Chapter III. Numerical Integration for Ship Forms

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Chapter III. Numerical Integration forShip
Forms
Review of CVEN 302
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• Data of Ship forms
• Discrete data (Line drawings, stations, water
  plane etc)
• Evenly distributed (most times)

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• Methods of Numerical Integration
• Trapezoidal rule (linear)
• Sinpsons 1/3 rule (quadratic)
• Simpsons 3/8 rule (cubic)
• Multiple applications
• Tchebycheffs (similar to Gauss Quadrature) rule
• -applied to a continues function

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• fn (x) can be linear
• fn (x) can be quadratic

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• fn (x) can also be cubic or other higher-order
  polynomials

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Trapezoidal Rule (single Application)

• Linear approximation

f(x)
L(x)
x
x0
x1
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Multiple Applications of Trapezoidal Rule
f(x)
x
x0
x1
x2
h
h
x3
h
h
x4
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Simpsons 1/3-Rule (single application)

• Approximate the function by a parabola

L(x)
f(x)
x
x0
x1
x2
h
h
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Multiple Applications of Simpsons 1/3 Rule

• Applicable only if the number of segments is even

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Multiple Applications of Simpsons 1/3 Rule
n must be even
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Simpsons 3/8-Rule (single application)

• Approximate by a cubic polynomial

f(x)
L(x)
x
x0
x1
x2
h
h
x3
h
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Tchebycheffs rule

• See Table 4.3 at p58
• Positions of ordinates (stations) depending on
  how many ordinates are used
• Odd of ordinates is preferred