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AP3114/Computational Methods for
Physicists and Materials Engineers
Dr. Jun Fan
Jan. 14, 2014
1. Typical Problems in Physics
and Materials Engineering
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Problems in Math, Physics and
Materials Engineering
• 1. Linear equations• 2. Free Fall
• 3. Electric Current
• 4. Cantilever Beam
•
5. Diffusion• 6. Random Events
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1. Linear Equations
Scilab
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2. Free Fall
• t=0, v=0,• Case 1. No air resistance
• Case 2. Resistance proportional to the linear
power of velocity
• Case 3. Resistance proportional to the second
power of the velocity
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2. Free Fall
• What is the status of the particle at t=5?(Given k=0.2, g=9.81m/s2)
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2. Free Fall
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2. Free Fall
Computation Results
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3. Electric Current
A single, closed-loop electrical circuit
• The loop contains
– a voltage source V (battery),
– a resistor R (energy dissipation device),
– an inductor L (an energy storage device),
– A switch that is instantaneously closed at time t=0
Solution:
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3. Electric Current
V=1; R=1; L=1; t=0:0.01:6;
i=(V/R)*(1-exp(-R.*t/L));
figure;plot(t,i,'ro');
xlabel('time, t','fontsize',6),
ylabel('current, i','fontsize',6)
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4. Cantilever Beam
The Olive View Hospital
damaged in the 1971 San
Fernando earthquake (original
image from the EqIIS Collection)
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4. Cantilever Beam• For a uniformly loaded span of a cantilever beam attached to a wall
at x = 0 with the free end at x = L, the formula for the verticaldisplacement from y = 0 under the loaded condition, with y thecoordinate in the direction opposite that of the load, can be writtenas follows:
where
E is a material property known as
the modulus of elasticity;
I is a geometric property of thecross-section of the beam known
as the moment of inertia; L is the
length; w is the load per unit width
of the beam.
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4. Cantilever beam
x=0:0.01:1;
y=-(x.^4-4*x.^3+6*x.^2);
plot(x,y)
xlabel('x','fontsize',6)
ylabel('y','fontsize',6)
title('Cantilever Beam')
plot([0 1],[0 0],'linewidth',2)
legend('Uniformly loaded beam‘,’unloaded’)
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5.
cdt
dc 2
Diffusion equation:
1)0,0( t xc
0)0,( t xc
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5. Diffusion
x=1:1:100; dt=0.05;
c=zeros(100,1); c(1)=1;
figure;for t=1:1:3000
for i=2:1:99
d(i)=(c(i+1)+c(i-1)-2*c(i))*dt+c(i);
end
d(100)=d(99);
d(1)=1-sum(d(2:100));c=d;
if pmodulo(t,1000)==0 then
plot(x,c);
end
end
2 ),(2),1(),1(),()1,( x
nicnicnic
t
nicnic
c(x,t)
x
t1
t3
t2
Results
cdt
dc 2
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6. Random walk (Markov process)
• A drunk wanders down a street that has six intersections. His
home is at interaction 1, and his favorite Internet café is at
intersection 6.
• At each intersection other than his home or the café, he moves
in the direction of the café with probability 2/3 and in the
direction of his home with probability 1/3.
• We would like to know the chances of the student ending up at
home or in the café is he starts at a given corner.
Home Cafe
3 621 4 5P=1/3 P=2/3
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6. Random walk (Markov process)
• We represent the probabilities of being in these states a six-component state vector X(t), where Xi(t) is the probability of him
being at intersection I at moment t. The components of X(t)
must sum to 1, since he has to be in one of these states.
Home Cafe
3 621 4 5P=1/3 P=2/3
Transition probability matrix
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6. Random walk (Markov process)
Results: 48% at home, 52% at Cafe
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Scientific software
• Two categories of general scientific software
– Computer algebra systems that perform symbolic
computations: Maple, Mathematica, Maxima,
Axiom, and MuPad
– General purpose numerical systems performing
numerical computations and designed specifically
for scientific applications: Matlab, Scilab (free)
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Scilab
• A large software package containing 13,000 files, more than400,000 lines of source code, 70,000 lines of Scilab code,80,000 lines of online help, and 18,000 lines ofconfiguration files, including – Elementary functions of scientific calculation
– Linear algebra, spare matrices
– Polynomials and rational functions
– Classic and robust control, optimizations
– Nonlinear methods (optimization, ODE solver, Scicos – dynamicsystems modeler and simulator)
– Signal processing
– Random sampling and statisticas
– Graphs, graphics, animation
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Programmable Calculator
• Scilab
– www.scilab.org
• Download (select a version for your own
computer, Windows/Mac/Linux) & install
– http://www.scilab.org/download/5.4.1
Windows Mac Linux
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Scilab interface
• Start -> Scilab
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HelpTo get help, type “help”
> help derivative
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Editor
>editor
Save as *.sce file
To run: exec(filename.sce)
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Creating real variables
• Elementary mathematical operators
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Variable names
• Variable names may be as long as the userwants, but only the first 24 characters aretaken into account in Scilab.
• Case sensitive – “a” to “z”,
– “A” to “Z”,
–
“0” to “9”, – “%”, “_”, “#”, “!”, “$”, “?”
– Starting with “%” has special meaning, such as %pi
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Case sensitive Variables
• For example
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Comments and continuation lines•
Comments: – Any line which begins with two slashes
– “//”
•
Continuation lines: – any line which ends with two dots
– “..”
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List of Elementary Mathematical Functioins
• Trigonometry
•Other functions
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Elementary mathematical functions
• For example,
– Let’s test check it in Scilab:
1sincos 22 x x
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Pre-defined mathematical variables
• In Scilab, several mathematical variables are
pre-defined, whose name starts with a
percent “%” character.
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Booleans
• Boolean variables can store
– true (%t, or %T) or false (%f, or %F) values.
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Complex numbers
12i
In Scilab, %i represents the mathematical imaginarynumber i
?
“ ’ ” is used tocompute the
conjugate
“real ” returns
real part
“ imag ” returns
imaginary part
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Integers
• Integer type:
• Number of bits used to store an integer and
the range of value:
– n-bit signed integer: [-2n-1, 2n-1-1]
– n-bit unsigned integer: [0, 2n-1]
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Conversions between integers
• Types of integers returned by the inttype
function
• Scilab integer conversion functions:
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Circular integers and portability issues
Different from other mathematical packages, suchas Octave or Matlab: if an integer is at the upper
limit, the next integer stays at the upper limit
• The Scilab circular way allows for a greater flexibility in the
processing of integers, since it allows to write algorithms
with fewer if statements
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The ans variable
When we make a computation and do not store the result into
an output variable, the result is stored in the default ans
variable
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Strings
• Strings can be stored in variables, provided that they aredelimited by double quotes “ ”.
• “+” can operate strings.
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Dynamic type of variables
• Scilab allows to change the type of a variable dynamically. The following example shows that we first create a
real value, and then put a string variable in it:
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Exercises (I)
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Exercises (II)
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Exercise (III)
What will you get by executing the following?
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Exercise (IV)
• What will you get by executing the following?
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Exercise (V)
• What will you get by executing the following?
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Exercise (VI)
• What will you get by executing the following?
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Exercise(VII)
• Determine the value of the following
expressions yourself before checking your
answers using Scilab
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Exercise(VIII)
• Given that a=[1 0 2] and b=[0 2 2], determine
the value of the following expressions. Check
your answers with Scilab
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Reference
• http://www.scilab.org/resources/documentation/tuto
rials
– Introduction to Scilab (introscilab.pdf)
– Scilab for very beginners (Scilab_beginners.pdf)