Meshfree volume-averaged nodal pressure methods for incompressible elasticity

We present a generalisation of the meshfree method for incompressible elasticity developed in Ortiz et al. (10.1016/j.cma.2010.02.013). We begin with the classical u-p mixed formulation of incompressible elasticity before eliminating the pressure using a volume-averaged nodal projection technique. This results in a family of projection methods of the type Q_p/Q_p-1 where Q_p is an approximation space of polynomial order p. These methods are particularly robust on low-quality tetrahedral meshes. Our framework is generic with respects to the type meshfree basis function used and includes 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 Q_1+/Q_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 Q_1*/_Q_1, certain undesirable tendencies of the full bubble scheme are also eliminated. This has important applications in non-linear hyperelasticity. We also discuss some difficulties associated with moving to second-order maximum entropy shape functions associated with numerical integration errors.