An ellipsoidal Newton’s iteration method of nonlinear structural systems with uncertain-but-bounded parameters. (English) Zbl 1506.65195
Summary: This paper presents an ellipsoidal Newton’s iteration method for predicting the response of nonlinear structural systems with uncertain-but-bounded parameters. In the study, the uncertainty in parameters is expressed in terms of an ellipsoid set in an appropriate vector space and the bounds for the solution set of the nonlinear equations are aiming to be calculated effectively. In the framework of the convex set theory and Taylor series expansion, the ellipsoidal Newton’s iteration scheme is established. Two various models of the scheme depending on the different models of quantifying the region of the iterative solution in iterative calculation are discussed. The bounds of the solution are updated iteratively by using the maximum and minimum values of the solution increment, which can be obtained by solving the optimization problem. The convergence of the scheme is proved and the general procedure for its implementation is also presented. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method in evaluating the bounds of nonlinear structural systems with uncertain-but-bounded parameters in comparison with the Monte-Carlo Simulation and the point-based iteration method.
MSC:
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 70K25 | Free motions for nonlinear problems in mechanics |
Keywords:
ellipsoidal Newton’s iteration method; nonlinear structural systems; uncertain-but-bounded parameters; Taylor series expansion; optimization problemReferences:
| [1] | Hui, Y.; Law, S. S.; Zhu, W. D., Extended IHB method for dynamic analysis of structures with geometrical and material nonlinearities, Eng. Struct., 205, Article 110084 pp. (2020) |
| [2] | Wei, G. T.; Jin, Y.; Wu, L. Z., Geometric and material nonlinearities of sandwich beams under static loads, Acta Mech. Sinica, 36, 1, 97-106 (2020) |
| [3] | Ye, S. Q.; Mao, X. Y.; Ding, H., Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions, Int. J. Mech. Sci., 168, Article 105294 pp. (2020) |
| [4] | Wu, B. H.; Gao, W.; Wu, D., Probabilistic interval geometrically nonlinear analysis for structures, Struct. Saf., 65, 100-112 (2017) |
| [5] | Gerald, C. F.; Wheatley, P. O., Applied Numerical Analysis (2004), Pearson Addison Wesley |
| [6] | Burden, R. L.; Faires, J. D., Numerical Analysis (2010), Brooks/Cole Cengage Learning |
| [7] | Wang, L.; Xiong, C., A novel methodology of sequential optimization and non-probabilistic time-dependent reliability analysis for multidisciplinary systems, Aerosp. Sci. Technol., 94, Article 105389 pp. (2019) |
| [8] | Qiu, Z. P.; Liu, D. L., Safety margin analysis of buckling for structures with unknown but bounded uncertainties, Appl. Math. Comput., 367, Article 124759 pp. (2020) · Zbl 1433.74050 |
| [9] | Liu, Y. S.; Wang, X. J.; Wang, L., A dynamic evolution scheme for structures with interval uncertainties by using bidirectional sequential Kriging method, Comput. Methods Appl. Mech. Engrg., 348, 712-729 (2019) · Zbl 1440.65055 |
| [10] | Zheng, Y. N., Predicting stochastic characteristics of generalized eigenvalues via a novel sensitivity-based probability density evolution method, Appl. Math. Model., 88, 437-460 (2020) · Zbl 1481.70095 |
| [11] | Ni, B. Y.; Jiang, C.; Huang, Z. L., Discussions on non-probabilistic convex modelling for uncertain problems, Appl. Math. Model., 59, 54-85 (2018) · Zbl 1480.74011 |
| [12] | Wang, C.; Matthies, H. G., A modified parallelepiped model for non-probabilistic uncertainty quantification and propagation analysis, Comput. Methods Appl. Mech. Engrg., 369, Article 113209 pp. (2020) · Zbl 1506.74489 |
| [13] | Xiong, C.; Wang, L.; Liu, G., An iterative dimension-by-dimension method for structural interval response prediction with multidimensional uncertain variables, Aerosp. Sci. Technol., 86, 572-581 (2019) |
| [14] | Qiu, Z. P.; Liu, D. L., Safety margin analysis of buckling for structures with unknown but bounded uncertainties, Appl. Math. Comput., 367, Article 124759 pp. (2020) · Zbl 1433.74050 |
| [15] | Luo, Z. X.; Wang, X. J.; Liu, D. L., Prediction on the static response of structures with large-scale uncertain-but-bounded parameters based on the adjoint sensitivity analysis, Struct. Multidiscip. Optim., 61, 123-139 (2020) |
| [16] | Wang, C.; Matthies, H. G., A comparative study of two interval-random models for hybrid uncertainty propagation analysis, Mech. Syst. Signal Process., 136, Article 106531 pp. (2020) |
| [17] | Ben-Haim, Y.; Elishakoff, I., Convex Models of Uncertainty in Applied Mechanics (1990), Elsevier: Elsevier Amsterdam · Zbl 0703.73100 |
| [18] | Elishakoff, I.; Eliseeff, P.; Glegg, S., Convex modelling of material uncertainty in vibrations of a viscoelastic structure, AIAA J., 32, 843-849 (1994) · Zbl 0804.73079 |
