Related papers: A derivation of Griffith functionals from discrete…
Linear fracture mechanics (or at least the initiation part of that theory) can be framed in a variational context as a minimization problem over a SBD type space. The corresponding functional can in turn be approximated in the sense of…
In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce…
This paper is concerned with the $\Gamma$-limits of Ambrosio-Tortorelli-type functionals, for maps $u$ defined on an open bounded set $\Omega\subset\mathbb R^n$ and taking values in the unit circle $\mathbb S^1\subset\mathbb R^2$. Depending…
We propose and study two variants of the Ambrosio-Tortorelli functional where the first-order penalization of the edge variable $v$ is replaced by a second-order term depending on the Hessian or on the Laplacian of $v$, respectively. We…
We define a notion of quasi-static 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…
We propose a $\Gamma$-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a…
The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah functional that has been widely used for image segmentation. The approximation has the advantages of being easy to implement, maintaining the segmentation…
The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation $GSBD^p(\Omega)$, $p\in(1,\infty)$, their treatment is however hindered by the very low…
We consider a nonlinear, frame indifferent Griffith model for nonsimple brittle materials where the elastic energy also depends on the second gradient of the deformations. In the framework of free discontinuity and gradient discontinuity…
We propose a phase-field model of dynamic fracture based on the Ambrosio--Tortorelli's approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in…
We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the…
In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we analyzed the $\Gamma$-convergence of non-homogeneous Dirichlet boundary problem for…
We study the integral representation of $\Gamma$-limits of $p$-coercive integral functionals of the calculus of variations in the spirit of \cite{dalmaso-modica86}. We use infima of local Dirichlet problems to characterize the limit…
Given an image $u_0$, the aim of minimising the Mumford-Shah functional is to find a decomposition of the image domain into sub-domains and a piecewise smooth approximation $u$ of $u_0$ such that $u$ varies smoothly within each sub-domain.…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
We propose a first rigorous homogenisation procedure in image-segmentation models by analysing the relative impact of (possibly random) fine-scale oscillations and phase-field regularisations for a family of elliptic functionals of Ambrosio…
This paper develops a functional-analytic framework for approximating the push-forward induced by an analytic map from finitely many samples. Instead of working directly with the map, we study the push-forward on the space of locally…
This work is devoted to the variational derivation of a reduced model for brittle membranes in finite elasticity. The main mathematical tools we develop for our analysis are: (i) a new density result in $GSBV^{p}$ of functions satisfying a…
Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the…
The Ambrosio-Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it {\Gamma}-converges to a Mumford-Shah image segmentation functional depending on the weight $\omega dx$, where $\omega\in…