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MSc Systems Engineering Homework: Modeling & Simulation Exercises, Exercises of Mathematical Modeling and Simulation

National Institute of Industrial EngineeringMathematical Modeling and Simulation

Problem sets from the msc systems engineering course, focusing on modeling and simulation. Topics include calculating averages and standard deviations using monte carlo methods, simulating ellipsoids and moment of inertia, testing gaussian distributions, and simulating radioactive decay and binomial processes. Students are expected to write programs and perform simulations to understand the concepts.

Typology: Exercises

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Modeling & Simulation
MSc Systems Engineering, First Semester, 2004
Home Work Problem Set 1
6.1 Use a random number generator of your choice to generate about ten thousand
numbers between zero and one. Calculate the average and standard deviation
after developing about 10 classes in the interval of equal width. Compare the
simulated average with expected mean of 0.5.
6.2 The equation of an ellipsoid centered on the origin is given by the following:
1
2
2
2
c
z
b
y
a
x
.
Simulate the solution for volume using Monte Carlo procedure. The ellipsoid
can be inscribed in a solid with six rectangular faces.
6.3 The moment of Inertia of a sphere about z-axis is defined as:
dxdydz)yx(Iz 22
.
Find the value of Iz for a given (say 2, using Monte Carlo procedure).
6.4 Write a program to test Gaussian distribution of random numbers. How you
are going to establish that the simulated distribution is a normal distribution?
6.5 Simulate a model of a radioactive element which decays into a radioactive
daughter which subsequently decays into a stable element. Use initial number
of parent isotope as 6000 and no daughter elements. Assume the pp = 0.07 and pd
= 0.02. Let your domain and range be [0, 300] and [0, 6000] respectively. Explain
the following systems through simulations
a. How the results change when pd is varied from 0.5pp to 0.2pp?
b. What happens when pd = pp (say pp = 0.01)?
c. What will be the effect on results when pd = 5pp?
6.6 A drunk person starts off at a lamppost and takes 36 steps each of length 2 and
each in a random direction where
oo 3600
. Use Monte Carlo method to
find his most probable distance from the lamppost after 36 steps. Compare it
with correct answer, 12. What is the average distance after 100 steps? Also check
the result at 1000, and 10000 steps.
6.7 Consider a binomial process in a system with the probability of success equal to
0.1 and probability of failure equal to 0.9. Calculate the variance and standard
deviation of number of successes after 100, 1000 and 10000 trials respectively.
6.8 Suppose that you are given with a probability distribution function of the
following form in the domain [0, 1]:
101 2 x),x(a)x(p
.
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Modeling & Simulation MSc Systems Engineering, First Semester, 2004

Home Work Problem Set 1

6.1 Use a random number generator of your choice to generate about ten thousand numbers between zero and one. Calculate the average and standard deviation after developing about 10 classes in the interval of equal width. Compare the simulated average with expected mean of 0.5.

6.2 The equation of an ellipsoid centered on the origin is given by the following:

1

2 2 2 ^  

c

z b

y a

x.

Simulate the solution for volume using Monte Carlo procedure. The ellipsoid can be inscribed in a solid with six rectangular faces.

6.3 The moment of Inertia of a sphere about z-axis is defined as:

Iz    (x^2  y^2 )dxdydz.

Find the value of Iz for a given  (say 2, using Monte Carlo procedure).

6.4 Write a program to test Gaussian distribution of random numbers. How you are going to establish that the simulated distribution is a normal distribution?

6.5 Simulate a model of a radioactive element which decays into a radioactive daughter which subsequently decays into a stable element. Use initial number of parent isotope as 6000 and no daughter elements. Assume the pp = 0.07 and pd = 0.02. Let your domain and range be [0, 300] and [0, 6000] respectively. Explain the following systems through simulations a. How the results change when pd is varied from 0.5pp to 0.2pp? b. What happens when pd = pp (say pp = 0.01)? c. What will be the effect on results when pd = 5pp?

6.6 A drunk person starts off at a lamppost and takes 36 steps each of length 2 and

each in a random direction  where 0 o^  360 o. Use Monte Carlo method to find his most probable distance from the lamppost after 36 steps. Compare it with correct answer, 12. What is the average distance after 100 steps? Also check the result at 1000, and 10000 steps.

6.7 Consider a binomial process in a system with the probability of success equal to 0.1 and probability of failure equal to 0.9. Calculate the variance and standard deviation of number of successes after 100, 1000 and 10000 trials respectively.

6.8 Suppose that you are given with a probability distribution function of the following form in the domain [0, 1]: p(x)a( 1  x^2 ), 0  x  1.

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It is zero elsewhere. Find the cumulative probability distribution function. Using uniform random numbers and inverse transform method find the distribution of x.

6.9 Find central difference approximation for a general point indexed as (i, j) to the following partial differential model:

G(x,y) f(x,y) x

G(x,y) x

2

2 2

2

6.10 Develop a normal distributed noise coupled with a sinusoidal function of the following form: f (t)AsinwtBn(t) , in the range 1  t  10. Then use it in Simulink after developing a block model for first, second and third order filters to study the effect of filters. Compare efficiencies of each filter and perform parametric studies over weighting factors.

6.11 A random variable X has a normal distribution if the probability distribution function is

, x.

(x ) fX (x) exp    

2

2 2

and is denoted N(  , ^2 ). Here,  is the mean and  is the variance. Since X = 

+  Z , where Z is the standard normal variable denoted by N(0, 1). We consider

only generation from N(0, 1). The inverse transform method cannot be applied to the normal distribution, therefore, suggest some alternative procedure to be employed.

6.12 A two-dimensional picture has normal random noise in it. Design an algorithm that will satisfactorily restore the image and reduce the noise to an acceptable level. Formulate the model and algorithm using MATLAB-Simulink.

6.13 Write down a function program in MATLAB language to test the Hoshen- Kopelman algorithm for cluster counting in a three by three and four by four sized system.

6.14 Simulate for an 8 x 8 lattice based on Ising model with periodic boundary conditions using Metropolis Monte Carlo method. Assume system is at a given temperature T. Plot the specific heat, internal energy and order parameter (M) as a function of T. Assume only nearest neighbor interactions.

6.15 Consider an L x L square lattice with L = 8, 16 and 32 size. Develop a computer program to study the percolation. Compute percolation probability for site percolation as a function of probability (p) of sites thrown on empty lattice. How you will estimate the percolation threshold?

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