Abstract
Finite cell method is known as a combination of finite element method and fictitious domain approach in order to reduce the difficulties associated with mesh generation so that it can successfully handle complex geometries. This study proposes a stochastic extension of finite cell method, as a novel computational framework, for uncertainty quantification of structures. For this purpose, stochastic finite cell method (SFCM) is developed as a new efficient method, including the features of finite cell method, for computational stochastic mechanics considering complicated geometries arising from computer-aided design (CAD). Firstly, finite cell method is formulated for solving the Fredholm integral equation of the second kind used for Karhunen-Loève expansion in order to decompose the random field within a physical domain having arbitrary boundaries. Then, the SFCM is formulated based on Karhunen-Loève and polynomial chaos expansions for the stochastic analysis. Several numerical examples consisting of benchmark problems are provided to demonstrate the efficiency, accuracy and capability of the proposed SFCM.
Similar content being viewed by others
References
Ghanem RG, Spanos PD (2003) Stochastic Finite elements: a spectral approach. Courier Dover Publications, Mineola
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
Kaminski M (2013) The stochastic perturbation method for computational mechanics. Wiley, Hoboken
Anders M, Hori M (2001) Three-dimensional stochastic finite element method for elasto-plastic bodies. Int J Numer Meth Eng 51:449–478. https://doi.org/10.1002/nme.165
Mishra S, Schwab C, Šukys J (2016) Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium. J Comput Phys 312:192–217. https://doi.org/10.1016/j.jcp.2016.02.014
Zakian P, Khaji N (2018) A stochastic spectral finite element method for wave propagation analyses with medium uncertainties. Appl Math Model 63:84–108. https://doi.org/10.1016/j.apm.2018.06.027
Zakian P, Khaji N (2019) A stochastic spectral finite element method for solution of faulting-induced wave propagation in materially random continua without explicitly modeled discontinuities. Comput Mech 64:1017–1048. https://doi.org/10.1007/s00466-019-01692-5
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
Khaji N, Zakian P (2017) Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method. Mech Adv Mater Struct 24:1030–1042. https://doi.org/10.1080/15376494.2016.1202359
Pokusiński B, Kamiński M (2019) Lattice domes reliability by the perturbation-based approaches vs. semi-analytical method. Comput Struct 221:179–192. https://doi.org/10.1016/j.compstruc.2019.05.012
Szafran J, Juszczyk K, Kamiński M (2020) Reliability assessment of steel lattice tower subjected to random wind load by the stochastic finite-element method. ASCE-ASME J Risk Uncertain Eng Syst Part A: Civil Eng 6:04020003. https://doi.org/10.1061/AJRUA6.0001040
Ghanem R, Dham S (1998) Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp Porous Media 32:239–262. https://doi.org/10.1023/A:1006514109327
Maitre OL, Knio OM (2010) Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer, Netherlands
Laz PJ, Browne M (2010) A review of probabilistic analysis in orthopaedic biomechanics. Proc Inst Mech Eng [H] 224:927–943. https://doi.org/10.1243/09544119jeim739
Arregui-Mena JD, Margetts L, Mummery PM (2016) Practical application of the stochastic finite element method. Archives Comput Methods Eng 23:171–190. https://doi.org/10.1007/s11831-014-9139-3
Zakian P, Khaji N (2016) A novel stochastic-spectral finite element method for analysis of elastodynamic problems in the time domain. Meccanica 51:893–920. https://doi.org/10.1007/s11012-015-0242-9
Komatitsch D, Tromp J (1999) Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys J Int 139:806–822. https://doi.org/10.1046/j.1365-246x.1999.00967.x
Komatitsch D, Vilotte J-P, Vai R, Castillo-Covarrubias JM, Sánchez-Sesma FJ (1999) The spectral element method for elastic wave equations—application to 2-D and 3-D seismic problems. Int J Numer Meth Eng 45:1139–1164. https://doi.org/10.1002/(sici)1097-0207(19990730)45:9%3c1139::aid-nme617%3e3.0.co;2-t
Zakian P, Khaji N, Kaveh A (2017) Graph theoretical methods for efficient stochastic finite element analysis of structures. Comput Struct 178:29–46. https://doi.org/10.1016/j.compstruc.2016.10.009
Bathe KJ (1996) Finite element procedures, 1st edn. Prentice Hall; 2nd ed KJ Bathe, Watertown, MA, 2014
Kaveh A (2013) Computational structural analysis and finite element methods. Springer, Switzerland
Kaveh A (2006) Optimal structural analysis, 2nd edn. Wiley, Chichester
Kaveh A (2013) Optimal analysis of structures by concepts of symmetry and regularity. Springer, Vienna
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. https://doi.org/10.1016/S0045-7825(96)01078-X
Liu GR (2009) Mesh free methods: moving beyond the finite element method. 2nd Edition, CRC Press, Boca Raton
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195. https://doi.org/10.1016/j.cma.2004.10.008
Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41:121–133. https://doi.org/10.1007/s00466-007-0173-y
Zhang L, Bathe KJ (2017) Overlapping finite elements for a new paradigm of solution. Comput Struct 187:64–76. https://doi.org/10.1016/j.compstruc.2017.03.008
Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2012) The finite cell method for bone simulations: verification and validation. Biomech Model Mechanobiol 11:425–437. https://doi.org/10.1007/s10237-011-0322-2
Schillinger D, Düster A, Rank E (2012) The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Meth Eng 89:1171–1202. https://doi.org/10.1002/nme.3289
Duczek S, Joulaian M, Düster A, Gabbert U (2014) Numerical analysis of Lamb waves using the finite and spectral cell methods. Int J Numer Meth Eng 99:26–53. https://doi.org/10.1002/nme.4663
Joulaian M, Duczek S, Gabbert U, Düster A (2014) Finite and spectral cell method for wave propagation in heterogeneous materials. Comput Mech 54:661–675. https://doi.org/10.1007/s00466-014-1019-z
Li K, Gao W, Wu D, Song C, Chen T (2018) Spectral stochastic isogeometric analysis of linear elasticity. Comput Methods Appl Mech Eng 332:157–190. https://doi.org/10.1016/j.cma.2017.12.012
Li K, Wu D, Gao W, Song C (2019) Spectral stochastic isogeometric analysis of free vibration. Comput Methods Appl Mech Eng 350:1–27. https://doi.org/10.1016/j.cma.2019.03.008
Mohammadi S (2012) XFEM fracture analysis of composites. Wiley, Hoboken
Spanos PD, Ghanem R (1989) Stochastic finite element expansion for random media. J Eng Mech 115:1035–1053. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:5(1035)
Oliveira SP, Azevedo JS (2014) Spectral element approximation of Fredholm integral eigenvalue problems. J Comput Appl Math 257:46–56. https://doi.org/10.1016/j.cam.2013.08.016
Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, New Jersey
Huang S, Mahadevan S, Rebba R (2007) Collocation-based stochastic finite element analysis for random field problems. Probab Eng Mech 22:194–205. https://doi.org/10.1016/j.probengmech.2006.11.004
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zakian, P. Stochastic finite cell method for structural mechanics. Comput Mech 68, 185–210 (2021). https://doi.org/10.1007/s00466-021-02026-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-021-02026-0