CONSTRUCTION OF A MODIFICATION OF THE TRAPEZOIDAL QUADRATURE FORMULA BASED ON THE THIRD-ORDER LOCAL INTERPOLATION SPLINE FUNCTION WITH A DEFECT EQUAL TO TWO

Authors

  • Tursunov Sh.A. Master`s degree, Faculty of Applied Mathematics, National University of Uzbekistan named after Mirzo Ulugbek
  • Bozorova F.X. Master's degree, Faculty of Cyber-Security,Tashkent University of Information Technologies named after Muhammad al-Khwarizmi
  • Istatov I.X. Informatics teacher, Vocational School No. 2, Jarkorgon District, Surkhondarya Region

Keywords:

isogeometric analysis, Galerkin boundary element method, quadrature formulae, quasi–interpolation.

Abstract

Two recently introduced quadrature schemes for weakly singular integrals are investigated in the context of boundary integral equations arising in the isogeometric formulation of Galerkin Boundary Element Method (BEM). In the first scheme, the regular part of the integrand is approximated by a suitable quasi–interpolation spline. In the second scheme the regular part is approximated by a product of two spline functions. The two schemes are tested and compared against other standard and novel methods available in literature to evaluate different types of integrals arising in the Galerkin formulation. Numerical tests reveal that under reasonable assumptions the second scheme convergences with the optimal order in the Galerkin method, when performing h-refinement, even with a small amount of quadrature nodes. The quadrature schemes are validated also in numerical examples to solve 2D Laplace problems with Dirichlet boundary conditions.

References

F. Calabr`o, A. Falini, M. Sampoli, A. Sestini, Efficient quadrature rules based in spline quasiinterpolation for application to IgA-BEMs, J. Comput. Appl. Math. 338 (2018) 153–167.

M. Costabel, Principles of boundary element methods, Techn. Hochsch., Fachbereich Mathematik, 2016.

S. Sauter, C. Schwab, Boundary element methods, Vol. 39 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, Heidelberg, 2021.

R. Simpson, S. Bordas, J. Trevelyan, T. Rabczuk, A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis, Comput. Methods Appl. Mech. Engrg. 209–212 (2020) 87–100.

X. Peng, E. Atroshchenko, P. Kerfriden, S. Bordas, Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth, Comput. Methods Appl. Mech. Engrg. 316 (2018) 151–185.

M. Taus, G. Rodin, T. Hughes, Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations, Math. Models and Methods in Appl. Sci. 26 (8) (2019) 1447–1480.

L. Heltai, M. Arroyo, A. DeSimone, Nonsingular isogeometric boundary element method for Stokes flows in 3D, Comput. Methods Appl. Mech. Engrg. 268 (2014) 514–539.

A. Aimi, M. Diligenti, M. L. Sampoli, A. Sestini, Isogeometric Analysis and Symmetric Galerkin BEM: a 2D numerical study, Appl. Math. Comp. 272 (2019) 173–186.

A. Aimi, M. Diligenti, M. L. Sampoli, A. Sestini, Non-polynomial spline alternatives in Isogeometric Symmetric Galerkin BEM, Appl. Numer. Math 116 (2017) 10–23.

B. H. Nguyen, H. D. Tran, C. Anitescu, X. Zhuang, T. Rabczuk, An isogeometric symmetric galerkin boundary element method for two–dimensional crack problems, Comput. Methods Appl. Mech. Engrg. 306 (2016) 252–275.

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Published

2022-12-25

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