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Structural vibration analysis with random fields using the hierarchical finite element method

  • A. T. FabroEmail author
  • N. S. Ferguson
  • B. R. Mace
Technical Paper
  • 17 Downloads

Abstract

Element-based techniques, like the finite element method, are the standard approach in industry for low-frequency applications in structural dynamics. However, mesh requirements can significantly increase the computational cost for increasing frequencies. In addition, randomness in system properties starts to play a significant role and its inclusion in the model further increases the computational cost. In this paper, a hierarchical finite element formulation is presented which incorporates spatially random properties. Polynomial and trigonometric hierarchical functions are used in the element formulation. Material and geometrical spatially correlated randomness are represented by the Karhunen–Loève expansion, a series representation for random fields. It allows the element integration to be performed only once for each term of the series which has benefits for a sampling scheme and can be used for non-Gaussian distributions. Free vibration and forced response statistics are calculated using the proposed approach. Compared to the standard h-version, the hierarchical finite element approach produces smaller mass and stiffness matrices, without changing the number of nodes of the element, and tends to be computationally more efficient. These are key factors not only when considering solutions for higher frequencies but also in the calculation of response statistics using a sampling method such as Monte Carlo simulation.

Keywords

Hierarchical finite element Random field Karhunen–Loève Structural vibration 

Notes

Acknowledgements

The authors gratefully acknowledge the financial support of the Brazilian National Council of Research (CNPq) Process number 445773/2014-6, the Federal District Research Foundation (FAPDF) Process number 0193001040/2015 and the Royal Society for the Newton International Exchanges Fund reference number IE140616.

