AP3114 Lecture1 Introduction Jan14 2014

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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



Problems in Math, Physics and

Materials Engineering

• 1. Linear equations• 2. Free Fall

• 3. Electric Current

• 4. Cantilever Beam

•

5. Diffusion• 6. Random Events



1. Linear Equations

Scilab



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



2. Free Fall

• What is the status of the particle at t=5?(Given k=0.2, g=9.81m/s2)



2. Free Fall



2. Free Fall

Computation Results



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:



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)



4. Cantilever Beam

The Olive View Hospital

damaged in the 1971 San

Fernando earthquake (original

image from the EqIIS Collection)



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.



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’)



5.

cdt

dc 2

Diffusion equation:

1)0,0( t xc

0)0,( t xc



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



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




• 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




Results: 48% at home, 52% at Cafe



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)



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



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

http://www.scilab.org/

http://www.scilab.org/download/5.4.1

http://www.scilab.org/download/5.4.1

http://www.scilab.org/



Scilab interface

• Start -> Scilab



HelpTo get help, type “help”

> help derivative



Editor

>editor

Save as *.sce file

To run: exec(filename.sce)



Creating real variables

• Elementary mathematical operators



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



Case sensitive Variables

• For example



Comments and continuation lines•

Comments: – Any line which begins with two slashes

– “//”

•

Continuation lines: – any line which ends with two dots

– “..”



List of Elementary Mathematical Functioins

• Trigonometry

•Other functions



Elementary mathematical functions

• For example,

– Let’s test check it in Scilab:

1sincos 22 x x



Pre-defined mathematical variables

• In Scilab, several mathematical variables are

pre-defined, whose name starts with a

percent “%” character.



Booleans

• Boolean variables can store

– true (%t, or %T) or false (%f, or %F) values.



Complex numbers

12i

In Scilab, %i represents the mathematical imaginarynumber i

?

“ ’ ” is used tocompute the

conjugate

“real ” returns

real part

“ imag ” returns

imaginary part



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]



Conversions between integers

• Types of integers returned by the inttype

function

• Scilab integer conversion functions:



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



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



Strings

• Strings can be stored in variables, provided that they aredelimited by double quotes “ ”.

• “+” can operate strings.



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:



Exercises (I)



Exercises (II)



Exercise (III)

What will you get by executing the following?



Exercise (IV)

• What will you get by executing the following?



Exercise (V)




Exercise (VI)




Exercise(VII)

• Determine the value of the following

expressions yourself before checking your

answers using Scilab



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



Reference

• http://www.scilab.org/resources/documentation/tuto

rials

– Introduction to Scilab (introscilab.pdf)

– Scilab for very beginners (Scilab_beginners.pdf)

http://www.scilab.org/resources/documentation/tutorials






Documents

AP3114 Lecture1 Introduction Jan14 2014