# A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition

## TL;DR

This paper introduces a new nodal integration scheme for meshfree Galerkin methods based on virtual element decomposition, improving accuracy and robustness in linear and nonlinear solid mechanics simulations.

## Contribution

The paper develops a novel nodal integration approach using virtual element decomposition, applicable to any linear meshfree approximant, enhancing stability and accuracy.

## Key findings

- Achieves optimal convergence in benchmark problems
- Demonstrates improved reliability over standard methods
- Effective in linear and nonlinear solid mechanics simulations

## Abstract

In this paper, we present a novel nodal integration scheme for meshfree Galerkin methods that draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the bilinear form is decomposed into a consistency part and a stability part that ensure consistency and stability of the method. The performance of the proposed nodal integration scheme is assessed through benchmark problems in linear and nonlinear analyses of solids for small displacements and small-strain kinematics. Numerical results are presented for linear elastostatics and linear elastodynamics, and viscoelasticity. We demonstrate that the proposed nodally integrated meshfree method is accurate, converges optimally, and is more reliable and robust than a standard cell-based Gauss integrated meshfree method.