On the Variational Approximation of Free-Discontinuity Problems in the Vectorial Case

Abstract

We provide a variational approximation for quasiconvex energies defined on vector valued special functions with bounded variation. We extend the Ambrosio–Tortorelli's construction to the vectorial case.

Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures

“…In fact all the previous arguments are based on Theorem 2.3 concerning the elliptic approximation and on Theorem 2.4 about the density of piecewise smooth functions with respect to the elastic energy. An elliptic approximation of Ambrosio-Tortorelli type of (7.1) has been proved in [14], while a density result of piecewise smooth functions with respect to non-isotropic surface energies has been proved in [10]. We conclude that all the previous theorems can be modified in order to treat the more general energy (7.1).…”

“…As in [14], letφ n be a cut-off function associated tow n , such that We are now in a position to conclude the proof. We pose v n := min{w n , w i n , i = 1, .…”

Asymptotic Analysis of Ambrosio--Tortorelli Energies in Linearized Elasticity

“…To prove such assertions we establish first the equi-coercivity in the space BD of the energies F k in (1.3) (see (4.7)). Given this, we use a global technique introduced by Ambrosio in [3] (see also [24,25]) to gain coercivity in the space SBD. To this aim we construct a new sequence of displacements, with SBV regularity, by cutting around suitable sublevel sets of v in order to decrease the elastic contribution of the energy at the expense of introducing a surface term that can be kept controlled (see (4.14)).…”

“…The quoted result has been later extended in several directions with different aims: to approximate energies arising in the theory of nematic liquid crystals [9], general free discontinuity functionals defined over vector-valued fields [24,25], the Blake and Zisserman second order model in computer vision [5], or fracture models for brittle linearly elastic materials [16,17,29]; to provide a common framework for curve evolution and image segmentation [33,1,2]; to study the asymptotic behavior of gradient damage models under different regimes [22,28]; and to give a regularization of variational models for plastic slip [7].…”

An Adaptive Finite Element Approximation of a Generalized Ambrosio–tortorelli Functional

“…Although previous numerical schemes have focused on approximating the Francfort-Marigo energy by the standard Ambrosio-Tortorelli functional, there exists, in fact, an entire family of generalized approximating functionals [10,24]. In this paper we consider the minimization of a generalized functional together with a new method for implementing crack irreversibility, which is based on the monotonicity condition proposed by Giacomini [27].…”