Analysis of Cyclic Fatigue Failure Data

School of Chemical and BioMedical

Engineering

Division of Bioengineering

BG2802 Bioengineering Year 2 Lab

Lab Report

Analysis of Cyclic Fatigue Failure Data

using Least-square Regression

Author:

Yang

WANG

a a

Matriculation Number: U1220560J

1 Background and Introduction . . . 2

2 Objectives . . . 2

3 Theories . . . 2

3.1 Euler’s Method . . . 2

3.2 4th _{order Runge-Kutta method . . . . 2}

4 Experiment Procedures . . . 3

5 Results and Analysis . . . 9

5.1 Euler’s Method . . . 9

5.2 4th _{order Runge-Kutta Method . . . . 10}

5.3 Comparison and Comments . . . 11

5.4 Epidemic Model . . . 12

6 Conclusion . . . 12

1

Background and Introduction

Solving differential equations is frequently required in handling engineering problems, however solving those equations by hands is not usually that easy. In many cases, there is no analytical solutions to some differential equations, ordinary or partial. Numerical methods to approximate the analytical solutions therefore is widely applied to get better results and simulations. In this experiment on computational methods for numerical analysis, we will use Matlab to apply Euler’s method and 4th _{order Runge-Kutta method for some data}

approximation. By comparing the outcomes of these methods we might able to tell the advantages and limitations in both cases. The third part of this experiment is to apply these methods to model a epidemic incident. By the aforementioned methods we will get a better understanding on the numerical analysis algorithms and apply these understandings into solving real cases in the future.

2

Objectives

There are three objectives for this experiment.

First

Learn to use Matlab to compute solution of differential equations using Euler’s method and 4th_{order Runge-Kutta method.}

Second

Study the effect of step size used in Euler’s method and 4th_{order Runge-Kutta method}

have on the solution of ODEs. Third

Apply the numerical methods on a simple epidemic spreading model.

3

Theories

3.1 Euler’s Method

For a general initial value problem:

y′₌ dy

dt =f(t, y), y(0) =y0

for a stepping size h, the Euler’s method for approximation gives

yk+1=yk+hf(tk, yk),

k∈ {N∩[0, M−1)},

h=b−a

M

by this, a series of points can be plotted, which is the approximation of the original function curve.

3.2 4th

order Runge-Kutta method

dy

dt =f(t, y) y(0) =y0

the 4th_{order Runge-Kutta method approximation uses the formula}

yk+1 =yk+

by expanding with 4th_{order Taylor series}

yi+1=yi+

and the general solution set is



1. Matlab Program for Euler’s Method (a) Euler.m

function [tout, yout] = Euler(FunFcn, tspan, y0, ssize)

% FunFcn is the function to be solved

5 % tspan is the interval

% y0 is the initial conditions

% ssize is the step size

% Initialization

15 c l f

t0 = tspan(1); %Start point

tfinal = tspan(2); %End point

20

i f (nargin < 4), ssize = (tfinal - t0)/100; end %calculate step size if

h = ssize;

25

t = t0;

y = y0(:);

30 tout = t;

yout = y.’;

35

% The main loop

while (t < tfinal)

40 i f t + h > tfinal, h = tfinal - t; end

% Compute the slope

s1 = f e v a l(FunFcn, t, y); s1 = s1(:); % s1=f(t(k),y(k))

45

t = t + h;

y = y + h*s1; % y(k+1) = y(k) + h*f(t(k),y(k))

50 tout = [tout; t];

yout = [yout; y.’];

end;

(b) f Euler.m

function z = f(t, y)

z = (t - y)/2; %DonŠt forget the ; at the end of the line

5 return

(c) Run Euler.m

%Define the parameters

tspan = [0 3];

5 y0 = 1;

%Call the Euler solver

10

[tout, yout] = Euler(’f’, tspan, y0, h);

fi g u re(2);

15 plot(tout,yout)

t i t l e([’Euler s Method Method using h=’, num2str(h), ’ by Stephen’]);

%To show the value of yout at t = 3 in the command window

20

[m n] = s i z e(yout);

yout(m)

2. Matlab program for 4th_{order Tunge-Kutta Method}

(a) rk4.m

function [tout, yout] = rk4(FunFcn, tspan, y0, ssize)

