KMU 206 Numerical Analysis with MATLAB®
Hacettepe University
Department of Chemical Engineering
2009-2010 Spring Semester
Selis Önel, PhD
2Selis Önel, PhD
KMU206 Staff: Instructor
Selis Önel, PhD
Room: 14, 2nd floor
E-mail: [email protected]
Web: http://yunus.hacettepe.edu.tr/~selis
Office hours: Mon. 9:00-10:00
3Selis Önel, PhD
KMU206 Staff: Assistant
Aslıhan Bayraktar
Room: Z01-Basement
E-mail: [email protected]
Office hours:
Mon. 11:00-12:00
Wed. 11:00-12:00
4Selis Önel, PhD
Importance of Numerical Analysis
Numerical methods?
Techniques used for solving mathematical
problems that cannot be solved or are
difficult to solve analytically
Analytical solution →Exact answer
Numerical solution →Approximate value
Numerical solutions can be very accurate!
Calculations are iterated until desired
accuracy is achieved
Start
Read r
Read V
),(1 rVf
V
),(2 rVf
r
End
<ε
>ε
<ε
>ε
5Selis Önel, PhD
Energy Optimization of the Yankee Hood Dryerin collaboration with
İpek Kağıt A.Ş., Yalova
and
Pulp and Paper Foundation of Turkey
Master’s Thesis Advisor: Dr. Güniz Gürüz, Professor
Department of Chemical Engineering
Middle East Technical University, Ankara, Turkey
My Research Background, MS
Macro Scale Thermodynamic Systems
6Selis Önel, PhD
Mathematical Modeling of the Thermal Drying
System NaturalGas
Steam in
Fresh air
Dry paper out
Fresh airFresh air
Exhaust air
Wet
hoodDry
hood
System 1
Wet paper in
TransferAir
Recycle air
Blowingair
Blowingair
Steam+condensate out
Yankee Speed 1250-1650 m/min
Production 2.5-3.6 ton/h
Paper width 2.7 m
Paper basis weight 15-20 g/m2
Paper moisture content at
Yankee entrance 60 %
Yankee exit 4-7 %
7Selis Önel, PhD
My Research Background, PhD
Comparison and Extension of Free Dendritic Growth
Models through Application to Ag-15%Cu Alloy
PhD Thesis Advisor: Dr. Teiichi Ando, Professor
Advanced Materials Processing Lab (AMPL)
Department of Mechanical and Industrial Engineering
Northeastern University, Boston, MA, USA
Nano Scale Thermodynamic Systems
8Selis Önel, PhD
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-7.6
-7.5
-7.4
-7.3
-7.2
-7.1
-7
-6.9
-6.8x 10
4
Mole fraction of Cu
Gib
bs
Fre
e E
ner
gy,
G (
J/m
ol)
Molar Free Energy Curves for Ag-Cu at 1060 K
CLeqC
Seq
C0
G*
CS* C
L*
2L
1L
2'
1'
GL
G
G shifted by Gr
Start
Read r
),(1 rVf
),(2 rVf
End
<ε
>ε
<ε
>ε
rr TGT ,,*
*
)(
0
*
*
,
)(
)(
LL
rL
eqL
eqS
CdC
TTd
LC
C
rL
eq
S
rL
eq
L
mk
TTCC
TTCC
** ,, SL CkC ),,,,,,,,,( **** BABAGCCTG LLL
rLS rLS GCCT ,,, ***
),,,,,,(
),,,,,,(
),,,,,,(
BABACT
BABATC
BABATC
LLL
LL
LLL
S
LLL
L
Read V
r
V
100
101
102
10-9
10-8
10-7
10-6
10-5
10-4
T (K)
r (m
)
Total Supercooling vs. Tip Radius
LKT
TLK
Modified TLK
BCT
DA
New
Liquid
Vr
CL
*
kT *
T
C0
z
Solid
Paraboloid
of
Revolution
ΔT
C0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1200
400
600
800
1000
1200
1400
1600
Composition (mol.fr. Cu)
Tem
pera
ture
(K
)
Metastable Phase Diagram for Ag-Cu System
1052.1 K
Eutectic point
(0.399,1052.1 K)
Critical point (0.61,1366.45 K)
0.9510.399
o Experimental data by Heycock et al.
