An enhanced BEM approach applied to potential problems: a comparative study

Document Type : Research

Authors

1 Department of Civil Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran.

2 Department of Civil Engineering, Zanjan Branch, Islamic Azad University, Zanjan, Iran.

Abstract

In the present paper, a hybrid analytical/numerical method capable of integrating singular logarithmic functions, as an essential part of the Boundary Element Method (BEM) process, is presented. The proposed scheme provides a more practical approach through the reduction of the computational effort of the analytical method. For this purpose, the singular function is divided into two parts of singular and non-singular. The non-singular part is numerically integrated, while the singular part is analytically integrated and the result of both parts is combined. The capabilities and accuracy of the proposed scheme are investigated through various elemental and potential examples. The results of numerical comparisons indicate the ability of the proposed scheme to reduce the computational effort of the analytical solutions, which develops an appropriate alternative for the simple analytical solution of the potential problems that can be used in practical modeling problems such as heat transfer.

Keywords

References

  1. R. J. Astley and W. Eversman, "A finite element method for transmission in non-uniform ducts without flow: comparison with the method of weighted residuals," Journal of Sound and Vibration, vol. 57, pp. 367-388, 1978. [DOI:10.1016/0022-460X(78)90317-6]
  2.  B. Finlayson and L. Scriven, "The method of weighted residuals-a review," Appl. Mech. Rev, vol. 19, pp. 735-748, 1966.
  3.  B. A. Finlayson, The method of weighted residuals and variational principles vol. 73: SIAM, 2013. [DOI:10.1137/1.9781611973242]
  4.  A. Haji-Sheikh, E. Sparrow, and W. Minkowycz, "Heat transfer to flow through porous passages using extended weighted residuals method-a Green's function solution," International journal of heat and mass transfer, vol. 48, pp. 1330-1349, 2005. [DOI:10.1016/j.ijheatmasstransfer.2004.10.002]
  5.  M. Ganz, "Finite Element Method I," 2010.
  6. T. Yang, Finite element structural analysis vol. 2: Prentice-Hall Englewood Cliffs, NJ, 1986.
  7.  G. Beer, Programming the boundary element method: John Wiley & Sons, Inc., 2000.
  8. G. Beer, I. Smith, and C. Duenser, The boundary element method with programming: for engineers and scientists: Springer Science & Business Media, 2008.
  9.  C. A. Brebbia and J. Dominguez, Boundary elements: an introductory course: WIT press, 1994.
  10.  T. A. Cruse, Boundary element analysis in computational fracture mechanics vol. 1: Springer Science & Business Media, 2012.
  11.  L. Gaul, M. Kögl, and M. Wagner, Boundary element methods for engineers and scientists: an introductory course with advanced topics: Springer Science & Business Media, 2013.
  12.  A. Nohegoo-Shahvari, M. Kamalian, and M. Panji, "A hybrid time-domain half-plane FE/BE approach for SH-wave scattering of alluvial sites," Engineering Analysis with Boundary Elements, vol. 105, pp. 194-206, 2019. [DOI:10.1016/j.enganabound.2019.04.020]
  13.  A. Sohrabi-Bidar, M. Kamalian, and M. K. Jafari, "Seismic response of 3-D Gaussian-shaped valleys to vertically propagating incident waves," Geophysical Journal International, vol. 183, pp. 1429-1442, 2010. [DOI:10.1111/j.1365-246X.2010.04792.x]
  14. H. Alielahi and M. Adampira, "Evaluation of 2D seismic site response due to hill-cavity interaction using boundary element technique," Journal of Earthquake Engineering, vol. 22, pp. 1137-1167, 2018. [DOI:10.1080/13632469.2016.1277437]
  15.  M. Panji, H. Koohsari, M. Adampira, H. Alielahi, and J. A. Marnani, "Stability analysis of shallow tunnels subjected to eccentric loads by a boundary element method," Journal of Rock Mechanics and Geotechnical Engineering, vol. 8, pp. 480-488, 2016. [DOI:10.1016/j.jrmge.2016.01.006]
  16.  A. Abdollahipour, M. F. Marji, A. Y. Bafghi, and J. Gholamnejad, "Time-dependent crack propagation in a poroelastic medium using a fully coupled hydromechanical displacement discontinuity method," International Journal of Fracture, vol. 199, pp. 71-87, 2016. [DOI:10.1007/s10704-016-0095-9]
