Over the past several decades, meshfree methods have proved advantageous over the traditional finite element methods in large-deformation problems such as manufacturing, fragment-impact, and natural disasters. However, instability in nodal integration of Galerkin meshfree methods is an inherent obstacle that must be addressed for effective dynamic simulations [1]. Both stabilized conforming nodal integration (SCNI) [2] and direct nodal integration (DNI) [3] suffer from low energy modes of node-to-node oscillation due to the under integrated internal energy in Galerkin meshfree methods, which can grow dramatically and destroy the simulation solution, particularly in high-strain-rate and highly nonlinear problems. While SCNI can alleviate the instability in DNI and provide better accuracy, severe oscillations can still occur when the surface to volume ratio of the domain is sufficiently low, or when the discretization is sufficiently fine.In this research work, a naturally stabilized conforming nodal integration method is proposed for the meshfree modelling of nonlinear explicit dynamic problems. The Lagrangian meshfree shape functions built under material configuration provides the efficiency in meshfree modelling, and the use of Lagrangian smoothed conforming gradients guarantees linear exactness (i.e. pass the linear patch test). Then, the Taylor series expansion of strain and stress are employed, which finally lead to a supplement of internal energy leading to a naturally stabilized nodal integration. As a result, the second order gradients of meshfree shape function are required, but are instead approximated by smoothed first order implicit gradients, which entails far less effort than computing the standard second order meshfree gradients. In fact, the formulation is devoid of direct derivatives altogether.The Taylor series expansion approach to stabilization entails differentiation of stress, which is not straightforward for arbitrary constitutive laws. In particular, commercial explicit dynamic codes often have a wide variety of material models, and do not compute material response tangents. To remedy this issue, a novel implicit stress gradient method is proposed for stabilization of arbitrary material models with high nonlinearity and softening. The effectiveness of proposed methodology is tested with several benchmark problems including elastic, nearly incompressible and plastic materials, as well as the problems with softening.References[1] Hillman, M., &amp; Chen, J. S. (2016). An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics.&nbsp;International Journal for Numerical Methods in Engineering,&nbsp;107(7), 603-630.[2] Chen, J. S., Wu, C. T., Yoon, S., &amp; You, Y. (2001). A stabilized conforming nodal integration for Galerkin mesh‐free methods.&nbsp;International journal for numerical methods in engineering,&nbsp;50(2), 435-466.[3] Beissel, S., &amp; Belytschko, T. (1996). Nodal integration of the element-free Galerkin method. Computer methods in applied mechanics and engineering, 139(1-4), 49-74.

A Naturally Stabilized Conforming Nodal Integration for Nonlinear Explicit Dynamics

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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