Abstract
This research explores the usage of classification approaches in order to facilitate the accurate estimation of probabilistic constraints in optimization problems under uncertainty. The efficiency of the proposed framework is achieved with the combination of a conventional topology optimization method and a classification approach- namely, probabilistic neural networks (PNN). Specifically, the implemented framework using PNN is useful in the case of highly nonlinear or disjoint failure domain problems. The effectiveness of the proposed framework is demonstrated with three examples. The first example deals with the estimation of the limit state function in the case of disjoint failure domains. The second example shows the efficacy of the proposed method in the design of stiffest structure through the topology optimization process with the consideration of random field inputs and disjoint failure phenomenon, such as buckling. The third example demonstrates the applicability of the proposed method in a practical engineering problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atlas L, Cole R et al (1990) Performance comparisons between backpropagation networks and classification trees on three real-world applications. Adv Neural Inf Process Syst 2:622–629
Basudhar A, Missoum S et al (2008) Limit state function identification using Support Vector Machines for discontinous responses and disjoint failure domains. Probab Eng Mech 23:1–11
Brown DE, Corruble V et al (1993) A comparison of decision tree classifiers with backpropagation neural networks for multimodal classification problems. Pattern Recogn 26:953–961
Bendsøe M, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224
Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin
Cacoullos T (1966) Estimation of a multivariate density. Ann Math Stat 18(2):179–189
Conti S, Held H, Pach M et al (2008) Shape optimization under uncertainty – A stochastic programming perspective. SIAM J Optim 19(4):1610–1632
Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscipl Optim 41(4):507–524
Chien YT, Fu KS (1967) On the generalized Karhunen-Loeve expansion. IEEE Trans Inf Theory 13:518–520
Choi S, Grandhi RV et al (2006) Reliability- based design optimization. London, Springer
Chtioui Y, Bertrand D et al (1997) Comparison of multilayer perceptron and probabilistic neural networks in artificial vision. Application to the discrimination of seeds. J Chemom 11(2):111–129
Curram SP, Mingers J (1994) Neural networks, decision tree induction and discriminant analysis: an emperical comparison. J Oper Res Soc 45(4):440–450
Cybenko G (1989) Approximation by superposition of a sigmoidal function. Math Control Signals Syst 2:303–314
Dietterich TG, Bakiri G (1995) Solving multiclass learning problems via error-correcting output codes. J Artif Intell Res 2:263–286
Dobbs MW, Felton LP (1969) Optimization of truss geometry. J Struct Div ASCE 95:2105–2118
Duda PO, Hart PE (1973) Pattern classification and scene analysis. New York, Wiley
Frecker M, Ananthasuresh GK, Nishiwaki S, Kikuchi N, Kota S (1997) Topological synthesis of compliant mechanisms using multi-criteria optimization. ASME J Mech Des 119(2):238–245
Fukunaga K, Koontz WLG (1970) Application of the Karhunen-Loève Expansion to feature selection and ordering. IEEE Trans Comput C-19(4):311–318
Hornik K (1991) Approximation capabilities of multilayer feedforward networks. Neural Netw 4:251–257
Hornik K, Stinchcombe M et al (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2:359–366
Hotelling H (1933) Analysis of a complex of statistical variables into principal components. J Educ Psychol 24:498–520
Huang MS, Lippmann RP (1987) Comparisons between neural net and conventional classifiers. In: IEEE 1st international conference on neural networks. San Diego, CA, pp 485–493
Hurtado JE, Alvarez DA (2003) Classification approach for reliability analysis with stochastic finite-element modeling. J Struct Eng 129(8):1141–1149
Kharmanda G, Olhoff N et al (2004) Reliability-based topology optimization. Struct Multidiscipl Optim 26(5):295–307
Kirsch U (1990) On singular topologies in optimum structural design. Struct Multidiscipl Optim 2:133–142
Kogiso N, Ahn W, Nishiwaki S et al (2008) Robust topology optimization for compliant mechanisms considering uncertainty of applied loads. J Adv Mech Des Syst Manuf 2(1):96–107
Maute K, Frangopol DM (2003) Reliability-based design of MEMS mechanisms by topology optimization. Comput Struct 81(8–11):813–824
McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245
Michelle AGM (1904) The limits of economy of material in frame structures. Philos Mag 8(47):589–597
Michie D, Spiegelhalter DJ et al (1994) Machine learning, neural, and statistical classification. London, UK
Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33:1065–1076
Patel J, Choi S-K (2009) Optimal synthesis of mesostructured materials under uncertainty. In: 50th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and material conference and 5th AIAA multidisciplinary design optimization specialist conference. Palm Springs, CA
Patwo E, Hu MY et al (1993) Two-group classification using neural networks. J Decis Sci 24(4):825–845
Richard MD, Lippmann R (1991) Neural network classifiers estimate Bayesian a posteriori probabilities. Neural Comput 3:461–483
Rozvany GIN, Bendsøe MP, Kirsch U (1995) Layout optimization of structures. Appl Mech Rev 48:41–119
Seepersad CC, Allen JK, McDowell DL, Mistree F (2006) Robust design of cellular materials with topological and dimensional imperfections. Mech Des 128:1285–1297
Sigmund O (1995) Tailoring materials with prescribed elastic properties. Mech Mater 20:351–368
Specht DF (1967) Generation of polynomial discriminant functions for pattern recognition. IEEE Trans Electron Comput EC-16:308–319
Specht DF (1990) Probabilistic neural networks. Neural Netw 3:109–118
Sved G, Ginos Z (1968) Structural optimization under multiple loading. Int J Mech Sci 8:803–805
Tsui KL (1992) An overview of Taguchi method and newly developed statistical methods for robust design. IIE Trans 24(5):44–57
Tu J, Choi KK (1997) A performance measure approach in reliability- based structural optimization. Technical Report R97–02, University of Iowa
Wang Y, Adali T et al (1998) Quantification and segmentation of brain tissues for MR images: a probabilistic neural network approach. IEEE Trans Image Process 7(8):1165–1181
Wang MY, Chen SK, Wang X et al (2005) Design of multi-material compliant mechanisms using level set methods. ASME J Mech Des 127(5):941–956
Witten IH, Frank E (2005) Data mining: practical machine learning tools and techniques. Morgan Kauffman Publications, San Francisco
Zhang GP (2000) Neural networks for classification: a survey. IEEE Trans Syst Man Cybern C Appl Rev 30(4):451–462
Zhou M (1996) Difficulties in truss topology optimization with stress and local buckling constraints. Struct Multidiscipl Optim 11:134–136
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Patel, J., Choi, SK. Classification approach for reliability-based topology optimization using probabilistic neural networks. Struct Multidisc Optim 45, 529–543 (2012). https://doi.org/10.1007/s00158-011-0711-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-011-0711-2