| [19] | Qiu, Z. P.; Elishakoff, I., Anti-optimization technique-a generalization of interval analysis for nonprobabilistic treatment of uncertainty, Chaos Solitons Fractals, 12, 9, 1747-1759 (2001) · Zbl 1172.74322 |
| [20] | Mcwilliam, S., Anti-optimisation of uncertain structures using interval analysis, Comput. Struct., 79, 4, 421-430 (2001) |
| [21] | Chen, S. H.; Lian, H. D.; Yang, X. W., Interval static displacement analysis for structures with interval parameters, Int. J. Comput. Math., 53, 2, 393-407 (2002) · Zbl 1112.74505 |
| [22] | Qiu, Z. P.; Elishakoff, I., Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis, Comput. Methods Appl. Mech. Engrg., 152, 3-4, 361-372 (1998) · Zbl 0947.74046 |
| [23] | Zhou, Y. T.; Jiang, C.; Han, X., Interval and subinterval analysis methods of the structural analysis and their error estimations, Int. J. Comput. Math., 3, 2, 229-244 (2006) · Zbl 1198.65088 |
| [24] | Xia, B. Z.; Yu, D. Z.; Liu, J., Interval and subinterval perturbation methods for a structural-acoustic system with interval parameters, J. Fluids Struct., 38, 146-163 (2013) |
| [25] | Qiu, Z. P.; Wang, X. J.; Chen, J. Y., Exact bounds for the static response set of structures with uncertain-but-bounded parameters, Int. J. Solids Struct., 43, 21, 6574-6593 (2006) · Zbl 1120.74624 |
| [26] | Qiu, Z. P.; Lv, Z., The vertex solution theorem and its coupled framework for static analysis of structures with interval parameters, Internat. J. Numer. Methods Engrg., 112, 7, 711-736 (2017) · Zbl 07867229 |
| [27] | Qi, W. C.; Qiu, Z. P., A collocation interval analysis method for interval structural parameters and stochastic excitation, Sci. China Phys. Mech. Astron., 55, 1, 66-77 (2012) |
| [28] | Wu, J. L.; Zhang, Y. Q.; Chen, L. P., A chebyshev interval method for nonlinear dynamic systems under uncertainty, Appl. Math. Model., 37, 6, 4578-4591 (2013) · Zbl 1269.93031 |
| [29] | Xu, M. H.; Qiu, Z. P., A dimension-wise method for the static analysis of structures with interval parameters, Sci. China Phys. Mech. Astron., 57, 10, 1934-1945 (2014) |
| [30] | Zhu, L. P.; Elishakoff, I.; Starnes, J. H., Derivation of multi-dimensional ellipsoidal convex model for experimental data, Math. Comput. Modelling, 24, 2, 103-114 (1996) · Zbl 0857.73081 |
| [31] | Polyak, B. T.; Nazin, S. A.; Durieu, C. C., Ellipsoidal parameter or state estimation under model uncertainty, Automatica, 40, 7, 1171-1179 (2004) · Zbl 1056.93063 |
| [32] | Qiu, Z. P., Comparison of static response of structures using convex models and interval analysis method, Internat. J. Numer. Methods Engrg., 56, 12, 1735-1753 (2003) · Zbl 1068.74083 |
| [33] | Kang, Z.; Zhang, W. B., Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data, Comput. Methods Appl. Mech. Engrg., 300, 461-489 (2016) · Zbl 1425.74376 |
| [34] | Wang, C.; Matthies, H. G., Random model with fuzzy distribution parameters for hybrid uncertainty propagation in engineering systems, Comput. Methods Appl. Mech. Engrg., 359, Article 112673 pp. (2020) · Zbl 1441.62922 |
| [35] | Yang, C.; Tangaramvong, S.; Tin-Loi, F., Influence of interval uncertainty on the behavior of geometrically nonlinear elastoplastic structures, J. Struct. Eng., 143, 1, Article 04016147 pp. (2017) |
| [36] | Wen, B. C.; Zhang, Y. M.; Liu, Q. L., Response of uncertain nonlinear vibration systems with 2D matrix functions, Nonlinear Dynam., 15, 2, 179-190 (1998) · Zbl 0904.73032 |
| [37] | Chen, J. B.; Li, J., Dynamic response and reliability analysis of non-linear stochastic structures, Probabilistic Eng. Mech., 20, 1, 33-44 (2005) |
| [38] | Li, J.; Chen, J. B., The probability density evolution method for dynamic response analysis of non-linear stochastic structures, Internat. J. Numer. Methods Engrg., 65, 6, 882-903 (2006) · Zbl 1179.74054 |
| [39] | Guo, X. Y.; Lee, Y. Y.; Mei, C., Non-linear random response of laminated composite shallow shells using finite element modal method, Internat. J. Numer. Methods Engrg., 67, 10, 1467-1489 (2006) · Zbl 1113.74030 |
| [40] | Qiu, Z. P.; Ma, L. H.; Wang, X. J., Non-probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty, J. Sound Vib., 319, 1-2, 531-540 (2009) |
| [41] | Lv, Z.; Liu, H., Nonlinear bending response of functionally graded nanobeams with material uncertainties, Int. J. Mech. Sci., 134, 123-135 (2017) |
| [42] | Lv, Z.; Qiu, Z. P., An iteration method for predicting static response of nonlinear structural systems with non-deterministic parameters, Appl. Math. Model., 68, 48-65 (2019) · Zbl 1481.74714 |
| [43] | Bararaa, M. S.; Sherali, H. D.; Shetly, C. M., Nonlinear Programming: Theory and Algorithms (1993), Wiley: Wiley New York · Zbl 0774.90075 |
| [44] | Qiu, Z. P.; Müller, P. C.; Frommer, A., Ellipsoidal set-theoretic approach for stability of linear state-space models with interval uncertainty, Math. Comput. Simulation, 57, 1-2, 45-59 (2001) · Zbl 0985.65077 |
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