Analytic and semi-analytic integration of logarithmic and Newtonian potentials and their gradients over line segments and rectilinear panels

Authors

DOI:

https://doi.org/10.26089/NumMet.v23r209

Keywords:

logarithmic potential, Newtonian potential, potential gradient, integral equation, integration over line segment, integration over triangle, singularity extraction

Abstract

The integrals are considered that arise when solving boundary integral equations, which kernels are logarithmic or Newtonian potentials or their gradients, in the case when the solution is considered to be piecewise-constant over panels, which are rectilinear segments in plane problems, and flat triangles in spatial problems. Integrals over one panel are considered which are calculated in the framework of collocations method, and the calculation technique is developed for repeated integrals over two panels arising in the Galerkin method. In plane problems for all the integrals exact analytical expressions suitable for practical usage are presented; the same applies to the integrals over one panel in three-dimensional problems. For repeated integrals in 3D case, a hybrid numericalanalytical scheme is proposed, which involves the extraction of singularities in the integrands and their analytical integration, as well as the numerical integration of smooth functions.

Author Biographies

Ilia K. Marchevsky

Sophia R. Serafimova

References

  1. P. K. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, London, 1981; Mir, Moscow, 1984).
  2. C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering (Springer, Berlin, 1984; Mir, Moscow, 1987).
  3. J. T. Katsikadelis, Boundary elements: Theory and Applications (Elsevier, New York, 2002; ASV, Moscow, 2007).
  4. V. G. Maz’ya, “Boundary Integral Equations,” in Analysis-4 (VINITI, Moscow, 1988), Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Fundam. Napr., Vol. 27, pp. 131-228.
  5. L. N. Sretenskii, The Theory of Newtonian Potential (Gostekhizdat, Moscow, 1946) [in Russian].
  6. I. K. Lifanov, The Method of Singular Integral Equations and Numerical Experiment in Mathematical Physics, Aerodynamics, Elasticity Theory and Wave Diffraction (Janus, Moscow, 1995) [in Russian].
  7. A. L. Cauchy, Leçons de Calcul Différentiel et de Calcul Intégral , Tome 2: Calcul Intégral (De L’École Polytechnique, Paris, 1844).
  8. J. Hadamard, Le Problème de Cauchy et les Équations aux Dérivées Partielles Linéaires Hyperboliques (Hermann, Paris, 1932).
  9. Yu. V. Gandel’, Introduction to Methods for Calculating Singular and Hypersingular Integrals (Kharkov National University, Kharkov, 2001) [in Russian].
  10. S. N. Kempka, M. W. Glass, J. S. Peery, et al., Accuracy Considerations for Implementing Velocity Boundary Conditions in Vorticity Formulations, SANDIA Report SAND96-0583 UC-700 (Sandia Labs, Albuquerque, 1996).
    doi 10.2172/242701.
  11. K. S. Kuzmina, I. K. Marchevskii, and V. S. Moreva, “Vortex Sheet Intensity Computation in Incompressible Flow Simulation Around an Airfoil by Using Vortex Methods,” Mat. Model. 29 (10), 20-34 (2017) [Math. Models Comput. Simul. 10 (3), 276-287 (2018).]
    doi 10.1134/S2070048218030092.
  12. I. K. Marchevskii and G. A. Shcheglov, “The Algorithm of the Vortex Sheet Intensity Determining in 3D Incompressible Flow Simulation around a Body,” Mat. Model. 31 (11), 21-35 (2019) [Math. Models Comput. Simul. 12 (4), 464-473 (2020).]
    doi 10.1134/S2070048220040122.
  13. V. A. Antonov, I. I. Nikiforov, and K. V. Kholshevnikov, Elements of Gravitational Potential Theory and Some Cases of Its Explicit Expression (St. Petersburg State University, St. Petersburg, 2008) [in Russian].
  14. A. van Oosterom and J. Strackee, “The Solid Angle of a Plane Triangle,” IEEE Trans. Biomed. Eng. 30 (2), 125-126 (1983).
    doi 10.1109/TBME.1983.325207.
  15. H. Dodig, M. Cvetković, and D. Poljak, “On the Computation of Singular Integrals over Triangular Surfaces in R^3,” WIT Trans. Eng. Sci. 122, 95-105 (2019).
    doi 10.2495/BE410091.
  16. G. R. Cowper, “Gaussian Quadrature Formulas for Triangles,” Int. J. Numer. Methods Eng. 7 (3), 405-408 (1973).
    doi 10.1002/nme.1620070316.
  17. M. T. H. Reid, J. K. White, and S. G. Johnson, “Generalized Taylor -- Duffy Method for Efficient Evaluation of Galerkin Integrals in Boundary Element Method Computations,” IEEE Trans. Antennas Propag. 63 (1), 195-209 (2015).
    doi 10.1109/TAP.2014.2367492.

Published

08-06-2022

How to Cite

Марчевский И. К., Серафимова С. Р. Analytic and Semi-Analytic Integration of Logarithmic and Newtonian Potentials and Their Gradients over Line Segments and Rectilinear Panels // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2022. 23. 137-152. doi 10.26089/NumMet.v23r209

Issue

Section

Methods and algorithms of computational mathematics and their applications