Problem of crack perturbation based on level sets and velocities

We describe cracks with the help of a given velocity as zero‐level sets of a non‐negative function satisfying a transport equation. For smooth velocities this description is equivalent to the coordinate transformation of a domain containing the crack inside. Analytical examples of cracks described by smooth as well as discontinuous velocities are presented in 2D and 3D domains. Based on a level‐set formulation we consider the crack perturbation problem subject to a non‐penetration condition and derive the formula for the shape derivative.