Chapter III. Numerical Integration for Ship Forms Review of CVEN 302.
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Chapter III. Numerical Integration for Ship Forms Review of CVEN 302
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Transcript of Chapter III. Numerical Integration for 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 RuleTrapezoidal Rule (single Application)(single Application)
• Linear approximation
)()(
)()()()(
10
1100i
1
0ii
b
a
xfxf2
h
xfcxfcxfcdxxf
x0 x1x
f(x)
L(x)
Multiple Applications of Trapezoidal RuleMultiple Applications of Trapezoidal Rule
)()()()()(
)()()()()()(
)()()()(
n1ni10
n1n2110
x
x
x
x
x
x
b
a
xfxf2x2fxf2xf2
h
xfxf2
hxfxf
2
hxfxf
2
h
dxxfdxxfdxxfdxxfn
1n
2
1
1
0
x0 x1x
f(x)
x2h h x3h h x4
n
abh
Simpson’s 1/3-Rule (single application)Simpson’s 1/3-Rule (single application)• Approximate the function by a
parabola
)()()(
)()()()()(
210
221100i
2
0ii
b
a
xfxf4xf3
h
xfcxfcxfcxfcdxxf
x0 x1x
f(x)
x2h h
L(x)
Multiple Applications of Simpson’s 1/3 RuleMultiple Applications of Simpson’s 1/3 Rule
Applicable only if the number of segments is even
Multiple Applications of Simpson’s 1/3 RuleMultiple Applications of Simpson’s 1/3 Rule
n
abh
1n
531i
2n
642jnji0 xfxf2xf4xf
n3
abI
,, ,,
)()()()()(
6
xfxf4xfh2
6
xfxf4xfh2
6
xfxf4xfh2I
n1n2n
432210
)()()(
)()()()()()(
n must be even
Simpson’s 3/8-Rule (single application)Simpson’s 3/8-Rule (single application)
Approximate by a cubic polynomial
)()()()(
)()()()()()(
3210
33221100i
3
0ii
b
a
xfxf3xf3xf8
h3
xfcxfcxfcxfcxfcdxxf
x0 x1x
f(x)
x2h h
L(x)
x3h
Tchebycheff’s rule
Sum of ordinates (stations)Length
# of ordinatesI
See Table 4.3 at p58
•Positions of ordinates (stations) depending on how many ordinates are used
•Odd # of ordinates is preferred
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