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.
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References
Petyt M (2010) Introduction to finite element vibration analysis, 2nd edn. Cambridge University Press, New York
Zienkiewicz OC, Morgan K (2006) Finite elements and approximation, Dover Ed edn. Dover Publications, Mineola
Ohayon R, Soize C (1998) Structural acoustic and vibration: mechanical models, variational formulations and discretization. Academic Press, San Diego
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
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
Zehn MW, Saitov A (2003) How can spatially distributed uncertainties be included in FEA and in parameter estimation for model updating? Shock Vib 10:15–25
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
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
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
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
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
Vanmarcke E (2010) Random field: analysis and synthesis, 2nd revised and expanded. Word Scientific, Cambridge
Ghanem R, Spanos PD (2012) Stochastic finite elements: a spectral approach, Revised edn. Dover Publications, Minneola
Kaminski MM (2005) Computational mechanics of composite materials: sensitivity, randomness and multiscale behaviour. Springer, London
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
Ostoja-Starzewski M (2011) Stochastic finite elements: Where is the physics? Theor Appl Mech 38:379–396. https://doi.org/10.2298/TAM1104379O
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
Der Kiureghian A, Ke J-B (1988) The stochastic finite element method in structural reliability. Probab Eng Mech 3:83–91
Sudret B, Der Kiuereghian A (2000) Stochastic finite element methods and reliability: a state-of-art report. University of California, Berkeley
Haldar A, Mahadevan S (2000) Reliability assessment using stochastic finite element analysis. Wiley, New York
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
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
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
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
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
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
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
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
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
Wachulec M, Kirkegaard PH (2001) Energy flow in plate assembles by hierarchical version of finite element method. Department of Civil Engineering, Aalborg University, p 22
Wachulec M (2001) Power flow and structure-borne noise in medium frequency range. PhD Thesis, Department of Civil Engineering, Aalborg University
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
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
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
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
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
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
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
Rubinstein RY, Kroese DP (2007) Simulation and the Monte Carlo method, 2nd edn. Wiley, Hoboken
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
Soize C (2017) Uncertainty quantification—an accelerated course with advanced applications in computational engineering, 1st edn. Elsevier, Amsterdam
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
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)
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
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.
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Technical Editor: Marcelo A. Trindade.
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Fabro, A.T., Ferguson, N.S. & Mace, B.R. Structural vibration analysis with random fields using the hierarchical finite element method. J Braz. Soc. Mech. Sci. Eng. 41, 80 (2019). https://doi.org/10.1007/s40430-019-1579-0
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DOI: https://doi.org/10.1007/s40430-019-1579-0