References

  1. 1.
    Petyt M (2010) Introduction to finite element vibration analysis, 2nd edn. Cambridge University Press, New YorkCrossRefGoogle Scholar
  2. 2.
    Zienkiewicz OC, Morgan K (2006) Finite elements and approximation, Dover Ed edn. Dover Publications, MineolazbMATHGoogle Scholar
  3. 3.
    Ohayon R, Soize C (1998) Structural acoustic and vibration: mechanical models, variational formulations and discretization. Academic Press, San DiegoGoogle Scholar
  4. 4.
    Deraemaeker A, Babuška I, Bouillard P (1999) Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int J Numer Methods Eng 46:471–499.  https://doi.org/10.1002/(sici)1097-0207(19991010)46:4%3c471:aid-nme684%3e3.0.co;2-6 CrossRefzbMATHGoogle Scholar
  5. 5.
    Hinke L, Pichler L, Pradlwarter HJ, Mace BR, Waters TP (2011) Modelling of spatial variations in vibration analysis with application to an automotive windshield. Finite Elem Anal Des 47:55–62.  https://doi.org/10.1016/j.finel.2010.07.013 CrossRefGoogle Scholar
  6. 6.
    Zehn MW, Saitov A (2003) How can spatially distributed uncertainties be included in FEA and in parameter estimation for model updating? Shock Vib 10:15–25CrossRefGoogle Scholar
  7. 7.
    Gangadhar M, Zehn MW (2007) A methodology to model spatially distributed uncertainties in thin-walled structures. ZAMM J Appl Math Mechanics Z Für Angew Math Mech 87:360–376.  https://doi.org/10.1002/zamm.200610321 CrossRefzbMATHGoogle Scholar
  8. 8.
    Guilleminot J, Soize C, Kondo D, Binetruy C (2008) Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites. Int J Solids Struct 45:5567–5583.  https://doi.org/10.1016/j.ijsolstr.2008.06.002 CrossRefzbMATHGoogle Scholar
  9. 9.
    Fabro AT, Ferguson NS, Gan JM, Mace BR, Bickerton S, Battley M (2015) Estimation of random field material properties for chopped fibre composites and application to vibration modelling. Compos Struct 125:1–12.  https://doi.org/10.1016/j.compstruct.2015.01.036 CrossRefGoogle Scholar
  10. 10.
    Machado MR, Adhikari S, Dos Santos JMC, Arruda JRF (2018) Estimation of beam material random field properties via sensitivity-based model updating using experimental frequency response functions. Mech Syst Signal Process 102:180–197.  https://doi.org/10.1016/j.ymssp.2017.08.039 CrossRefGoogle Scholar
  11. 11.
    Hu Z, Mahadevan S (2017) Uncertainty quantification and management in additive manufacturing: current status, needs, and opportunities. Int J Adv Manuf Technol 93:2855–2874.  https://doi.org/10.1007/s00170-017-0703-5 CrossRefGoogle Scholar
  12. 12.
    Vanmarcke E (2010) Random field: analysis and synthesis, 2nd revised and expanded. Word Scientific, CambridgeCrossRefGoogle Scholar
  13. 13.
    Ghanem R, Spanos PD (2012) Stochastic finite elements: a spectral approach, Revised edn. Dover Publications, MinneolazbMATHGoogle Scholar
  14. 14.
    Kaminski MM (2005) Computational mechanics of composite materials: sensitivity, randomness and multiscale behaviour. Springer, LondonGoogle Scholar
  15. 15.
    Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21:112–132.  https://doi.org/10.1016/j.probengmech.2005.07.007 CrossRefGoogle Scholar
  16. 16.
    Ostoja-Starzewski M (2011) Stochastic finite elements: Where is the physics? Theor Appl Mech 38:379–396.  https://doi.org/10.2298/TAM1104379O MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Savvas D, Stefanou G, Papadrakakis M (2016) Determination of RVE size for random composites with local volume fraction variation. Comput Methods Appl Mech Eng 305:340–358.  https://doi.org/10.1016/j.cma.2016.03.002 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Der Kiureghian A, Ke J-B (1988) The stochastic finite element method in structural reliability. Probab Eng Mech 3:83–91CrossRefGoogle Scholar
  19. 19.
    Sudret B, Der Kiuereghian A (2000) Stochastic finite element methods and reliability: a state-of-art report. University of California, BerkeleyGoogle Scholar
  20. 20.
    Haldar A, Mahadevan S (2000) Reliability assessment using stochastic finite element analysis. Wiley, New YorkGoogle Scholar
  21. 21.
    Schuëller GI, Pradlwarter HJ (2009) Uncertain linear systems in dynamics: retrospective and recent developments by stochastic approaches. Eng Struct 31:2507–2517.  https://doi.org/10.1016/j.engstruct.2009.07.005 CrossRefGoogle Scholar
  22. 22.
    Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198:1031–1051.  https://doi.org/10.1016/j.cma.2008.11.007 CrossRefzbMATHGoogle Scholar
  23. 23.
