MAE550 PROJECTMAE550 PROJECT

By: LinBy: Lin TomTom YanYan Lv Lv

1 Introduction1 Introduction OSA is a method of estimating the OSA is a method of estimating the

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

There are two general approaches to the There are two general approaches to the OSA problem:OSA problem:

o (1) Base on Kuhn-Tucker conditions.(1) Base on Kuhn-Tucker conditions.o (2) Use the concept of a feasible direction.(2) Use the concept of a feasible direction.

In this project, we use the first one to obtain In this project, we use the first one to obtain estimated optimum solution of cantilevered estimated optimum solution of cantilevered beam problem shown below (in the book, beam problem shown below (in the book, Page-184).Page-184).

OSA Algorithm:OSA Algorithm:

We only consider the active constraintWe only consider the active constraint

Jj

jj XgXF 0)()(

0)( Xg j Jj

0j

)()(22

XgXX

XFXX

A jkiJj

jki

ik

)(XgX

B ji

ij

Jj

)()(22

XgpX

XFpX

c jiJj

ji

i

)(Xgp

d jj

Jj

p

X

p

Xp

X

X

n

.

.

.

2

1

p

p

p

J

.

.

.

2

1

Then we can form the matrix:Then we can form the matrix:

At last, we derive the optimum solution:At last, we derive the optimum solution:

j

jjp

Jj

00 1

1

J

n

JJnJ

Jnnn

d

cX

B

BA

)(*

**oldnewoldnew PP

p

XXX

)()( *

**oldnewoldnew PP

dp

XdFFF

2 Cantilevered Beam Problem2 Cantilevered Beam Problem

Figure 2-1Figure 2-1

Minimize:Minimize:

N=5, for conveniences, N=5, for conveniences, we assume eachwe assume each

In this problem, we use In this problem, we use optimum solution from optimum solution from Dot program (method 3) Dot program (method 3) which is more accurate which is more accurate than the Master due to than the Master due to direct handling issues.direct handling issues.

N

iiii lhbV

1

100l

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)

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(13)=1-X(3)

G(14)=1-X(4)G(14)=1-X(4)

G(15)=1-X(5)G(15)=1-X(5)

G(16)=5-X(6)G(16)=5-X(6)

G(17)=5-X(7)G(17)=5-X(7)

G(18)=5-X(8)G(18)=5-X(8)

G(19)=5-X(9)G(19)=5-X(9)

G(20)=5-X(10)G(20)=5-X(10)

G(21)=(0.032*P*L*L*L/(X(5)*X(10)*X(10)*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

3 OSA Calculation3 OSA Calculation

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

Load Load P=55,000 (10%)

Actual OSA Error

b1 3.1312 3.14428 0.42%

b2 2.8801 2.89848 0.64%

b3 2.5774 2.60529 1.08%

b4 2.2046 2.27562 3.22%

b5 1.7497 1.80648 3.25%

h1 64.4486 62.8856 2.43%

h2 59.2811 57.9696 2.21%

h3 53.051 52.1058 1.78%

h4 46.2957 45.5124 1.69%

h5 36.7411 36.1297 1.66%

F(X) 67562 67034.1 0.78%

NPorigin 000,50

P=60,000 (20%)

Actual OSA Error

b1 3.1312 3.19396 2.00%

b2 2.8801 2.95362 2.55%

b3 2.5774 2.68423 4.14%

b4 2.2046 2.3427 6.26%

b5 1.7497 1.8594 6.27%

h1 66.2736 63.8793 3.61%

h2 60.9597 59.0724 3.10%

h3 54.5533 53.6843 1.59%

h4 48.5002 46.8539 3.39%

h5 38.4907 37.1881 3.38%

F(X) 69796 70151.9 0.51%

P=65,000 (30%)

Actual OSA Error

b1 3.1312 3.26595 2.00%

b2 2.8801 3.03147 2.55%

b3 2.5774 2.75455 4.14%

b4 2.2046 2.40606 6.26%

b5 1.7497 1.90972 6.27%

h1 68.0986 65.319 3.61%

h2 62.6384 60.6293 3.10%

h3 56.0555 55.0909 1.59%

h4 50.7048 48.1213 3.39%

h5 40.2402 38.1943 3.38%

F(X) 72030 73759.8 0.51%

Elastic modulusElastic modulus E=22,000,000 (10%)

Actual OSA Error

b1 3.1312 2.99133 4.47%

b2 2.8801 2.77801 3.55%

b3 2.5774 2.52343 2.09%

b4 2.2046 2.20481 0.01%

b5 1.7497 1.75002 0.02%

h1 60.0112 59.8265 0.31%

h2 55.1995 55.5601 0.65%

h3 49.3984 50.4686 2.17%

h4 44.0911 44.0962 0.01%

h5 34.9915 35.0004 0.03%

F(X) 63263 61913.6 2.13%

27 /100.2 cmNEorigin

E=24,000,000 (20%)

