Num Integration

Numerical IntegrationPengantar Teknologi Kelautan

Adi Wirawan Husodo

Why numerical integration?

◦ Ship is complex and its shape cannot usually be presented by mathematical equation.

◦ Numerical scheme, therefore, should be used to calculate the ship’s geometrical properties.

Which numerical method ?

◦ Trapezoidal rule◦ Simpson’s 1st rule◦ Simpson’s 2nd rule

Contoh-contoh

Contoh-contoh

Contoh-contoh

Trapezoidal rule (skip)

- uses 2 data points - assume linear curve

x1 x2 x3 x4s s s

y1 y2 y3

y4

A1 A2 A3

: y=ax+b

Total Area = A1+A2+A3 = s/2 (y1+2y2+2y3+y4)

A1=s/2 (y1+y2)A2=s/2 (y2+y3)A3=s/2 (y3+y4)

Simpson’s 1st Rule

- uses 3 data points - assume 2nd order polynomial curve

Area : )4(3

321

3

1yyy

sdxydAA

x

x

x1 x3

y(x)=ax²+bx+c

x

y

A

dx

x1 x2 x3s

y1 y2 y3

x

y

AdA

Mathematical Integration Numerical Integration

x2s

y(x)=ax²+bx+c

Simpson’s 1st Rule (cont)

x1 x2 x3

s

y1 y2 y3

x

y

x4 x5 x6 x7 x8 x9

y4y5

y6 y7y8 y9

Gen. Eqn.

Odd number

)y4y2y...2y4y(y3

sA n1n2n321

)4242424(3

)4(3

)4(3

)4(3

)4(3

987654321

987765

543321

yyyyyyyyys

yyys

yyys

yyys

yyys

A

- uses 4 data points - assume 3rd order polynomial curve

x1 x2 x3s s

y1 y2 y3 y(x)=ax³+bx²+cx+d

x

y

Area : )33(8

34321 yyyy

sA

A

x4

y4

Simpson’s 2nd Rule (skip)

Application of Numerical Integration

• Application

- Waterplane Area

- Sectional Area

- Submerged Volume

- LCF

- VCB

- LCB• Scheme

- Simpson’s 1st Rule

Numerical Calculation• Calculation Steps

1. Start with a picture of what you are about to integrate.

2. Show the differential element you are using.

3. Properly label your axis and drawing.

4. Write out the generalized calculus equation written in

the same symbols you used to label your picture .

5. Write out Simpson’s equation in generalized form.

6. Substitute each number into the generalized Simpson’s

equation.

7. Calculate final answer.

Not optional ! Always follow the above steps!

Waterplane Area

y

x

dxFPAP

y(x)

area

LppWP dxxydAA

0 )( 2 2

) width(aldifferenti

)(at breadth)-foffset(hal )(

)area( aldifferenti

)area( planewater 2

2

ftdx

ftxyxy

ftdA

ftAWP

Factor for Symmetric W.A.

Waterplane Area(cont.)

• Generalized Simpson’s Equation

..24y 3

1 2 210 yyxAWP

stations between distancex

y

x

FP AP0 1 2 3 4 5 6

x

Sectional Area

• Sectional Area : Numerical integration of half-breadth as a function of draft

WL

z

y

dz

y(z)T

area

Tt dzzydAA

0sec )( 2 2

) width(aldifferenti

)z(at breadth)-foffset(hal )(

)area( aldifferenti

)( toup area sectional2

2sec

ftdz

ftyzy

ftdA

ftzA t

Sectional Area(cont.)• Generalized Simpson’s equation

s waterlinebetween distancez

nn

area

T

t

yyyyz

dzzydAA

1210

0sec

4..24y 3

1 2

)( 2 2

z

y

WL

T

0

24

68

z

Submerged Volume : Longitudinal Integration

• Submerged Volume : Integration of sectional area over the length of ship

• Scheme z

x

y)(xAs

Submerged Volume

• Sectional Area Curve

• Calculus equation

volume

L

tssubmerged

pp

dxxAdVV0

sec )(

x

As

FP AP

dx

)(sec xA t

• Generalized equation

nns yyyyx 1210 4..24y 3

1

stations between distancex

Longitudinal Center of Floatation (LCF)

• LCF - Centroid of waterplane area - Distance from reference point to center of floatation - Referenced to amidships or FP - Sign convention of LCF

+

+-

FP

WL

Center of Flotation Merupakan titik berat dari luas bidang garis

air (water plane area). Suatu titik dimana kapal mengalami heel

atau trim. Titik ini terletak pada centre line (dalam

arah memanjang), disekitar midship (bisa di depan atau dibelakang midship).

contoh

Disebut juga dengan KB (Keel to Buoyancy)

Merupakan titik berat dari volume displacement kapal

KB atau VCB =

Vertical center of buoyancy (VCB)

ntdisplaceme vol.

keel about themoment total

Contoh KB