Compared with the interval model, the ellipsoidal convex model can describe the correlation of the uncertain parameters through a multidimensional ellipsoid, and whereby excludes extreme combination of uncertain parameters and avoids over-conservative designs. In this paper, we attempt to propose an efficient multi-objective optimization method for uncertain structures based on ellipsoidal convex model. Firstly, each uncertain objective function is transformed into deterministic optimization problem by using nonlinear interval number programming (NINP) method and a possibility degree of interval number is applied to deal with the uncertain constraints. The penalty function method is suggested to transform the uncertain optimization problem into deterministic non-constrained optimization problem. Secondly, the approximation model based on radial basis function (RBF) is applied to improve computational efficiency. For ensuring the accuracy of the approximation models, a local-densifying approximation technique is suggested. Then, the micro multi-objective genetic algorithm (μMOGA) is used to optimize design parameters in the outer loop and the intergeneration projection genetic algorithm (IP-GA) is used to treat uncertain vector in the inner loop. Finally, two numerical examples and an engineering example are investigated to demonstrate the effectiveness of the present method.
Uncertainty structures Multi-objective optimization Ellipsoidal convex model Local-densifying approximation technique
This is a preview of subscription content, log in to check access.
The authors would also like to thank anonymous reviewers for their valuable comments.
This work is supported by the National Natural Science Foundation of China (Grant No.51775057), the Hunan Provincial Natural Science Foundation of China (No.2017JJ3323), the Research Foundation of Education Department of Hunan Province (No.16B014), and the Science Foundation of State Key Laboratory of Mechanical Transmissions (No.SKLMT-KFKT-201609).
Adduri PR, Penmetsa RC (2007) Bounds on structural system reliability in the presence of interval variables. Comput Struct 85:320–329CrossRefGoogle Scholar
Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainties in applied mechanics. Elsevier Science Publisher, AmsterdamzbMATHGoogle Scholar
Bobby S, Suksuwan A, Spence SMJ, Kareem A (2017) Reliability-based topology optimization of uncertain building systems subject to stochastic excitation. Struct Saf 66:1–16CrossRefGoogle Scholar
Deb K (2001) Multi-objective optimization using evolutionary algorithms. John Wiley &Sons Ltd., EnglandzbMATHGoogle Scholar
Du XP (2007) Interval reliability analysis, in: ASME 2007 Design Engineering Technical Conference & Computers and Information in Engineering Conference (DETC2007), Las Vegas, Nevada, USAGoogle Scholar
Dubourg V, Sudret B, Bourinet JM (2011) Reliability-based design optimization using kriging surrogates and subset simulation. Struct. Multidisc. Optim. 44(5):673–690CrossRefGoogle Scholar
Fang H, Rais-Rohani M, Liu Z, Horstemeyer MF (2005) A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Comput Struct 83:2121–2136CrossRefGoogle Scholar
Hawchar L, El-Soueidy CP, Schoefs F (2018) Global kriging surrogate modeling for general time-variant reliability-based design optimization problems. Struct. Multidisc. Optim. 58(3):955–968MathSciNetCrossRefGoogle Scholar
Jiang C, Han X, Liu GP (2008a) Uncertain optimization of composite laminated plates using a nonlinear interval number programming method. Comput Struct 86:1696–1703CrossRefGoogle Scholar
Jiang C, Han X, Liu GR (2008b) A nonlinear interval number programming method for uncertain optimization problems. Eur J Oper Res 188:1–13MathSciNetCrossRefGoogle Scholar
Jiang C, Han X, Lu GY, Liu J, Zhang Z, Bai YC (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Methods Appl Mech Eng 200:2528–2546CrossRefGoogle Scholar
Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodelling techniques under multiple modeling criteria. Struct Multidisc Optim 23:1–13CrossRefGoogle Scholar
Kang Z, Luo Y (2010) Reliability-based structural optimization with probability and convex set hybrid models. Struct. Multidisc. Optim. 42(1):89–102CrossRefGoogle Scholar
Kaushik S (2007) Reliability-based multi-objective optimization for automotive crashworthiness and occupant safety. Struct. Multidisc. Optim. 33:255–268CrossRefGoogle Scholar
Lagaros ND, Plevris V, Papadrakakis M (2005) Multi-objective design optimization using cascade evolutionary computations. Comput Methods Appl Mech Eng 194(30):3496–3515CrossRefGoogle Scholar
Li F, Luo Z, Rong J, Zhang N (2013) Interval multi-objective optimisation of structures using adaptive kriging approximations. Comput Struct 119(4):68–84CrossRefGoogle Scholar
Liang JH, Mourelatos ZP, Nikolaidis E (2007) A single-loop approach for system reliability-based design optimization. ASME J Mech Des 129:1215–1224CrossRefGoogle Scholar
Lin J, Luo Z, Tong L (2010) A new multi-objective programming scheme for topology optimization of compliant mechanisms. Struct Multidisc Optim 40(40):241–255CrossRefGoogle Scholar
Liu GR, Han X (2003) Computational inverse techniques in nondestructive evaluation. CRC Press, FloridaCrossRefGoogle Scholar
Liu X, Zhang ZY (2014) A hybrid reliability approach for structure optimization based on probability and ellipsoidal convex models. J Eng Design 25(4–6):238–258CrossRefGoogle Scholar
Liu GP, Han X, Jiang C (2008) A novel multi-objective optimization method based on an approximation model management technique. Comput Methods Appl Mech Eng 197:2719–2731CrossRefGoogle Scholar
Liu X, Zhang ZY, Yin LR (2017) A multi-objective optimization method for uncertain structures based on nonlinear interval number programming method. Mech Based Des Struc 45(1):25–42CrossRefGoogle Scholar
Luo Y, Kang Z, Luo Z, Li A (2009) Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct. Multidisc. Optim. 39(3):297–310MathSciNetCrossRefGoogle Scholar
Moore RE (1979) Methods and applications of interval analysis. Prentice-Hall Inc., LondonCrossRefGoogle Scholar
Qiu Z, Elishakoff I (1998) Anti-optimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput Methods Appl Mech Eng 152(3):361–372CrossRefGoogle Scholar