, Volume 54, Issue 1–2, pp 47–70 | Cite as

A study on the stress gradient reconstruction in finite elements problems with application of radial basis function networks

  • Giorgio Previati
  • Massimiliano Gobbi
  • Federico BalloEmail author


The recovery of the stress gradient in finite elements problems is a widely discussed topic with many applications in the design process. The stress gradient is related to the second derivative (Hessian) of the nodal displacements and numerical techniques are required for its calculation. Particular difficulties are encountered in the reconstruction of the stress gradient in the boundary regions of the domain. This is of particular concern in most applications, especially in mechanical components, where the maximum values of stresses are often located in these regions and the stress gradient has a strong influence on the fatigue life of the component. This paper presents a comparison between some already published, partially modified, recovery techniques and a different approach based on radial basis function networks. The aim of the paper is to compare the performances of the different approaches for a number of element types with particular focus on the boundary regions. Some examples of mechanical interest are considered.


Stress recovery Gradient recovery Hessian recovery Radial basis function 


Compliance with ethical standards

Conflict of interest

The authors declare that they don’t have conflict of interests.


  1. 1.
    ALTAIR (2014) Optistruct user’s guideGoogle Scholar
  2. 2.
    Benedetti A, de Miranda S, Ubertini F (2006) A posteriori error estimation based on the superconvergent recovery by compatibility in patches. Int J Numer Methods Eng 67:108–131CrossRefzbMATHGoogle Scholar
  3. 3.
    Björck Å (1996) Numerical methods for least squares problems. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefzbMATHGoogle Scholar
  4. 4.
    Blacker T, Belytschko T (1994) Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements. Int J Numer Methods Eng 37(3):517–536. MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Booromand B, Zienkievicz OC (1997) Recovery by equilibrium in patches (REP). Int J Numer Methods Eng 40:137–164MathSciNetCrossRefGoogle Scholar
  6. 6.
    Broomhead DS, Lowe D (1988) Multivariable functional interpolation and adaptive networks. Complex Syst 2:321–355MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen J, Li CJ (2014) A quadrilateral spline element for couple stress/strain gradient elasticity. Comput Struct 138:133–141ADSCrossRefGoogle Scholar
  8. 8.
    Duy NM, Cong TT (2003) Approximation of function and its derivatives using radial basis function networks. Appl Math Model 27(3):197–220. CrossRefzbMATHGoogle Scholar
  9. 9.
    Ettehad M, Abu Al-Rub RK (2015) On the numerical implementation of the higher-order strain gradient-dependent plasticity theory and its non-classical boundary conditions. Finite Elem Ana Des 93:50–59MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ferreira AJ, Roque CM, Carrera E, Cinefra M, Polit O (2013) Bending and vibration of laminated plates by a layerwise formulation and collocation with radial basis functions. Mech Adv Mater Struct 20(8):624–637. CrossRefGoogle Scholar
  11. 11.
    Gan X, Akin JE (2014) Super-convergent second derivative recovery for lower-order strain gradient plasticity. Comput Struct 135:118–127CrossRefGoogle Scholar
  12. 12.
    Grätsch T, Bathe KJ (2005) A posteriori error estimation techniques in practical finite element analysis. Comput Struct 72:235–265MathSciNetCrossRefGoogle Scholar
  13. 13.
    Guo H, Zhang Z, Zhao R (2014) Hessian recovery for finite element methods, pp pp 1–19. arXiv:14063108
  14. 14.
    Han CS, Wriggers P (2000) An h-adaptive method for elastoplastic shell problems. Comput Methods Appl Mech Eng 189:651–671ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Han CS, Ma A, Roters F, Raabe D (2007) A finite element approach with patch projection for strain gradient plasticity formulations. Int J Plast 23:690–710CrossRefzbMATHGoogle Scholar
  16. 16.
    Haykin S (1999) Neural networks: a comprehensive foundation. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  17. 17.
    Kahrobaiyan MH, Asghari M, Ahmadian MT (2013) Strain gradient beam element. Finite Elem Anal Des 68:63–75MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Karageorghis A, Chen CS, Smyrlis YS (2007) A matrix decomposition RBF algorithm: approximation of functions and their derivatives. Appl Numer Math 57(3):304–319. MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kim J, Bathe KJ (2013) The finite element method enriched by interpolation covers. Comput Struct 116:35–49CrossRefGoogle Scholar
  20. 20.
    Lee T, Park HC, Lee SW (1997) A superconvergent stress recovery technique with equilibrium constraint. Numer Methods Eng 40:1139–1160CrossRefzbMATHGoogle Scholar
  21. 21.
    Li B, Zhang Z (1999) Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements. Numer Methods Partial Differ Equ 15:151–167MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lin R, Zhan Z (2008) Natural superconvergent points in three-dimensional finite elements. SIAM J Numer Anal 46:1281–1297MathSciNetCrossRefGoogle Scholar
  23. 23.
    LMS SIEMENS (2015) LMS virtual lab durability theory manualGoogle Scholar
  24. 24.
    Lo SH, Lee CK (1998) On using different recovery procedures for the construction of smoothed stress in finite element method. Int J Numer Methods Eng 43:1223–1252CrossRefzbMATHGoogle Scholar
  25. 25.
    Martinez-Paneda E, del Busto S, Niordson CF, Betegon C (2016) Strain gradient plasticity modeling of hydrogen diffusion to the crack tip. Int J Hydrogen Energy 41:10265–10274CrossRefGoogle Scholar
