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On the random differential quadrature (RDQ) method: consistency analysis and application in elasticity problems

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Abstract

Differential Quadrature (DQ) is an efficient derivative approximation technique but it requires a regular domain with uniformly arranged nodes. This restricts its application for a regular domain only discretized by the field nodes in a fixed pattern. In the presented random differential quadrature (RDQ) method however this restriction of the DQ method is removed and its applicability is extended for a regular domain discretized by randomly distributed field nodes and for an irregular domain discretized by uniform or randomly distributed field nodes. The consistency analysis of the locally applied DQ method is carried out, based on it approaches are suggested to obtain the fast convergence of function value by the RDQ method. The convergence studies are carried out by solving 1D, 2D and elasticity problems and it is concluded that the RDQ method can effectively handle regular as well as irregular domains discretized by random or uniformly distributed field nodes.

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References

  1. Lucy LB (1977) A numerical approach to testing the fission hypothesis. Astronom J 8(12): 1013–1024

    Article  Google Scholar 

  2. Gingold RA, Monaghan JJ et al (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Notices Roy Astronom Soc 181: 375–389

    MATH  Google Scholar 

  3. Nayroles B, Touzot G, Villon P et al (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10: 307–318

    Article  MATH  Google Scholar 

  4. Belytschko T, Gu L, Lu YY et al (1994) Fracture and crack growth by element free Galerkin method. Modell Simul Mater Sci Eng 2: 519–534

    Article  Google Scholar 

  5. Lu YY, Belytschko T, Gu L et al (1994) A new implementation of the element free Galerkin method. Comp Methods Appl Mech Eng 113: 397–414

    Article  MATH  MathSciNet  Google Scholar 

  6. Belytschko T, Lu YY, Gu L et al (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37: 229–256

    Article  MATH  MathSciNet  Google Scholar 

  7. Braun J, Sambridge M et al (1995) A numerical method for solving partial differential equations on highly irregular evolving grids. Nature 376: 655–660

    Article  Google Scholar 

  8. Liu WK, Jun S et al (1998) Multiple scale reproducing kernel particle methods for large deformation problems. Int J Numer Methods Eng 41: 1339–1362

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu WK, Chen Y, Uras RA, Chang CT et al (1996) Generalized multiple scale reproducing kernel particle method. Comp Methods Appl Mech Eng 139: 91–157

    Article  MATH  MathSciNet  Google Scholar 

  10. Liu WK, Jun S, Zhang YF et al (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20: 1081–1106

    Article  MATH  MathSciNet  Google Scholar 

  11. Melenk JM, babushka I et al (1996) The partition of unity finite element method: basic theory and applications. Comp Methods Appl Mech Eng 139: 289–314

    Article  MATH  Google Scholar 

  12. Zhu T, Zhang JD, Atluri SN et al (1998) A local boundary integral equation (LBIE) method in computational mechanics and a meshless discretization approach. Comput Mech 21: 223–235

    Article  MATH  MathSciNet  Google Scholar 

  13. Atluri SN, Sladek J, Sladek V, Zhu T et al (2000) The local boundary integral equation (LBIE) and its meshless implementation for linear elasticity. Comput Mech 25: 180–189

    Article  MATH  Google Scholar 

  14. Atluri SN, Zhu T et al (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22: 117–127

    Article  MATH  MathSciNet  Google Scholar 

  15. Li H, Wang QX, Lam KY et al (2004) Development of a novel meshless Local Kriging (LoKriging) method for structural dynamics analysis. Comput Methods Appl Mech Eng 193: 2599–2619

    Article  MATH  Google Scholar 

  16. Liu GR, Gu YT et al (2001) A point interpolation method for two-dimensional solids. Int J Numer Eng 50: 937–951

    Article  MATH  Google Scholar 

  17. Onate E, Idelsohn EO, Zienkiewicz OC, Taylor RL et al (1996) A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Eng 39: 3839–3866

    Article  MATH  MathSciNet  Google Scholar 

  18. Li H, Ng TY, Cheng JQ, Lam KY et al (2003) Hermite–Cloud: a novel true meshless method. Comput Mech 33: 30–41

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu GR, Zhang J, Lam KY, Li H, Xu G, Zhong ZH, Li GY, Han X et al (2008) A gradient smoothing method (GSM) with directional correction for solid mechanics problems. Comput Mech 41: 457–472

    Article  MATH  Google Scholar 

  20. Sukumar N, Moran B, Semenov AY, Belicov VV et al (2001) Natural neighbour Galerkin methods. Int J Numer Methods Eng 50: 1–27

    Article  MATH  Google Scholar 

  21. Li H, Yew YK, Ng TY, Lam KY et al (2005) Meshless steady-state analysis of chemo-electro-mechanical coupling behaviour of pH-sensitive hydrogel in buffered solution. J Electroanal Chem 580: 161–172

    Article  Google Scholar 

  22. Sukumar N, Moran B, Belytschko T et al (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43: 839–887

    Article  MATH  MathSciNet  Google Scholar 

  23. Li H, Cheng JQ, Ng TY, Chen J, Lam KY et al (2004) A meshless Hermite–cloud method for nonlinear fluid–structure analysis of near-bed submarine pipelines under current. Eng Struct 26: 531–542

    Article  Google Scholar 

  24. Idelsohn SR, Onate E, Calvo N, Pin FD et al (2003) The meshless finite element method. Int J Numer Methods Eng 58: 893–912

