Skip to main content
SpringerLink
Log in

A Hermite DRK interpolation-based collocation method for the analyses of Bernoulli–Euler beams and Kirchhoff–Love plates

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A Hermite differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving fourth-order differential equations where the field variable and its first-order derivatives are regarded as the primary variables. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the Hermite DRK interpolation, without directly differentiating it. In addition, the shape function of this interpolation at each sampling node is separated into a primitive function possessing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied for the field variable and its first-order derivatives. A weighted least-squares collocation method based on this interpolation is developed for the static analyses of classical beams and plates with fully simple and clamped supports, in which its accuracy and convergence rate are examined, and some guidance for using this method is suggested.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139: 3–47

    Article  MATH  Google Scholar 

  2. Atluri SN, Shen S (2002) The meshless local Petrov-Galerkin (MLPG) method. Tech Science Press, Encino

    MATH  Google Scholar 

  3. Liu WK, Chen Y, Jun S, Chen JS, Belytschko T, Pan C, Uras RA, Chang CT (1996) Overview and applications of the reproducing kernel particle methods. Arch Comput Methods Eng 3: 3–80

    Article  MathSciNet  Google Scholar 

  4. Li S, Liu WK (2004) Meshfree particle methods. Springer, Berlin

    MATH  Google Scholar 

  5. Li S, Liu WK (2002) Meshfree particle methods and their applications. Appl Mech Rev 55: 1–34

    Article  Google Scholar 

  6. Liu GR, Gu YT (2005) Meshfree methods moving beyond the finite element method. CRC Press, New York

    Google Scholar 

  7. Monaghan JJ (1988) An introduction to SPH. Comput Phys Commun 48: 89–96

    Article  MATH  Google Scholar 

  8. Lucy L (1977) A numerical approach to testing the fission hypothesis. Astron J 82: 1013–1024

    Article  Google Scholar 

  9. Liszka TJ, Duarte CAM, Tworzydlo WW (1996) hp-Meshless cloud method. Comput Methods Appl Mech Eng 139: 263–288

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Õnate E, Perazzo F, Miquel J (2001) A finite point method for elasticity problems. Comput Struct 79: 2151–2163

    Article  Google Scholar 

  12. Boroomand B, Najjar M, Õnate E (2009) The generalized finite point method. Comput Mech 44: 173–190

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu X, Tai K (2006) Point interpolation collocation method for the solution of partial differential equations. Eng Anal Bound Elem 30: 598–609

    Article  MATH  Google Scholar 

  14. Zhang X, Liu XH, Song KZ, Lu MW (2001) Least-squares collocation meshless method. Int J Numer Methods Eng 51: 1089–1100

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhang X, Song KZ, Lu MW, Liu X (2000) Meshless methods based on collocation with radial basis functions. Comput Mech 26: 333–343

    Article  MATH  Google Scholar 

  16. Wang QX, Li H, Lam KY (2005) Development of a new meshless-point weighted least squares (PWLS) method for computational mechanics. Comput Mech 35: 170–181

    Article  MATH  MathSciNet  Google Scholar 

  17. Aluru NR (2000) A point collocation method based on reproducing kernel approximations. Int J Numer Methods Eng 47: 1083–1121

    Article  MATH  Google Scholar 

  18. Wu CP, Chiu KH, Wang YM (2008) A meshfree DRK-based collocation method for the coupled analysis of functionally graded magneto-electro-elastic shells and plates. Comput Model Eng Sci 35: 181–214

    MATH  MathSciNet  Google Scholar 

  19. Wu CP, Wang JS, Wang YM (2009) A DRK interpolation-based collocation method for the analysis of functionally graded piezoelectric hollow cylinders under electro-mechanical loads. Comput Model Eng Sci 52: 1–37

    Google Scholar 

  20. Yang SW, Wang YM, Wu CP, Hu HT (2010) A meshless collocation method based on the differential reproducing kernel approximation. Comput Model Eng Sci 60: 1–39

    MathSciNet  Google Scholar 

  21. Wang YM, Chen SM, Wu CP (2010) A meshless collocation method based on the differential reproducing kernel interpolation. Comput Mech 45: 585–606

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Liu WK, Chen Y, Uras RA, Chang CT (1996) Generalized multiple scale reproducing kernel particle methods. Comput Methods Appl Mech Eng 139: 91–157

    Article  MATH  MathSciNet  Google Scholar 

  24. Liu WK, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods (I) methodology and convergence. Comput Methods Appl Mech Eng 143: 113–154

    Article  MATH  MathSciNet  Google Scholar 

  25. Li S, Liu WK (1996) Moving least-square reproducing kernel method (II) Fourier analysis. Comput Methods Appl Mech Eng 139: 159–193

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Atluri SN, Kim HG, Cho JY (1999) A critical assessment of the truly meshless local Petro-Galerkin (MLPG) and local boundary integral equation (LBIE) methods. Comput Mech 22: 348–372

