Breaking Ice with Air Cushion Vehicle Using Newton Raphson Method

Introduction

In numerical analysis, Newton\'s method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. It is one example of a root-finding algorithm.

In this project we will perform the minimum cushion pressure needed to break a given thickness of ice using an air cushion vehicle, Muller (“Ice Breaking with an Air Cushion Vehicle”) in Mathematical Modeling: Classroom Notes in Applied Mathematics, SIAM 1987) derived the equation

i.e.,

Code

#Newton-Raphson Method By Harsh Chimmani
import matplotlib.pyplot as plt
import math
import numpy as np

# Function
def f(p,h):
   return (pow(p,3)*(1-pow(beta,2))) + (((0.4*h*pow(beta,2))-(sigma*pow(h,2)/pow(r,2)))
*pow(p,2)) + (((pow(sigma,2)*pow(h,4))/(3*pow(r,4)))*p)-(pow((sigma*pow(h,2)/(3*pow(r,2)))
,3))

#Derivation of Function 
def fprime(p,h):
   return (3*(pow(p,2)*(1-pow(beta,2)))) + (2*p*((0.4*h*pow(beta,2))-(sigma*pow(h,2)/po
w(r,2)))) + ((pow(sigma,2)*pow(h,4))/(3*pow(r,4)))

#Performing Newton Raphson Method Iteration
#Assigning Values
# Width of the ice wedge
beta = 0.5
# Size of the air cushion (in feet)
r = 40
# Tensile strength of the ice (psi)
sigma = 150
# Thickness of the ice field
height = [0.6,1.2,1.8,2.4,3.0,3.6,4.2]
# Cushion Pressure
p_guess = 0.2
# Relaxation Factor
alpha = 0.8
#Tolerance Error
tol = 1e-4
# Pressure
Pressure = []
iter = 1
  for h in height :
      while(abs(f(p_guess,h))>tol):
           p_guess = p_guess - ((f(p_guess,h)/fprime(p_guess,h))*alpha)
           iter=iter +1
      Pressure.append(p_guess)
for i in range(0,7) :
      if i == 0:
           print (\" : height : Pressure \" )
      print(\" : \" , height[i] , \" : \" , Pressure[i] )

plt.plot(height,Pressure)
plt.xlabel(\'Height (feet)\')
plt.ylabel(\'Pressure (psi)\')
plt.title(\'Pressure Vs Height - Newton-Raphson Method Solution\')
plt.xlim([0,5])
plt.ylim([0,0.3])
plt.show()

Results

Graph

Observation

The minimum pressure of 0.033 Psi is required for breaking the ice at height of 0.6 feet .