Transcript Numerical Integration of Ship Forms (Review of CVEN 302)

Chapter III.
Numerical Integration for
Ship Forms
Review of CVEN 302
Data of Ship forms
• Discrete data (Line drawings, stations, water plane
etc)
• Evenly distributed (most times)
Methods of Numerical Integration
•Trapezoidal rule (linear)
•Sinpson’s 1/3 rule (quadratic)
•Simpson’s 3/8 rule (cubic)
•Multiple applications
•Tchebycheff’s (similar to Gauss Quadrature) rule
-applied to a continues function
 fn (x) can be linear
 fn (x) can be quadratic
 fn (x) can also be cubic or other
higher-order polynomials
Trapezoidal Rule (single Application)
• Linear approximation

b
a
1
f ( x )dx   c i f ( x i )  c0 f ( x0 )  c 1 f ( x1 )
i 0
h
  f ( x0 )  f ( x 1 )
2
f(x)
L(x)
x0
x1
x
Multiple Applications of Trapezoidal Rule

b
a
x1
x2
xn
x0
x1
xn  1
f ( x )dx   f ( x )dx   f ( x )dx    
f ( x )dx
h
 f ( x0 )  f ( x 1 )   h  f ( x 1 )  f ( x 2 )     h  f ( x n 1 )  f ( x n ) 
2
2
2
h
  f ( x0 )  2 f ( x1 )    2f ( x i )    2 f ( x n1 )  f ( x n )
2

f(x)
ba
h
n
x0
h
x1
h
x2
h
x3
h
x4
x
Simpson’s 1/3-Rule (single application)
• Approximate the function by a
parabola
2
b
c i f ( x i )  c 0 f ( x0 )  c 1 f ( x 1 )  c 2 f ( x 2 )
a f ( x )dx  
i 0

h
 f ( x0 )  4 f ( x 1 )  f ( x 2 )
3
L(x)
f(x)
x0
h
x1
h
x2
x
Multiple Applications of Simpson’s 1/3 Rule
 Applicable only if the number of segments is even
Multiple Applications of Simpson’s 1/3 Rule
ba
h
n
n must be even
f ( x0 )  4 f ( x 1 )  f ( x 2 )
f ( x2 )  4 f ( x3 )  f ( x4 )
 2h
6
6
f ( x n 2 )  4 f ( x n 1 )  f ( x n )
   2h
6
I  2h
n 1
n 2

( b  a) 
I
 f ( x0 )  4  f ( x i )  2  f ( x j )  f ( x n ) 
3n 
i  1, 3 , 5
j 2 , 4 ,6

Simpson’s 3/8-Rule (single application)
 Approximate by a cubic polynomial

b
a
3
f ( x )dx   c i f ( x i )  c0 f ( x0 )  c 1 f ( x1 )  c 2 f ( x 2 )  c 3 f ( x 3 )
i 0

3h
 f ( x0 )  3 f ( x 1 )  3 f ( x 2 )  f ( x 3 ) 
8
L(x)
x0
h
f(x)
x1
h
x2
h
x3
x
Tchebycheff’s rule
Sum of ordinates (stations)
I  Length 
# of ordinates
See Table 4.3 at p58
•Positions of ordinates (stations) depending on
how many ordinates are used
•Odd # of ordinates is preferred