Addressing Rank Deficiency in the Finite Element Analysis of Incompressible Materials With QR Decomposition and a Nullspace Approach

摘要

The finite element analysis of incompressible materials has received considerable attention by researchers in both applied mechanics and mathematics. This activity has been motivated, in part, by potential applications in biomechanics. Although robust modelling of soft tissue ultimately includes consideration of properties such as time-dependence, anisotropy, geometric non-linearity, and material non-linearity, an understanding of incompressibility is requisite to biomechanical studies of skin, blood vessels, and other biological structures.A key issue is the numerical instability associated with the physical nature of incompressibility. As the structure is deformed, the volume change is zero or negligible regardless of the loading conditions, and therefore the hydrostatic pressure component of the stress field is indeterminate from the constitutive law. As a result, mixed formulations have been developed where the hydrostatic component, p, is removed from the constitutive relationships and treated, like the displacement field, u, as an unknown to be determined on the basis of a weak formulation of equilibrium and boundary conditions. A number of these u-p formulations, together with competing methods, have been tested in detail. See, for example, [1].