Abstract
This paper focuses on the development of an optimization tool with the aim to obtain robust and reliable designs in short computational time. The robustness measures considered here are the expected value and standard deviation of the performance function involved in the optimization problem. When using these robustness measures combined, the search of optimal design appears as a robust multiobjective optimization (RMO) problem. Reliable design addresses uncertainties to restrict the structural probability of failure. The mathematical formulation for the reliability based robust design optimization (RBRDO) problem is obtained by adding a reliability based constraint into the RMO problem. As both, statistics calculations and the reliability analysis could be very costly, approximation technique based on reduced-order modeling (ROM) is also incorporated in our procedure. The selected ROM is the proper orthogonal decomposition (POD) method, with the aim to produce fast outputs considering structural non-linear behavior. Moreover, to obtain RBRDO designs with reduced CPU time we propose others developments to be added in the integrated tool. They are: Probabilistic Collocation Method (PCM) to evaluate the statistics of the structural responses and, also, an approximated reliability constraints procedure based on the Performance Measure Approach (PMA) for reliability constraint assessment. Finally, Normal-Boundary Intersection (NBI) or Normalized Normal-Constraint (NNC) multiobjective optimization techniques are employed to obtain fast and even spread Pareto robust designs. To illustrate the application of the proposed tool, optimization studies are conducted for a linear (benchmark) and nonlinear trusses problems. The nonlinear example consider different loads level, exploring the material plasticity. The integrated tool prove to be very effective reducing the computational time by up to five orders of magnitude, when compared to the solutions obtained via classical standard approaches.
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References
Afonso S, Lyra P, Albuquerque T, Motta R (2010) Structural analysis and optimization in the framework of reduced-basis method. Struct Multidiscip Optim 40(1–6):177–199
Arora J, Wang Q (2005) Review of formulations for structural and mechanical system optimization. Struct Multidiscip Optim 30(4):251–272
Arora JS, Messac A, Mullur AA (2007) Optimization of structural and mechanical system. World Scientific Pub Co Inc, University of Iowa, USA
Beyer H-G, Sendhoff B (2007) Robust optimization - a comprehensive survey. Comput Methods Appl Mech Eng 196(33):3190–3218
Bucher C (2009) Computational analysis of randomness in structural mechanics: structures and infrastructures book series. Structures and infrastructures. Taylor & Francis
Burkardt J, Gunzburger M, Lee HC (2006) Pod and cvt-based reduced-order modeling of navier-stokes flows. Comput Methods Appl Mech Eng 196:337–355
Cavazzuti M (2012) Optimization methods: from theory to scientific design and technological aspects in mechanics. Springer Science & Business Media
Collette Y, Siarry P (2004) Multiobjective optimization: principles and case studies. Springer
Crisfield M.A. (2000) Non-linear finite element analysis of solids and structures: essentials, vol 1. Wiley, Chichester
Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3):631–657
Deb K, Gupta S, Daum D, Branke J, Mall AK, Padmanabhan D (2009) Reliability-based optimization using evolutionary algorithms. IEEE Trans Evol Comput 13(5):1054–1074
Doltsinis I, Kang Z (2004) Robust design of structures using optimization methods. Comput Methods Appl Mech Eng 193(23):2221–2237
Engelund S, Rackwitz R (1993) A benchmark study on importance sampling techniques in structural reliability. Struct Saf 12:255–276
Haftka RT, Sobieszczanski-Sobieski J. (2009) Structural optimization: history structural optimization: history, in encyclopedia of optimization. Springer, pp 3834–3836
Hasofer A, Lind N (1974) Exact and invariant second-moment code format. ASCE J Eng Mech Div 100:111–121
Huang B, Fery P, Xue L, Wang Y (2008) Seeking the Pareto front for multiobjective spatial optimization problems. Int J Geogr Inf Sci 22(5):507–526
Hwang C-L, Paidy SR, Yoon K, Masud ASM (1980) Mathematical programming with multiple objectives: a tutorial. Comput Oper Res 7(1):5–31
Keane AJ, Nair PB (2005) Computational approaches for aerospace design: the pursuit of excellence. Wiley
Koch P, Yang R-J, Gu L (2004) Design for six sigma through robust optimization. Struct Multidiscip Optim 26(3–4):235–248
Lee I (2008) Reliability-based design optimization and robust design optimization using univariate dimension reduction method. PhD thesis, University of Iowa
Liang Y, Lee H, Lim S, Lin W, Lee K, Wu C (2002) Proper orthogonal decomposition and its applications-part i: theory. J Sound Vib 252(3):527–544
Lin PT, Gea HC, Jaluria Y (2011) A modified reliability index approach for reliability-based design optimization. J Mech Des 133(4):044501
Lopez RH, Beck AT (2012) Reliability-based design optimization strategies based on form: a review. J Braz Soc Mech Sci Eng 34(4):506–514
Marczyk J (2000) Stochastic multidisciplinary improvement: beyond optimization. In: Proceedings of 8th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization. Long Beach
