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 the convergence towards a generalized heat equation outside a time increasing crack set. In the spirit of gradient flows in metric spaces, a notion of curve of maximal unilateral slope is also investigated, and analogies with the unilateral slope of the Mumford–Shah functional are also discussed.
@article{AIHPC_2014__31_4_779_0, author = {Babadjian, Jean-Fran\c{c}ois and Millot, Vincent}, title = {Unilateral gradient flow of the {Ambrosio{\textendash}Tortorelli} functional by minimizing movements}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {779--822}, publisher = {Elsevier}, volume = {31}, number = {4}, year = {2014}, doi = {10.1016/j.anihpc.2013.07.005}, zbl = {1302.35051}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.005/} }
TY - JOUR AU - Babadjian, Jean-François AU - Millot, Vincent TI - Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 779 EP - 822 VL - 31 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.005/ DO - 10.1016/j.anihpc.2013.07.005 LA - en ID - AIHPC_2014__31_4_779_0 ER -
%0 Journal Article %A Babadjian, Jean-François %A Millot, Vincent %T Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 779-822 %V 31 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.005/ %R 10.1016/j.anihpc.2013.07.005 %G en %F AIHPC_2014__31_4_779_0
Babadjian, Jean-François; Millot, Vincent. Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, pp. 779-822. doi : 10.1016/j.anihpc.2013.07.005. http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.005/
[1] Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995), 191 -246 | MR | Zbl
,[2] Energies in SBV and variational models in fracture mechanics, Homogenization and Applications to Material Sciences, Nice, 1995, GAKUTO Internat. Ser. Math. Sci. Appl. vol. 9 , Gakkotosho (1995), 1 -22 | MR | Zbl
, ,[3] Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press (2000) | MR | Zbl
, , ,[4] Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich , Birkhäuser Verlag, Basel (2008) | MR | Zbl
, , ,[5] On the approximation of free discontinuity problems, Boll. Unione Mat. Ital. 7 (1992), 105 -123 | MR | Zbl
, ,[6] Approximation of functionals depending on jumps by elliptic functionals by Γ-convergence, Comm. Pure Appl. Math. 43 (1990), 999 -1036 | MR | Zbl
, ,[7] Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements, http://fr.arxiv.org/abs/1207.3687 (2012) | Numdam | MR | Zbl
, ,[8] Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids 48 (2000), 797 -826 | MR | Zbl
, , ,[9] The variational approach to fracture, J. Elasticity 9 (2008), 5 -148 | MR | Zbl
, , ,[10] A handbook of Gamma-convergence, , (ed.), Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 3, Elsevier (2006) | MR
,[11] A relaxation result for energies defined on pairs set-function and applications, ESAIM Control Optim. Calc. Var. 13 (2007), 717 -734 | EuDML | Numdam | MR | Zbl
, , ,[12] Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, American Elsevier, Amsterdam–London, New York (1973) | MR | Zbl
,[13] Minimizing movements of the Mumford–Shah functional, Discrete Contin. Dyn. Syst. A 3 (1997), 153 -174 | MR | Zbl
, ,[14] Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. III, Paris, 1980/1981, Res. Notes in Math. vol. 70 , Pitman, Boston, MA (1982), 154 -178 | MR | Zbl
, ,[15] Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal. 176 (2005), 165 -225 | MR | Zbl
, , ,[16] Existence for wave equations on domains with arbitrary growing cracks, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9) Mat. Appl. 22 (2011), 387 -408 | MR | Zbl
, ,[17] Γ-limits of obstacles, Ann. Mat. Pura Appl. 128 (1981), 1 -50 | MR | Zbl
, ,[18] A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Rational Mech. Anal. 162 (2002), 101 -135 | MR | Zbl
, ,[19] A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci. 12 (2002), 1773 -1800 | MR | Zbl
, ,[20] On a notion of unilateral slope for the Mumford–Shah functional, NoDEA 13 (2007), 713 -734 | MR | Zbl
, ,[21] New problems on minimizing movements, , (ed.), Boundary Value Problems for PDE and Applications, Masson, Paris (1993), 81 -98 | Zbl
,[22] Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal. 108 (1989), 195 -218 | MR | Zbl
, , ,[23] Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980), 180 -187 | MR | Zbl
, , ,[24] Vector Measures, Mathematical Surveys vol. 15 , American Mathematical Society, Providence, RI (1977) | MR | Zbl
, ,[25] Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003), 1465 -1500 | MR | Zbl
, ,[26] Critical Points of Ambrosio–Tortorelli converge to critical points of Mumford–Shah in the one-dimensional Dirichlet case, ESAIM Control Optim. Calc. Var. 15 (2009), 576 -598 | EuDML | Numdam | MR | Zbl
, , ,[27] Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), 1319 -1342 | MR | Zbl
, ,[28] Analysis of gradient flow of a regularized Mumford–Shah functional for image segmentation and image inpainting, ESAIM: M2AN 38 (2004), 291 -320 | EuDML | Numdam | MR | Zbl
, ,[29] On the variational approximation of free discontinuity problems in the vectorial case, Math. Models Methods Appl. Sci. 11 (2001), 663 -684 | MR | Zbl
,[30] Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Differential Equations 22 (2005), 129 -172 | MR | Zbl
,[31] Gradient flow for the one-dimensional Mumford–Shah functional, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 145 -193 | EuDML | Numdam | MR | Zbl
,[32] Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics vol. 24 , Pitman (Advanced Publishing Program), Boston, MA (1985) | MR | Zbl
,[33] Epsilon-stable quasi-static brittle fracture evolution, Comm. Pure Appl. Math. 63 (2010), 630 -654 | MR | Zbl
,[34] Existence of solutions to a regularized model of dynamic fracture, Math. Models Methods Appl. Sci. 20 (2010), 1021 -1048 | MR | Zbl
, , ,[35] Convergence results for critical points of the one-dimensional Ambrosio–Tortorelli functional with fidelity term, Adv. Differential Equations 15 (2010), 255 -282 | MR | Zbl
,[36] Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 (2005), 73 -99 | MR | Zbl
, ,[37] Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 17 (1989), 577 -685 | MR | Zbl
, ,[38] Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var. 12 (2006), 564 -614 | EuDML | Numdam | MR | Zbl
, ,[39] Gamma-convergence of gradient flows with applications to Ginzburg–Landau, Comm. Pure Appl. Math. 57 (2004), 1627 -1672 | MR | Zbl
, ,[40] Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. A 31 (2011), 1427 -1451 | MR | Zbl
,Cited by Sources: