Computation & Simulation

Assignment TwoNumerical methods for solving ODEs

Group 5Xiang Wang Issraa Alhiti Padraig O'Dwyer

Tutor: Dr. Conor Brennan

Use of Numerical Solvers

1-Car companies can improve crash safety of their models› This article describes a virtual crash test dummy simulation in

which the crash effects on two software generated android models were analysed. The crash simulation resulted in a series of ODEs which represented dynamic effects on these virtual dummies. By solving the ODEs using numerical methods (Runge–Kutta) the authors were able to compute the generalized coordinates, their time derivatives and the body forces of the two models。

› Bud Fox, Leslie S. Jennings, and Albert Y. Zomaya.1999. Numerical computation of differential-algebraic

equations for nonlinear dynamics of multibody android systems in automobile crash simulation.› IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 46, NO. 10, OCTOBER 1999. Available from Inspec (EI

Engineering Village) database .http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=790496&isnumber=17164

› [Accessed 18 March 2009].

Use of Numerical Solvers

2- Approximation of artificial satellite trajectories and planetary ephemerides.› ODEs are solved using an integration subroutine which

generates the body's coordinates and time derivatives and hence produces the planetary ephemerides and satellite trajectories for a time interval. Computer simulation and graphical output indicates the satellite and planetary positions.

› Fox, B.; Jennings, L.S.; Zomaya, A.Y.2000. Numerical computation of differential-algebraic equations for the approximation of artificial satellite trajectories and planetary ephemerides.

› Transactions of the ASME. Journal of Applied Mechanics, v 67, n 3, 574-80, Sept. 2000› Available from Inspec (EI Engineering Village) database .› http://www.engineeringvillage2.org/controller/servlet/Controller?SEARCHID=da37721201089cf30M488

6prod2data2&CID=quickSearchAbstractFormat&DOCINDEX=6&database=3&format=quickSearchAbstractFormat

› [Accessed 18 March 2009].

Use of Numerical Solvers Animation of a body

› Modelling each leg segment as a separate pendulum, and calculating its movements continuously. What you see here are not cartoon animations, each move is calculated from first principles, using differential equations.

› http://palaeo.gly.bris.ac.uk/dinosaur/Allostick.gif› http://palaeo.gly.bris.ac.uk/dinosaur/animation.html [Accessed 18 March 2009].

David O’Meara, CEO, Havoc:“The quality and quantity of Irish computer science graduates are too low…”

• Quality?– Not according to Peter Brabazon, Dir., Discover Science & Engineering

– “One of the best in EU for percentage of 3rd level graduates in science, maths & computing”

[Keenan,B. 2008. 'INVISIBLE HAND' IS GUIDING STUDENTS IN RIGHT DIRECTION. The Sunday Independent (Ireland) August 24, 2008

– O’Meara acknowledges there are plenty of people with ideas but that graduates are more attracted to multinational companies. Why? Something for CEOs of smaller companies to address in terms of incentives for graduates?

– Possible need to develop business and entrepreneurial skills in graduates

– Perhaps O’Meara and his cohorts could provide more mentoring and advice rather than criticism of maths and science teaching!

David O’Meara, CEO, Havoc:“The quality and quantity of Irish computer science graduates are too low…” Quantity?› 3rd level computer course intake not up to pre-2002 levels…

› Why?

› Accepted wisdom: Teaching standards in maths and science and 2nd level facilities are so poor…etc, etc.

› But unlikely that standards and facilities have deteriorated so much so quickly! So why the fall off?

› Students voting with their feet based on job prospects? Business services predicted as the growth area.

[Keenan,B. 2008. 'INVISIBLE HAND' IS GUIDING STUDENTS IN RIGHT DIRECTION. The Sunday Independent (Ireland) August 24, 2008]

› Even ESRI Medium Term Review predicts steady fall in manufacturing jobs from 2010 with hi-tech sector shedding 2.5% of jobs per year

http://www.esri.ie/UserFiles/publications/20080515155545/MTR11.pdf (p.65-68)

› Chicken and egg situation?

Projectile Simulation – Matlab Code•Objectives

– Comparison of analytical and numerical methods– Adding realism – air friction, bouncing etc.– Exploiting Matlab – iterative methods, fast calculations, animation

•Issues– Analytical method => position can be determined at any point in time. BUT computer can only sample time at discrete intervals. –Numerical methods. Which one?

