Engineering Optimization: Theory and Practice, Fourth Edition

14.3 Fast Reanalysis Techniques 743 Thus the equilibrium equations of the structure can be expressed as [K]Y=P (E 3 ) where Y=  ...

744 Practical Aspects of Optimization Table 14.2 ExactY 0 =      0 .116462E− 02 0 .232923E− 02 0 .514654E− 01 − 0 .7032 ...

14.4 Derivatives of Static Displacements and Stresses 745 By premultiplying Eq. (14.27) by [Y]Twe obtain [K ̃] r×r c r× 1 = P ̃ ...

746 Practical Aspects of Optimization analysis for computing the values of the objective function and/or constraint functions at ...

14.5 Derivatives of Eigenvalues and Eigenvectors 747 14.5 Derivatives of Eigenvalues and Eigenvectors Let the eigenvalue problem ...

748 Practical Aspects of Optimization 14.5.2 Derivatives ofYi The differentiation of Eqs. (14.39) and (14.45) with respect toxjr ...

14.6 Derivatives of Transient Response 749 Table 14.3 Derivatives of Eigenvalues [14.4] i Eigenvalue,λi 10 −^9 ∂λi ∂x 1 10 −^9 ∂ ...

750 Practical Aspects of Optimization Note that if the undamped natural modes of vibration are used as basis functions and if [C ...

14.7 Sensitivity of Optimum Solution to Problem Parameters 751 derivative,∂yj/∂xi, can be computed using Eq. (14.53) as ∂yj ∂xi ...

752 Practical Aspects of Optimization 14.7.1 Sensitivity Equations Using Kuhn–Tucker Conditions The Kuhn–Tucker conditions satis ...

14.7 Sensitivity of Optimum Solution to Problem Parameters 753 ∂X ∂p =      ∂x 1 ∂p .. . ∂xn ∂p      , ∂λ ∂p =   ...

754 Practical Aspects of Optimization Similarly, a currently inactive constraint will become critical due topif the new value o ...

14.8 Multilevel Optimization 755 If the vectorSis normalized by dividing its components bysn+ 1 , Eq. (14.86) gives λ=pand henc ...

756 Practical Aspects of Optimization 14.8.2 Method Let the optimization problem be stated as follows: FindX= {x 1 x 2 · · ·xn}T ...

14.8 Multilevel Optimization 757 Y(l)≤Y≤Y(u) Z(l)k ≤Zk≤Z(u)k , k= 1 , 2 ,... , K (14.97) Similarly, the objective functionf (X)c ...

758 Practical Aspects of Optimization 3.Solve the second-level optimization problem stated in Eqs. (14.101) and find a new vecto ...

14.8 Multilevel Optimization 759 We treat the bars 1 and 2 as subsystems 1 and 2, respectively, withyas the coordination variabl ...

760 Practical Aspects of Optimization Findywhich minimizes f= 76 , 500 z∗ 1 √ y^2 + 63 + 76 , 500 z∗ 2 √ y^2 + 1 = 109. 2857 y^2 ...

14.10 Multiobjective Optimization 761 If a multilevel (decomposition) approach is used, the optimization of various subsystems ( ...

762 Practical Aspects of Optimization Initialize node i Data from the host node Randomly perturb one variable out of S(i) Change ...