Optimisation of Trapezoidal Channel

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  • Topic: Golden ratio, Golden section search, Continued fraction
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  • Published : May 4, 2013
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Analytical Techniques for Engineers
Optimisation

1/22/2013
H8869939
Derek Grieveson

Optimisation
This technique according to Chapra (2004) is similar to root location. Instead of searching for a point of a function that intersects the x-axis, the idea is to find the minimum or maximum point of the function.

Fig. 1 Difference between root and optima finding A popular general optimisation technique is the Golden-section method. After guessing the interval containing the optima, two points X1 and X2 are calculated using the golden section formulae. Like the bisection method one section is discarded and becomes the new interval boundary. Further iterations will converge on the optima until an iteration limit or error condition is met. The trapezoidal channel has one minimum therefore it is unimodal. The golden section method will be used to find the minimum wetted area by optimising the angle and lengths. The initial guesses can be derived from the understanding that the maximum Area is 4m². This means the maximum length of the wetted area (figure 2) at 90° will be 6 metres. As the angle of the sides decreases the length of the sides will decrease then increase (see fig.2). This will give a suitable curve where the interval guesses will be XL = 0° and XU = 90°. 4m²

2mm
2mm
90°
Lower Guess = 0 rads
Upper Guess = (90°)
90x(π/180)=1.5708rads


4m²

2mm
2mm
90°
Lower Guess = 0 rads
Upper Guess = (90°)
90x(π/180)=1.5708rads


Fig.2 Logic for initial guesses Formula derivation for angle vs wetted perimeter.
Variables: Angle and length of wetted side S cos α
S sin α
Length - metres
Angles - Degrees
S cos α
S sin α
Length - metres
Angles - Degrees
B
B

Fig. 3 Trapezoid channel Formulate Area
(1) Area of trapezoid = 0.5(B+b)h (2) B= b+2 S cos α (3) h= S sine α Sub in (2) & (3) into (1).

A = 0.5(b + b + 2 S cos α)S sine α → A = (b + S cos α) S sine α If A = 4m² (4) 4 = (b + S cos α) S sine α Constraints:

The sides of the wetted perimeter (L) are equal
.·. S = b (5) .·. L = 3b → b = (L/3) Multiply brackets out and Sub (5) in (4).

4 = (L/3)² sine α (1 + cos α)
Solve for L
(6) L = 3 √4/ sine α( 1 + cos α)

To find the minimum wetted perimeter the Golden section technique uses two interior points (X1, X2) calculated from the golden ratio:

After each iteration the value of f1(x) and f2(x) are evaluated. A graph of the function will give a parabolic like curve with the minimum at the lowest point. To converge on the minimum the following rules are used:

If : f1 < f2 → X2 = XL
f1 < f2 So, X2 becomes the new XL
Further iterations will converge on the optima.
f1 < f2 So, X2 becomes the new XL
Further iterations will converge on the optima.
f2 < f1 → X1 = Xu
F1
F1
F2
F2

XL X2 X1 XU XL X2 X1 XU

Fig 4. Golden Search Method Spread sheet
The spread sheet follows the steps above. Inclusion of Degrees, Evaluation and Error are not necessary but can be used in deciding the outcome.

Fig 5 Design ‘A’ spread sheet
After twenty iterations the...
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