•Euler•Predictor-Corrector•Runge-Kutta

–Recursion–Accuracy v Speed–Code structure

•Dealing with time variable events – hitting the ground or other object

Projectile Simulation – Matlab Code Code Structure

› Setup starting parameters – firing velocity, angle etc.› Step size : Analytical and Numerical› Matrix arrays to store multiple relative values (e.g. x and y co-ords, time

value)› WHILE loop

Analytical Method: simply uses equations of motion to calculate the exact position of the projectile at each point in time.

x(ct,1)=v_x0*t(ct,1); y(ct,1)=v_y0*t(ct,1)-0.5*g*power(t(ct,1),2);

Numerical Method: successively estimates the position based on the previous estimate and the application of a specific numerical method (Euler in this case)

t(ct,1)=t_start+(ct-1)*delta_t; %Calculate velocities in x and y dir numerically if (t(ct,1)~=t_start)

v_x(ct,1)=v_x(ct-1,1)+0*delta_t; v_y(ct,1)=v_y(ct-1,1)-g*delta_t;

else v_x(ct,1)=v_x0;

v_y(ct,1)=v_y0; end %Calculate x and y positions based on velocities x(ct+1,1)=x_0+x(ct,1)+v_x(ct,1)*delta_t; y(ct+1,1)=y(ct,1)+v_y(ct,1)*delta_t;

Loop break out flag – when a certain event occurs if (y(ct+1,1)<threshold)

Projectile Simulation – Matlab Code

Plot results› Within loop ensures each iteration plots its result

Repeat loop for different situations› Projectile without air friction applied (analytically and

numerically)› Projectile with air friction applied (numerically with different

air friction constants)

Projectile Simulation – Matlab Code

Plot:

Caparison of moving ball with/without air friction

Challenge One

Runge Kutta is used instead of Euler series Advantage

› Increasing the speed of matlab code› Smaller error when delta time getting bigger and bigger

› Euler series : delta time = 0.2s› Elapsed time is 2.150858s

› Runge Kutta : delta time = 0.4s› Elapsed time is 1.031713s

Runge Kutta vs. Euler Series

Challenge Two--hitting

Balls Hitting

Two balls being thrown at the same time› Velocities are fixed for both› the throwing angle is fix for upper one› The initial throwing angle is 90 degree for bottom one, it

will reduce the angle if it cannot hit the upper ball.

Hitting condition› Time, x distance and y distance should be the same

Balls Hitting

Tips in matlab code› Two flags are used in our code b &c

Flags b : speed up of finding the hitting point If both objects did not collide and either one hits the

ground – b=1; if the objects do collide – b=m+n If velocity y is great than a number i.e. 1m/s m=0, n=0 If velocity y is less than 1m/s m=1, n=1

Flags c : fixed the angle for bottom one When two balls collide, flags c is changed to 1, matlab

code will jump out of the reducing angle loop

Challenge Three -- Tank

Throwing Out of the Tank

Matlab Code

Compare position of ball with tank wall

Change the velocities when ball hits tank wall

Using flags to switch off the comparison method

If ball is thrown out of the tank, it work as normal, it will hit ground and bound several times

ODEs

We are asked to numerically solve (in Matlab) the ordinary differential equation

The exact solution is y(x)=exp(x)-x-1

We want to solve this problem as quick as possible and also the error is less than 0.1%, therefore Runge Kutta method is used to solve this problem.

Matlab Code

Problem› We are asked to find the error at x=10 and keep the

error to be less than 0.1%, however, if we want to speed up the code , if 10 is not divisible by h, then we can not obviously show the error at x=10.

Solutions› If the x plus h is greater than 10, which means the next

h is out of the range and the difference between x and 10 is equal to 10-x, this is the new step for Runge Kutta. If using the new step to run Runge Kutta, we will calculate y value when x=10.

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X increases unit step except last point, when x is close to 10, the increasing step is changed to (10-x)

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Matlab Code

Step h = 0.3565

when x=10, › Runge Cutta : y=2.1993e4› Exact : y=2.2015e4› error =0.099958%

time take =0.5s