Meshfree Volume-Averaged Nodal Projection Method for Incompressible Media Problems
11th World Congress on Computational Mechanics (WCCM XI)
5th European Conference on Computational Mechanics (ECCM V)
6th European Conference on Computational Fluid Dynamics (ECFD VI)
July 20 – 25, 2014, Barcelona, Spain
Alejandro Ortiz-Bernardin¹, Jack S. Hale² and Christian J. Cyron³
1 Department of Mechanical Engineering, University of Chile, Av. Beauchef 850, Santiago 8370448, Chile, firstname.lastname@example.org
2 Faculty of Science, Technology and Communication, Research Unit in Engineering Science, University of Luxembourg, rue Richard Coudenhove-Kalergi L-1359, Luxembourg, email@example.com
3 Department of Biomedical Engineering, Yale University, Malone Engineering Center, PO box 208260, New Haven, CT 06520-8260, firstname.lastname@example.org
Key Words: Meshfree methods, Nodal Projection Methods, Incompressible Media.
We present a generalization of the meshfree method for incompressible elasticity developed in Ortiz et al. . We begin with the classical u-p mixed formulation of incompressible elasticity and proceed to eliminate the pressure parameter using a volume-averaged nodal projection technique. This results in a family of projection methods of the type T_p/T_p-1, where T_p is an approximation space of polynomial order p over a background mesh of triangles or tetrahedra for integration of the weak form integrals. These methods are particularly robust on low-order tetrahedral meshes. Our framework is generic with respect to the type of meshfree basis function used and reduces to various types of existing finite element methods such as B-bar and nodal-pressure techniques.
As a particular example, we use maximum-entropy basis functions to build a scheme T_1+/T_1 with the displacement field being enriched with bubble-like functions for stability. The flexibility of the nodal placement in meshfree methods allows us to demonstrate the importance of this bubble-like enrichment for stability; with no bubbles the pressure field is liable to oscillations, whilst with bubbles the oscillation is eliminated. Interestingly, however, with half the bubbles removed, a scheme we call T_1*/T_1, certain undesirable tendencies of the full bubble scheme in the numerical integration of the weak integrals are also eliminated. For high-order approximations, we use the RPIM basis functions up to order three in the scheme T_p*/T_p-1, where the effect of the bubble is highlighted as a mechanism for pressure oscillation stabilization. The so-devised method has important applications in linear and nonlinear incompressible elasticity as well as in incompressible fluids.
A. Ortiz, M. A. Puso and N. Sukumar, Maximum-entropy meshfree method for compressible and near-incompressible elasticity, Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 1859–1871, 2010.