×

A multi-fidelity boundary element method for structural reliability analysis with higher-order sensitivities. (English) Zbl 1464.74300

Summary: A novel multi-fidelity modelling methodology for structural reliability analysis using the Boundary Element Method (BEM) with an Implicit Differentiation Method (IDM) is presented. Reliability analyses are conducted with methods such as Monte Carlo Simulation (MCS) and the First-Order Reliability Method (FORM). The higher-order sensitivities of the elastostatic Boundary Element Method equations with respect to changes in several geometric variables have been derived for the first time for use with the IDM for the purpose of conducting reliability analyses with the Second-Order Reliability Method (SORM), a more accurate alternative to FORM for problems with non-linear limit state functions. Multi-fidelity formulations involving the IDM have also been derived for the first time, making use of the metamodelling technique Kriging. The use of multi-fidelity modelling enables the creation of a model that has similar accuracy to a high-fidelity model, but with a computational cost similar to that of a low-fidelity model. By combining the accuracy of the IDM with the efficiency of multi-fidelity modelling the proposed methodology has the capability to be very effective when used for structural reliability analysis. The IDM is validated through a numerical example for which the analytical solution is known. A further two numerical examples featuring an I-beam section and a triangular support bracket with a large number of variables are also investigated. Results show that the employed multi-fidelity models were up to 6000 times faster in terms of CPU-time than the high-fidelity model, while also providing probabilities of failure that were up to 2225 times more accurate than the low-fidelity model. Overall, it has been shown that the use of the proposed IDM/multi-fidelity modelling methodology significantly improved the efficiency and accuracy of the above reliability analysis techniques when applied to complex problems involving a large number of random variables under high levels of uncertainty.

MSC:

74S15 Boundary element methods applied to problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
74S10 Finite volume methods applied to problems in solid mechanics

Software:

