1. MAE550 PROJECT By: Lin Tom Tom Yan Yan Lv Lv

2. 1 Introduction OSA is a method of estimating the approximate effect that some change in problem parameters has on the optimum design. For example, if materials or design requirements are changed after we have already found an optimum solution to the original problem, we wish to estimate the effect that this will have on the design without actually solving the optimization problem over again. OSA is a method of estimating the approximate effect that some change in problem parameters has on the optimum design. For example, if materials or design requirements are changed after we have already found an optimum solution to the original problem, we wish to estimate the effect that this will have on the design without actually solving the optimization problem over again.

3. There are two general approaches to the OSA problem: There are two general approaches to the OSA problem: o (1) Base on Kuhn-Tucker conditions. o (2) Use the concept of a feasible direction. In this project, we use the first one to obtain estimated optimum solution of cantilevered beam problem shown below (in the book, Page-184). In this project, we use the first one to obtain estimated optimum solution of cantilevered beam problem shown below (in the book, Page-184).

4. OSA Algorithm: OSA Algorithm: We only consider the active constraint

6. Then we can form the matrix: At last, we derive the optimum solution:

7. 2 Cantilevered Beam Problem Figure 2-1 Minimize: Minimize: N=5, for conveniences, we assume each N=5, for conveniences, we assume each In this problem, we use optimum solution from Dot program (method 3) which is more accurate than the Master due to direct handling issues. In this problem, we use optimum solution from Dot program (method 3) which is more accurate than the Master due to direct handling issues.

8. Core Code List (Dot): Core Code List (Dot): P= 50000 E= 20000000 L=500 Sigma=14000 Y=2.5 OBJ=0 DO 50 i=1,5 50 OBJ=OBJ+100*x(i)*x(i+5) G(1)=6.*P*L/(X(1)*X(6)*X(6)*Sigma)-1 G(2)=6.*P*(L-100)/(X(2)*X(7)*X(7)*Sigma)-1 G(3)=6.*P*(L-200)/(X(3)*X(8)*X(8)*Sigma)-1 G(4)=6.*P*(L-300)/(X(4)*X(9)*X(9)*Sigma)-1 G(5)=6.*P*(L-400)/(X(5)*X(10)*X(10)*Sigma)-1 G(6)=X(6)-20*X(1) G(7)=X(7)-20*X(2) G(8)=X(8)-20*X(3)

9. G(9)=X(9)-20*X(4) G(10)=X(10)-20*X(5) G(11)=1-X(1) G(12)=1-X(2)G(13)=1-X(3)G(14)=1-X(4)G(15)=1-X(5)G(16)=5-X(6)G(17)=5-X(7)G(18)=5-X(8)G(19)=5-X(9)G(20)=5-X(10)G(21)=(0.032*P*L*L*L/(X(5)*X(10)*X(10)*X(10))+ *0.144*P*L*L*L/(X(4)*X(9)*X(9)*X(9))+ *0.144*P*L*L*L/(X(4)*X(9)*X(9)*X(9))+ *0.608*P*L*L*L/(X(3)*X(8)*X(8)*X(8))+ *0.608*P*L*L*L/(X(3)*X(8)*X(8)*X(8))+ *1.184*P*L*L*L/(X(2)*X(7)*X(7)*X(7))+ *1.184*P*L*L*L/(X(2)*X(7)*X(7)*X(7))+ *1.936*P*L*L*L/(X(1)*X(6)*X(6)*X(6)))/Y/E-1 *1.936*P*L*L*L/(X(1)*X(6)*X(6)*X(6)))/Y/E-1

10. 3 OSA Calculation We consider four factors(P, E, Sigma and Y),respectively, as the parameter P in OSA problem to see how they perform with approximation to objective function and to the constraints. We consider four factors(P, E, Sigma and Y),respectively, as the parameter P in OSA problem to see how they perform with approximation to objective function and to the constraints.