    Article  MATH  Google Scholar 

  25. Onate E, Idelsohn EO, Zienkiewicz OC, Taylor RL, Sacco C et al (1996) A stabilized finite point method for analysis of fluid mechanics problems. Comp Methods Appl Mech Eng 139: 315–346

    Article  MATH  MathSciNet  Google Scholar 

  26. Atluri SN, Cho JY, Kim HG et al (1999) Analysis of thin beams, using meshless local Petrov–Galerkin method, with generalized moving least square interpolations. Comput Mech 24: 334–347

    Article  MATH  Google Scholar 

  27. Liu GR, Zhang J, Li H, Lam KY, Bernard BT Kee et al (2006) Radial point interpolation based finite difference method for mechanics problems. Int J Numer Methods Eng 68: 728–754

    Article  MATH  Google Scholar 

  28. Atluri SN, Kim HG, Cho JY et al (1999) A critical assessment of the truly meshless Local Petrov–Galerkin (MLPG), and local boundary integral equation (LBIE) methods. Comput Mech 24: 348–372

    Article  MATH  Google Scholar 

  29. Gilhooley DF, Xiao JR, Batra RC, McCarthy MA, Gillespie JW et al (2008) Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions. Comput Materials Sci 41: 467–481

    Article  Google Scholar 

  30. Aluru NR, Li G (2001) Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation. Int J Numer Methods Eng 50: 2373–2410

    Article  MATH  Google Scholar 

  31. Aluru NR (2000) A point collocation method based on Reprod Kernel Approx. Int J Numer Methods Eng 47: 1083–1121

    Article  MATH  Google Scholar 

  32. Li G, Paulino GH, Aluru NR et al (2003) Coupling of the mesh-free finite cloud method with the boundary element method: a collocation approach. Computer Methods Appl Mech Eng 192: 2355–2375

    Article  MATH  Google Scholar 

  33. Jin X, Li G, Aluru NR et al (2005) New approximations and collocation schemes in the finite cloud method. Comp Struct 83: 1366–1385

    Article  MathSciNet  Google Scholar 

  34. Aluru NR (1999) A reproducing kernel particle method for meshless analysis of microelectromechanical systems. Comput Mech 23: 324–338

    Article  MATH  Google Scholar 

  35. Jin X, Li G, Aluru NR et al (2004) Positivity conditions in meshless collocation methods. Comp Methods Appl Mech Eng 193: 1171–1202

    Article  MATH  MathSciNet  Google Scholar 

  36. Bellman RE, Kashef BG, Casti J et al (1972) Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J Comput Phys 10: 40–52

    Article  MATH  MathSciNet  Google Scholar 

  37. Quan JR, Chang CT et al (1989) New insights in solving distributed system equations by the quadrature method—I. Analysis. Comp Chem Eng 13: 779–788

    Article  Google Scholar 

  38. Quan JR, Chang CT et al (1989) New insights in solving distributed system equations by the quadrature method—II. Numerical experiments. Comp Chem Eng 13: 1017–1024

    Article  Google Scholar 

  39. Shu C, Khoo BC, Yeo KS et al (1994) Numerical solutions of incompressible Navier–Stokes equations by generalized differential quadrature. Finite Elem Analy Des 18: 83–97

    Article  Google Scholar 

  40. Ding H, Shu C, Yeo KS, Xu D et al (2006) Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method. Comp Methods Appl Mech Eng 195: 516–533

    Article  MATH  Google Scholar 

  41. Naadimuthu G, Bellman R, Wang KM et al (1984) Differential quadrature and partial differential equations: some numerical results. J Math Anal Appl 98: 220–235

    Article  MATH  MathSciNet  Google Scholar 

  42. Chang S (2000) Differential quadrature and its application in engineering. Springer, London

    MATH  Google Scholar 

  43. Nayroles B, Touzot G, Villon P et al (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10: 307–318

    Article  MATH  Google Scholar 

  44. John S (1989) Finite difference schemes and partial differential equations. Wadsworth and Brooks, California

    MATH  Google Scholar 

  45. Christoph U (1997) Numerical computation : methods, software, and analysis. Springer, Germany

    MATH  Google Scholar 

  46. Sarra SA (2006) Chebyshev interpolation: an interactive tour. J Online Math Appl 6: 1–13

    Google Scholar 

  47. Mukherjee YX, Mukherjee S (1997) On boundary conditions in the element-free Galerkin method. Comput Mech 19: 264–270

    Article  MATH  MathSciNet  Google Scholar 

  48. Edward S (2006) C++ for mathematicians: an introduction for students and professionals. CRC Press, USA

    MATH  Google Scholar 

  49. Fish J, Belytschko T (2007) A first course in finite elements. Wiley, Chichester

    Book  MATH  Google Scholar 

  50. Liu GR, Quek SS (2003) The finite element method: a practical course. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  51. Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New york

    MATH  Google Scholar 

  52. Zhilun Xu (1992) Applied elasticity. New Age International, India

    Google Scholar 

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Authors and Affiliations

  1. School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Republic of Singapore

    Shantanu S. Mulay & Hua Li

  2. Asia Pacific Science and Technology Center, Sun Microsystems, Inc., 1 Magazine Road, Singapore, 059567, Singapore

    Simon See

Authors
  1. Shantanu S. Mulay
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  2. Hua Li
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  3. Simon See
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Correspondence to Hua Li.

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Mulay, S.S., Li, H. & See, S. On the random differential quadrature (RDQ) method: consistency analysis and application in elasticity problems. Comput Mech 44, 563–590 (2009). https://doi.org/10.1007/s00466-009-0393-4

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  • DOI: https://doi.org/10.1007/s00466-009-0393-4

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