The Weighted Ambrosio--Tortorelli Approximation Scheme

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.

Ambrosio-Tortorelli Segmentation of Stochastic Images

We present an extension of the classical Ambrosio-Tortorelli approximation of the Mumford-Shah approach for the segmentation of images with uncertain gray values resulting from measurement errors and noise. Our approach yields a reliable precision estimate for the segmentation result, and it allows to quantify the robustness of edges in noisy images and under gray value uncertainty. We develop ...

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

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...

Asymptotic Analysis of Ambrosio-Tortorelli Energies in Linearized Elasticity

We provide an approximation result in the sense of Γ-convergence for energies of the form ˆ Ω Q1(e(u)) dx+ aH(Ju) + b ˆ Ju Q 1/2 0 ([u] νu) dH n−1, where Ω ⊂ Rn is a bounded open set with Lipschitz boundary, Q0 and Q1 are coercive quadratic forms on Mn×n sym , a, b are positive constants, and u runs in the space of fields SBD2(Ω) , i.e., it’s a special field with bounded deformation such that i...