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A gradient reproducing kernel collocation method for high order differential equations

  • Ashkan Mahdavi
  • Sheng-Wei ChiEmail author
  • Huiqing Zhu
Original Paper
  • 74 Downloads

Abstract

The High order Gradient Reproducing Kernel in conjunction with the Collocation Method (HGRKCM) is introduced for solutions of 2nd- and 4th-order PDEs. All the derivative approximations appearing in PDEs are constructed using the gradient reproducing kernels. Consequently, the computational cost for construction of derivative approximations reduces tremendously, basis functions for derivative approximations are smooth, and the accumulated error arising from calculating derivative approximations are controlled in comparison to the direct derivative counterparts. Furthermore, it is theoretically estimated and numerically tested that the same number of collocation points as the source points can be used to obtain the optimal solution in the HGRKCM. Overall, the HGRKCM is roughly 10–25 times faster than the conventional reproducing kernel collocation method. The convergence of the present method is estimated using the least squares functional equivalence. Numerical results are verified and compared with other strong-form-based and Galerkin-based methods.

Keywords

Strong form collocation Weighted collocation method Gradient reproducing kernel Reproducing kernel collocation method 

Notes

Acknowledgements

Research reported in this paper was partially supported by DoD SERDP under contract number W912HQ18C0099.

References

  1. 1.
    Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10(5):307–318MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139(1–4):3–47CrossRefzbMATHGoogle Scholar
  3. 3.
    Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37(155):141–158MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen J-S, Pan C, Wu C-T, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1–4):195–227MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9):1081–1106MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liu W-K, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods (i) methodology and convergence. Comput Methods Appl Mech Eng 143(1–2):113–154MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Babuska I, Melenk JM (1995) The partition of unity finite element method. tech. rep., DTIC DocumentGoogle Scholar
  9. 9.
    Sukumar N (1998) The natural element method in solid mechanics. Ph.D. thesis, Northwestern UniversityGoogle Scholar
  10. 10.
    Atluri S, Cho J, Kim H-G (1999) Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations. Comput Mech 24(5):334–347CrossRefzbMATHGoogle Scholar
  11. 11.
    Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181(1):43–69MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel element method. Part I: theoretical formulation. Comput Methods Appl Mech Eng 193(12):933–951CrossRefzbMATHGoogle Scholar
  13. 13.
    Chen J-S, Wang H-P (2000) New boundary condition treatments in meshfree computation of contact problems. Comput Methods Appl Mech Eng 187(3–4):441–468MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193(12–14):1257–1275MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moës N, Bechet E, Tourbier M (2005) Imposing essential boundary conditions in the extended finite element method, In: VIII international conference on computational plasticity. Citeseer, BarcelonaGoogle Scholar
  16. 16.
    Fonseca A, Viana S, Silva E, Mesquita R (2008) Imposing boundary conditions in the meshless local Petrov–Galerkin method. IET Sci Meas Technol 2(6):387–394CrossRefGoogle Scholar
  17. 17.
    Boyce B, Kramer S, Bosiljevac T, Corona E, Moore J, Elkhodary K, Simha C, Williams B, Cerrone A, Nonn A et al (2016) The second sandia fracture challenge: predictions of ductile failure under quasi-static and moderate-rate dynamic loading. Int J Fract 198(1–2):5–100CrossRefGoogle Scholar
  18. 18.
    Liu G-R, Zhang G, Gu Y, Wang Y (2005) A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Comput Mech 36(6):421–430MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hu H-Y, Li Z-C (2006) Collocation methods for Poisson’s equation. Comput Methods Appl Mech Eng 195(33):4139–4160MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li Z-C, Lu T-T, Hu H-Y, Cheng AH (2008) Trefftz and collocation methods. WIT Press, AshurstzbMATHGoogle Scholar
  21. 21.
    Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915CrossRefGoogle Scholar
  22. 22.
    Hardy RL (1990) Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988. Comput Math Appl 19(8–9):163–208MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19(8):147–161MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hon Y, Schaback R (2001) On unsymmetric collocation by radial basis functions. Appl Math Comput 119(2):177–186MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kansa E, Hon Y (2000) Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput Math Appl 39(7–8):123–137MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Aluru N (2000) A point collocation method based on reproducing kernel approximations. Int J Numer Methods Eng 47(6):1083–1121CrossRefzbMATHGoogle Scholar
  27. 27.
    Kim DW, Kim Y (2003) Point collocation methods using the fast moving least-square reproducing kernel approximation. Int J Numer Methods Eng 56(10):1445–1464MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Onate E, Idelsohn S, Zienkiewicz O, Taylor R (1996) A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Eng 39(22):3839–3866MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hu H-Y, Chen J-S, Hu W (2011) Error analysis of collocation method based on reproducing kernel approximation. Numer Methods Partial Differ Equ 27(3):554–580MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hu H-Y, Lai C-K, Chen J-S (2009) A study on convergence and complexity of reproducing kernel collocation method. National Science Council Tunghai University Endowment Fund for Academic Advancement Mathematics Research Promotion Center, Taichung CityCrossRefGoogle Scholar
  31. 31.
    Hu H, Chen J, Hu W (2007) Weighted radial basis collocation method for boundary value problems. Int J Numer Methods Eng 69(13):2736–2757MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang D, Wang J, Wu J (2018) Superconvergent gradient smoothing meshfree collocation method. Comput Methods Appl Mech Eng 340:728–766MathSciNetCrossRefGoogle Scholar
  33. 33.
    Chi S-W, Chen J-S, Hu H-Y, Yang JP (2013) A gradient reproducing kernel collocation method for boundary value problems. Int J Numer Methods Eng 93(13):1381–1402MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, part I—formulation and theory. Int J Numer Methods Eng 45(3):251–288CrossRefzbMATHGoogle Scholar
  35. 35.
    Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, part II—applications. Int J Numer Methods Eng 45(3):289–317CrossRefGoogle Scholar
  36. 36.
    Chen J-S, Zhang X, Belytschko T (2004) An implicit gradient model by a reproducing kernel strain regularization in strain localization problems. Comput Methods Appl Mech Eng 193(27):2827–2844CrossRefzbMATHGoogle Scholar
  37. 37.
    Li S, Liu WK (1998) Synchronized reproducing kernel interpolant via multiple wavelet expansion. Comput Mech 21(1):28–47MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Chen J-S, Wu C-T, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50(2):435–466CrossRefzbMATHGoogle Scholar
  39. 39.
    Chen J-S, Hillman M, Chi S-W (2017) Meshfree methods: progress made after 20 years. J Eng Mech 143(4):04017001CrossRefGoogle Scholar
  40. 40.
    Han W, Meng X (2001) Error analysis of the reproducing kernel particle method. Comput Methods Appl Mech Eng 190(46–47):6157–6181MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Mirzaei D, Schaback R, Dehghan M (2012) On generalized moving least squares and diffuse derivatives. IMA J Numer Anal 32(3):983–1000MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Timoshenko SP, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-hill, New YorkzbMATHGoogle Scholar
  43. 43.
    Auricchio F, Da Veiga LB, Hughes T, Reali A, Sangalli G (2010) Isogeometric collocation methods. Math Models Methods Appl Sci 20(11):2075–2107MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Reali A, Gomez H (2015) An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates. Comput Methods Appl Mech Eng 284:623–636 (Isogeometric Analysis Special Issue)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Qi D, Wang D, Deng L, Xu X, Wu C-T (2019) Reproducing kernel mesh-free collocation analysis of structural vibrations. Eng Comput 36(3):734–764CrossRefGoogle Scholar
  46. 46.
    Guan P, Chi S, Chen J, Slawson T, Roth M (2011) Semi-Lagrangian reproducing kernel particle method for fragment-impact problems. Int J Impact Eng 38(12):1033–1047CrossRefGoogle Scholar
  47. 47.
    Siriaksorn T, Chi S-W, Foster C, Mahdavi A (2018) u-p semi-Lagrangian reproducing kernel formulation for landslide modeling. Int J Numer Anal Methods Geomech 42(2):231–255CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Civil and Material EngineeringUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of Southern MississippiHattiesburgUSA

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