The Virtual Element Decomposition: A New Paradigm for Developing Nodal Integration Schemes for Meshfree Galerkin Methods
THE VIRTUAL ELEMENT DECOMPOSITION: A NEW PARADIGM FOR DEVELOPING NODAL INTEGRATION SCHEMES FOR MESHFREE GALERKIN METHODS
R. SILVA-VALENZUELA a, A. ORTIZ-BERNARDIN a, N. SUKUMAR b AND E. ARTIOLI c
Abstract. In meshfree Galerkin methods to solve partial diﬀerential equations, a cloud of nodes is used to discretize the domain. On using the nodal data, smooth, compactly-supported, non-polynomial basis functions are constructed to form the trial and test functions. Instead of using Gauss cubature points to compute the weak form integrals, use of nodal integration  (material state variables are stored at the nodes thereby avoiding the need for remapping) is attractive for meshfree Lagrangian simulations; however, stability of meshfree nodal integration schemes remains an unsolved problem. This work presents a new paradigm for developing nodal integration schemes for meshfree Galerkin methods via the virtual element decomposition [2, 3] on Voronoi cells that are associated with a node. In doing so, both consistency and stability of the meshfree method are ensured. A few benchmark problems in two-dimensional linear elastostatics and elastodynamics will be presented to demonstrate the accuracy and robustness of the nodal integration method.
Keywords: meshfree Galerkin methods, nodal integration, Voronoi cells, consistency and stability, virtual element decomposition.
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a Department of Mechanical Engineering, Universidad de Chile, Av. Beauchef 851, Santiago 8370456, Chile
b Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA
c Department of Civil Engineering and Computer Science, University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy