Adaptive analysis of plates and laminates using natural neighbor Galerkin meshless method

Abstract

In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner–Mindlin plate theory. The interpolation functions employed here were developed by Sibson and based on natural neighbor coordinates. An adaptive refinement strategy based on recovery energy norm which is in turn based on natural neighbors is employed for analysis of plates. The present adaptive procedure is applied to classical plate problems subjected to in-plane loads. In addition to that the laminated composite plates with cutouts subjected to transverse loads are investigated. Influence of the location of the cutout and the boundary conditions of the plate on the results have been studied. The results obtained with present adaptive analysis are accurate at lower computational effort when compare to that of no adaptivity. Further, the adaptive analysis provided accurate magnitude of maximum stresses and their locations in the laminate plates with and without cutout subjected to transverse loads. Additionally, failure prone areas in the geometry of the plates subjected to loads are revealed with the adaptive analysis.

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References

  1. 1.

    Belytschko T, Lu YY, Gu Y (1994) Element free Galerkin method. Int J Numer Methods Eng 37:229–256. https://doi.org/10.1002/nme.1620370205

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Atluri SN, Zhu T (1998) A new meshless local Petrov–Galerkin approach in computational mechanics. Comput Mech 22(2):117–127. https://doi.org/10.1007/s004660050346

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and applications to non-spherical stars. Mon Not R Astron Soc 181:375–389. https://doi.org/10.1093/mnras/181.3.375

    Article  MATH  Google Scholar 

  4. 4.

    Bathe KJ, De S (2001) Towards an efficient meshless computational technique: the method of finite spheres. Eng Comput 18:170–192. https://doi.org/10.1108/02644400110365860

    Article  MATH  Google Scholar 

  5. 5.

    Sambridge MS, Braun J, McQueen H (1995) Geophysical parametrization and interpolation of irregular data using natural neighbours. Geophys J Int 122(1):837–857. https://doi.org/10.1111/j.1365-246X.1995.tb06841.x

    Article  Google Scholar 

  6. 6.

    Sibson R (1980) A vector identity for the Dirichlet tessellation. Math Proc Camb Philos Soc 87:151–155. https://doi.org/10.1017/S0305004100056589

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43(5):839–887. http://doi.org/10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R

  8. 8.

    Reddy JN (2004) Mechanics of laminated composite plates and shells theory and analysis, 2nd edn. CRC Press, Boca Raton

    Google Scholar 

  9. 9.

    Chate A, Makinen K (1994) Plane finite element for static and free vibration analysis of sandwich plates. Mech Compos Mater 30(2):168–176. https://doi.org/10.1007/BF00635849

    Article  Google Scholar 

  10. 10.

    Lee LJ, Fan YJ (1996) Bending and vibration analysis of composite sandwich plates. Comput Struct 60(1):103–112. https://doi.org/10.1016/0045-7949(95)00357-6

    Article  MATH  Google Scholar 

  11. 11.

    Reddy JN, Cho WC (1981) Non-linear bending of thick rectangular, laminated composite plates. Int J Non Linear Mech 16(3&4), 291–301. https://doi.org/10.1016/0020-7462(81)90042-1

  12. 12.

    Wung PM, Reddy JN (1991) A transverse deformation theory of laminated composite plates. Comput Struct 41(4):821–833. https://doi.org/10.1016/0045-7949(91)90191-N

    Article  MATH  Google Scholar 

  13. 13.

    Reddy JN (1984) A simple higher order theory for laminated composite plates. J Appl Mech T ASME 51:745–752. https://doi.org/10.1115/1.3167719

    Article  MATH  Google Scholar 

  14. 14.

    Reddy JN, Phan ND (1985) Stability and vibration of isotropic, orthotropic and laminated plates according to a higher order shear deformation theory. J Sound Vib 98(2):157–170. https://doi.org/10.1016/0022-460X(85)90383-9

    Article  MATH  Google Scholar 

  15. 15.

    Lim SP, Lee KH, Chow ST, Senthilnathan NR (1988) Linear and non-linear bending of shear deformable plates. Comput Struct 30(4):945–952. https://doi.org/10.1016/0045-7949(88)90132-0

    Article  MATH  Google Scholar 

  16. 16.

    Madhukar S, Singha MK (2013) Geometrically nonlinear finite element analysis of sandwich plates using normal deformation theory. Compos Struct 97:84–90. https://doi.org/10.1016/j.compstruct.2012.10.034

    Article  Google Scholar 

  17. 17.

    Nayak AK, Moy SSJ, Shenoi RA (2002) Free vibration analysis of composite sandwich based on Reddy’s higher order theory. Compos Part B Eng 33:505–519. https://doi.org/10.1016/S1359-8368(02)00035-5

    Article  Google Scholar 

  18. 18.

    Pandya BN, Kant T (1988) A refined higher order generally orthotropic C 0 plate bending element. Comput Struct 28(2):119–133. https://doi.org/10.1016/0045-7949(88)90031-4

    Article  MATH  Google Scholar 

  19. 19.

    Dinis LMJS., Jorge RMN, Belinha J (2007) Analysis of 3D solids using the natural neighbor radial point interpolation method. Comput Methods Appl Mech Eng 196:2009–2028. https://doi.org/10.1016/j.cma.2006.11.002

    Article  MATH  Google Scholar 

  20. 20.

    Dinis LMJS., Jorge RMN, Belinha J (2008) Analysis of plates and laminates using the natural neighbor radial point interpolation method. Eng Anal Bound Elem 32:267–279. https://doi.org/10.1016/j.enganabound.2007.08.006

    Article  MATH  Google Scholar 

  21. 21.

    Dinis LMJS., Jorge RMN, Belinha J (2010) A 3D shell-like approach using a natural neighbor meshless method: isotropic and orthotropic thin structures. Compos Struct 92:1132–1142. https://doi.org/10.1016/j.compstruct.2009.10.014

    Article  Google Scholar 

  22. 22.

    Dinis LMJS., Jorge RMN, Belinha J (2011) A natural neighbor meshless method with a 3D shell-like approach in the dynamic analysis of thin 3D structures. Thin Wall Struct 49:185–196. https://doi.org/10.1016/j.tws.2010.09.023

    Article  Google Scholar 

  23. 23.

    Leung AYT (1991) An unconstrained third order plate theory. Comput Struct 40(4):871–875. https://doi.org/10.1016/S0045-7949(03)00290-6

    Article  MATH  Google Scholar 

  24. 24.

    Dinis LMJS., Jorge RMN, Belinha J (2010) An unconstrained third order plate theory applied to functionally graded plates using a meshless method. Mech Adv Mater Struct 17:108–133. https://doi.org/10.1080/15376490903249925

    Article  Google Scholar 

  25. 25.

    Li SL, Liu KY, Long SY, Li GY (2011) The natural neighbor Petrov–Galerkin method for thick plates. Eng Anal Bound Elem 35:616–622. https://doi.org/10.1016/j.enganabound.2010.11.003

    MathSciNet  Article  Google Scholar 

  26. 26.

    Zienkiewicz OC, Taylor RL (2000) The finite element method, vol 1. BH Publications, Clarens, Oxford, England

  27. 27.

    Babuska I, Rheinboldt WC (1978) A posteriori error estimates for the finite element method. Int J Numer Methods Eng 12:1597–1615. https://doi.org/10.1002/nme.1620121010

    Article  MATH  Google Scholar 

  28. 28.

    Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 24:337–357. https://doi.org/10.1002/nme.1620240206

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Mohite PH, Upadhyay CS (2002) Local quality of smoothening based a posteriori error estimator for laminated plates under transverse loading. Comput Struct 80:1447–1488. https://doi.org/10.1016/S0045-7949(02)00099-8

    Article  Google Scholar 

  30. 30.

    Mohite PM, Upadhyay CS (2003) Focused adaptivity for laminated plates. Comput Struct 81:287–298. https://doi.org/10.1016/S0045-7949(02)00448-0

    Article  Google Scholar 

  31. 31.

    Mohite PM, Upadhyay CS (2006) Accurate computation of critical local quantities in composite laminated plates under transverse loading. Comput Struct 84:657–675. https://doi.org/10.1016/j.compstruc.2005.11.004

    Article  Google Scholar 

  32. 32.

    Mohite PM, Upadhyay CS (2007) Region by region modeling of laminated composite plates. Comput Struct 85:1808–1827. https://doi.org/10.1016/j.compstruc.2007.04.005

    Article  Google Scholar 

  33. 33.

    Tabarraei A, Sukumar N (2005) Adaptive computation on conforming quadtree meshes. Finite Elem Anal Des 41:686–702. https://doi.org/10.1016/j.finel.2004.08.002

    Article  Google Scholar 

  34. 34.

    Ullah Z, Augarde CE (2013) Finite deformation elasto-plastic modeling using an adaptive meshless method. Comput Struct 118:39–52. https://doi.org/10.1016/j.compstruc.2012.04.001

    Article  Google Scholar 

  35. 35.

    Cai Y, Zhu H (2004) A meshless local natural neighbor interpolation method for stress analysis of solids. Eng Anal Bound Elem 28:607–613. https://doi.org/10.1016/j.enganabound.2003.10.001

    Article  MATH  Google Scholar 

  36. 36.

    Yvonnet J, Coffignal G, Ryckelynck D, Lorong P, Chinesta F (2006) A simple error indicator for meshfree methods based on natural neighbors. Comput Struct 84:1301–1312. https://doi.org/10.1016/j.compstruc.2006.04.002

    MathSciNet  Article  Google Scholar 

  37. 37.

    Madhukar S, Rajagopal A (2014) Meshless natural neighbor Galerkin method for the bending and vibration analysis of composite plates. Compos Struct 111:138–146. https://doi.org/10.1016/j.compstruct.2013.12.023

    Article  Google Scholar 

  38. 38.

    Mohite PM, Upadhyay CS (2015) Adaptive finite element based shape optimization in laminated composite plates. Comput Struct 153:19–35. https://doi.org/10.1016/j.compstruc.2015.02.020

    Article  Google Scholar 

  39. 39.

    Stein E, Rüter M, Ohnimus S (2004) Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends. Int J Numer Methods Eng 60(1):103–38. https://doi.org/10.1002/nme.956

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Stein E, Rüter M, Ohnimus S (2007) Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity. Comput Methods Appl Mech Eng 196(37):3598–3613. https://doi.org/10.1016/j.cma.2006.10.032

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Rüter M, Stein E (2006) Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Comput Methods Appl Mech Eng 15(4):251–278. https://doi.org/10.1016/j.cma.2004.05.032 195) .

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Roylance D (2001) Introduction to fracture mechanics. Massachusetts Institute of Technology, Cambridge, pp 1–2

    Google Scholar 

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Correspondence to Madhukar Somireddy.

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Kumar, B., Somireddy, M. & Rajagopal, A. Adaptive analysis of plates and laminates using natural neighbor Galerkin meshless method. Engineering with Computers 35, 201–214 (2019). https://doi.org/10.1007/s00366-018-0593-7

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Keywords

  • Meshless methods
  • Natural neighbors
  • Adaptive strategy
  • Laminates
  • Plate theory