Alessandro Giacomini
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 fractures in linearly elastic bodies.
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The Ambrosio-Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it Γ-converges to a Mumford-Shah image segmentation functional depending on the weight $ωdx$, where $ω\in SBV(Ω)$, and on its value $ω^-$.
Motivated by models of fracture mechanics, this paper is devoted to the analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. In the spirit of gradient flows in metric spaces, such evolutions are defined in terms of curves of maximal unilateral slope, and are constructed by means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah regime is also carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set.
This work is devoted to study the asymptotic behavior of critical points $\{(u_\varepsilon,v_\varepsilon)\}_{\varepsilon>0}$ of the Ambrosio-Tortorelli functional. Under a uniform energy bound assumption, the usual $Γ$-convergence theory ensures that $(u_\varepsilon,v_\varepsilon)$ converges in the $L^2$-sense to some $(u_*,1)$ as $\varepsilon\to 0$, where $u_*$ is a special function of bounded variation. Assuming further the Ambrosio-Tortorelli energy of $(u_\varepsilon,v_\varepsilon)$ to converge to the Mumford-Shah energy of $u_*$, the later is shown to be a critical point with respect to i…
Existence, uniqueness and regularity of solutions to the parabolic Ambrosio-Tortorelli system
研究任意维安布罗西奥 - 托尔托雷利泛函梯度流初边值问题，用时间离散欧拉格式证弱梯度流存在及最大原理，确定唯一性空间，证明时空域内部光滑，在额外条件下证解直至时空边界的最优正则性。
Critical points of the one dimensional Ambrosio-Tortorelli functional with an obstacle condition
该研究针对带障碍条件的一维Ambrosio-Tortorelli泛函临界点展开，分析其正则性、继承性质等特征，证明其极限为继承障碍序列诱导不连续性的Mumford-Shah能量临界点，为裂纹不可逆演化的离散模型分析提供理论支撑。
Critical points of the two-dimensional Ambrosio-Tortorelli functional with convergence of the phase-field energy
本文研究了二维Ambrosio-Tortorelli功能临界点的收敛性，证明了在相场能量收敛到Mumford-Shah功能长度项时，极限解是Mumford-Shah功能的临界点。
On jump minimizing liftings for $\mathbb S^1$-valued maps and connections with Ambrosio-Tortorelli-type $Γ$-limits
该研究刻画 $\mathbb S^1$ 值映射的 Ambrosio-Tortorelli 型泛函在不同定义域下的两类 $Γ$-极限，并通过提升序列的紧性结果建立非局部极限与跳跃最小化提升存在性的联系。
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 show that both the variants as above provide an elliptic approximation of the Mumford-Shah functional in the sense of $Γ$-convergence. In particular the variant with the Laplacian penalization can be implemented without any difficulties compared to the standard Ambrosio-Tortorelli functional. The computational results indicate several advantages however. First …
Existence, uniqueness and $L^2_t (H_x ^2) \cap L^\infty_t (H^1_x) \cap H^1_t (L^2_x) $ regularity of the gradient flow of the Ambrosio-Tortorelli functional
本文研究二维固定参数下 Ambrosio-Tortorelli 泛函的梯度流，借助新的关键 $L^2_t(H_x^2)$ 估计技术，证明其全局存在性、唯一性及所需混合时空正则性。
We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing $δ$, the discretized functionals $Γ$-converge to the Mumford-Shah functional only if $δ\ll\varepsilon$, $\varepsilon$ being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic and isotropic random lattices we prove this $Γ$-convergence result also for $δ\sim\varepsilon$, a regime at which the discretization with respect to …
$Γ$-Convergence of an Ambrosio-Tortorelli approximation scheme for image segmentation
本文证明了Ambrosio-Tortorelli逼近在分片光滑设置下对Mumford-Shah泛函的$\Gamma$-收敛，从而为图像分割中该逼近的极小元收敛性提供了理论保证。
We provide a variational approximation of Ambrosio-Tortorelli type for brittle fracture energies of piecewise-rigid solids. Our result covers both the case of geometrically nonlinear elasticity and that of linearised elasticity.
In this paper we introduce a new phase field approximation of the Mumford-Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an $H^1$-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio-Tortorelli approximation, where the phase fiel…
$Γ$-convergence of free discontinuity problems for circle-valued maps in the linear regime
本文研究了圈值映射在线性增长情形下的$Γ$-收敛性，揭示了非局部极限的形成机制，并扩展了之前二次情形的分析。
In this paper we propose a notion of irreversibility for the evolution of cracks in presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models. We investigate its applicability to the construction of a quasi-static evolution in a simple one-dimensional model. The cohesive fracture model arises naturally via Gamma-convergence from a phase-field model of the generalized Ambrosio-Tortorelli type, which may be used as regularization for numerical simulations.
We study the $Γ$-limit of Ambrosio-Tortorelli-type functionals $D_\varepsilon(u,v)$, whose dependence on the symmetrised gradient $e(u)$ is different in $\mathbb{A} u$ and in $e(u)-\mathbb{A} u$, for a $\mathbb{C}$-elliptic symmetric operator $\mathbb{A}$, in terms of the prefactor depending on the phase-field variable $v$. This is intermediate between an approximation for the Griffith brittle fracture energy and the one for a cohesive energy by Focardi and Iurlano. In particular we prove that $G(S)BD$ functions with bounded $\mathbb{A}$-variation are $(S)BD$.
An elliptic approximation for phase separation in a fractured material
本文研究弹性材料中相分离与断裂耦合的变分问题，提出用修正的 Cahn-Hilliard 能量与 Ambrosio-Tortorelli 型逼近相结合，通过 Gamma 收敛恢复尖锐的自由边界和自由不连续能量。
Convergence of critical points for a phase-field approximation of 1D cohesive fracture energies
本文研究一维内聚断裂能量相场近似的临界点收敛性，证明相场能量临界点收敛到极限泛函的一类选定临界点，且该类中每个临界点均可由相场泛函临界点族逼近。
Existence of solutions to a phase-field model of dynamic fracture with a crack-dependent dissipation
该研究提出含裂纹尖端速度耗散效应的Ambrosio–Tortorelli型动态断裂相场模型，通过调整已有时间离散格式证明满足Griffith能量耗散平衡的裂纹演化解存在，并分析了无耗散的相关动态相场模型。
We analyze a finite-difference approximation of a functional of Ambrosio-Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $δ$ is smaller than the ellipticity parameter $\varepsilon$, we show the $Γ$-convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $L^p$ fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit f…
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