Today's Agenda Optimization methods have been combined with design synthesis and parametric analysis for ca.40 years ■MDO definition Traditionally used graphical methods to find maximum or minimum of a multivariate function("carpet plot"),but.... Optimization problem formulation MDO in the design process "peaks" Graphics break down above 3-4 dimensions ■MDO challenges Where is max J(x)? Caution:local extrema 一Where is min J(x)? Design variable x1 System Engineering by J.J.Gao ③cLe0 国倒 13 System Engineering by J.J.Gao ③cLco 国 14 Combinatorial Explosion The General Model Any design can be defined by a vector in Quantitative side of the design problem may be formulated multidimensional space,where each design as a problem of Nonlinear Programming (NLP) variable represents a different dimension minJ x.p This is the problem formulation For n>3 a combinatorial "explosion"takes place that we will discuss this semester. and the design space cannot be computed and s.tg(k,p)≤0 plotted in polynomial time h(K,P)=0 where J=[J,x…J.x了 B≤X,≤xB（G=l…m) X=x1…x…Xm Numerical optimization offers an alternative to the graphical approach and"brute force"evaluation g=[8(…8()] h=[h()…h(x)] During past three decades much progress has been made in numerical optimization System Engineering by J.J.Gao ③coo 15 System Engineering by .J.Gao ③ctse 16+ - 13 Today䇻s Agenda System Engineering by J. J. Gao MDO definition Optimization problem formulation MDO in the design process MDO challenges + - System Engineering by J. J. Gao 14 + - System Engineering by J. J. Gao 15 Combinatorial Explosion + - System Engineering by J. J. Gao 16 The General Model