    Article  Google Scholar 

  28. Atluri SN, Zhu T (1998) A new meshless local Petro-Galerkin (MLPG) approach in computational mechanics. Comput Mech 22: 117–127

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Gu YT, Liu GR (2001) A local point interpolation method for static and dynamic analysis of thin beams. Comput Methods Appl Mech Eng 190: 5515–5528

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  32. De S, Bathe KJ (2000) The method of finite spheres. Comput Mech 25: 329–345

    Article  MATH  MathSciNet  Google Scholar 

  33. Li S, Liu WK (1998) Synchronized reproducing kernel interpolant via multiple wavelet expansion. Comput Mech 21: 28–47

    Article  MATH  MathSciNet  Google Scholar 

  34. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity. Part I—formulation and theory. Int J Numer Methods Eng 45: 251–288

    Article  MATH  Google Scholar 

  35. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity. Part II—applications. Int J Numer Methods Eng 45: 289–317

    Article  Google Scholar 

  36. 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: 933–951

    Article  MATH  MathSciNet  Google Scholar 

  37. Li S, Lu H, Han W, Liu WK, Simkins DC (2004) Reproducing kernel element method, Part II: Globally conforming I m/C n hierarchies. Comput Methods Appl Mech Eng 193: 953–987

    Article  MATH  MathSciNet  Google Scholar 

  38. Lu H, Li S, Simkins DC, Liu WK, Cao J (2004) Reproducing kernel element method. Part III: generalized enrichment and applications. Comput Methods Appl Mech Eng 193: 989–1011

    Article  MATH  Google Scholar 

  39. Simkins DC Jr, Li S, Lu H, Liu WK (2004) Reproducing kernel element method. Part IV: globally compatible C n(n ≥ 1) triangular hierarchy. Comput Methods Appl Mech Eng 193: 1013–1034

    Article  MATH  MathSciNet  Google Scholar 

  40. Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139: 195–227

    Article  MATH  MathSciNet  Google Scholar 

  41. Chen JS, Pan C, Wu CT (1997) Large deformation analysis of rubber based on a reproducing kernel particle method. Comput Mech 19: 211–227

    Article  MATH  MathSciNet  Google Scholar 

  42. Liew KM, Ng TY, Wu YC (2002) Meshfree method for large deformation analysis—a reproducing kernel particle approach. Eng Struct 24: 543–551

    Article  Google Scholar 

  43. Zhao X, Liew KM, Ng TY (2003) Vibration analysis of laminated composite cylindrical panels via a meshfree approach. Int J Solids Struct 40: 161–180

    Article  MATH  Google Scholar 

  44. Cheng R, Liew KM (2009) The reproducing kernel particle method for two-dimensional unsteady heat conduction problems. Comput Mech 45: 1–10

    Article  MATH  MathSciNet  Google Scholar 

  45. Kim DW, Kim Y (2003) Point collocation methods using the fast moving least-square reproducing kernel approximation. Int J Numer Methods Eng 56: 1445–1464

    Article  MATH  Google Scholar 

  46. Lee C, Kim DW, Park SH, Kim HK, Im CH, Jung HK (2008) Point collocation mesh-free using FMLSRKM for solving axisymmetric Laplace equations. IEEE Trans Magn 44: 1234–1237

    Article  Google Scholar 

  47. Oñate E, Idelsohn S, Zienkiewicz OC, Taylor RL (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  Google Scholar 

  48. Oñate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) A stabilized finite point method for analysis of fluid mechanics problems. Comput Methods Appl Mech Eng 139: 315–347

    Article  MATH  Google Scholar 

  49. Oñate E, Idelsohn S (1988) A meshfree finite point method for advective-diffusive transport and fluid flow problems. Comput Mech 21: 283–292

    Google Scholar 

  50. Wu CP, Chiu KH, Wang YM (2008) A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells. Comput Mater Continua 8: 93–132

    Google Scholar 

  51. Wu CP, Chiu KH, Wang YM (2011) RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates. Compos Struct 93: 923–943

    Article  Google Scholar 

  52. Krongauz Y, Belytschko T (1996) Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput Methods Appl Mech Eng 131: 133–145

    Article  MATH  MathSciNet  Google Scholar 

  53. Timoshenko SP, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, National Cheng Kung University, Tainan, 70101, Taiwan, ROC

    Syuan-Mu Chen, Chih-Ping Wu & Yung-Ming Wang

Authors
  1. Syuan-Mu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chih-Ping Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yung-Ming Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chih-Ping Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, SM., Wu, CP. & Wang, YM. A Hermite DRK interpolation-based collocation method for the analyses of Bernoulli–Euler beams and Kirchhoff–Love plates. Comput Mech 47, 425–453 (2011). https://doi.org/10.1007/s00466-010-0552-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-010-0552-7

Keywords

Search

Navigation