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Tau-lines: A new hybrid approach to the numerical treatment of crack problems based on the Tau method. (English) Zbl 0576.73095
A new hybrid computaional technique based on Ortiz’ recursive formulation of the Tau method is introduced in this paper and applied to some model singular boundary value problems which are relevant to fracture mechanics (modes I and III). This technique, which we call Tau-lines, combines the method of lines with the Tau method. The former is used in the construction of a system of coupled ordinary differential equations which is the discretized model of a given partial differential equation; the latter is used to find an accurate approximation of the solution of such a system which involves no further discretization. Recent theoretical results on the Tau method show that its error is optimal in the sense that, for a given degree n, it has the same order of error as the best uniform approximation of the exact solution by algebraic polynomials of degree n. The present work may be considered as an encouraging first step towards the development of the Tau-lines approach into a useful and efficient computational tool for the numerical treatment of problems in fracture mechanics.

MSC:
74R05 Brittle damage
65N40 Method of lines for boundary value problems involving PDEs
65J99 Numerical analysis in abstract spaces
65N99 Numerical methods for partial differential equations, boundary value problems
74S99 Numerical and other methods in solid mechanics
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[1] Rothe, H., Zweidimensionale parabolische randwertaufgaben als grenzfall eindimensionaler randwertaufgaben, Math. ann., 102, 650-670, (1929) · JFM 56.1076.02
[2] Fadeeva, V.N., The method of lines applied to some boundary value problems, Trudy mat. inst. Steklov, 28, 73-103, (1949)
[3] Smirnov, V.I., (), 739-742
[4] Berezin, I.S.; Zhidkov, N.P., (), 580-607
[5] Rektoris, K., The method of discretization in time and partial differential equations, (1982), Reidel Dordrecht
[6] Holt, M., Numerical methods in fluid dynamics, (1977), Springer Berlin · Zbl 0357.76009
[7] Malik, S.N.; Fu, S.L., The method of lines applied to crack problems including the plasticity effect, Comput. & structures, 10, 447-457, (1979) · Zbl 0395.73089
[8] Mendelson, A.; Alam, J., The use of the method of lines in 3-D fracture mechanics analysis with applications to compact tension specimens, Internat. J. fracture, 22, 105-116, (1983)
[9] Lanczos, C., Trigonometric interpolation of empirical and analytical functions, J. math. phys., 17, 123-199, (1938) · Zbl 0020.01301
[10] Lanczos, C., Applied analysis, (), 438-507 · Zbl 0111.12403
[11] Ortiz, E.L., The tau method, SIAM J. numer. anal., 6, 480-492, (1969) · Zbl 0195.45701
[12] Ortiz, E.L., Canonical polynomials in Lanczos’ tau method, (), 73-93
[13] Rivlin, T.J., The Chebyshev polynomials, (1974), Wiley New York · Zbl 0291.33012
[14] Ortiz, E.L., Step by step tau method: piecewise polynomial approximations, Comput. math. appl., 1, 381-392, (1975) · Zbl 0356.65006
[15] Ortiz, E.L., On the numerical solution of nonlinear and functional differential equations with the tau method, (), 127-139 · Zbl 0387.65053
[16] Ortiz, E.L.; Samara, H., An operational approach to the tau method for the numerical solution of nonlinear diffferential equations, Computing, 27, 15-25, (1981) · Zbl 0449.65053
[17] Onumanyi, P.; Ortiz, E.L., Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the tau method, Math. comp., 43, 189-203, (1984) · Zbl 0574.65091
[18] Ortiz, E.L.; Pham Ngoc Dinh, A., On the convergence of the tau method for nonlinear differential equations of Riccati’s type, Nonlinear anal., 9, 53-60, (1985) · Zbl 0551.65050
[19] Ortiz, E.L.; Dinh, A.Pham Ngoc, Linear recursive schemes associated with some nonlinear partial differential equations in one dimension, SIAM J. math. anal., (1986), to appear. · Zbl 0619.35017
[20] Freilich, J.H.; Ortiz, E.L., Numerical solution of systems of differential equations with the tau method: an error analysis, Math. comp., 39, 467-479, (1982) · Zbl 0501.65042
[21] Crisci, M.R.; Russo, E., An extension of Ortiz’ recursive formulation of the tau method to certain linear systems of ordinary differential equations, Math. comp., 41, 27-42, (1983) · Zbl 0526.65054
[22] Ortiz, E.L.; Samara, H., Numerical solution of partial differential equations with variable coefficients with the tau method, Comput. math. appl., 10, 5-13, (1984) · Zbl 0575.65118
[23] Ortiz, E.L.; Pun, K.S., Numerical solution of nonlinear partial differential equations with the tau method, J. comput. appl. math., 12-13, 511-516, (1985) · Zbl 0579.65124
