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# Stru Des Sens AnaOpt EGM 6365

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This 18 page Class Notes was uploaded by Rollin Mann DVM on Saturday September 19, 2015. The Class Notes belongs to EGM 6365 at University of Florida taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/207077/egm-6365-university-of-florida in Engineering & Applied Science at University of Florida.

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Date Created: 09/19/15

Sequential linear approximation Jaco F Sch utte EGM 6365 Structural Optimization Fall 2005 Introduction Using local approximations advantageous when high computational cost is associated with f g Ag Ax Usually cost of optimization algorithm operations is negligible compared to above Sequential Linear Programming Inimilnize f X subject to gjx 2 0 j 1 H779 Start with initial trial design x0 Obtain linear approximations around X0 offand g using 1st order Taylor expansion u a 111iui1uizo j39lxm J39 a 11 UI quot Kn 11 I 1 subject to mix A E 1711 J i ll j It V l l uquot Tl quot X0 and m ltf r 7139 1 ll Sequential Linear Programming con nued Move limits ah am must be small to maintain accuracy in approximation f ll al C One dimen3ona example 6f 5x x0 xL Sequential Linear Programming con nued Method iterates by replacing x0 by XL from previous step and constructing a new Hneanza on Algorithm terminated when either rate of change stopping criterion is met KuhnTucker optimality conditions are adequately satisfied Example 631 111ini1nize fX 7quot I 7 11 subject to g1 r 2 I 1 Starting point Move Ilmlts 61Lx all11OocOx Example 631 continued Constraint functions and derivatives evaluated at xu yield 91X025717123 92XO77117 72m 72 W1 72 Valixo 2 i 72 72 Vgg 21f V92XO 2 Example 631 continued 1St order Taylor expansion linearization gives the following equations for constraints Example 631 continued om 1000 h I W 39I I 3 8m soillutio without VLmDveM39 ES 5 I quot 32L I 6013 6 I E L g 3Equot r 4 I 5 5 Ew 4040 a r 7 IFIX luv rayt I E 18 200 Z Kquot I i 7 7 2 397 77 14 3 quot 3 000 203 ALDO 600 800 I 100 Example 631 continued 20 First iteration with move limits yield x1 2 0 f 60 39 85 If move limits were not imposed X1 5 0 f 22 o Ignoring move limits result in large constraint violations and inaccurate values forfif cost function is nonlinear Sequential Linear Programming con nued Advantages LP packages are readily available in most mathematical software packages Matlab Numerical recipes in C Octave etc Simple implementation Requires only 1St order derivatives which are easily obtained by finite difference methods Objective function software considerations Computational cost Sequential Linear Programming con nued Disadvantages Choice of move limits are important because Too large will cause inaccuracies and prevent convergence Too small translates to large a large number of iterations gt excessive computational effort Optimum is to have initially large move limits which are reduced as optimum is approached Reduction rate is problem dependent and can be obtained by monitoring the fitness value Rate is reduced 1050 iffl 2f s Typical starting values of move limits are 1030 of design variable range Sequential Linear Programming con nued Disadvantages continued lf starting design x0 is infeasible a combination of approximation and move limits may result in algorithm remaining in infeasible region lf solution of problem does not lie at a vertex of a constraint set it may alternate between to fixed points This may be solved by appropriate reduction of move limits Relaxation of constraints If an infeasible starting point is obtained it becomes necessary to relax constraints to allow algorithm to escape from infeasible region minilnize fx0 Z ri 7 Tm dip In 13971 X0 017i n 01739 r subject to gJXOZJT7TUZ 33 J j 177g 1 71 1 X0 L11 g 1391 7 17m g um and 3 Z 0 where 16 is an additional design variable and k is a scaling weight to ensure that the minimization of 16 receives priority overf Example 632 Uywngioquot 773 X 10 4 E 0y compression 4833 X 10 4 E Allowable 6y 3 X 1031 1351quot f2 Pa f34117 f4723p Example 632 continued By nondimensionalizing design variables 3103 1 103 1 t 11 AlE 2 AZE We can formulate the optimization problem as lnlnllnlze fr11 g i 1 1 1 2 subject to 1811 G l g S 3 005 g 11 3 01546 t 005 g 42 3 01395 t with lowerbounds on X1 and X2 005 Example 632 continued First iteration x50101 gt 75 132 feasible am 117 00L 239 1 2 07 300 1732 an I luiuimizc fL subjevt to 1811 001 009 This problem is solved to yield 1 1 010316 1 2 011 and f 446410

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