    Arregui-Mena JD, Margetts L, Mummery PM (2016) Practical application of the stochastic finite element method. Arch Comput Methods Eng 23:171–190.  https://doi.org/10.1007/s11831-014-9139-3 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ostoja-Starzewski M, Woods A (2003) Spectral finite elements for vibrating rods and beams with random field properties. J Sound Vib 268:779–797.  https://doi.org/10.1016/S0022-460X(03)00037-3 CrossRefGoogle Scholar
  25. 25.
    Adhikari S (2011) Doubly spectral stochastic finite-element method for linear structural dynamics. J Aerosp Eng 24:264–276.  https://doi.org/10.1061/(ASCE)AS.1943-5525.0000070 CrossRefGoogle Scholar
  26. 26.
    Yang L, Zhou Y, Zhou J, Wang M (2013) Hierarchical stochastic finite element method for structural analysis. Acta Mech Solida Sin 26:189–196.  https://doi.org/10.1016/S0894-9166(13)60018-X CrossRefGoogle Scholar
  27. 27.
    Bardell NS (1992) The free vibration of skew plates using the hierarchical finite element method. Comput Struct 45:841–874.  https://doi.org/10.1016/0045-7949(92)90044-Z CrossRefGoogle Scholar
  28. 28.
    Houmat A (1997) An alternative hierarchical finite element formulation applied to plate vibrations. J Sound Vib 206:201–215.  https://doi.org/10.1006/jsvi.1997.1076 CrossRefGoogle Scholar
  29. 29.
    Han W, Petyt M (1996) Linear vibration analysis of laminated rectangular plates using the hierarchical finite element method—I. Free vibration analysis. Comput Struct 61:705–712.  https://doi.org/10.1016/0045-7949(95)00379-7 CrossRefzbMATHGoogle Scholar
  30. 30.
    Wachulec M, Kirkegaard PH (2001) Energy flow in plate assembles by hierarchical version of finite element method. Department of Civil Engineering, Aalborg University, p 22Google Scholar
  31. 31.
    Wachulec M (2001) Power flow and structure-borne noise in medium frequency range. PhD Thesis, Department of Civil Engineering, Aalborg UniversityGoogle Scholar
  32. 32.
    Bardell NS, Gange GJ (1994) An efficient static analysis of sandwich beams. Compos Struct 29:107–117.  https://doi.org/10.1016/0263-8223(94)90040-X CrossRefGoogle Scholar
  33. 33.
    Yu Z, Guo X, Chu F (2010) A multivariable hierarchical finite element for static and vibration analysis of beams. Finite Elem Anal Des 46:625–631.  https://doi.org/10.1016/j.finel.2010.03.002 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Giunta G, Belouettar S, Nasser H, Kiefer-Kamal EH, Thielen T (2015) Hierarchical models for the static analysis of three-dimensional sandwich beam structures. Compos Struct 133:1284–1301.  https://doi.org/10.1016/j.compstruct.2015.08.049 CrossRefGoogle Scholar
  35. 35.
    Boukhalfa A, Hadjoui A (2010) Free vibration analysis of an embarked rotating composite shaft using the hp-version of the FEM. Lat Am J Solids Struct 7:105–141.  https://doi.org/10.1590/S1679-78252010000200002 CrossRefGoogle Scholar
  36. 36.
    Han W, Petyt M (1996) Linear vibration analysis of laminated rectangular plates using the hierarchical finite element method—II. Forced vibration analysis. Comput Struct 61:713–724.  https://doi.org/10.1016/0045-7949(96)00213-1 CrossRefzbMATHGoogle Scholar
  37. 37.
    Giunta G, Crisafulli D, Belouettar S, Carrera E (2011) Hierarchical theories for the free vibration analysis of functionally graded beams. Compos Struct 94:68–74.  https://doi.org/10.1016/j.compstruct.2011.07.016 CrossRefGoogle Scholar
  38. 38.
    Hui Y, Giunta G, Belouettar S, Huang Q, Hu H, Carrera E (2017) A free vibration analysis of three-dimensional sandwich beams using hierarchical one-dimensional finite elements. Compos Part B Eng 110:7–19.  https://doi.org/10.1016/j.compositesb.2016.10.065 CrossRefGoogle Scholar
  39. 39.
    Rubinstein RY, Kroese DP (2007) Simulation and the Monte Carlo method, 2nd edn. Wiley, HobokenCrossRefGoogle Scholar
  40. 40.
    Betz W, Papaioannou I, Straub D (2014) Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion. Comput Methods Appl Mech Eng 271:109–129.  https://doi.org/10.1016/j.cma.2013.12.010 CrossRefzbMATHGoogle Scholar
  41. 41.
    Huang SP, Quek ST, Phoon KK (2001) Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes. Int J Numer Methods Eng 52:1029–1043.  https://doi.org/10.1002/nme.255 CrossRefzbMATHGoogle Scholar
  42. 42.
    Phoon KK, Huang SP, Quek ST (2002) Simulation of second-order processes using Karhunen–Loeve expansion. Comput Struct 80:1049–1060.  https://doi.org/10.1016/S0045-7949(02)00064-0 MathSciNetCrossRefGoogle Scholar
  43. 43.
    Phoon KK, Huang HW, Quek ST (2005) Simulation of strongly non-Gaussian processes using Karhunen–Loeve expansion. Probab Eng Mech 20:188–198.  https://doi.org/10.1016/j.probengmech.2005.05.007 CrossRefGoogle Scholar