Actual OSA Error

b1 3.1312 2.99259 4.43%

b2 2.8801 2.77751 3.56%

b3 2.5774 2.52415 2.07%

b4 2.2046 2.20484 0.01%

b5 1.7497 1.74992 0.01%

h1 57.3988 59.8518 4.27%

h2 52.7965 55.5501 5.22%

h3 47.248 50.483 6.85%

h4 44.0911 44.0965 0.01%

h5 34.9915 34.9983 0.02%

F(X) 61199 61930 1.19%

E=26,000,000 (30%)

Actual OSA Error

b1 3.1312 2.99258 4.43%

b2 2.8801 2.77751 3.56%

b3 2.5774 2.52417 2.07%

b4 2.2046 2.20483 0.00%

b5 1.7497 1.75004 0.02%

h1 54.7864 59.8516 9.25%

h2 50.3936 55.55 10.23%

h3 45.0976 50.4834 11.94%

h4 44.0911 44.0964 0.01%

h5 34.9915 35.0007 0.02%

F(X) 59135 61930.8 4.73%

StressStress Sigma=15,400 (10%)

Actual OSA Error

b1 3.1312 3.0972 1.09%

b2 2.8801 2.85368 0.92%

b3 2.5774 2.55366 0.92%

b4 2.2046 2.13579 3.12%

b5 1.7497 1.69498 3.13%

h1 63.411 61.9439 2.31%

h2 58.3267 57.0736 2.15%

h3 52.197 51.0731 2.15%

h4 41.8865 42.7159 1.98%

h5 33.2419 33.8997 1.98%

F(X) 65158 63383.8 2.72%

2/000,14 cmNorigin Sigma=16,800 (20%)

Actual OSA Error

b1 3.1312 3.11778 0.43%

b2 2.8801 2.87189 0.29%

b3 2.5774 2.56912 0.32%

b4 2.2046 2.07457 5.90%

b5 1.7497 1.64658 5.89%

h1 64.1984 62.3556 2.87%

h2 59.0509 57.4378 2.73%

h3 52.8451 51.3825 2.77%

h4 39.682 41.4914 4.56%

h5 31.4924 32.9317 4.57%

F(X) 64988 63167.6 2.80%

Sigma=18,200 (30%)

Actual OSA Error

b1 3.1312 3.1318 0.02%

b2 2.8801 2.89236 4.26%

b3 2.5774 2.58479 2.87%

b4 2.2046 2.02939 7.95%

b5 1.7497 1.60325 8.37%

h1 64.9858 62.636 3.62%

h2 59.7752 57.8471 3.23%

h3 53.4933 51.6959 3.36%

h4 37.4774 40.5877 8.30%

h5 29.7428 32.0649 7.81%

F(X) 64818 63087.7 2.67%

DeflectionDeflection Y=2.75 (10%)

Actual OSA Error

b1 3.1312 2.99133 4.47%

b2 2.8801 2.77801 3.55%

b3 2.5774 2.52343 2.09%

b4 2.2046 2.20481 0.01%

b5 1.7497 1.75002 0.02%

h1 60.0548 59.8265 0.38%

h2 55.1732 55.5601 0.70%

h3 49.3749 50.4686 2.22%

h4 44.0911 44.0962 0.01%

h5 34.9915 35.0004 0.03%

F(X) 63263 61913.6 2.13%

cmyorigin 5.2

Y=3 (20%)

Actual OSA Error

b1 3.1312 2.9927 4.42%

b2 2.8801 2.77783 3.55%

b3 2.5774 2.52434 2.01%

b4 2.2046 2.2046 0.00%

b5 1.7497 1.74984 0.01%

h1 57.486 59.8539 4.12%

h2 52.744 55.5566 5.33%

h3 47.2009 50.4869 6.96%

h4 44.0911 44.092 0.00%

h5 34.9915 34.9968 0.02%

F(X) 61199 61934.1 1.20%

Y=3.25 (30%)

Actual OSA Error

b1 3.1312 2.99261 4.43%

b2 2.8801 2.7775 3.56%

b3 2.5774 2.52417 2.07%

b4 2.2046 2.20482 0.01%

b5 1.7497 1.75003 0.02%

h1 54.9172 59.8522 8.99%

h2 50.3148 55.55 10.40%

h3 45.027 50.4834 12.12%

h4 44.0911 44.0964 0.01%

h5 34.9915 35.0007 0.02%

F(X) 59135 61931 4.73%