A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation

  • Eugeniusz Zieniuk
  • Krzysztof Szerszeń
Open Access
Original Paper


The paper introduces the analytical modification of the classic boundary integral equation (BIE) for Stokes equation in 3D. The performed modification allows us to obtain separation of the approximation boundary shape from the approximation of the boundary functions. As a result, the equations, called the parametric integral equation system (PIES) with formal separation between the boundary geometry and the boundary functions, are obtained. It enables us to independently choose the most convenient methods of boundary geometry modeling depending on its complexity without any intrusion into the approximation of boundary functions and vice versa. Furthermore, we investigated the possibility of the modeling of the domains bounded by rectangular and triangular parametric Bézier patches. The boundary functions are approximated by generalized Chebyshev series. In addition, numerical techniques for solving the obtained PIES have been developed. The effectiveness of the presented strategy for boundary representation by surface patches in connection with PIES has been studied in numerical examples.


Parametric integral equation system (PIES) Boundary integral equation (BIE) Stokes equation Bézier surface patches 


  1. 1.
    Schlichting, H.: Boundary Layer Theory. McGraw-Hill, New York (1979)MATHGoogle Scholar
  2. 2.
    Youngren, G.K., Acrivos, A.: Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69(2), 377–403 (1975)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bringley, T.T., Peskin, C.S.: Validation of a simple method for representing spheres and slender bodies in an immersed boundary method for Stokes flow on an unbounded domain. J. Comput. Phys. 227(11), 5397–5425 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cortez, R., Hoffmann, F.: A fast numerical method for computing doubly-periodic regularized Stokes flow in 3D. J. Comput. Phys. 258, 1–14 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Pironneau, O.: On optimum profiles in Stokes flow. J. Fluid Mech. 59(1), 117–128 (1973)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Butterworth-Heinemann, Oxford (2000)MATHGoogle Scholar
  7. 7.
    Backer, A.T., Brenner, S.C.: A mixed finite element method for the Stokes equations based a weakly over-penalized symmetric interior penalty approach. J. Sci. Comput. 58(2), 290–307 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hou, L.S.: Error estimates for semidiscrete finite element approximations of the Stokes equations under minimal regularity assumptions. J. Sci. Comput. 16(3), 287–317 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fang, J., Parriaux, A., Rentschler, M., Ancey, C.: Improved SPH methods for simulating free surface flows of viscous fluids. Appl. Numer. Math. 59(2), 251–271 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Zhang, L., Ouyang, J., Zhang, X.H.: On a two-level element-free Galerkin method for incompressible fluid flow. Appl. Numer. Math. 59(8), 1894–1904 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Oñate, E., Sacco, C., Idelsohn, S.: A finite point method for incompressible flow problems. Comput. Vis. Sci. 3(1), 67–75 (2000)CrossRefMATHGoogle Scholar
  12. 12.
    Wu, X.H., Tao, W.Q., Shen, S.P., Zhu, X.W.: A stabilized MLPG method for steady state incompressible fluid flow simulation. J. Comput. Phys. 229(22), 8564–8577 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Tan, F., Zhang, Y., Li, Y.: Development of a meshless hybrid boundary node method for Stokes flows. Eng. Anal. Bound. Elem. 37(6), 899–908 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Brebbia, C.A., Telles, J.C., Wrobel, L.C.: Boundary Element Techniques, Theory and Applications in Engineering. Springer, New York (1984)CrossRefMATHGoogle Scholar
  15. 15.
    Becker, A.A.: The Boundary Element Method in Engineering: a Complete Course. McGraw-Hill Book Companies, Cambridge (1992)Google Scholar
  16. 16.
    Beskos, D.E.: Boundary Element Methods in Mechanics. North-Holland, Amsterdam (1987)MATHGoogle Scholar
  17. 17.
    Power, H., Wrobel, L.C.: Boundary integral methods in fluid mechanics. Computational Mechanics Publications (1995)Google Scholar
  18. 18.
    Muldowney, G.P., Higdon, J.J.L.: A spectral boundary element approach to three-dimensional Stokes flow. J. Fluid Mech. 298, 167–192 (1995)CrossRefMATHGoogle Scholar
  19. 19.
    Zieniuk, E.: Computational method PIES for solving boundary value problems. PWN Warsaw. (in Polish) (2013)Google Scholar
  20. 20.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39), 4135–4195 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009)CrossRefMATHGoogle Scholar
  22. 22.
    Scott, M.A., Simpson, R.N., Evans, J.A., Lipton, S., Bordas, S., Hughes, T.J.R., Sederberg, T.W.: Isogeometric boundary element analysis using unstructured T-splines. Comput. Methods Appl. Mech. Eng. 254, 197–221 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zieniuk, E., Szerszeń, K.: The PIES for solving 3D potential problems with domains bounded by rectangular bézier patches. Eng. Comput. 31(4), 791–809 (2014)CrossRefGoogle Scholar
  24. 24.
    Zieniuk, E., Szerszeń, K.: Triangular bézier surface patches in modeling shape of boundary geometry for potential problems in 3D. Eng. Comput. 29(3), 517–527 (2013)CrossRefGoogle Scholar
  25. 25.
    Zieniuk, E., Szerszeń, K.: Triangular bézier patches in modeling smooth boundary in exterior Helmholtz problems solved by PIES. Archives of Acoustics 34(1), 51–61 (2009)MATHGoogle Scholar
  26. 26.
    Zieniuk, E., Szerszen, K., Kapturczak, M.: A Numerical Approach to the Determination of 3D Stokes Flow in Polygonal Domains Using PIES. Lecture Notes in Computer Sciences 7203, part I, pp 112–121. Springer, Berlin (2012)Google Scholar
  27. 27.
    Farin, G.: Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann Publishers, Burlington (2001)Google Scholar
  28. 28.
    Pozrikidis, C.: A Practical Guide to Boundary Element Methods with the Software Library BEMLIB. CRC Press, Boca Raton (2002)CrossRefMATHGoogle Scholar
  29. 29.
    Kokkinos, F.T., Reddy, J.N.: BEM And penalty FEM models for viscous incompressible fluids. Comput. Struct. 56(5), 849–859 (1995)CrossRefMATHGoogle Scholar
  30. 30.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)CrossRefMATHGoogle Scholar
  31. 31.
    Stroud, A.H.: Gaussian Quadrature Formulas. Prentice-Hall, New Jersey (1966)MATHGoogle Scholar
  32. 32.
    Rathod, H.T., Nagaraja, K.V., Venkatesudu, B.: Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface. Appl. Math. Comput. 188(1), 865–876 (2007)MathSciNetMATHGoogle Scholar
  33. 33.
    Rong, J., Wen, L., Xiao, J.: Efficiency improvement of the polar coordinate transformation for evaluating BEM singular integrals on curved elements. Eng. Anal. Bound. Elem. 38, 83–93 (2014)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Guiggiani, M., Krishnaswamy, G., Rudolphi, T.J., Rizzo, F.J.: A general algorithm for the numerical solution of hypersingular boundary integral equations. J. Appl. Mech. 59(3), 604–614 (1992)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Young, D.L., Jane, S.C., Lin, C.Y., Chiu, C.L., Chen, K.C.: Solution of 2D and 3D Stokes laws using multiquadrics method. Eng. Anal. Bound. Elem. 28 (10), 1233–1243 (2004)CrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BialystokBiałystokPoland

Personalised recommendations