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Computation & Simulation. Assignment Two Numerical methods for solving ODEs Group 5 Xiang Wang Issraa Alhiti Padraig O'Dwyer Tutor: Dr. Conor Brennan. Use of Numerical Solvers. 1-Car companies can improve crash safety of their models - PowerPoint PPT Presentation

Computation & Simulation

Computation & SimulationAssignment 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 modelsThis 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 (RungeKutta) 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. 2000Available from Inspec (EI Engineering Village) database .http://www.engineeringvillage2.org/controller/servlet/Controller?SEARCHID=da37721201089cf30M4886prod2data2&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.gifhttp://palaeo.gly.bris.ac.uk/dinosaur/animation.html[Accessed 18 March 2009].

David OMeara, 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

OMeara 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 OMeara and his cohorts could provide more mentoring and advice rather than criticism of maths and science teaching!

David OMeara, CEO, Havoc:The quality and quantity of Irish computer science graduates are too lowQuantity? 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 pooretc, 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 yearhttp://www.esri.ie/UserFiles/publications/20080515155545/MTR11.pdf (p.65-68)

Chicken and egg situation?

Projectile Simulation Matlab CodeObjectives 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?

EulerPredictor-CorrectorRunge-Kutta

RecursionAccuracy v SpeedCode structure

Dealing with time variable events hitting the ground or other object

Projectile Simulation Matlab CodeCode StructureSetup starting parameters firing velocity, angle etc.Step size : Analytical and NumericalMatrix arrays to store multiple relative values (e.g. x and y co-ords, time value)WHILE loopAnalytical 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)