Stochastic BEM with spectral approach in elastostatic and elastodynamic problems with geometrical uncertainty.

*(English)*Zbl 1182.74218Summary: The present paper proposes a method for stochastic problems that have uncertainty in the boundary geometry. The method is developed by applying the spectral stochastic approach to the boundary element method and is called the spectral stochastic boundary element method (SSBEM). In the SSBEM, the uncertainty in the boundary geometry is represented by the Karhunen-Loève expansion. It is shown that, by utilizing material derivative, variation of boundary element matrices associated with the geometrical fluctuation of the boundary can be approximated by the Taylor expansion. The solution is represented by a stochastic process expressed in the form of polynomial chaos expansion. The stochastic equation is then projected on a homogeneous chaos space. This procedure reduces a stochastic equation to an ordinary linear matrix equation that can be solved by conventional schemes. The SSBEM can estimate not only mean values and variances of the solutions but also their probability density functions. In order to examine the performance, the SSBEM is applied to two-dimensional elastostatic and elastodynamic problems with geometrical boundary uncertainty. Computation results of the SSBEM exhibit good agreement with those obtained by Monte Carlo simulation. The efficiency of the SSBEM is verified by comparison of their computation times.

##### MSC:

74S15 | Boundary element methods applied to problems in solid mechanics |

74S60 | Stochastic and other probabilistic methods applied to problems in solid mechanics |

74S25 | Spectral and related methods applied to problems in solid mechanics |

74B15 | Equations linearized about a deformed state (small deformations superposed on large) |

##### Keywords:

stochastic BEM; spectral stochastic approach; geometrical uncertainty; homogeneous chaos; polynomial chaos
PDF
BibTeX
XML
Cite

\textit{R. Honda}, Eng. Anal. Bound. Elem. 29, No. 5, 415--427 (2005; Zbl 1182.74218)

Full Text:
DOI

##### References:

[1] | Liu, P.L.; Kiureghian, D.A., Finite element reliability of geometrically nonlinear uncertain structures, J eng mech, ASCE, 117, 8, 1806-1825, (1991) |

[2] | Ghanem, R.; Brzakala, W., Stochastic finite-element analysis of soil layers with random interface, J eng mech, ASCE, 122, 4, (1996) |

[3] | Haldar, A.; Mahadevan, S., Reliability assessment using stochastic finite element analysis, (2000), Wiley New York |

[4] | Ghanem, R.G.; Spanos, P.D., Stochastic finite elements—a spectral approach, (1991), Springer New York · Zbl 0722.73080 |

[5] | Dasgupta, G.; Papusha, A.N.; Malsch, E., First order stochasticity in boundary geometry: a computer algebra BE development, Eng anal bound elements, 25, 741-751, (2001) · Zbl 0991.65126 |

[6] | Nakagaki, S.; Suzuki, K.; Hisada, T., Stochastic boundary element method applied to stress analysis, (), 439-448 |

[7] | Kaminski, M., Stochastic second-order BEM perturbation formulation, Eng anal bound elements, 23, 123-129, (1999) · Zbl 0958.74076 |

[8] | Kaljević, I.; Saigal, S., Stochastic boundary elements in elastostatics, Comput methods appl mech eng, 109, (1993) · Zbl 0846.73078 |

[9] | Burczyński, T.; Skrzypczyk, J., Theoretical and computational aspects of the stochastic boundary element method, Comput methods appl mech eng, 168, 321-344, (1999) · Zbl 0955.74071 |

[10] | Manolis, G.D.; Karakostas, C.Z., A Green’s function method to SH-wave motion in a random continuum, Eng anal bound elements, 27, 93-100, (2003) · Zbl 1080.74561 |

[11] | Honda, R., Wave propagation analysis in the random media by spectral stochastic FEM, J struct earthquake eng, JSCE, 689/I-57, 321-331, (2001), [in Japanese] |

[12] | Ghanem, R.G.; Kruger, R.M., Numerical solution of spectral stochastic finite element analysis, Comput methods appl mech eng, 129, 289-303, (1996) · Zbl 0861.73071 |

[13] | Ghanem, R., Stochastic finite elements with multiple random non-Gaussian properties, J eng mech, ASCE, 125, 1, (1999) |

[14] | Anders, M.; Hori, M., Stochastic finite element method for elasto-plastic body, Int J numer methods eng, 46, 1897-1916, (1999) · Zbl 0967.74058 |

[15] | Anders, M.; Hori, M., Three-dimensional stochastic finite element method for elasto-plastic bodies, Int J numer methods eng, 51, 449-478, (2001) · Zbl 1015.74055 |

[16] | Spanos, P.; Ghanem, R., Stochastic finite element expansion for random media, J eng mech, ASCE, 115, 5, 1035-1053, (1989) |

[17] | Wiener, N., The homogeneous chaos, Am J math, 60, 4, 897-936, (1938) · JFM 64.0887.02 |

[18] | Cameron, R.H.; Martin, W.T., The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann math, second ser, 48, 2, 385-392, (1947) · Zbl 0029.14302 |

[19] | Brebbia, C.A., The boundary element method for engineers, (1978), Wiley New York · Zbl 0414.65060 |

[20] | Brebbia, C.A.; Telles, J.C.F.; Wrobel, L.C., The boundary element techniques, theory and applications in engineering, (1984), Springer Berlin · Zbl 0556.73086 |

[21] | Choi, J.H.; Kwak, B.M., Shape design sensitivity analysis of elliptic problems in boundary integral equation formulation, Mech struct Mach, 16, 2, (1988) |

[22] | Zhao, Z., Shape design sensitivity and optimization using the boundary element method, (1991), Springer Berlin |

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.