zbMATH — the first resource for mathematics

Recursive approach for random response analysis using non-orthogonal polynomial expansion. (English) Zbl 1398.74338
Summary: Using non-orthogonal polynomial expansions, a recursive approach is proposed for the random response analysis of structures under static loads involving random properties of materials, external loads, and structural geometries. In the present formulation, non-orthogonal polynomial expansions are utilized to express the unknown responses of random structural systems. Combining the high-order perturbation techniques and finite element method, a series of deterministic recursive equations is set up. The solutions of the recursive equations can be explicitly expressed through the adoption of special mathematical operators. Furthermore, the Galerkin method is utilized to modify the obtained coefficients for enhancing the convergence rate of computational outputs. In the post-processing of results, the first- and second-order statistical moments can be quickly obtained using the relationship matrix between the orthogonal and the non-orthogonal polynomials. Two linear static problems and a geometrical nonlinear problem are investigated as numerical examples in order to illustrate the performance of the proposed method. Computational results show that the proposed method speeds up the convergence rate and has the same accuracy as the spectral finite element method at a much lower computational cost, also, a comparison with the stochastic reduced basis method shows that the new method is effective for dealing with complex random problems.

74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
65C05 Monte Carlo methods
42C15 General harmonic expansions, frames
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
74H50 Random vibrations in dynamical problems in solid mechanics
Full Text: DOI
[1] Shinozuka M (1972) Monte-Carlo solution of structural dynamics. Comput Struct 2: 855–874 · doi:10.1016/0045-7949(72)90043-0
[2] Joy C, Boyle PP, Tan KS (1996) Quasi-Monte Carlo methods in numerical finance. Manag Sci 42(6): 926–938 · Zbl 0880.90006 · doi:10.1287/mnsc.42.6.926
[3] Melchers RE (1989) Importance sampling in structural systems. Struct Saf 6: 3–10 · doi:10.1016/0167-4730(89)90003-9
[4] Melchers RE (1990) Radial importance sampling for structural reliability. J Eng Mech ASCE 114(8): 189–203 · doi:10.1061/(ASCE)0733-9399(1990)116:1(189)
[5] Schuëller GI (1989) On efficient computational schemes to calculate structural failure probabilities. In: Lin YK, Schueller GI (eds) Stochastic structural mechanics. Lecture notes in Engineering. Springer, Heidelberg
[6] Nie J, Ellingwood BR (2000) Directional methods for structural reliability analysis. Struct Saf 22: 233–249 · doi:10.1016/S0167-4730(00)00014-X
[7] Stein ML (1987) Large sample properties of simulations using latin hypercube sampling. Technometrics 29: 143–150 · Zbl 0627.62010 · doi:10.2307/1269769
[8] Proppe C, Pradlwarter HJ, Schuëller GI (2003) Equivalent linearization and Monte Carlo simulation in stochastic dynamics. Probab Eng Mech 18(1): 1–15 · doi:10.1016/S0266-8920(02)00037-1
[9] Spanos PD, Zeldin BA (1998) Monte Carlo treatment of random fields: a broad perspective. Appl Mech Rev 51(3): 219–237 · doi:10.1115/1.3098999
[10] Grigoriu M (1991) Statistically equivalent solutions of stochastic mechanics problems. J Eng Mech ASCE 117: 1906–1918 · Zbl 0738.62042 · doi:10.1061/(ASCE)0733-9399(1991)117:8(1906)
[11] Li J, Chen J-B (2005) Dynamic response and reliability analysis of structures with uncertain parameters. Int J Numer Meth Eng 62: 289–315 · Zbl 1179.74062 · doi:10.1002/nme.1204
[12] Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19: 393–408 · doi:10.1016/j.probengmech.2004.04.003
[13] Liu WK, Belytschko T, Mani A (1986) Random field finite elements. Int J Numer Meth Eng 23: 1831–1845 · Zbl 0597.73075 · doi:10.1002/nme.1620231004
