Locking-Proof Tetrahedra

Mihail Francu, Árni Gunnar Ásgeirsson, Erleben, Kenny, Mads J. L. Rønnow

The simulation of incompressible materials suffers from locking when us-ing the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio𝜈gets close to0.5and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating in-compressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean and we focus on adapting it to our method. For𝜈=0.5we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking which in turn affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis

Locking-Proof Tetrahedra