The minimum value of pressure on Ice - Hovercraft using the newton raphson method in python.

Problem Statement:

For this challenge, you will be writing a python script to solve for the minimum value of pressure using the newton raphson method.

where p denotes the cushion pressure, h the thickness of the ice field, r the size of the air cushion, σ the tensile strength of the ice and β is related to the width of the ice wedge.

Take β = 0.5, r = 40 feet and σ = 150 pounds per square inch (psi).

Do not convert to the metric system, however, make sure the units are consistent before proceeding further with the problem.

You need to,

1. Use the newton-raphson method to find out the value of pressure for h = 0.6 ft.

2. Find the optimal relaxation factor for this problem with the help of a suitable plot.

3. Tabulate the results of p for h = [0.6,1.2,1.8,2.4,3,3.6,4.2] assuming a suitable relaxation factor.

Theory:

In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f ′, and an initial guess x₀ for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then

is a better approximation of the root than x₀. Geometrically, (x₁, 0) is the intersection of the x-axis and the tangent of the graph of f at (x₀, f (x₀)): that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as

until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.

Python Program:

"""
Newton - Raphson method using python.

"""
#Author- Asad Salah
#importing modules

import matplotlib.pyplot as plt

#function defination

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)*p/(3*pow(r,4))-pow((sigma*pow(h,2))/(3*pow(r,2)),3)


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

#inputs
p_guess=10000
alpha=1
tol=1e-3
beta=0.5
r=40
sigma=150
pressure=[]
Thickness=[0.6,1.2,1.8,2.4,3.0,3.6,4.2]

iter=1

for h in Thickness:
	while(abs(f(p_guess,h))>tol):
		p_guess = p_guess  - alpha*(f(p_guess,h)/fprime(p_guess,h))
		iter = iter + 1
	pressure.append(p_guess)

for i in range(0,7):
	if i == 0:
		print(" : Thickness : pressure")
	print(" : ", Thickness[i]," : ", pressure[i])

print("Number of iterations = ",iter)
plt.plot(Thickness,pressure)
plt.title('Pressure Vs Thickness')
plt.xlabel('Thickness')
plt.ylabel('Pressure')
plt.show()	

The above program will solve for the value of the cushion pressure for the different thickness of the ice field. It is using the Newton-Raphson method to find the roots of the equation which indicates the pressure of the ice upon the hovercraft. The Newton-Raphson method is an iterative method.

This method requires us to give an initial value, which is basically a guess value given by the solver (In this program it is the P_guess = 10000).

This program uses two functions(,i.e,f(p,h) and fprime(p,h)), which calculates the equation and its derivative for the inputed values,which are used for evaluation by the Newton-Raphson method.

The graph on the left shows the variation of the pressure relative to the thickness, and the right shows the output for the above-given program.

The graph delineates that the pressure upon the air cushion decreases as the thickness of the ice field decreases. At a certain point on the graph onwards, the value of the pressure is constant or with insignificant variations. From the graph, we can get that the minimum pressure applied on the air cushion is 0.066(psi). And this observed pressure is constant for a range of the variations in the thickness of the ice field. This variation ranges approximately from 2.4 feet to 0.6 feet.

The output at right shows a table that depicts the values of the pressure for the different values of the thickness of the ice field. The value of the minimum pressure approximated by us from the graph conforms with the value output at the table. A total of 35 iterations have taken place until a constant pressure value is reached for a unity relaxation factor.

So from these results, we can conclude that the minimum pressure will depend on the minimum thickness of the ice field. And for a unity relaxation factor, it seems that the deterministic values are varying gradually and its under-relaxing.

Python Program:

"""
Optimizing the relaxation factor

"""
#Author - Asad Salah (B.E Mechanical)

import matplotlib.pyplot as plt


#function defination
def f(p,h,beta,sigma,r):
	t1=(1-pow(beta,2))*pow(p,3)
	t2=(0.4*h*pow(beta,2)-(sigma*pow(h,2))/(pow(r,2)))*pow(p,2)
	t3=(p*pow(sigma,2)*pow(h,4))/(3*pow(r,4))
	t4=pow((sigma*pow(h,2))/(3*pow(r,2)),3)
	t=t1+t2+t3-t4
	return t


def fprime(p,h,beta,sigma,r):

	tp1=3*pow(p,2)*(1-pow(beta,3))
	tp2=2*p*(0.4*h*pow(beta,2)-(sigma*pow(h,2))/(pow(r,2)))
	tp3=pow(sigma,2)*pow(h,4)/(3*pow(r,4))
	tp=tp1+tp2+tp3 
	return tp

# Input Values
h = 0.6				
beta = 0.5			
r = 40			
sigma = 150	
p_guess = 10000
Pressure =[]
iterations = []
tol = 1e-3
alpha_array =[]

#looping
for alpha in range(1,90):
	alpha = alpha*0.07
	p_guess = 10000
	alpha_array.append(alpha)

	iter = 1
	while (abs(f(p_guess,h,beta,sigma,r))>tol):		 
		p_guess = p_guess - alpha*(f(p_guess,h,beta,sigma,r)/fprime(p_guess,h,beta,sigma,r))	
		iter = iter + 1
	iterations.append(iter)
	Pressure.append(p_guess)

#plotting 
plt.plot(alpha_array, iterations)
plt.xlabel('Relaxation Factor')
plt.ylabel('No. of iterations')
plt.show()

The above program is used for optimizing the relaxation factor used.

For evaluating the optimum relaxation factor we are comparing the relaxation factor relative to the number of iterations for the corresponding relaxation factor.

In this program, there is a constant value usage of the thickness of the ice field.

Output:

The above graph describes the relationship between the relaxation factor relative to its corresponding number of iterations.

The curve depicted in the graph is an exponentially decreasing curve. From the graph we can understand that the no. of iterations is very high for lower values of the relaxation factor, therefore it is acting as an under-relaxation entity(,i.e, it is taking very small interval values for the iterations), but as the value of the relaxation factor increases there is a rapid drop in the no. of iterations(,i.e the values are taken with larger interval gap) and as the relaxation factor reaches an approximate value of 2.5 there is a slight increase again in the no. of iterations, this is indicating the overthrow nature of the interation depending on the alpha value.

Therefore for this case, the optimum value for the relaxation factor is 3.5, inferring from the graph. (,i.e, for a relaxation value of 3.5 the no.of iterations is two)

References:

https://en.wikipedia.org/wiki/Newton's_method