A new semi-Gaussian quadrature rule for finite-part integrals in crack problems with a second-order singularity

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2018
• Θέματα: Two-dimensional plane cracks; Three-dimensional elasticity; Singular integrals; Hypersingular integrals; Finite-part integrals; Hadamard-type integrals; Quadrature rules; Numerical integration; Gaussian integration; Nodes and weights; Second-order singularity; Kutt's quadrature rule; Διδιάστατες επίπεδες ρωγμές; Τριδιάστατη ελαστικότητα; Ιδιόμορφα ολοκληρώματα; Υπεριδιόμορφα ολοκληρώματα; Ολοκληρώματα πεπερασμένου μέρους; Ολοκληρώματα τύπου Hadamard; Κανόνες αριθμητικής ολοκληρώσεως; Αριθμητική ολοκλήρωση; Γκαουσιανή ολοκλήρωση; Κόμβοι και βάρη; Ιδιομορφία δευτέρας τάξεως; Κανόνας αριθμητικής ολοκληρώσεως του Kutt

Finite-part (hypersingular or Hadamard-type) integrals appear, naturally, in two-dimensional crack and additional problems in three-dimensional elasticity. Their computation along the radial direction leads to a one-dimensional finite-part integral on the interval [0,1] with a second-order singularity at the left end, x=0, of this interval. Kutt's Gauss–Jacobi-equivalent quadrature rule (on [0,1]) is the natural approach to the computation of this finite-part integral, but the appearance of two complex conjugate nodes (and analogous weights) outside the integration interval is a physically serious disadvantage to its use. Although the subtraction of the singularity is completely possible, here a new, semi-Gaussian quadrature rule is suggested for n = 4, 5, . . . , 10 nodes, where no node lies outside the integration interval [0,1] (exactly as in classical Gaussian quadrature rules) and, moreover, no derivative of the integrand appears in the approximation to the integral on [0,1]. The polynomial accuracy of the present quadrature rule for finite-part integrals is very high and equal to 2n-3. The proposed quadrature rule seems to be a reasonable choice (as an alternative both to Kutt's rule and to the subtraction of the singularity approach) during the evaluation of two-dimensional finite-part integrals in crack and related problems. The computation of the nodes and the weights (for n = 4, 5, . . . , 10 nodes) is described in brief and numerical results for both of these quantities are displayed together with experimental numerical results, which illustrate both the accuracy and the rapid convergence of the present quadrature rule.