Yu. V. Khalevitsky, A. V. Konovalov, A. S. Partin

CONVERGENCE PATTERNS OF KRYLOV SUBSPACE SOLVERS IN THE SIMULATION OF LARGE ELASTIC-PLASTIC DEFORMATIONS OF HETEROPHASE MEDIA

Simulating the process of heterophase medium deformation is essential for the scientific substantiation of composite material forming. Representative-volume models of composite materials are generally represented as a conglomerate of elastic and elastic-plastic bodies. Usually, composite deformations are numerically simulated by the finite element method, which requires solving a series of simultaneous linear equations. When applying a fine mesh, an iterative Krylov subspace solver is usually used.

We present a comparison of convergence for several iterative solvers in a two-dimensional finite-element model problem on the deformation of a heterophase representative volume of a composite. The volume contains 13,000 three-node triangular finite elements. The composite is based on the AMg6 alloy, and it contains 10 vol% of silicon carbide reinforcement.

Six methods are compared within the scope of the article: the biconjugate gradient stabilized method (BiCGStab), the generalized minimal residual method (GMRES), the conjugate gradient squared method (CGS), the quasi minimal residual method (QMR) and variants of the BiCGStab and QMR methods, namely BiCGStab(L) and TFQMR. The best timing is shown by a relatively rare QMR method.

Acknowledgment: The work was supported from the federal budget (government registration No. АААА-А18-118020790140-5) in terms of studying linear solver convergence in elastoplastic problems and partially supported by the Russian Science Foundation (Project 14-19-01358) in the part of developing a finite element suite for simulating composite structure deformation.

References:

1. Mishnaevsky L. Computational mesomechanics of composites: Numerical analysis of the effect of microstructures of composites of strength and damage resistance, London, John Wiley and Sons, 2007, 294 p. ISBN: 978-0-470-02764-6.

2. Schmauder S., Mishnaevsky L.  Micromechanics and nanosimulation of metals and composites: Advanced methods and theoretical concepts, Heidelberg, Springer, 2008, 420 p., ISBN 978-3-540-78678-8.

3. Pugacheva N.B., Michurov N.S., Bykova T.M. The structure and properties of the 30AL-70SIC metal matrix composite material. Diagnostics, Resource and Mechanics of materials and structures, 2015, iss. 6, pp. 6–18. DOI: 10.17804/2410-9908.2015.6.006-018.

4. Smirnov S.V., Konovalov A.V., Myasnikova M.V., Khalevitskiy Yu.V., Smirnov A.S., Igumnov A.S. A Computational Model of V95/SiCp (7075/ SiCp) Aluminum Matrix Composite Applied to Stress-Strain State Simulation under Tensile, Compressive and Shear Loading Conditions. Diagnostics, Resource and Mechanics of materials and structures, 2017, iss. 6, pp. 16–27. DOI: 10.17804/2410-9908.2017.6.016-027.

5. Pozdeev А.А., Trusov P.V., Nyashin Yu.I. Bolshie uprugoplasticheskie deformatsii: teoriya, algoritmy, prilozheniya [Large elasticplastic deformation: theory, algorithms and applications]. Moscow, Nauka, 1986, 232 p. (In Russian).

6. Yousef S. Iterative Methods for Sparse Linear Systems, second ed., SIAM Publishing, 2003, 547 p. DOI: 10.1137/1.9780898718003.

7. Driscoll Tobin A., Chuan Toh Kim, Trefethen Lloyd Nicholas. Matrix iterations: the six gaps between potential theory and convergence: Theory Center technical report, Cornell Theory Center, Cornell University, 1996. Available at: https://books.google.ru/books?id=HbFQAAAAYAAJ

8. Khalevitsky Yu.V., Konovalov A.V., and Partin A.S. On convergence of various iterative linear solvers in heterophase elastoplastic media deformation models. In: AIP Conference Proceedings, 2018, vol. 2053, pp. 030026. DOI: 10.1063/1.5084387.

9. Noël M.N., Satish C.R., Lloyd N.T. How Fast are Nonsymmetric Matrix Iterations?  SIAM Journal on Matrix Analysis and Applications, 1992, vol. 13, no. 3, pp. 778–795. DOI: 10.1137/0613049.

10. Ilyin V.P. Finite Element Methods and Technologies. In: Parallelnye vychislitelnye tekhnologii (PaVT’2009): trudy mezhdunarodnoy konferentsii [Parallel Computational Technologies: International Conference Proceedings]. Novosibirsk, ICM&MG Publ., 2007. (In Russian).

11. Smirnov A.S., Shveikin V.P., Smirnova E.O., Belozerov G.A., Konovalov Anatoly V., Vichuzhanin Dmitry I., Muizemnek Olga Yu. Effect of silicon carbide particles on the mechanical and plastic properties of the AlMg6/10% SiC metal matrix composite. Journal of Composite Materials, 2018, vol. 52 (24), pp. 3351–3363. DOI: 10.1177/0021998318765622.

12. Askeland D., Fulay P., Wright W. The Science and Engineering of Materials, Cengage Learning, 2010. ISBN 978-1-4899-2895-5.

13. Van der Vorst H.A. Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM J. Sci. and Stat. Comput., 1992, vol. 13, iss. 2, pp. 631–644. DOI: 10.1137/0913035.

14. Youcef S., Martin H.S. GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. and Stat. Comput., 1986, vol. 7, iss. 3, pp. 856–869. DOI: 10.1137/0907058.

15. Freund R.W., Nachtigal N.M. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik, 1991, vol. 60, iss. 1, pp. 315–339. DOI: 10.1007/BF01385726.

16. Sonneveld Peter. CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems. SIAM Journal on Scientific Computing, 1989, vol. 10, no. 1, pp. 36–52.

17. Sleijpen G.L.G., Fokkema D.R. BiCGstab(L) for linear equations involving matrices with complex spectrum. Electron. Trans. Numer. Anal., 1993, no 1, pp. 11–32.

18. Freund R.W. A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Stat. Comput., 1993, vol. 14, pp. 470–482.

19. Zhang J. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Computer Methods in Applied Mechanics and Engineering, 2000, vol. 189, iss. 3, pp. 825–840.

20. Khalevitsky Yu.V., Burmasheva N.V., Konovalov A V., and Partin A.S. Comparative study of Krylov Subspace method implementations for a GPU cluster in elastoplastic problems. In: AIP Conference Proceedings, 2016, vol. 1785, pp. 040024. DOI: 10.1063/1.4967081.

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