Bézier Shell Finite Element for Interactive Surgical Simulation

Abstract

There is a strong need, in surgical simulations, for physically based deformable model of thin or hollow structures. The use of shell theory allows to have a well-founded formulation resulting from continuum mechanics of thin objects. However, this formulation asks for second order spatial derivatives so requires the use of complex elements. In this paper, we present a new way of building the interpolation: First, we use the trianular cubic Bezier shell to allow for a good continuity inside and between the elements and second, we build a kinematic mapping to reduce the degrees of freedom of the element from 10 control points with 3 Degrees of Freedom ($=30$ DOFs) to only 3 nodes with 6 DOFs ($=18$ DOFs). This reduction allows for good computation performance. This new shell model description is also used to map a smooth surface (for the collision detection and response) on a coarse mechanical mesh to account for the complex contacts that take place during surgical procedures. We demonstrate the convergence and the computational efficiency of our approach as well as its use in two different simulation cases: the planning of surgery for congenital heart disease correction and a preliminary simulation of childbirth.