SIMPSON’S RULE vs MATHEMATICAL INTEGRATION
for Volume, Area and other calculations for Ship Shape
Traditionally Naval Architects have been using Simpson’s rule [1-3-1, 1-3-3-1] for hydrostatic calculations which was a great tool for the times when computing was not evolved. Lets check the error that got introduced with the usage of Simpson’s rule in today’s era by comparing the results with the direct curve fitting and integration of the curve over the waterlines.
Consider the below set of cordinates for a station. WL on X-Axis and half breadths on Y-Axis.
x (waterline) = [0.155, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.5, 16, 18, 20, 22, 24.2]
y (half breadth) = [0, 1.3056236068274, 2.16, 2.55, 2.778, 2.891, 2.894, 2.784, 2.569, 2.231, 1.76, 1.385, 1.116, 1.279, 1.685, 2.532, 4.15, 6.237, 8.448, 11.2]
Pic: Curve plotted by finding a mathematical equation passing through each of the x,y cordinates
Area calculated by simpson’s rule: 85.67678287568746 m2
Area calculated by Integration: 85.50639852678901 m2 (after fitting a mathematical spline curve)
error : 85.67678287568746 – 85.50639852678901 = 0.170384349 m2
% error = [0.170384349/85.50639852678901]*100 = 0.199265%
Simpson’s area calculation % error is ~0.2% just for one station.
Imagine the error in GZ curve or Hydrostatics or Cross Curves or Damage Stability using Simpson’s method....!!!!
The naval architects of navalarchitect-design.com have decades of experience in Existing Ship surveys in India, China, US and Canada and new building surveys of Ships while working at IACS classification societies. And in the entire careers have never sighted a Stability Booklet or Loading Manual where it is specifically made a disclosure which mathematical route was taken for arriving at the calculations Simpson’s Rule or the more accurate Mathematical Integration.
Mathematical Integration is a computer resource hungry method (RAM and processor) where the end results of calculation (thousands of mathematical integrations) might take from a minute to an hour depending on the number of mathematical curves generated and integrations over the limits. And therefore the swiftness with which the modern day onboard Lodicators and Stability Programs calculate the results begs to answer which method was used to calculate the results. Those onboard computers are the same home/office use desktops.
Naval architects at navalarchitect-design.com use direct mathematical integrations for ship hydrostatics calculations and committed to using mathematical methods instead of using shorthcuts aka approximations and assumptions.