The Virtual Element Decomposition: A New Paradigm for Developing Nodal Integration Schemes for Meshfree Galerkin Methods
- Post by: alejandro
- January 24, 2019
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A talk given by A. Ortiz-Bernardin at WONAPDE 2019, January 22, 2019, Concepción, Chile.
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 [1] (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.
References
[1] M. A. Puso, J. S. Chen, E. Zywicz, W. Elmer. Meshfree and finite element nodal integration methods. International Journal for Numerical Methods in Engineering. 74(3): 416–446, 2008.
[2] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01):199–214, 2013.
[3] A. Ortiz-Bernardin, A. Russo, N. Sukumar. Consistent and stable meshfree Galerkin methods using the virtual element decomposition. International Journal for Numerical Methods in Engineering. 112(7): 655– 684, 2017.
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