  44. 44.
    Li LB, Phoon KK, Quek ST (2007) Comparison between Karhunen–Loève expansion and translation-based simulation of non-Gaussian processes. Comput Struct 85:264–276.  https://doi.org/10.1016/j.compstruc.2006.10.010 CrossRefGoogle Scholar
  45. 45.
    Charmpis DC, Schuëller GI, Pellissetti MF (2007) The need for linking micromechanics of materials with stochastic finite elements: a challenge for materials science. Comput Mater Sci 41:27–37.  https://doi.org/10.1016/j.commatsci.2007.02.014 CrossRefGoogle Scholar
  46. 46.
    Bardell NS (1991) Free vibration analysis of a flat plate using the hierarchical finite element method. J Sound Vib 151:263–289.  https://doi.org/10.1016/0022-460X(91)90855-E CrossRefGoogle Scholar
  47. 47.
    Allaix DL, Carbone VI (2009) Discretization of 2D random fields: a genetic algorithm approach. Eng Struct 31:1111–1119.  https://doi.org/10.1016/j.engstruct.2009.01.008 CrossRefGoogle Scholar
  48. 48.
    Shang S, Yun GJ (2013) Stochastic finite element with material uncertainties: implementation in a general purpose simulation program. Finite Elem Anal Des 64:65–78.  https://doi.org/10.1016/j.finel.2012.10.001 MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ma Y, Zhang Y, Kennedy D (2016) Energy flow analysis of mid-frequency vibration of coupled plate structures with a hybrid analytical wave and finite element model. Comput Struct 175:1–14.  https://doi.org/10.1016/j.compstruc.2016.06.007 CrossRefGoogle Scholar
  50. 50.
    Wester ECN, Mace BR (2005) Wave component analysis of energy flow in complex structures—part I: a deterministic model. J Sound Vib 285:209–227.  https://doi.org/10.1016/j.jsv.2004.08.025 CrossRefGoogle Scholar
  51. 51.
    Wester ECN, Mace BR (2005) Wave component analysis of energy flow in complex structures—part II: ensemble statistics. J Sound Vib 285:229–250.  https://doi.org/10.1016/j.jsv.2004.08.026 CrossRefGoogle Scholar
  52. 52.
    Legault J, Woodhouse J, Langley RS (2014) Statistical energy analysis of inhomogeneous systems with slowly varying properties. J Sound Vib 333:7216–7232.  https://doi.org/10.1016/j.jsv.2014.08.026 CrossRefGoogle Scholar
  53. 53.
    Fabro AT, Ferguson NS, Jain T, Halkyard R, Mace BR (2015) Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability. J Sound Vib 343:20–48.  https://doi.org/10.1016/j.jsv.2015.01.013 CrossRefGoogle Scholar
  54. 54.
    Li C, Der Kiureghian A (1993) Optimal discretization of random fields. J Eng Mech 119:1136–1154.  https://doi.org/10.1061/(ASCE)0733-9399(1993)119:6(1136) CrossRefGoogle Scholar
  55. 55.
    Koutsourelakis PS, Pradlwarter HJ, Schuëller GI (2004) Reliability of structures in high dimensions, part I: algorithms and applications. Probab Eng Mech 19:409–417.  https://doi.org/10.1016/j.probengmech.2004.05.001 CrossRefGoogle Scholar
  56. 56.
    Han W, Petyt M (1997) Geometrically nonlinear vibration analysis of thin, rectangular plates using the hierarchical finite element method—I: the fundamental mode of isotropic plates. Comput Struct 63:295–308.  https://doi.org/10.1016/S0045-7949(96)00345-8 CrossRefzbMATHGoogle Scholar
  57. 57.
    Houmat A (2012) Nonlinear free vibration of a composite rectangular specially-orthotropic plate with variable fiber spacing. Compos Struct 94:3029–3036.  https://doi.org/10.1016/j.compstruct.2012.05.006 CrossRefGoogle Scholar
  58. 58.
    Soize C (2017) Uncertainty quantification—an accelerated course with advanced applications in computational engineering, 1st edn. Elsevier, AmsterdamzbMATHGoogle Scholar
  59. 59.
    Schevenels M, Lombaert G, Degrande G (2004) Application of the stochastic finite element method for Gaussian and non-Gaussian systems. In: Proceedings of ISMA2004, Leuven. http://bwk.kuleuven.be/apps/bwm/papers/scheip04a.pdf. Accessed 31 Jan 2017
  60. 60.
    Olsson AMJ, Sandberg GE (2002) Latin hypercube sampling for stochastic finite element analysis. J Eng Mech 128:121–125.  https://doi.org/10.1061/(asce)0733-9399(2002)128:1(121) CrossRefGoogle Scholar
  61. 61.
    Zeldin BA, Spanos PD (1998) On random field discretization in stochastic finite elements. J Appl Mech 65:320–327.  https://doi.org/10.1115/1.2789057 CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Departmento de Engenharia Mecânica, Faculdade de TecnologiaUniversidade de Brasília – Campus Darcy RibeiroBrasíliaBrazil
  2. 2.ISVRUniversity of SouthamptonSouthamptonUK
  3. 3.Acoustics Research Centre, Department of Mechanical EngineeringUniversity of AucklandAucklandNew Zealand

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