Engineering Engineering Computation and Computation and

SimulationSimulationConor BrennanConor Brennan

Dublin City University Dublin City University brennanc@eeng.dcu.ie brennanc@eeng.dcu.ie

EE317

GROUP 4 :-

Manpreet Singh Nanda

Varun Raina

Kanika Poply

Numerical Methods

What are Numerical Solvers?What are Numerical Solvers? Used to compute the solution of a complex model of a Used to compute the solution of a complex model of a

physical systemphysical system Available in Simulation Software or algorithms available Available in Simulation Software or algorithms available

in Mathematical software packages like Matlab.in Mathematical software packages like Matlab.

Riemann SolverRiemann Solver: Used heavily in Computational Fluid : Used heavily in Computational Fluid Dynamics and Magnetohydrodynamics.Dynamics and Magnetohydrodynamics.

Roe SolverRoe Solver:: Linearization of the Jacobian Linearization of the Jacobian HLLC SolverHLLC Solver: : Used to restore missing rarefaction wavesUsed to restore missing rarefaction waves

Some Examples of Numerical Some Examples of Numerical Solvers:Solvers:

Some Numerical Solvers in Some Numerical Solvers in Matlab:Matlab:

Two Basic Categories of Solvers in Matlab:Two Basic Categories of Solvers in Matlab:

Fixed Step Solver:Fixed Step Solver: Solves the model at regular time Solves the model at regular time intervals from beginning to the end of interval.intervals from beginning to the end of interval.

Variable Step Solver:Variable Step Solver: Vary the step size during Vary the step size during simulation changing the step size to increase accuracy simulation changing the step size to increase accuracy according to how model’s states are changingaccording to how model’s states are changing

ODE 45 methodODE 45 method: Inbuilt Function in Matlab for solving : Inbuilt Function in Matlab for solving ODE's. It is based on the Runge-Kutta method.ODE's. It is based on the Runge-Kutta method.

Handles general equations of the form Handles general equations of the form

‘‘t’ is the independent variablet’ is the independent variable ‘‘y’ is the dependent variabley’ is the dependent variable Varies the size of the step of the independent variable in Varies the size of the step of the independent variable in

order to meet the accuracy we specify at any particular order to meet the accuracy we specify at any particular point along the solution.point along the solution.

Numerical Solvers in Numerical Solvers in Engineering Examples:Engineering Examples:

Example Problem of Numerical Example Problem of Numerical SolverSolver

Kinematics ProblemKinematics Problem : :

Consider a Paratrooper of mass 80 kg falling from a Consider a Paratrooper of mass 80 kg falling from a height of 600 meters.height of 600 meters.

He is accelerated by gravity and then decelerated by He is accelerated by gravity and then decelerated by Drag Force.Drag Force.

V is velocity in m/s.V is velocity in m/s. Governing Equation: dV/dt = -mg + 4/15(V^2)/mGoverning Equation: dV/dt = -mg + 4/15(V^2)/m Reference: (Book: Numerical Computing in Matlab)Reference: (Book: Numerical Computing in Matlab)

Example to illustrate Matlab’s Example to illustrate Matlab’s Inbuilt SolverInbuilt Solver

F.MFile

FUNC..MCode that calls the ODE45 Function

Example to illustrate Matlab’s Example to illustrate Matlab’s Inbuilt Solver: ResultInbuilt Solver: Result

Ordinary Differential Equation

An Ordinary differential (or ODE) is a relation that contains function of only one independent variable and one or more of its derivatives with respect to that variable.

Three Methods to Solve ODE's

• Euler Method: It is based on finite difference approximations to the derivative.

f (x+h) = f (x) + h*f ‘(x)

• Predictor Corrector Method : yn+1 = yn +1/2*h (f ( xn , yn) + f(xn+1,y*n+1))

where y*n+1 = yn + f( xn, yn)

• Range - Kutta Method :yn+1 = yn + 1/6 ( A1+ 2A2 + 2A3 + A4)*h

where A1 = f (xn , yn)A2 = f (xn + h/2 , (yn+h/2)A1)A3 = f (xn + h/2 , (yn+h/2)A2)A4 = f (xn +h, yn+hA3)

Solution for Ordinary Differential Equation dy/dx = x +y

Graph Showing the Relative Error

Relative Error and Elapsed Time

Algorithm for Projectile codeAlgorithm for Projectile code

Setting the Position (x , y) and Velocity (Vy, Vx) Vectors Setting the Position (x , y) and Velocity (Vy, Vx) Vectors and parameters( g, time, friction etc).and parameters( g, time, friction etc).

Running for fixed number of Steps (N).Running for fixed number of Steps (N). Without FrictionWithout Friction: Vx is constant and Vy is : Vx is constant and Vy is

Dependent( on g).Dependent( on g). Run the loop to generate projectile parabola. ( Vy, Vx, x Run the loop to generate projectile parabola. ( Vy, Vx, x

and y)and y) Concept of small interval time frame( Realistic Effect).Concept of small interval time frame( Realistic Effect).

Algorithm Cont:Algorithm Cont:

Testing loop condition: Start condition and threshold Testing loop condition: Start condition and threshold conditioncondition

Since, it keeps on moving ( Vx friction independent).Since, it keeps on moving ( Vx friction independent).

Vx Constant

Frictionless ProjectileFrictionless ProjectilePlot(Euler_x) increasing gradually

Plot(Euler_Vy) Decreasing with threshold loop

Plot(Euler_y)

Problems facedProblems facedRe-bouncing even on Friction present.

First Projectile obtained

Projectile Fired From the Origin (0,0)

Projectile with Friction Friction with k = 0.02Friction with k = 0.02 Values for Euler_vy, Values for Euler_vy,

Euler_x and Projectile Euler_x and Projectile generated.generated.

Projectile with Friction with diff “k”

Different values of k = Different values of k = 0.04, 0.08 and 0.1 0.04, 0.08 and 0.1

Effects on Projectile.Effects on Projectile. Due to constant air Due to constant air

friction.friction.

K = 0.2

Realistic ReflectionsRealistic Reflections Written code with smaller time frame to compare to the Written code with smaller time frame to compare to the

thresholdthreshold..Friction with Realistic Reflections

Ireland Lacks Quality Graduates?Ireland Lacks Quality Graduates? I believe that instead of just emphasizing on negatives he should take the I believe that instead of just emphasizing on negatives he should take the

Lead Role in providing a solution to this problem.Lead Role in providing a solution to this problem. Steps that could be taken at two different levels:Steps that could be taken at two different levels:

School LevelSchool Level Organizations should interact more with the student community during Organizations should interact more with the student community during

the development stage by for e.g. organizing more quiz competitions the development stage by for e.g. organizing more quiz competitions etc. etc.

Children between the age-group of 10-13 years should be encouraged Children between the age-group of 10-13 years should be encouraged to take up maths and science subjects and hence should be provided to take up maths and science subjects and hence should be provided with the necessary platform or resourceswith the necessary platform or resources

University LevelUniversity Level More opportunities should be given to aspiring graduates through More opportunities should be given to aspiring graduates through

means of work placements, regular industry visits and Awards to means of work placements, regular industry visits and Awards to recognize and develop next generation of talent.recognize and develop next generation of talent.

Organizations should take an active role and advice the universities Organizations should take an active role and advice the universities regularly about the changing needs of the industry.regularly about the changing needs of the industry.

Thank You !!

QUESTIONS ???