11. Load Load P=55,000 (10%) ActualOSAError b1 3.13123.144280.42% b2 2.88012.898480.64% b3 2.57742.605291.08% b4 2.20462.275623.22% b5 1.74971.806483.25% h1 64.448662.88562.43% h2 59.281157.96962.21% h3 53.05152.10581.78% h4 46.295745.51241.69% h5 36.741136.12971.66% F(X) 6756267034.10.78%

12. P=60,000 (20%) ActualOSAError b1 3.13123.193962.00% b2 2.88012.953622.55% b3 2.57742.684234.14% b4 2.20462.34276.26% b5 1.74971.85946.27% h1 66.273663.87933.61% h2 60.959759.07243.10% h3 54.553353.68431.59% h4 48.500246.85393.39% h5 38.490737.18813.38% F(X) 6979670151.90.51%

13. P=65,000 (30%) ActualOSAError b1 3.13123.265952.00% b2 2.88013.031472.55% b3 2.57742.754554.14% b4 2.20462.406066.26% b5 1.74971.909726.27% h1 68.098665.3193.61% h2 62.638460.62933.10% h3 56.055555.09091.59% h4 50.704848.12133.39% h5 40.240238.19433.38% F(X) 7203073759.80.51%

14. Elastic modulus Elastic modulus E=22,000,000 (10%) ActualOSAError b1 3.13122.991334.47% b2 2.88012.778013.55% b3 2.57742.523432.09% b4 2.20462.204810.01% b5 1.74971.750020.02% h1 60.011259.82650.31% h2 55.199555.56010.65% h3 49.398450.46862.17% h4 44.091144.09620.01% h5 34.991535.00040.03% F(X) 6326361913.62.13% E=24,000,000 (20%) ActualOSAError b1 3.13122.992594.43% b2 2.88012.777513.56% b3 2.57742.524152.07% b4 2.20462.204840.01% b5 1.74971.749920.01% h1 57.398859.85184.27% h2 52.796555.55015.22% h3 47.24850.4836.85% h4 44.091144.09650.01% h5 34.991534.99830.02% F(X) 61199619301.19%

15. E=26,000,000 (30%) ActualOSAError b1 3.13122.992584.43% b2 2.88012.777513.56% b3 2.57742.524172.07% b4 2.20462.204830.00% b5 1.74971.750040.02% h1 54.786459.85169.25% h2 50.393655.5510.23% h3 45.097650.483411.94% h4 44.091144.09640.01% h5 34.991535.00070.02% F(X) 5913561930.84.73%

17. Stress Stress Sigma=15,400 (10%) ActualOSAError b1 3.13123.09721.09% b2 2.88012.853680.92% b3 2.57742.553660.92% b4 2.20462.135793.12% b5 1.74971.694983.13% h1 63.41161.94392.31% h2 58.326757.07362.15% h3 52.19751.07312.15% h4 41.886542.71591.98% h5 33.241933.89971.98% F(X) 6515863383.82.72% Sigma=16,800 (20%) ActualOSAError b1 3.13123.117780.43% b2 2.88012.871890.29% b3 2.57742.569120.32% b4 2.20462.074575.90% b5 1.74971.646585.89% h1 64.198462.35562.87% h2 59.050957.43782.73% h3 52.845151.38252.77% h4 39.68241.49144.56% h5 31.492432.93174.57% F(X) 6498863167.62.80%

18. Sigma=18,200 (30%) ActualOSAError b1 3.13123.13180.02% b2 2.88012.892364.26% b3 2.57742.584792.87% b4 2.20462.029397.95% b5 1.74971.603258.37% h1 64.985862.6363.62% h2 59.775257.84713.23% h3 53.493351.69593.36% h4 37.477440.58778.30% h5 29.742832.06497.81% F(X) 6481863087.72.67%

20. Deflection Deflection Y=2.75 (10%) ActualOSAError b1 3.13122.991334.47% b2 2.88012.778013.55% b3 2.57742.523432.09% b4 2.20462.204810.01% b5 1.74971.750020.02% h1 60.054859.82650.38% h2 55.173255.56010.70% h3 49.374950.46862.22% h4 44.091144.09620.01% h5 34.991535.00040.03% F(X) 6326361913.62.13% Y=3 (20%) ActualOSAError b1 3.13122.99274.42% b2 2.88012.777833.55% b3 2.57742.524342.01% b4 2.2046 0.00% b5 1.74971.749840.01% h1 57.48659.85394.12% h2 52.74455.55665.33% h3 47.200950.48696.96% h4 44.091144.0920.00% h5 34.991534.99680.02% F(X) 6119961934.11.20%

21. Y=3.25 (30%) ActualOSAError b1 3.13122.992614.43% b2 2.88012.77753.56% b3 2.57742.524172.07% b4 2.20462.204820.01% b5 1.74971.750030.02% h1 54.917259.85228.99% h2 50.314855.5510.40% h3 45.02750.483412.12% h4 44.091144.09640.01% h5 34.991535.00070.02% F(X) 59135619314.73%