Accurate displacement derivatives for structural optimization using approximate reanalysis Uri Kirsch * ,1 , Panos Y. Papalambros Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, USA Received 21 January 2000 Abstract A uni®ed approach for accurate approximations of displacements and displacement derivatives with respect to design variables is presented.Thesolutionprocedureisbasedonresultsofasingleexactanalysisataninitialdesign.Unlikecommonapproximationsof thestructuralresponse,theapproachpresentedisnotbasedoncalculationofderivatives.Rather,approximationsofdisplacementsare usedtoevaluatedisplacementderivativesatvariousmodi®eddesigns.Itisshownthatsimilarcomputationalproceduresareemployed for evaluating displacements, ®rst- and second-order derivatives of displacements. The procedure presented is suitable for dierent types of design variables and structures. Numerical results show that accurate approximations of derivatives can be achieved with a small computational eort for designs obtained by very large changes in the initial structure. Ó 2001 Elsevier Science B.V. All rights reserved. 1. Introduction Thederivativesoftheresponsevectorwithrespecttothesystemparametersareusuallyreferredtoasthe sensitivity coecients. Sensitivity analysis is used to: · Predict the changes in the system response due to changes in the structure parameters. · Select a search direction in design optimization problems. · Construct explicit approximations of the constraint functions in terms of the structure parameters e.g. ®rst-order Taylor series approximations). · Generate approximations for the response of a modi®ed system, including approximate reanalysis models. · Assess the eects of uncertainties in the structure properties material properties, geometric parameters and other parameters of the computational model) on the system response. It has been noted [1] that a predominant contributor to the cost and time of the optimization of large structural systems is the calculation of ®rst-order derivatives. Moreover, calculation of the latter derivatives for a given design involves structural analysis of the design. As a consequence, there has been much interest in ecient computation procedures for sensitivity analysis. Early and recent de- velopments in methods for sensitivity analysis are discussed in [1±4]. Ecient evaluation of approximate ®rst-order derivatives by various sensitivity analysis methods has been demonstrated in a previous study [5]. www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 190 2001) 3945±3956 * Corresponding author. Fax: +734-647-8403. E-mail address: kirsch@tx.technion.ac.il U. Kirsch). 1 On leave from the Department of Civil Engineering, Technion ± Israel Institute of Technology, Haifa 32000, Israel. 0045-7825/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII:S0045-782500)00306-6