[24] Ortiz, E.L.; Pun, K.S., Numerical solution of Burgers’ nonlinear partial differential equation with a multi-dimensional formulation of the tau method, Comput. math. appl., (1986), to appear. · Zbl 0631.65120
[25] El Misiery, A.E.M.; Ortiz, E.L., Numerical solution of regular and singular biharmonic problems with the tau-lines method, Comm. appl. numer. meth., 1, 281-285, (1986) · Zbl 0591.65083
[26] Chaves, T.; Ortiz, E.L., On the numerical solution of two point boundary value problems for linear differential equations, Z. angew. math. mech., 48, 415-418, (1968) · Zbl 0172.19601
[27] Ortiz, E.L.; Samara, H., Numerical solution of differential eigenvalue problems with an operational approach to the tau method, Computing, 31, 95-103, (1983) · Zbl 0508.65045
[28] Liu, K.M.; Ortiz, E.L., Approximation of eigenvalues defined by ordinary differential equations with the tau method, (), 90-102
[29] Liu, K.M.; Ortiz, E.L., Numerical solution of eigenvalue problems for partial differential equations with the tau-lines method, Comput. math. appl., (1986), to appear. · Zbl 0626.65109
[30] Liu, K.M.; Ortiz, E.L.; Pun, K.-S., Numerical solution of Steklov’s partial differential eigenvalue problem, (), 244-249
[31] Fletcher, C.A.J., Computational Galerkin methods, (1984), Springer Berlin · Zbl 0533.65069
[32] Namasivayam, S.; Ortiz, E.L., Best approximation and the numerical solution of partial differential equations, Portugal. math., 40, 97-119, (1985) · Zbl 0563.65061
[33] Namasivayam, S.; Ortiz, E.L., Dependence of the local truncation error on the choice of perturbation term in the step by step tau method for systems of differential equations, () · Zbl 0769.65045
[34] Ortiz, E.L.; Purser, W.F.C.; Ca├▒izares, F.J.Rodriguez, Automation of the tau method, ()
[35] Kondriat’ev, V.A., Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow math. soc., 227-313, (1967) · Zbl 0194.13405
[36] Onumanyi, P.; Ortiz, E.L., Numerical solution of high order boundary value problems for ordinary differential equations with an estimation of the error, Internat. J. numer. meths. engrg., 18, 775-781, (1982) · Zbl 0478.65049
[37] Symm, G.T., Treatment of singularities in the solution of Laplace’s equation by an integral equation method, National physical laboratory, report NAC, (1973) · Zbl 0364.65097
[38] Papamichael, N.; Symm, G.T., Numerical techniques for two dimensional Laplacian problems, Comput. meths. appl. mech. engrg., 6, 175-194, (1975) · Zbl 0324.65050
[39] Xanthis, L.S.; Bernai, M.; Atkinson, C., The treatment of singularities in the calculation of stress intensity factors using the boundary integral equation method, Comput. meths. appl. mech. engrg., 26, 285-304, (1981) · Zbl 0461.73066
[40] Whiteman, J.R., Numerical treatment of a problem from linear fracture mechanics, (), 128-138 · Zbl 0423.73081
[41] Motz, H., Treatment of singularities in partial differential equations by relaxations methods, Quart. appl. math., 4, 371-377, (1946) · Zbl 0030.40002
[42] Woods, L.C., The relaxation treatment of singular points in Poisson’s equation, Quart. J. mech. appl. math., 6, 163-185, (1953) · Zbl 0050.34901
[43] Gross, B.; Srawley, J.E.; Brown, W.F., Stress intensity factors for a single-edge notch tension specimen by boundary collocation of a stress function, ()
[44] Fix, G., High-order Rayleigh-Ritz approximations, J. math. mech., 18, 645-657, (1969) · Zbl 0234.65095
[45] Barnhill, R.E.; Whiteman, J.R., Error analysis of Galerkin methods for Dirichlet problems containing singularities, Internat. J. numer. meths. engrg., 15, 121-125, (1975) · Zbl 0294.65060
[46] Griffith, D.F., A numerical study of a singular elliptic boundary value problem, J. inst. math. appl., 19, 59-69, (1977) · Zbl 0344.65050
[47] Akin, J.E., Generation of elements with singularities, Internat. J. numer. meths. engrg., 10, 1249-1259, (1976) · Zbl 0341.65078
[48] Henschell, R.H.; Shaw, K.G., Crack tip finite elements are unnecessary, Internat. J. numer. meths. engrg., 9, 495-507, (1975) · Zbl 0306.73064
[49] Wait, R., Singular isoparametric finite elements, J. inst. math. appl., 20, 133-141, (1977) · Zbl 0363.65086
[50] Gregory, J.A.; Fishelov, D.; Schiff, B.; Whiteman, J.R., Local mesh refinement with finite elements for elliptic problems, J. comput. phys., 28, 133-140, (1978) · Zbl 0393.65045
[51] Schiff, B.; Whiteman, J.R.; Fishelov, D., Determination of a stress intensity factor using local mesh refinement, () · Zbl 0441.73125
[52] Mangasarian, O.L., Numerical solution of the first biharmonic problem by linear programming, Internat. J. engrg. sci., 1, 231-240, (1963) · Zbl 0137.13904
[53] Williams, M.L., Stress singularities resulting from various boundary conditions in angular corners or plates in extension, J. appl. mech., 24, 526-528, (1952)
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