  17.  M. Lak, M. F. Marji, A. Y. Bafghi, and A. Abdollahipour, "A coupled finite difference-boundary element method for modeling the propagation of explosion-induced radial cracks around a wellbore," Journal of Natural Gas Science and Engineering, vol. 64, pp. 41-51, 2019. [DOI:10.1016/j.jngse.2019.01.019]
  18.  M. F. Marji, "Numerical analysis of quasi-static crack branching in brittle solids by a modified displacement discontinuity method," International Journal of Solids and Structures, vol. 51, pp. 1716-1736, 2014. [DOI:10.1016/j.ijsolstr.2014.01.022]
  19.  M. F. Marji, "Simulation of crack coalescence mechanism underneath single and double disc cutters by higher order displacement discontinuity method," Journal of Central South University, vol. 22, pp. 1045-1054, 2015. [DOI:10.1007/s11771-015-2615-6]
  20.  M. Panji, M. Kamalian, J. Asgari Marnani, and M. Jafari, "Analysing seismic convex topographies by a half-plane time-domain BEM," Geophysical Journal International, vol. 197, pp. 591-607, 2014. [DOI:10.1093/gji/ggu012]
  21.  M. Panji, M. Kamalian, M. J. ASGARI, and M. Jafari, "Antiplane seismic response from semi-sine shaped valley above embedded truncated circular cavity: a time-domain half-plane BEM," 2014.
  22. M. Panji, M. Kamalian, J. A. Marnani, and M. Jafari, "Transient analysis of wave propagation problems by half-plane BEM," Geophysical Journal International, vol. 194, pp. 1849-1865, 2013. [DOI:10.1093/gji/ggt200]
  23.  J. A. Crow, "Quadrature of integrands with a logarithmic singularity," mathematics of computation, vol. 60, pp. 297-301, 1993. [DOI:10.1090/S0025-5718-1993-1155572-3]
  24.  R. Smith, "Simplifying integration for logarithmic singularities," WIT Transactions on Modelling and Simulation, vol. 14, 1970.
  25.  V. Sladek, J. Sladek, and M. Tanaka, "Numerical integration of logarithmic and nearly logarithmic singularity in BEMs," Applied Mathematical Modelling, vol. 25, pp. 901-922, 2001. [DOI:10.1016/S0307-904X(01)00021-X]
  26.  M. F. Marji, H. Hosseini-Nasab, and A. H. Kohsary, "A new cubic element formulation of the displacement discontinuity method using three special crack tip elements for crack analysis," JP J. Solids Struct, vol. 1, pp. 61-91, 2007.
  27.  H. Haeri, K. Shahriar, M. FatehiMarji, and P. Moarefvand, "On the crack propagation analysis of rock like Brazilian disc specimens containing cracks under compressive line loading," Latin American journal of solids and structures, vol. 11, pp. 1400-1416, 2014. [DOI:10.1590/S1679-78252014000800007]
  28.  C. W. Clenshaw and A. R. Curtis, "A method for numerical integration on an automatic computer," Numerische Mathematik, vol. 2, pp. 197-205, 1960. [DOI:10.1007/BF01386223]
  29.  B. Birgisson, E. Siebrits, and A. P. Peirce, "Elastodynamic direct boundary element methods with enhanced numerical stability properties," International journal for numerical methods in engineering, vol. 46, pp. 871-888, 1999. https://doi.org/10.1002/(SICI)1097-0207(19991030)46:6<871::AID-NME698>3.0.CO;2-6 [DOI:10.1002/(SICI)1097-0207(19991030)46:63.0.CO;2-6]
  30.  A. Tadeu, P. Santos, and E. Kausel, "Closed-form integration of singular terms for constant, linear and quadratic boundary elements. Part 1. SH wave propagation," Engineering Analysis with Boundary Elements, vol. 23, pp. 671-681, 1999. [DOI:10.1016/S0955-7997(99)00016-8]
  31.  M. Fratantonio and J. Rencis, "Exact boundary element integrations for two-dimensional Laplace equation," Engineering Analysis with Boundary Elements, vol. 24, pp. 325-342, 2000. [DOI:10.1016/S0955-7997(00)00005-9]
  32.  A. P. Selvadurai, Partial differential equations in mechanics 2: the biharmonic equation, Poisson's equation: Springer Science & Business Media, 2013.
  33.  S. W. C. A. BREBBIA, Boundary Element Techniques in Engineering. LONDON, BOSTON, Sydney, Wellington, Durban,Toronto: Camelot Press Ltd, 1980.
Numerical Methods in Civil Engineering
Volume 5, Issue 1
September 2020
Pages 51-59

Files

History

  • Receive Date: 07 May 2020
  • Revise Date: 07 July 2020
  • Accept Date: 07 September 2020

Share

How to cite

Statistics