# Level Set Method

**Introduction**

Within the XFEM framework, the discontinuities are modeled independent of the finite element mesh. This creates the problem of how one goes about keeping track of the evolution of these discontinuities as they are not explicitly defined by the finite element mesh. The level set method has been used to tracking cracks, voids and holes. The level set method was originally introduced by Osher and Sethian^{1} for tracking the evolution of moving boundaries and later adopted and modified for use in XFEM. The narrow band level set method introduced by Adalsteinsson and Sethian^{2} can be used to reduce the computational burden associated with the level set method. The level set function for a closed curve takes the following form.

**Cracks**

The level set method was originally introduced for tracking the evolution of closed boundaries. The method was then extended for tracking the evolution of open segments by Stolarska^{3} so that the evolution of a crack could be tracked. This is accomplished through the use of two orthogonal level set functions. The psi level set is used to track the crack body, while the phi level set is used to track the crack tip. For the case when there are multiple crack tips, multiple phi level set functions are used. The i^{th} crack tip is found to be the intersection of the zero level set of psi and the zero level set of the i^{th} phi. Specifically, the phi and psi level set functions are defined such in the following way.

As the crack grows, the phi and psi level set functions evolve as shown in the animation above. Both the full and narrow band level set representations of the crack are given. If the animation does not play you may need to install Adobe Flash to your computer.

**Inclusions and Voids**

The level set method can also be used to track the evolution of voids and inclusions^{4} using the original version of the level set method for closed boundaries. In particular for a circular void or inclusion the cooresponding level set function is given by

where (x_{o},y_{o}) is the center and r_{o} is the radius of the inclusion.

**References**

1. Osher, S., Sethian, J.A. (1988) "Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations," *Journal of Computational Physics*, 79, 12-49.

2. Adalsteinsson, D., Sethian, J.A. (1995) "A fast level set method for propagating interfaces," *Journal of Computational Physics*, 118, 269-277.

3. Stolarska, M., Chopp, D.L., Moes, N., Belytschko, T. (2001) "Modelling crack growth by level sets in the extended finite element method," *International Journal for Numerical Methods in Engineering*, 51, 943-960.

4. Sukumar, N., Chopp, D.L., Moes, N., Belytschko, T. (2001) "Modeling holes and inclusions by level sets in the extended finite-element method," *Computer Methods in Applied Mechanics and Engineering*, 190, 6183-6200.