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  • 1
    Electronic Resource
    Electronic Resource
    Numerical determination of a class of generalized stress intensity factors (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 949-959 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A modification of the collocation method for the numerical solution of Cauchy-type singular integral equations appearing in plane elasticity and, especially, crack problems is proposed. This modification, based on a variable transformation, applies to the case when the unknown function of the singular integral equation behaves like A(x - c)α + B(x - c)β, where α 〈 0, 0 〈 β - α 〈 1, near an endpoint c of the integration interval. In plane elasticity such a point is either a crack tip or a corner point of the boundary of the elastic medium. Thus the method seems to be quite efficient for the numerical evaluation of generalized stress intensity factors near such points. A successful application of the method to the classical plane elasticity problem of an antiplane shear crack terminating at a bimaterial interface was also made.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620140702
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  • 2
    Electronic Resource
    Electronic Resource
    A remark on the numerical evaluation of stress intensity factors by the method of singular integral equations (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 1710-1714 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A modification of the numerical techniques of solution of Cauchy-type singular integral equations and determination of stress intensity factors at crack tips in plane elastic media is proposed. This technique presents some advantages under appropriate geometry conditions.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620141112
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  • 3
    Electronic Resource
    Electronic Resource
    On the numerical evaluation of two-dimensional principal value integrals (1980)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 15 (1980), S. 629-634 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620150414
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  • 4
    Electronic Resource
    Electronic Resource
    A superconvergence result for the natural extrapolation formula for the numerical determination of stress intensity factors (1985)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 21 (1985), S. 1391-1401 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The best approach for the numerical determination of stress intensity factors at crack tips in plane and antiplane elasticity problems is frequently the numerical solution of the corresponding Cauchy-type singular integral equation by the Gauss-Chebyshev method, followed by the application of the natural extrapolation formula for the numerical determination of the stress intensity factors. It is shown here that this approach converges for Hölder-continuous and discontinuous (with jump discontinuities) loading distributions along the crack (or cracks) and that in all cases the rate of convergence is greater than that believed up to now. This superconvergence result is based on a theorem on the numerical equivalence of the Gauss-Chebyshev direct method to a relevant indirect method for the numerical solution of Cauchy-type singular integral equations, also proved here. Numerical results in various crack problems corroborate the theoretical ones.
    Additional Material: 4 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620210804
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  • 5
    Electronic Resource
    Electronic Resource
    A remark on the direct numerical determination of stress intensity factors at crack tips (1982)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 18 (1982), S. 1416-1419 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss-Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto-Chebyshev method for the numerical solution of the same class of equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620180913
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  • 6
    Electronic Resource
    Electronic Resource
    The Gauss-Laguerre quadrature rule for finite-part integrals (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 439-450 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The classical Gauss-Laguerre quadrature rule for the semi-infinite integration interval [0,∞] is modified and applied to the case of the weight function exp(-x)/x corresponding to finite-part (or, equivalently, hypersingular) integrals. The new set of orthogonal polynomials is constructed and it is seen to consist of linear combinations of the classical Laguerre polynomials with appropriately determined coefficients. The zeros of these modified Laguerre polynomials are seen to be distinct, but one of these lies outside the integration interval. Formulae for the corresponding weights are also given and numerical values and results are presented. The present results generalize the corresponding results for the finite interval [0,1] to semi-infinite intervals and they are applicable to a variety of applied mechanics and related problems, where finite-part integrals appear in a natural way.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090509
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  • 7
    Electronic Resource
    Electronic Resource
    BEAMS ON TENSIONLESS ELASTIC FOUNDATION: APPROXIMATE QUANTIFIER ELIMINATION WITH CHEBYSHEV SERIES (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 663-686 