% FunFcn is the function to be solved % tspan is the interval

% y0 is the initial conditions

5 % ssize is the step size

15 % We need to compute the number of steps.

dt = abs(tfinal - t0); N = f l o o r(dt/ssize) + 1;

i f (N-1)*ssize < dt N = N + 1;

20 end

% Initialize the output.

tout = zeros(N,1);

% Compute the slopes

s1 = f e v a l(FunFcn, t, y); s1 = s1(:);

s2 = f e v a l(FunFcn, t + h/2, y + h*s1/2); s2=s2(:);

s4 = f e v a l(FunFcn, t + h, y + h*s3); s4=s4(:); y = y + h*(s1 + 2*s2 + 2*s3 +s4)/6;

t = tout(k); yout(k,:) = y.’;

45 end;

(b) f rk.m

function z = f(t, y)

z = (t - y)/2; %DonŠt forget the ; at the end of the line

5 return

(c) Run rk4.m

%Define the parameters and constants

tspan = [0 3];

5 y0 = 1;

h = 1/16;

%To call the rk4 solver

10

[tout, yout] = rk4(’f’, tspan, y0, h); %Results from rk4 is stored in

fi g u re(2);

15 plot(tout,yout)

t i t l e([’Runge Kutta Method using h=’, num2str(h), ’ by Stephen’]);

%To show the value of yout at t = 3 in the command window

20

[m n] = s i z e(yout);

yout(m)

3. Epidemic Model

(a) Parameter adjustment Euler P5.m and rk4 P5.m , define new function f2.m

function [tout, yout] = Euler(FunFcn, tspan, y0, ssize)

% FunFcn is the function to be solved

5 % tspan is the interval

% y0 is the initial conditions

% ssize is the step size

10

% Initialization

15 c l f

tfinal = tspan(2); %End point

20

i f (nargin < 4), ssize = (tfinal - t0)/100; end %calculate step size if

h = ssize;

while (t < tfinal)

40 i f t + h > tfinal, h = tfinal - t; end

% Compute the slope

s1 = f e v a l(FunFcn, t, y); s1 = s1(:); % s1=f(t(k),y(k))

function [tout, yout] = rk4(FunFcn, tspan, y0, ssize)

% FunFcn is the function to be solved % tspan is the interval

% y0 is the initial conditions

5 % ssize is the step size

15 % We need to compute the number of steps.

dt = abs(tfinal - t0); N = f l o o r(dt/ssize) + 1;

i f (N-1)*ssize < dt N = N + 1;

20 end

% Initialize the output.

tout(1) = t;

% Compute the slopes

s1 = f e v a l(FunFcn, t, y); s1 = s1(:);

(b) Run Euler approximation Run Euler P5.m

%Define the parameters

tspan = [0 20];

5 y0 = 250;

h = 0.2;

%Call the Euler solver

10

[tout, yout] = Euler(’f2’, tspan, y0, h);

fi g u re(2);

15 plot(tout,yout)

t i t l e([’Euler s Method Method using h=’, num2str(h), ’ by Stephen’]);

%To show the value of yout at t = 3 in the command window

20

[m n] = s i z e(yout);

yout(m)

(c) Run 4th_{order Tunge-Kutta approximation Run rk4 P5.m}

tspan = [0 20];

5 y0 = 250;

h = 0.2;

%To call the rk4 solver

10

[tout, yout] = rk4(’f2’, tspan, y0, h); %Results from rk4 is stored in

fi g u re(2);

15 plot(tout,yout)

t i t l e([’Runge Kutta Method using h=’, num2str(h), ’ by Stephen’]);

%To show the value of yout at t = 3 in the command window

20

[m n] = s i z e(yout);

yout(m)

5

Results and Analysis

5.1 Euler’s Method

h t_{= 3}

5.2 4th

order Runge-Kutta Method

h t_{= 3}

5.3 Comparison and Comments

Obviously 4th_{order Runge-Kutta Method provided a better and faster simulation than}

Euler’s Method 4th

order Runge-Kutta Method

t_{= 20 2}._3625×104 ₂._3702×104

5.4 Epidemic Model

6

Conclusion

This experiment practice some fundamental approximation using Matlab on Euler’s method and 4th _{order Runge-Kutta method. In real cases there are more complicated}