An enriched quadtree/octree implicit boundary finite element method for the simulation of incompressible hyperelastic materials

Octree (and quadtree) representations of computational geometry are particularly well suited to modelling domains that are defined implicitly, such as those generated by image segmentation algorithms applied to medical scans. In this work we consider the simulation of soft-tissue which can be modelled with a hyperelastic constitutive law. We include the effects of both non-linear geometry and material properties.

Similarly to Legrain et al. (10.1002/nme.3070) and Moumnassi et al. (10.1016/j.cma.2010.10.002) we use the implicitly designed level set functions as the basis for a partition of unity enrichment to more accurately represent the domain boundary. Furthermore we use traditional extended finite element (XFEM) ideas to introduce arbitrary cuts and discontinuities in the domain.

We explore the use of a two-field u-p mixed approach as well as a smoothed finite element method (SFEM) to deal with the problem of volumetric-locking in the incompressible limit.

We will discuss the extension of our method towards both traditional parallel and GPU implementation. We aim to solve extremely large problems as well as produce snapshots to feed into model order reduction methods for real-time surgical simulations.