메뉴 건너뛰기




Volumn 48, Issue 3, 2010, Pages 980-1012

An adaptive finite element approximation of a variational model of brittle fracture

Author keywords

Adaptive finite element method; Ambrosio Tortorelli functional; Brittle fracture; Free discontinuity problem

Indexed keywords

ADAPTIVE FINITE ELEMENT; ADAPTIVE FINITE ELEMENT APPROXIMATIONS; ADAPTIVE FINITE ELEMENT METHODS; ADAPTIVE MESH REFINEMENT; AMBROSIO-TORTORELLI FUNCTIONAL; DISCONTINUITY PROBLEMS; LOCAL MINIMIZER; NEWTON-TYPE METHODS; THEORETICAL RESULT; VARIATIONAL MODELS;

EID: 77956051522     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/080741033     Document Type: Article
Times cited : (132)

References (38)
  • 1
    • 0001400376 scopus 로고
    • A compactness theorem for a new class of functions of bounded variation
    • 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.
    • (1989) Boll. Un. Mat. Ital. B , vol.3 , Issue.7 , pp. 857-881
    • Ambrosio, L.1
  • 2
    • 0000465804 scopus 로고
    • Existence theory for a new class of variational problems
    • L. AMBROSIO, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., 111 (1990), pp. 291-322.
    • (1990) Arch. Rational Mech. Anal. , vol.111 , pp. 291-322
    • Ambrosio, L.1
  • 4
    • 34748925841 scopus 로고
    • Approximation of junctionals depending on jumps by elliptic junctionals via Γ-convergence
    • L. AMBROSIO AND V. M. TORTORELLI, Approximation of junctionals depending on jumps by elliptic junctionals via Γ-convergence, Comm. Pure Appl. Math., 43 (1990), pp. 999-1036.
    • (1990) Comm. Pure Appl. Math. , vol.43 , pp. 999-1036
    • Ambrosio, L.1    Tortorelli, V.M.2
  • 5
    • 0000654840 scopus 로고
    • On the approximation of free discontinuity problems
    • L. AMBROSIO AND V. M. TORTORELLI, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7), 6 (1992), pp. 105-123.
    • (1992) Boll. Un. Mat. Ital. B , vol.6 , Issue.7 , pp. 105-123
    • Ambrosio, L.1    Tortorelli, V.M.2
  • 7
    • 0040745183 scopus 로고
    • Discrete approximation of a free discontinuity problem
    • G. BELLETTINI AND A. COSCIA, Discrete approximation of a free discontinuity problem, Numer. Funct. Anal. Optim., 15 (1994), pp. 201-224.
    • (1994) Numer. Funct. Anal. Optim. , vol.15 , pp. 201-224
    • Bellettini, G.1    Coscia, A.2
  • 8
    • 35648980407 scopus 로고    scopus 로고
    • Numerical implementation of the variational formulation for quasi-static brittle fracture
    • B. BOURDIN, Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Bound., 9 (2007), pp. 411-430.
    • (2007) Interfaces Free Bound. , vol.9 , pp. 411-430
    • Bourdin, B.1
  • 9
    • 84861146076 scopus 로고    scopus 로고
    • The variational formulation of brittle fracture: Numerical implementation and extensions
    • Springer-Verlag, Dordrecht, The Netherlands
    • 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.
    • (2007) IUTAM Symposium on Discretization Methods for Evolving Discontinuities , pp. 381-393
    • Bourdin, B.1
  • 10
    • 0034412473 scopus 로고    scopus 로고
    • Implementation of an adaptive finite element approximation of the mumford-shah functional
    • B. BOURDIN AND A. CHAMBOLLE, Implementation of an adaptive finite element approximation of the Mumford-Shah functional, Numer. Math., 85 (2000), pp. 609-646.
    • (2000) Numer. Math. , vol.85 , pp. 609-646
    • Bourdin, B.1    Chambolle, A.2
  • 11
  • 12
    • 17144426836 scopus 로고    scopus 로고
    • Γ-convergence for beginners
    • Oxford Lecture Ser. Oxford University Press, Oxford, UK
    • A. BRAIDES, Γ-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University Press, Oxford, UK, 2002.
    • (2002) Math. Appl. , vol.22
    • Braides, A.1
  • 15
    • 54249162100 scopus 로고    scopus 로고
    • Quasi-optimal convergence rate for an adaptive finite element method
    • 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.
    • (2008) SIAM J. Numer. Anal. , vol.46 , pp. 2524-2550
    • Cascon, J.M.1    Kreuzer, C.2    Nochetto, R.H.3    Siebert, K.G.4
  • 16
    • 0033438682 scopus 로고    scopus 로고
    • Discrete approximation of the mumford-shah functional in dimension two
    • 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.
    • (1999) M2AN Math. Model. Numer. Anal. , vol.33 , pp. 651-672
    • Chambolle, A.1    Dal Maso, G.2
  • 17
    • 0002214041 scopus 로고
    • Maximum principle and uniform convergence for the finite element method
    • 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.
    • (1973) Comput. Methods Appl. Mech. Engrg. , vol.2 , pp. 17-31
    • Ciarlet, P.G.1    Raviart, P.-A.2
  • 19
    • 0036557535 scopus 로고    scopus 로고
    • A model for the quasi-static growth of brittle fractures: Existence and approximation results
    • DOI 10.1007/s002050100187
    • 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. (Pubitemid 36195113)
    • (2002) Archive for Rational Mechanics and Analysis , vol.162 , Issue.2 , pp. 101-135
    • Dal Maso, G.1    Toader, R.2
  • 20
    • 0000532812 scopus 로고
