From a Modified Ambrosio-Tortorelli to a Randomized Part Hierarchy Tree

Ambrosio-tortorelli Approximation of Quasi-static Evolution of Brittle Fractures

We define a notion of quasistatic evolution for the elliptic approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we prove that this regular evolution converges to a quasi static growth of brittle fractures in linearly elastic bodies.

An Estimation Theoretical View on Ambrosio-Tortorelli Image Segmentation

In this paper, we examine the Ambrosio-Tortorelli (AT) functional [1] for image segmentation from an estimation theoretical point of view. Instead of considering a single point estimate, i.e. the maximum-a-posteriori (MAP) estimate, we adopt a wider estimation theoretical view-point, meaning we consider images to be random variables and investigate their distribution. We derive an effective blo...

An Adaptive Finite Element Approximation of a Generalised Ambrosio–Tortorelli Functional

The Francfort–Marigo model of brittle fracture is posed in terms of the minimization of a highly irregular energy functional. A successful method for discretizing the model is to work with an approximation of the energy. In this work a generalised Ambrosio–Tortorelli functional is used. This leads to a bound-constrained minimization problem, which can be posed in terms of a variational inequali...

Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements

Motivated by models of fracture mechanics, this paper is devoted to the analysis of a unilateral gradient flow of the Ambrosio-Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. Solutions of such evolution are constructed by means of an implicit Euler scheme. An asymptotic analysis in the Mumford-Shah regime is then carried out. It shows...

Second-Order Edge-Penalization in the Ambrosio-Tortorelli functional