# Simulation of Jawbone Fracture with the Extended Finite Element Method

## Outline

Surgical simulations help training surgeons and improve the surgery workflow and outcome. This work is part of an interdisciplinary development of a surgical simulator emulating an initiation of fracture in jawbone in order to relocate parts of the jaw, whereby the fracture initiation causes a three dimensional crack propagation which is to be simulated. Non-planar 3D crack growth is modeled in conjunction with the extended finite element method (XFEM), which has proven to achieve excellent results in fracture mechanics. The implementation should have a real time response under a 6 DoF force feedback.

Overview The Extended Finite Element Method (XFEM) allows to predict how objects deform as cracks form and propagate through them. Here, we propose the use of XFEM to model the deformations resulting from a crack initiation through the jaw. A coupling of the XFEM to the Level-Set Method is required to represent the shape and location of the crack. The interaction integral method is used in order to extract the mixed modes stress intensity factors which are needed to compute the crack growth angle. As a start, two dimensional crack growth is realized in order to test examples and adjust expectations before extending to three dimensions.

Previous work–2D crack growth In 2D crack propagation, the crack is described by two orthogonal level set functions (signed distance lines/zero level lines). One describes the crack shape and the other one describes the location of the crack tip. Two types of enrichment functions are used: a sign-enrichment along the crack and four branch enrichment functions at the element containing the crack tip. The stiffness matrix from element contributions is then assembled whereby the enrichment in the matrix accounts for the presence of the crack. Then, the system of equations is solved for the displacement field at all nodes in the domain. The XFEM solution is then superimposed with an auxiliary solution in order to extract the stress intensity at the crack tip represented by two stress intensity factors (SIF) in 2D. The option of a suitable material model for the jawbone is not a choice in the 2D problem. Therefore, an assumption of homogeneous, linear, elastic material behaviour for the jawbone is made.

Expected work–3D crack growth In 3D crack analysis, the crack is as well described by two 2D level set functions in order to describe the crack surface and crack front shape and location. Multiple points are to be located on the crack front and treated as individual cracktips for the extraction of the three SIF. Yet, for the 3D crack propagation, an extension of the velocity field of the “tips” at the crack front to the whole domain is required in order to update the level set functions (LSF) and reintialize them to preserve the signed distance characteristic and orthogonalization. The design should consider special aspects to include the complex structure and geometry of the jawbone. Initially, a simplified constitutive model of linear elasticity is applied. Then various parameters representing the behavior of bone are to be included while keeping the simulation simple and efficient to allow for real-time simulations. Several parrallelization strategies are to be investigated depending on the desired domain decomposition approach to achieve high performance. Finally, a validation and verification of the simulation with medical personnel will take place.