A uniform nodal strain tetrahedron with isochoric stabilization

Abstract

SUMMARYA stabilized node-based uniform strain tetrahedral element is presented and analyzed for finite deformation elasticity. The element is based on linear interpolation of a classical displacement-based tetrahedral element formulation but applies nodal averaging of the deformation gradient to improve mechanical behavior, especially in the regime of near-incompressibility where classical linear tetrahedral elements perform very poorly. This uniform strain approach adopted here exhibits spurious modes as has been previously reported in the literature. We present a new type of stabilization exploiting the circumstance that the instability in the formulation is related to the isochoric strain energy contribution only and we therefore present a stabilization based on an isochoric-volumetric splitting of the stress tensor. We demonstrate that by stabilizing the isochoric energy contributions only, reintroduction of volumetric locking through the stabilization can be avoided. The isochoric-volumetric splitting can be applied for all types of materials with only minor restrictions and leads to a formulation that demonstrates impressive performance in examples provided.

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

“…Four different types of boundary conditions are considered: Fix (condition 1), Free (condition 2), Skew symmetric (condition 3) and Symmetric (condition 4). 6 field, contrary to the classical displacement based formulation [33,35,[37][38][39]. Crucially, the evolution of the fibre map (1b) must be advanced in time satisfying a set of compatibility conditions (known as involutions [62]), namely CURL(F ) = 0.…”

“…This is due to the fact that the amount of dissipation added into the JST algorithm can be tuned through the tailormade artificial dissipative parameter (see (35) in [1]). The upwind FVM, on the other hand, has a fixed physically based numerical dissipation contributed from Riemann solver, making it less convenient for problems where physical diffusion is present.…”

“…Extensive efforts have since been made to further develop this class of averaged nodal strain technologies with the use of various types of stabilisation [34][35][36][37][38][39]. Despite exhibiting geometric locking-free behaviours, the resulting formulation still suffers from spurious hydrostatic pressure fluctuations when simulating nearly incompressible materials.…”

A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

“…Reference [31] extended this application to the large strain regime with the idea of employing both a nodal average Jacobian and a nodal average deformation gradient in the calculation of the stress tensor. As reported in [32][33][34][35], the resulting formulation suffers from artificial mechanisms similar to hourglassing unless some form of stabilisation is used. Despite exhibiting very good behaviour in terms of displacements, this class of averaged nodal strain tetrahedral formulations tend to exhibit non-physical hydrostatic pressure fluctuations [32,36].…”

“…At present and to the best of our knowledge, most of the proposed schemes for linear tetrahedral elements are restricted to elastostatics [32,[40][41][42][43]. The development of an effective linear tetrahedral formulation in the range of fully and nearly incompressible large strain dynamics remains an open issue.…”