A stabilised mixed meshfree method for incompressible media: Application to linear elasticity and Stokes flow

This paper addresses some problems of modelling incompressible media using meshfree methods with a stabilised mixed formulation. The aims of this paper are twofold: to avoid the volumetric locking that occurs at the incompressible limit; and to circumvent the inf-sup condition while approximating both displacement and pressure with the same set of nodes and shape functions. We employ a modified polynomial pressure projection stabilisation method to stabilise and filter spurious pressure modes, where the gradient is corrected as proposed by Duan et al. (2014). The gradient correction is used to counteract the lack of variational consistency when numerical integration of the rational shape functions is performed, a threat that could undermine the stabilisation effort. The resulting novel method passes the mixed patch tests, and is demonstrated to be effective and optimally accurate in linear elasticity and Stokes flow test cases.