[14] Huang B, Li QS, Shi WH, Wu Z (2007) Eigenvalues of structures with uncertain elastic boundary restraints. Appl Acoust 68: 350–363 · doi:10.1016/j.apacoust.2006.01.012
[15] Ghanem R, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, Heidelberg · Zbl 0722.73080
[16] Ghanem RG (1999) Ingredients for a general purpose stochastic finite element formulation. Comp Meth Appl Mech Eng 168: 19–34 · Zbl 0943.65008 · doi:10.1016/S0045-7825(98)00106-6
[17] Xiu D, Karniadakis GE (2002) Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput Methods Appl Mech Engrg 191: 4927–4948 · Zbl 1016.65001 · doi:10.1016/S0045-7825(02)00421-8
[18] Xiu D, Karniadakis GE (2003) Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 187: 137–167 · Zbl 1047.76111 · doi:10.1016/S0021-9991(03)00092-5
[19] Yamazaki F, Shinozuka M (1988) Neumann expansion for stochastic finite element analysis. J Eng Mech ASCE 114: 1335–1354 · doi:10.1061/(ASCE)0733-9399(1988)114:8(1335)
[20] Spanos PD, Ghanem RG (1989) Stochastic finite element expansion for random media. J Eng Mech ASCE 115: 1035–1053 · doi:10.1061/(ASCE)0733-9399(1989)115:5(1035)
[21] Zhao L, Chen Q (2000) Neumann dynamic stochastic finite element method of vibration for structures with stochastic parameters to random excitation. Comput Struct 77: 651–657 · doi:10.1016/S0045-7949(00)00019-5
[22] Grigoriu M (2002) Stochastic calculus: applications in science and engineering. Springer, Heidelberg · Zbl 1015.60001
[23] Chung DB, Gutiérrez1 MA, Graham-Bradyand LL, Lingen F-J (2005) Efficient numerical strategies for spectral stochastic finite element models. Int J Numer Meth Eng 64: 1334–1349 · Zbl 1113.74065
[24] Sachdeva SK, Nair PB, Keane AJ (2006) Hybridization of stochastic reduced basis methods with polynomial chaos expansions. Probab Eng Mech 21: 182–192 · doi:10.1016/j.probengmech.2005.09.003
[25] Sachdeva SK, Nair PB, Keane AJ (2006) Comparative study of projection schemes for stochastic finite element analysis. Comput Methods Appl Mech Eng 195: 2371–2392 · Zbl 1142.74047 · doi:10.1016/j.cma.2005.05.010
[26] Genz A, Monahan J (1999) A stochastic algorithm for high-dimensional integrals over unbounded regions with Gaussian weight. J Comput Appl Math 112: 71–81 · Zbl 0943.65034 · doi:10.1016/S0377-0427(99)00214-9
[27] Xu H, Rahman S (2004) A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int J Numer Meth Eng 61: 1992–2019 · Zbl 1075.74707 · doi:10.1002/nme.1135
[28] Elishakoff I, Ren YJ, Shinozuka M (1995) Some exact solutions for bending of beams with spatially stochastic stiffness. Int J Solids Struct 32: 2315–2327 · Zbl 0920.73075 · doi:10.1016/0020-7683(94)00257-W
[29] Elishakoff I, Impollonia N, Ren YJ (1999) New exact solutions for randomly loaded beams with stochastic flexibility. Int J Solids Struct 36(16): 2325–2340 · Zbl 0930.74028 · doi:10.1016/S0020-7683(98)00113-9
[30] Huang B, Liu W, Qu W (2004) Recursive stochastic finite element method. In: Proceedings of the 8th international symposium on structural engineering for young experts. Science Press, pp 155–160
[31] Zhang D, Lu Z (2004) An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loeve and polynomial expansions. J Comput Phys 194: 773–794 · Zbl 1101.76048 · doi:10.1016/j.jcp.2003.09.015
[32] Vanmarcke E, Shinozuka M, Nakagiri S, Schueller GI, Grigoriu M (1986) Random fields and stochastic finite elements. Struct Saf 3(3): 143–166 · doi:10.1016/0167-4730(86)90002-0
[33] Zhu WQ, Ren YJ, Wu WQ (1992) Stochastic FEM based on local averages of random vector fields. J Eng Mech ASCE 118(3): 496–511 · doi:10.1061/(ASCE)0733-9399(1992)118:3(496)
[34] Li CC, Kiureghian ADer (1993) Optimal discretization of random fields. J Eng Mech ASCE 119: 1136–1154 · doi:10.1061/(ASCE)0733-9399(1993)119:6(1136)
[35] Zhang J, Ellingwood B (1994) Orthogonal series expansion of random fields in reliability analysis. J Eng Mech ASCE 120(12): 2660–2677 · doi:10.1061/(ASCE)0733-9399(1994)120:12(2660)
[36] Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia · Zbl 1031.65046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.