    Details
    ISSN: 0029-5981
    Keywords: beams ; bending ; Chebyshev approximations ; quantifier elimination ; Sturm sequences ; tensionless elastic foundation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The well-known Sturm's theorem (based on Sturm's sequences) for the determination of the number of distinct real zeros of polynomials in a finite or infinite real interval has been already used in elementary quantifier elimination problems including applied mechanics and elasticity problems. Here it is further suggested that this theorem can also be used for quantifier elimination, but in more complicated problems where the functions involved are not simply polynomials, but they may contain arbitrary transcendental functions. In this case, it is suggested that the related transcendental equations/inequalities can be numerically approximated by polynomial equations/inequalities with the help of Chebyshev series expansions in numerical analysis. The classical problem of a straight isotropic elastic beam on a tensionless elastic foundation, where the deflection function (incorporating both the exponential function and trigonometric functions) should be continuously positive (this giving rise to a quantifier elimination problem along the length of the beam) is used as an appropriate vehicle for the illustration of the present mixed (symbolic-numerical) approach. Two such elementary beam problems are considered in some detail (with the help of the Maple V computer algebra system) and the related simple quantifier-free formulae are established and seen to coincide with those already available in the literature for the same beam problems. More complicated problems, probably necessitating the use of more advanced computer algebra techniques (together with Sturm's theorem), such as the Collins well-known and powerful cylindrical algebraic decomposition method for quantifier elimination, can also easily be employed in the present approximate (because of the use of Chebyshev series expansions) symbolic-numerical computational environment.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Chebyshev approximations to stress intensity factors: An application of ‘DERIVE’ (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Communications in Applied Numerical Methods 7 (1991), S. 289-293 
    Details
    ISSN: 0748-8025
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Chebȳshev polynomials have been used extensively for the approximation of functions. Here we apply this approach (together with two Gauss-Chebyshev numerical integration rules) to the case of stress intensity factors. The problem of a simple straight crack under exponential loading is used as the vehicle for the illustration of this semi-analytical/numerical (SAN) approach. The computer algebra system DERIVE was found to be appropriate and was used in the present application. Sufficient SAN results (from DERIVE) are displayed. The present results are numerically superior to the corresponding results based on Taylor (or Maclaurin) polynomials.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1630070406
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  • 9
    Electronic Resource
    Electronic Resource
    Semi-numerical iterative series solution of linear algebraic equations with ‘MATHEMATICA’ (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Communications in Applied Numerical Methods 8 (1992), S. 421-429 
    Details
    ISSN: 0748-8025
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The computer algebra system MATHEMATICA is applied to the iterative solution of systems of linear algebraic equations where the matrix of the coefficients depends on a parameter. The solution is found in a Taylor-Maclaurin series form with respect to this parameter at an appropriate point and, therefore, the system can be solved only once and for all even if the parameter varies. All three popular iterative methods, that is, the Gauss-Jacobi, the Gauss-Seidel and the successive overrelaxation method are used in particular applications. The complete MATHEMATICA procedure for the last of these methods is also presented. Two elementary applications to structural mechanics are also made and further possibilities are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1630080702
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  • 10
    Electronic Resource
    Electronic Resource
    A natural interpolation formula for Prandtl's singular integrodifferential equation (1984)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 4 (1984), S. 283-290 
    Details
    ISSN: 0271-2091
    Keywords: Prandtl's Equation ; Singular Integrodifferential Equations ; Quadrature Method ; Natural Interpolation Formula ; Principal Value Integrals ; Gauss- and Lobatto-Chebyshev Quadrature Rules ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Prandtl's singular integrodifferential equation and related equations appear in problems of aerofoil and propeller theory in fluid mechanics. Here a natural interpolation formula for the approximation to the unknwon function of Prandtl's equation when this is solved numerically by the direct quadrature method, based on the Gauss- and Lobatto-Chebyshev quadrature rules, is proposed. This interpolation formula is analogous to Nyström's natural interpolation formula for Fredholm integral equations of the second kind and the corresponding formula for singular integral equations. Numerical applications of the same formula are also made.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650040306
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