MathWorks (2014) Optimization toolbox user’s guide: MATLAB. MathWorks Inc., USA
Melchers RE (1999) Structural reliability: analysis and prediction, 2nd edn. Wiley, Chichester
Messac A, Mattson CA (2004) Normal constraint method with guarantee of even representation of complete Pareto frontier. AIAA J 42(10):2101–2111
Miller I, Freund J, Johnson R (1990) Probability and statistics for engineers, 4th edn, Prentice-Hall, Englewood Cliffs
Motta RS, Afonso SMB (2011) Optimization of trusses under nonlinear conditions considering the proper orthogonal decomposition method. In: XXXII CILAMCE - Iberian Latin American congress on computational methods in engineering. Ouro Preto, Brazil
Motta RS, Afonso S, Lyra P (2009) Structural robust optimization considering reduced-basis method, Master’s thesis, Civil Engineering Department. UFPE, Recife-PE Brazil. in Portuguese
Motta RS, Afonso S, Lyra P (2012) A modified nbi and nc method for the solution of n-multiobjective optimization problems. Struct Multidiscip Optim 1:1–21
Motta RdS, Afonso SMB, Lyra PR, Willmersdorf RB (2015) Development of a computational efficient tool for robust structural optimization. Eng Comput 32(2):258–288
Nocedal J, Wright S (2006) Numerical optimization. Springer Science & Business Media
Papadrakakis M, Lagaros N, Plevris V (2005) Design optimization of steel structures considering uncertainties. Eng Struct 27(9):1408–1418
Paiva RM, Crawford C, Suleman A (2014) Robust and reliability-based design optimization framework for wing design. AIAA J - Special Section Multidiscip Des Optim 52:711–724
Rackwitz R, Fiessler B (1978) Structural reliability under combined load sequences. Comput Struct 9 (5):489–494
Rao RV, Savsani VJ (2012) Mechanical design optimization using advanced optimization techniques. Springer Science & Business Media
Ramamurthy D (2005) Smart simulation techniques for the evaluation of parametric uncertainties on black box systems. PhD thesis, Washington State University
Rathod V, Yadav O, Rathore A, Jain R (2011) Reliability-based robust design optimization: a comparative study. In: IEEE international conference on industrial engineering and engineering management (IEEM), 2011, pp 1558–1563
Robinson D, Atcitty C (1999) Comparison of quasi- and pseudo-monte carlo sampling for reliability and uncertainty analysis. In: 40th structures, structural dynamics, and materials conference and exhibit, American institute of aeronautics and astronautics
Schuëller GI, Jensen HA (2008) Computational methods in optimization considering uncertainties–an overview. Comput Methods Appl Mech Eng 198(1):2–13
Simpson T, Poplinski J, Koch PN, Allen J (2001) Metamodels for computer-based engineering design: Survey and recommendations. Eng Comput 17(2):129–150
Sirovich L (1987) Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Q Appl Math 45(3):561–571
Spillers WR, MacBain KM (2009) Structural optimization. Springer Science & Business Media
Stein M (1987) Large sample properties of simulations using latin hypercube sampling. Technometrics 29(2):143–151
Stoer J, Bulirsch R (1991) Introduction to numerical analysis, 2nd edn. Springer-Verlag, Heidelberg, Berlin
Sudret B, Der Kiureghian A (2000) Stochastic finite element methods and reliability: a state-of-the-art report. Department of Civil and Environmental Engineering, University of California
Tan BT, Damodaran M, Willcox K (2001) Proper orthogonal decomposition extensions and their applications in steady aerodynamics. Master’s thesis, Ho Chi Minh City University of Technology, Vietnam
Tu J, Choi KK, Park YH (1999) A new study on reliability-based design optimization. J Mech Des 121(4):557–564
Valdebenito MA, Schuëller GI (2010) A survey on approaches for reliability-based optimization. Struct Multidiscip Optim 42(5):645–663
Wu Y-T, Millwater H, Cruse T (1990) Advanced probabilistic structural analysis method for implicit performance functions. AIAA J 28(9):1663–1669
Yadav OP, Bhamare SS, Rathore A (2010) Reliability-based robust design optimization: a multi-objective framework using hybrid quality loss function. Qual Reliab Eng Int 26(1):27–41
Youn BD, Choi K (2004) An investigation of nonlinearity of reliability-based design optimization approaches. J Mech Des 126(3):403–411
Youn BD, Choi KK (2004) Selecting probabilistic approaches for reliability-based design optimization. AIAA J 42(1):124– 131
Youn BD, Xi Z (2009) Reliability-based robust design optimization using the eigenvector dimension reduction (edr) method. Struct Multidiscip Optim 37(5):475–492
Youn BD, Choi KK, Park YH (2003) Hybrid analysis method for reliability-based design optimization. J Mech Des 125(2):221–232
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The authors acknowledges the financial support given by the Brazilian research agency CNPq and Pernambuco state research agency FACEPE to the execution of the present work.
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Motta, R.S., Afonso, S.M.B. An efficient procedure for structural reliability-based robust design optimization. Struct Multidisc Optim 54, 511–530 (2016). https://doi.org/10.1007/s00158-016-1418-1
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DOI: https://doi.org/10.1007/s00158-016-1418-1