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Smoothed naturally stabilized RKPM for non-linear explicit dynamics with novel stress gradient update

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Abstract

A smoothed naturally stabilized conforming nodal integration (S-NSNI) for the reproducing kernel particle method (RKPM) is proposed for non-linear explicit dynamics. The Taylor series expansion of the nodal strains in the solid variational formulation is employed, which introduces stabilization by enriching the energy of the originally underintegrated system. As a result, the higher-order gradients of meshfree shape functions are required, and are approximated by using the gradient smoothing technology. In conjunction with implicit gradients, this results in a formulation devoid of computationally intensive differentiation of meshfree shape functions. In addition, when conforming smoothing domains are employed, the formulation is variationally consistent and converges optimally while passing the patch test. The smoothed framework further alleviates a numerical locking in the original NSNI by avoiding direct differentiation that leads to inaccuracies in stabilization terms. To enhance the nontrivial computation of the required stress gradients or estimates there of, a novel stress re-interpolation methodology is introduced, which is favorable for the implementation of arbitrary constitutive laws such as those in commercial codes, which are often plentiful. The framework is developed for both Lagrangian and semi-Lagrangian RKPM and is applicable to both moderate and extreme deformations. The effectiveness of the proposed methodology is demonstrated by the implementation into the KC-FEMFRE and MEGA codes and applied to several benchmark problems that include elastic, nearly incompressible, and plastic materials, as well as geomaterial modeling using continuum damage mechanics coupled with plasticity.

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Acknowledgements

The support of the work done at Penn State by Karagozian and Case Inc. is gratefully acknowledged. At Brown the work was supported through the ONR Grant No. N00014-21-1-2670. J. Wang was also supported by the Hibbitt Postdoctoral Fellowship in the School of Engineering at Brown.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, 16802, USA

    Jiarui Wang & Michael Hillman

  2. School of Engineering, Brown University, 184 Hope St., Providence, RI, 02912, USA

    Jiarui Wang & Yuri Bazilevs

  3. Karagozian & Case, 700 N Brand Blvd, # 700, Glendale, CA, 91203, USA

    Michael Hillman, Dominic Wilmes & Joseph Magallanes

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  1. Jiarui Wang
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Correspondence to Jiarui Wang.

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Wang, J., Hillman, M., Wilmes, D. et al. Smoothed naturally stabilized RKPM for non-linear explicit dynamics with novel stress gradient update. Comput Mech (2024). https://doi.org/10.1007/s00466-024-02494-0

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