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

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## Abstract

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.

## Keywords

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

- 1.Schlichting, H.: Boundary Layer Theory. McGraw-Hill, New York (1979)zbMATHGoogle Scholar
- 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 5.Pironneau, O.: On optimum profiles in Stokes flow. J. Fluid Mech.
**59**(1), 117–128 (1973)MathSciNetzbMATHGoogle Scholar - 6.Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Butterworth-Heinemann, Oxford (2000)zbMATHGoogle Scholar
- 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 11.Oñate, E., Sacco, C., Idelsohn, S.: A finite point method for incompressible flow problems. Comput. Vis. Sci.
**3**(1), 67–75 (2000)zbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 14.Brebbia, C.A., Telles, J.C., Wrobel, L.C.: Boundary Element Techniques, Theory and Applications in Engineering. Springer, New York (1984)zbMATHGoogle Scholar
- 15.Becker, A.A.: The Boundary Element Method in Engineering: a Complete Course. McGraw-Hill Book Companies, Cambridge (1992)Google Scholar
- 16.Beskos, D.E.: Boundary Element Methods in Mechanics. North-Holland, Amsterdam (1987)zbMATHGoogle Scholar
- 17.Power, H., Wrobel, L.C.: Boundary integral methods in fluid mechanics. Computational Mechanics Publications (1995)Google Scholar
- 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)zbMATHGoogle Scholar - 19.Zieniuk, E.: Computational method PIES for solving boundary value problems. PWN Warsaw. (in Polish) (2013)Google Scholar
- 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)MathSciNetzbMATHGoogle Scholar - 21.Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009)zbMATHGoogle Scholar
- 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)MathSciNetzbMATHGoogle Scholar - 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)Google Scholar - 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)Google Scholar - 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)zbMATHGoogle Scholar - 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.Farin, G.: Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann Publishers, Burlington (2001)Google Scholar
- 28.Pozrikidis, C.: A Practical Guide to Boundary Element Methods with the Software Library BEMLIB. CRC Press, Boca Raton (2002)zbMATHGoogle Scholar
- 29.Kokkinos, F.T., Reddy, J.N.: BEM And penalty FEM models for viscous incompressible fluids. Comput. Struct.
**56**(5), 849–859 (1995)zbMATHGoogle Scholar - 30.Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)zbMATHGoogle Scholar
- 31.Stroud, A.H.: Gaussian Quadrature Formulas. Prentice-Hall, New Jersey (1966)zbMATHGoogle Scholar
- 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 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)MathSciNetzbMATHGoogle Scholar - 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)zbMATHGoogle Scholar

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