    • Existence theorem for a minimum problem with free discontinuity set
    • 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.
    • (1989) Arch. Rational Mech. Anal. , vol.108 , pp. 195-218
    • De Giorgi, E.1    Carriero, M.2    Leaci, A.3
  • 21
    • 1542639888 scopus 로고    scopus 로고
    • A convergent adaptive algorithm for poisson's equation
    • W. DÖRFLER, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106-1124.
    • (1996) SIAM J. Numer. Anal. , vol.33 , pp. 1106-1124
    • Dörfler, W.1
  • 22
    • 0003346795 scopus 로고
    • Measure theory and fine properties of functions
    • CRC Press, Boca Raton, FL
    • L. C. EVANS AND R. F. GARIEPY, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992.
    • (1992) Stud. Adv. Math.
    • Evans, L.C.1    Gariepy, R.F.2
  • 23
    • 0035595682 scopus 로고    scopus 로고
    • On the variational approach of free discontinuity problems in the vectorial case
    • M. FOCARDI, On the variational approach of free discontinuity problems in the vectorial case, Math. Models Methods Appl. Sci., 11 (2001), pp. 663-684.
    • (2001) Math. Models Methods Appl. Sci. , vol.11 , pp. 663-684
    • Focardi, M.1
  • 24
    • 0042859940 scopus 로고    scopus 로고
    • Existence and convergence for quasi-static evolution in brittle fracture
    • 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.
    • (2003) Comm. Pure Appl. Math. , vol.56 , pp. 1465-1500
    • Francfort, G.A.1    Larsen, C.J.2
  • 25
    • 0032139947 scopus 로고    scopus 로고
    • Revisiting brittle fracture as an energy minimization problem
    • G. A. FRANCFORT AND J.-J. MARIGO, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), pp. 1319-1342.
    • (1998) J. Mech. Phys. Solids , vol.46 , pp. 1319-1342
    • Francfort, G.A.1    Marigo, J.-J.2
  • 26
    • 12144268240 scopus 로고    scopus 로고
    • Ambrosio-tortorelli approximation of quasi-static evolution of brittle fractures
    • A. GIACOMINI, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Differential Equations, 22 (2005), pp. 129-172.
    • (2005) Calc. Var. Partial Differential Equations , vol.22 , pp. 129-172
    • Giacomini, A.1
  • 27
    • 0000357002 scopus 로고
    • The phenomena of rupture and flow in solids
    • A. A. GRIFFITH, The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. London Ser. A, 221 (1921), pp. 163-198.
    • (1921) Phil. Trans. Roy. Soc. London Ser. A , vol.221 , pp. 163-198
    • Griffith, A.A.1
  • 28
    • 0039248648 scopus 로고    scopus 로고
    • Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle
    • 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. (Pubitemid 33574338)
    • (2001) Mathematics of Computation , vol.70 , Issue.233 , pp. 107-119
    • Korotov, S.1    Krizek, M.2    Neittaanmaki, P.3
  • 29
    • 0024883616 scopus 로고
    • A comparison of adaptive refinement techniques for elliptic problems
    • W. MITCHELL, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software, 15 (1989), pp. 326-247.
    • (1989) ACM Trans. Math. Software , vol.15 , pp. 326-247
    • Mitchell, W.1
  • 30
    • 84990602490 scopus 로고
    • Optimal approximation by piecewise smooth functions and associated variational problems
    • D. MUMFORD AND J. SHAH, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), pp. 577-685.
    • (1989) Comm. Pure Appl. Math. , vol.42 , pp. 577-685
    • Mumford, D.1    Shah, J.2
  • 31
    • 4744358509 scopus 로고    scopus 로고
    • Numerical minimization of the mumford-shah functional
    • M. NEGRI AND M. PAOLINI, Numerical minimization of the Mumford-Shah functional, Calcolo, 38 (2001), pp. 67-84.
    • (2001) Calcolo , vol.38 , pp. 67-84
    • Negri, M.1    Paolini, M.2
  • 32
    • 84940605935 scopus 로고    scopus 로고
    • Triangle: Engineering a 2D quality mesh generator and delaunay triangulator
    • Lecture Notes in Comput. Sci. M. C. Lin and D. Manocha, eds., Springer-Verlag, Berlin
    • 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.
    • (1996) Applied Computational Geometry: Towards Geometric Engineering , vol.1148 , pp. 203-222
    • Shewchuk, J.R.1
  • 33
    • 31244438495 scopus 로고    scopus 로고
    • Delaunay refinement algorithms for triangular mesh generation
    • J. R. SHEWCHUK, Delaunay refinement algorithms for triangular mesh generation, Comput. Geom., 22 (2002), pp. 21-74.
    • (2002) Comput. Geom. , vol.22 , pp. 21-74
    • Shewchuk, J.R.1
  • 35
    • 0003208045 scopus 로고
    • Galerkin finite element methods for parabolic problems
    • Springer-Verlag, Berlin
    • V. THOMÉE, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math. 1054, Springer-Verlag, Berlin, 1984.
    • (1984) Lecture Notes in Math. , vol.1054
    • Thomée, V.1
  • 37
    • 0033450371 scopus 로고    scopus 로고
    • Error estimates for some quasi-interpolation operators
    • R. VERFÜRTH, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 695-713.
    • (1999) M2AN Math. Model. Numer. Anal. , vol.33 , pp. 695-713
    • Verfürth, R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.