Over the past two decades, domain integration in Galerkin-based meshfree methods for high strain rate events remains a challenge due to numerical instability and sub-optimal convergence \[1\]. Many integration methods have been proposed to resolve the above issues. Some of the integration methods require information obtained from conforming subdomains. When they are applied to simulate large deformation problems using updated reference configurations, e.g. semi-Lagrangian formulation, high computational cost due to the need of conforming subdomains hinders their application for high strain rate simulations.  Although Natural Stabilized Nodal Integration (NSNI) with Variational Consistent Integration (VCI) \[2\] alleviates the above issue without the need of conforming subdomains, the resultant non-symmetric system increases the computational cost and some spurious modes may still be observed in the semi-Lagrangian problems. In this work, a novel robust integration method is introduced based on the extension of modified Simpson’s rule \[3\]. The integration method is derived such that the quadrature point relies only on the nodal point without auxiliary evaluation points from any conforming nodal presentation domain. To further achieve computational efficiency while maintaining accuracy, implicit gradient Reproducing Kernel is applied in the modified quadrature rule. It is shown that the proposed integration method has a slightly higher eigen-spectrum than the fully integration FEM, maintains numerical stability, and at the same time attains the optimal convergence. The resultant integration method does not rely on conforming subdomains and is free from VCI corrections, which is not straightforward to obtain when dealing with continuous evolving surfaces. Several numerical examples are studied to assess the stability, convergence, and effectiveness of the proposed method for non-linear large deformation problems using the semi-Lagrangian meshfree formulation. The proposed integration methods show superior performance over the other nodal integration method in the non-linear large deformation problems.

 

\[1\] J.S. Chen, M. Hillman, and S.W. Chi, (2017). Meshfree methods: progress made after 20 years. Journal of Engineering Mechanics 143(4), 04017001

 

\[2\] M. Hillman, J.S. Chen, S.W. Chi (2014). Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems. *Computational Particle Mechanics* 1 (3), 245-256.

 

\[3\] M. Hillman, and J. S. Chen, (2016). An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. *Int. J. Numer. Methods Eng*. 107 (7): 603–630.

 

\[4\] P. Cerone and S. S. Dragomir, (2000). Midpoint-type Rules from an Inequalities Point of View. *Handbook of Analytic-Computational Methods in Applied Mathematics*, *CRC Press*, 135–200.


A Novel Stabilized Integration in Meshfree Modeling for the Impact Problems

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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