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High-order extended finite element method for cracked domains. (English) Zbl 1181.74136
Summary: The aim of the paper is to study the capabilities of the extended finite element method (XFEM) to achieve accurate computations in non-smooth situations such as crack problems. Although the XFEM method ensures a weaker error than classical finite element methods, the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity. The difficulty can be overcome by modifying the enrichment of the finite element basis with the asymptotic crack tip displacement solutions as well as with the Heaviside function. Numerical simulations show that the modified XFEM method achieves an optimal rate of convergence (i.e. like in a standard finite element method for a smooth problem).

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74R10 Brittle fracture
Software:
Getfem++; XFEM
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References:
[1] Fracture Mechanics: Fundamentals and Applications (2nd edn). CRC Press: Boca Raton, FL, 1994.
[2] Fracture Mechanics of Concrete: Applications of Fracture Mechanics to Concrete, Rock and Other Quasi-Brittle Materials. Wiley: New York, 1995.
[3] Mechanics of Aircraft Structures. Wiley: New York, 1998.
[4] . An Analysis of the Finite Element Method. Prentice-Hall: Englewood Cliffs, NJ, 1973.
[5] Babuška, SIAM Journal on Numerical Analysis 31 pp 945– (1994)
[6] Melenk, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996)
[7] Babuška, International Journal for Numerical Methods in Engineering 40 pp 27– (1997) · Zbl 0949.65117
[8] Duarte, Computers and Structures 77 pp 219– (2000)
[9] Strouboulis, Computer Methods in Applied Mechanics and Engineering 181 pp 43– (2000)
[10] Strouboulis, Computer Methods in Applied Mechanics and Engineering 190 pp 4081– (2001)
[11] Babuška, Acta Numerica pp 1– (2003)
[12] Moës, International Journal for Numerical Methods in Engineering 46 pp 131– (1999)
[13] Belytschko, International Journal for Numerical Methods in Engineering 45 pp 601– (1999)
[14] Dolbow, Computer Methods in Applied Mechanics and Engineering 190 pp 6825– (2001)
[15] Sukumar, International Journal for Numerical Methods in Engineering 48 pp 1549– (2000)
[16] Belytschko, International Journal for Numerical Methods in Engineering 50 pp 993– (2000)
[17] Gravouil, International Journal for Numerical Methods in Engineering 53 pp 2569– (2002)
[18] Sukumar, Computer Methods in Applied Mechanics and Engineering 90 pp 6183– (2001)
[19] Moës, Engineering Fracture Mechanics 69 pp 813– (2002)
[20] Moës, International Journal for Numerical Methods in Engineering 53 pp 2549– (2002)
[21] Stazi, Computational Mechanics 31 pp 38– (2003)
[22] Chessa, International Journal for Numerical Methods in Engineering 57 pp 1015– (2003)
[23] Singularities in Boundary Value Problems. Masson: Paris, 1992.
[24] . Mechanics of Solid Materials. Cambridge University Press: Cambridge, 1994.
[25] Sukumar, International Journal of Solids and Structures 40 pp 7513– (2003)
[26] Béchet, International Journal for Numerical Methods in Engineering (2005)
[27] Yau, Journal of Applied Mechanics 47 pp 335– (1980)
[28] Destuynder, Mathematical Methods in the Applied Sciences 3 pp 70– (1981)
[29] . Getfem ++. An open source generic C++ it library for finite element methods, http://www-gmm.insa-toulouse.fr/getfem.
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