  26. 26.
    Maturi DA, Ferreira AJ, Zenkour AM, Mashat DS (2013) Analysis of laminated shells by Murakami’s zig-zag theory and radial basis functions collocation. J Appl Math 2013:1–14. CrossRefzbMATHGoogle Scholar
  27. 27.
    Matusov JB (1995) Multicriteria optimisation and engineering. Chapman & Hall, New YorkGoogle Scholar
  28. 28.
    Mcdonald DB, Grantham WJ, Tabor WL, Murphy MJ (2007) Global and local optimization using radial basis function response surface models. Appl Math Model 31(10):2095–2110. CrossRefzbMATHGoogle Scholar
  29. 29.
    MSC (2014) Nastran 2014 linear static analysis user’s guideGoogle Scholar
  30. 30.
    Naga A, Zhang Z (2004) A posteriori error estimates based on the polynomial preserving recovery. SIAM J Numer Anal 42:1780–1800MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Papadopoulos IV (1996) Invariant formulation of a gradient dependent multiaxial high-cycle fatigue criterion. Eng Fract Mech 55:513–528CrossRefGoogle Scholar
  32. 32.
    Payen DJ, Bathe KJ (2012) A stress improvement procedure. Comput Struct 112–113:311–326CrossRefGoogle Scholar
  33. 33.
    Picasso M, Alauzet F, Borouchaki H, George PL (2011) A numerical study of some hessian recovery techniques on isotropic and anisotropic meshes. SIAM J Sci Comput 33:1058–1076MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Polizzotto C (2016) Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models. Int J Solids Struct 80:405–419. CrossRefGoogle Scholar
  35. 35.
    Pouliot B, Fortin M, Fortin A, Chamberland E (2013) On a new edge-based gradient recovery technique. Int J Numer Methods Eng 93:52–65MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rankovic V, Radulovic J (2011) Prediction of magnetic field near power lines by normalized radial basis function network. Adv Eng Softw 42:934–938CrossRefGoogle Scholar
  37. 37.
    Ródenas JJ, Tur M, Fuenmayor FJ, Vercher A (2007) Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR-C technique. Int J Numer Methods Eng 70(6):705–727. CrossRefzbMATHGoogle Scholar
  38. 38.
    Schneider R (2013) A review of anisotropic refinement methods for triangular meshes in FEM. In: Apel T, Steinbach O (eds) Advanced finite element methods and applications. Springer, BerlinGoogle Scholar
  39. 39.
    Simulia (2014) ABAQUS analysis user’s guide, ver. 6.14Google Scholar
  40. 40.
    Soh AK, Wanji C (2001) Finite element formulations of strain gradient theory for microstructures and the \(C^{0-1}\) patch test. Int J Plast 50:1369–1388Google Scholar
  41. 41.
    Stein E, Ohnimus S (1997) Equilibrium method for postprocessing and error estimation in the finite element method. Comput Assist Mech Eng Sci 4:645–666zbMATHGoogle Scholar
  42. 42.
    Ubertini F (2004) Patch recovery based on complementary energy. Int J Numer Methods Eng 59:1501–1538CrossRefzbMATHGoogle Scholar
  43. 43.
    Vallet MG, Manole CM, Dompierre J, Dufour S, Guibault F (2007) Numerical comparison of some Hessian recovery techniques. Int J Numer Methods Eng 72:987–1007MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Varias AG, Massih AR (2000) Simulation of hydrogen embrittlement in zirconium alloys under stress and temperature gradients. J Nucl Mater 279:273–285ADSCrossRefGoogle Scholar
  45. 45.
    Wang BP (2004) Parameter optimization in multiquadric response surface approximations. Struct Multidiscip Optim 26(3–4):219–223. CrossRefGoogle Scholar
  46. 46.
    Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 54:1623–1648CrossRefzbMATHGoogle Scholar
  47. 47.
    Wang JG, Liu GR (2002) On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Comput Methods Appl Mech Eng 191:2611–2630ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Wilberg NE (1997) Superconvergent patch recovery—a key to quality assessed FE solutions. Adv Eng Softw 28:85–95CrossRefGoogle Scholar
  49. 49.
    Xing J, Zheng G (2015) Stress field gradient analysis technique using lower-order \(C_0\) elements. Math Probl Eng 2015:1–12. Google Scholar
  50. 50.
    Zervos A, Papanastasiou P, Vardoulakis I (2001) A finite element displacement formulation for gradient elastoplasticity. Int J Plast 50:1369–1388zbMATHGoogle Scholar
  51. 51.
    Zervos A, Papanicolopulos SA, Vardoulakis I (2009) Two finite element discretizations for gradient elasticity. J Eng Mech 135:203–213CrossRefzbMATHGoogle Scholar
  52. 52.
    Zhan Z, Lin R (2003) Ultraconvergence of ZZ patch recovery at mesh symmetry points. Numer Math 95:781–801MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Zhang Z (2000) Ultraconvergence of the patch recovery technique. Math Comput 69:141–158ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Zhang Z, Naga A (2005) A new finite element gradient recovery method: superconvergence property. SIAM J Sci Comput 26:1192–1213MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Zienkiewicz O, Zhu JZ (1992) The superconvergence patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int J Numer Methods Eng 33:1331–1364CrossRefzbMATHGoogle Scholar
  56. 56.
    Zienkiewicz O, Zhu JZ (1992) The superconvergence patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. Int J Numer Methods Eng 33:1365–1382CrossRefzbMATHGoogle Scholar
  57. 57.
    Zienkiewicz OC, Taylor RL (2000) The finite element method: the basis, 5th edn. Butterworth Heinemann, OxfordzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Giorgio Previati
    • 1
  • Massimiliano Gobbi
    • 1
  • Federico Ballo
    • 1
    Email author
  1. 1.Department of Mechanical EngineeringPolitecnico di MilanoMilanItaly

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