DACE
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Su, C.; Zhao, S.; Ma, H., Reliability analysis of plane elasticity problems by stochastic spline fictitious boundary element method, Eng Anal Bound Elem, 36, 2, 118-124 (2012) · Zbl 1245.74003
[2] X. Du, Probabilistic engineering design, 2005, http://web.mst.edu/ dux/repository/me360/me360_presentation.html; X. Du, Probabilistic engineering design, 2005, http://web.mst.edu/ dux/repository/me360/me360_presentation.html
[3] Haldar, A.; Mahadevan, S., Probability, reliability and statistical methods in engineering design (1999), John Wiley and Sons
[4] Melchers, R., Structural reliability analysis and prediction (1999), John Wiley and Sons
[5] Huang, X.; Aliabadi, M. H., A boundary element method for structural reliability, Key Eng Mater, 627, 453-456 (2015)
[6] Morse, L.; Sharif Khodaei, Z.; Aliabadi, M. H., Multi-fidelity modeling-based structural reliability analysis with the boundary element method, J Multiscale Model, 8, 3 (2017)
[7] Leonel, E. D.; Venturini, W. S., Probabilistic fatigue crack growth using BEM and reliability algorithms, WIT Transactions on Modelling and Simulation, 3-14 (2011)
[8] Huang, X.; Aliabadi, M. H., Probabilistic fracture mechanics by the boundary element method, Int J Fract, 171, 1, 51-64 (2011) · Zbl 1283.74061
[9] Leonel, E. D.; Chateauneuf, A.; Venturini, W. S., Probabilistic crack growth analyses using a boundary element model: applications in linear elastic fracture and fatigue problems, Eng Anal Bound Elem, 36, 6, 944-959 (2012) · Zbl 1351.74117
[10] Oliveira, H. L.; Chateauneuf, A.; Leonel, E. D., Probabilistic mechanical modelling of concrete creep based on the boundary element method, Adv Struct Eng, 22, 2, 337-348 (2018)
[11] Su, C.; Xu, J., Reliability analysis of reissner plate bending problems by stochastic spline fictitious boundary element method, Eng Anal Bound Elem, 51, 37-43 (2015) · Zbl 1403.74232
[12] Oliveira, H. L.; Chateauneuf, A.; Leonel, E. D., Boundary element method applied to decision-making problems involving geometric variabilities in topology optimization, Eng Anal Bound Elem, 85, 116-126 (2017) · Zbl 1403.74206
[13] Huang, X., Proabilistic fracture mechanics by boundary element method (2010), Thesis
[14] Won Kim, D.; Man Kwak, B., Reliability-based shape optimization of two-dimensional elastic problems using BEM, Comput Struct, 60, 5, 743-750 (1995) · Zbl 0919.73096
[15] Iott, J.; Haftka, R. T.; Adelman, H. M., Selecting step sizes in sensitivity analysis by finite difference, Report (1985), National Aeronautics and Space Administration
[16] Aliabadi, M. H., The Boundary Element Method: Applications in solids and structures, 2 (2002), John Wiley and Sons · Zbl 0994.74003
[17] Yu, L.; Das, P. K.; Zheng, Y., A response surface approach to fatigue reliability of ship structures, Ships Offshore Struct, 4, 3, 253-259 (2009)
[18] Gaspar, B.; Bucher, C.; Guedes Soares, C., Reliability analysis of plate elements under uniaxial compression using an adaptive response surface approach, Ships Offshore Struct, 10, 2, 145-161 (2014)
[19] Hassanien, S.; Kainat, M.; Adeeb, S.; Langer, D., On the use of surrogate models in reliability-based analysis of dented pipes, Proceedings of the 11th international pipeline conference (2016)
[21] Su, G.; Peng, L.; Hu, L., A gaussian process-based dynamic surrogate model for complex engineering structural reliability analysis, Struct Safety, 68, 97-109 (2017)
[22] Sun, Z.; Wang, J.; Li, R.; Tong, C., Lif: A new kriging based learning function and its application to structural reliability analysis, Reliab Eng Syst Saf, 157, 152-165 (2017)
[23] Papadrakakis, M.; Papadopoulos, V.; Lagaros, N. D., Structural reliability analysis of elastic-plastic structures using neural networks and monte carlo simulation, Comput Methods Appl Mech Eng, 136, 1-2, 145-163 (1996) · Zbl 0893.73079
[24] Papadrakakis, M.; Lagaros, N. D., Reliability-based structural optimization using neural networks and monte carlo simulation, Comput Methods Appl Mech Eng, 191, 32, 3491-3507 (2002) · Zbl 1101.74377
[25] T. Simpson, V. Toropov, V. Balabanov, F. Viana, Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come - or not (2008). 10.2514/6.2008-5802; T. Simpson, V. Toropov, V. Balabanov, F. Viana, Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come - or not (2008). 10.2514/6.2008-5802
[26] Bhosekar, A.; Ierapetritou, M., Advances in surrogate based modeling, feasibility analysis, and optimization: a review, Comput Chem Eng, 108, 250-267 (2018)
[29] Keane, A. J.; Sóbester, A.; Forrester, A. I.J., Multi-fidelity optimization via surrogate modelling, Proc R Soc A: Math Phys Eng Sci, 463, 2088, 3251-3269 (2007) · Zbl 1142.90489
[30] Perdikaris, P.; Venturi, D.; Royset, J. O.; Karniadakis, G. E., Multi-fidelity modelling via recursive co-Kriging and Gaussian-Markov random fields, Proc Math Phys Eng Sci, 471, 2179, 20150018 (2015)
[31] Rackwitz, R.; Fiessler, B., Structural reliability under combined random load sequences, Comput Struct, 9, 5, 484-494 (1978) · Zbl 0402.73071
[32] Breitung, K., Asymptotic approximations for multinormal integrals, J Eng Mech, 110, 3, 357-366 (1984) · Zbl 0571.73100
[33] Rackwitz, R., Reliability analysis - a review and some perspectives, Struct Saf, 23, 4, 365-395 (2001)
[34] F. Viana, Things you wanted to know about the latin hypercube design and were afraid to ask, in: Proceedings of the 10th world congress on structural and multidisciplinary optimization.; F. Viana, Things you wanted to know about the latin hypercube design and were afraid to ask, in: Proceedings of the 10th world congress on structural and multidisciplinary optimization.
[35] Simpson, T., Comparison of Response Surface and kriging Models in the Mutlidisiplinary Design or and Aerospike Nozzle, Report (1998), Institute for Computer Applications in Science and Engineering
[36] S.N. Lophaven, H.B. Nielsen, J. Sondergaard, Dace: A matlab Kriging toolbox, 2011, http://www2.imm.dtu.dk/projects/dace/; S.N. Lophaven, H.B. Nielsen, J. Sondergaard, Dace: A matlab Kriging toolbox, 2011, http://www2.imm.dtu.dk/projects/dace/
[37] Martin, J. D.; Simpson, T. W., Use of Kriging models to approximate deterministic computer models, AIAA J, 43, 4, 853-863 (2005)
[38] Benham, P. P.; Crawford, R. J.; Armstrong, C. G., Mechanics of engineering materials (1996), Prentice Hall
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.