3480935.
10. H. Antz et al., Ginkgo: A high performance numerical linear algebra library, Journal of Open Source Software, 5(52): 2260, 2020, doi: 10.21105/joss.02260.
11. R.J. Allemang, The modal assurance criterion – Twenty years of use and abuse, Sound and Vibration Magazine, August: 14–20, 2003,
12. Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, 2003.
13. D.R. Fokkema, G.L.G. Sleijpen, H.A. Van der Vorst, Generalized conjugate gradient squared, Journal of Computational and Applied Mathematics, 71(1): 125–146, 1996, doi: 10.1016/0377-0427(95)00227-8.
14. D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2007, doi: 10.1017/CBO9780511618635.
15. A. Greenbaum, L.N. Trefethen, GMRES/CR and Arnoldi/Lanczos as matrix approximation problems, SIAM Journal on Scientific Computing, 15(2): 359–368, 1994, doi: 10.1137/0915025.
16. A. Kalinkin, A. Anders, R. Anders, Intel® Math Kernel Library PARDISO* for Intel® Xeon Phi™ Manycore Coprocessor, Applied Mathematics, 6(8): 1276–1281, 2015, doi: 10.4236/am.2015.68121.
17. M.R. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49(6): 409–436, 1952, doi: 10.6028/jres.049.044.
18. S.C. Eisenstat, H.C. Elman, M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM Journal on Numerical Analysis, 20(2): 345–357, 1983, doi: 10.1137/0720023.
19. Y. Saad, M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 7(3): 856–869, 1986. doi: 10.1137/0907058.
Feb 28, 2024
How to Cite
GORECKI, Piotr et al. Accelerating Computation of a Reduced Order Model of a Structural System Resulting from Craig–Bampton Reduction Using GPU Programming. Computer Assisted Methods in Engineering and Science, [S.l.], v. 31, n. 1, p. 51–66, feb. 2024. ISSN 2956-5839. Available at: <>. Date accessed: 17 apr. 2024. doi:
CMM-SolMech 2022