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Efficient and flexible MATLAB implementation of 2D and 3D elastoplastic problems

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  • Čermák, M.
  • Sysala, S.
  • Valdman, J.

Abstract

Fully vectorized MATLAB implementation of various elastoplastic problems formulated in terms of displacement is considered. It is based on implicit time discretization, the finite element method and the semismooth Newton method. Each Newton iteration represents a linear system of equations with a tangent stiffness matrix. We propose a decomposition of this matrix consisting of three large sparse matrices representing the elastic stiffness operator, the strain-displacement operator, and the derivative of the stress-strain operator. The first two matrices are fixed and assembled once and only the third matrix needs to be updated in each iteration. Assembly times of the tangent stiffness matrices are linearly proportional to the number of plastic integration points in practical computations and never exceed the assembly time of the elastic stiffness matrix. MATLAB codes are available for download and provide complete finite element implementations in both 2D and 3D assuming von Mises and Drucker–Prager yield criteria. One can also choose several finite elements and numerical quadrature rules.

Suggested Citation

  • Čermák, M. & Sysala, S. & Valdman, J., 2019. "Efficient and flexible MATLAB implementation of 2D and 3D elastoplastic problems," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 595-614.
    • Handle: RePEc:eee:apmaco:v:355:y:2019:i:c:p:595-614
    DOI: 10.1016/j.amc.2019.02.054
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    References listed on IDEAS

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    1. Sysala, Stanislav, 2012. "Application of a modified semismooth Newton method to some elasto-plastic problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(10), pages 2004-2021.
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    Cited by:

    1. Moskovka, Alexej & Valdman, Jan, 2022. "Fast MATLAB evaluation of nonlinear energies using FEM in 2D and 3D: Nodal elements," Applied Mathematics and Computation, Elsevier, vol. 424(C).
    2. Vasileios E. Kontosakos, 2020. "Fast Quadratic Programming for Mean-Variance Portfolio Optimisation," SN Operations Research Forum, Springer, vol. 1(3), pages 1-15, September.
    3. Innerberger, Michael & Praetorius, Dirk, 2023. "MooAFEM: An object oriented Matlab code for higher-order adaptive FEM for (nonlinear) elliptic PDEs," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    4. Lukáš Pospíšil & Martin Čermák & David Horák & Jakub Kružík, 2020. "Non-Monotone Projected Gradient Method in Linear Elasticity Contact Problems with Given Friction," Sustainability, MDPI, vol. 12(20), pages 1-11, October.
    5. Meijun Zhou & Jiayu Qin & Zenan Huo & Fabio Giampaolo & Gang Mei, 2022. "epSFEM: A Julia-Based Software Package of Parallel Incremental Smoothed Finite Element Method (S-FEM) for Elastic-Plastic Problems," Mathematics, MDPI, vol. 10(12), pages 1-25, June.
    6. Dang-Bao Tran & Jaroslav Navrátil & Martin Čermák, 2021. "An Efficiency Method for Assessment of Shear Stress in Prismatic Beams with Arbitrary Cross-Sections," Sustainability, MDPI, vol. 13(2), pages 1-17, January.

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