Abstract

The energy of the Francfort–Marigo model of brittle fracture can be approximated, in the sense of $\Gamma$-convergence, by the Ambrosio–Tortorelli functional. In this work, we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm, we combine a Newton-type method with residual-driven adaptive mesh refinement. We present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the Ambrosio–Tortorelli functional.

MSC codes

  1. 65N30
  2. 65N50
  3. 74R10

Keywords

  1. adaptive finite element method
  2. brittle fracture
  3. free-discontinuity problem
  4. Ambrosio–Tortorelli functional

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. B (7), 3 (1989), pp. 857–881.
2.
L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., 111 (1990), pp. 291–322.
3.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
4.
L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence, Comm. Pure Appl. Math., 43 (1990), pp. 999–1036.
5.
L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7), 6 (1992), pp. 105–123.
6.
V. Barbu and T. Precopeanu, Convexity and Optimization in Banach Spaces, D. Reidel, Dordrecht, The Netherlands, 1986.
7.
G. Bellettini and A. Coscia, Discrete approximation of a free discontinuity problem, Numer. Funct. Anal. Optim., 15 (1994), pp. 201–224.
8.
B. Bourdin, Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Bound., 9 (2007), pp. 411–430.
9.
B. Bourdin, The variational formulation of brittle fracture: Numerical implementation and extensions, in IUTAM Symposium on Discretization Methods for Evolving Discontinuities, Springer-Verlag, Dordrecht, The Netherlands, 2007, pp. 381–393.
10.
B. Bourdin and A. Chambolle, Implementation of an adaptive finite element approximation of the Mumford-Shah functional, Numer. Math., 85 (2000), pp. 609–646.
11.
B. Bourdin, G. A. Francfort, and J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, 48 (2000), pp. 797–826.
12.
A. Braides, $\Gamma$-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University Press, Oxford, UK, 2002.
13.
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Springer-Verlag, New York, 1994.
14.
S. Burke, C. Ortner, and E. Süli, An Adaptive Finite Element Approximation of the Generalised Ambrosio–Tortorelli Functional, manuscript, 2010.
15.
J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524–2550.
16.
A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 651–672.
17.
P. G. Ciarlet and P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 2 (1973), pp. 17–31.
18.
G. Dal Maso, G. A. Francfort, and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), pp. 165–225.
19.
G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), pp. 101–135.
20.
E. De Giorgi, M. Carriero, and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal., 108 (1989), pp. 195–218.
21.
W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124.
22.
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992.
23.
M. Focardi, On the variational approach of free discontinuity problems in the vectorial case, Math. Models Methods Appl. Sci., 11 (2001), pp. 663–684.
24.
G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), pp. 1465–1500.
25.
G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), pp. 1319–1342.
26.
A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Differential Equations, 22 (2005), pp. 129–172.
27.
A. A. Griffith, The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. London Ser. A, 221 (1921), pp. 163–198.
28.
S. Korotov, M. Křížek, and P. Neittaanmäki, Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle, Math. Comp., 70 (2001), pp. 107–119.
29.
W. Mitchell, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software, 15 (1989), pp. 326–247.
30.
D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), pp. 577–685.
31.
M. Negri and M. Paolini, Numerical minimization of the Mumford-Shah functional, Calcolo, 38 (2001), pp. 67–84.
32.
J. R. Shewchuk, Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator, in Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Comput. Sci. 1148, M. C. Lin and D. Manocha, eds., Springer-Verlag, Berlin, 1996, pp. 203–222.
33.
J. R. Shewchuk, Delaunay refinement algorithms for triangular mesh generation, Comput. Geom., 22 (2002), pp. 21–74.
34.
G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice–Hall, Englewood Cliffs, NJ, 1973.
35.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math. 1054, Springer-Verlag, Berlin, 1984.
36.
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, Stuttgart, 1996.
37.
R. Verfürth, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 695–713.
38.
A. J. Weir, Lebesgue Integration and Measure, Cambridge University Press, London, New York, 1973.

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 980 - 1012
ISSN (online): 1095-7170

History

Submitted: 17 November 2008
Accepted: 26 April 2010
Published online: 1 July 2010

MSC codes

  1. 65N30
  2. 65N50
  3. 74R10

Keywords

  1. adaptive finite element method
  2. brittle fracture
  3. free-discontinuity problem
  4. Ambrosio–Tortorelli functional

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media