Calculated phase boundaries
0 20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T (K)
Fra
ctions o
f S
uperc
oolin
g (
K/K
)
Components of Total Supercooling (DA model)
Tr/T
Tt/T
Tc/T
Tk/T
Mathematical Modeling of
Crystal Growth
9Selis Önel, PhD
Rapid Solidification Problems
Thermal
spray
Rapid
solidificationProperties
& Quality
Uniform droplet size
and microstructure
Nanostructures
Metastable phases
Advanced materials: automotive, aerospace, semiconductor, electronic
industries
Purpose Controlling the nano-structures of advanced materials that form during
rapid solidification
Improves mechanical,
chemical, thermal,
electrical, magnetic,
optical properties
Biomedicine: “Cryobiology”!
Purpose Reducing the amount of
poisonous cryoprotectants and formation of
ice crystals detrimental to cells during the
freezing/vitrification of cells for
cryopreservation Cooling rate
Cell
via
bil
ity Solution effects
Intracellular ice
formation
Droplet
vitrification
Bulk vitrification
%100 Viability
10Selis Önel, PhD
Numerical Methods have a long history
Babylonians used (~3700 years ago )
Numerical solutions to quadratic equations
Approximations to square root of an integer
Linear interpolation to solve problems involving compound interest
Chinese Han Dynasty used (2000 years ago)
Matrix notation
What we call Gauss elimination to solve linear systems of equations
Successive approximations
Chinese Sung Dynasty used (1000 years ago) successive approximations to numerically solutions of higher degree equations
Indians used methods for obtaining square and cube roots
Greek mathematics used large number of regions of known area to estimate the desired area (letter from Archimedes to Eratosthenes)
Middle Eastern mathematicians such as Omar Khayyam and Jemshid Al-Kashi worked on algebra and solving cubic equations using iterative and trigonometric methods. The method we call Horner’s method was actually found by Jemshid Al-Kashi.
11Selis Önel, PhD
History of Numerical Methods:
Omar Khayyam (1048-1112)
Seljuk mathematician, philosopher, astronomer, poet
Lived in Khorasan, the Seljuk capital
Wrote Rubaiyat of Omar Khayyam
Wrote “Treatise on Demonstration of Problems of
Algebra” giving a geometric method for solving cubic
equations by intersecting a hyperbola with a circle:
“Whoever thinks algebra is a trick in obtaining unknowns
has thought it in vain. No attention
should be paid to the fact that algebra and geometry
are different in appearance. Algebras
are geometric facts which are proved by propositions
five and six of Book two of Elements.”
—Omar Khayyam
Cubic equation and
intersection of conic
sections" the first page
of two-chaptered
manuscript kept in
Tehran University
12Selis Önel, PhD
KMU206 Course Goals
Teach fundamentals of numerical methods
Enhance students’ programming skills using
the MATLAB environment to implement
algorithms
Teach the use of MATLAB as a tool (using
built-in functions) for solving problems in
science and engineering
13Selis Önel, PhD
In my time there were …
Hardly any PC’s in the department
Long lines for the public PC’s in the dorms
Slower processors and smaller memories
No memory sticks
Really slow inkjet printers
So take advantage of the high technology
14Selis Önel, PhD
My Expectations from You
Attend classes
Do homework !
(Group study is allowed as long as you submit
your own work)
Review your Math notes
Get familiar with MATLAB®
Turn homework in on time
Avoid plagiarism and copying!
15Selis Önel, PhD
Plagiarism ??
transitive verb to steal and pass off (the ideas
or words of another) as one's own : use (another's
production) without crediting the source
intransitive verb to commit literary theft
: present as new and original an idea or product
derived from an existing source
Reference:http://www.merriam-webster.com/dictionary/plagiarizing
16Selis Önel, PhD
Useful Textbooks
L. v. Fausett, Applied Numerical Analysis Using MATLAB® 2/E, Prentice Hall, ISBN: 0132397285
A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications, Prenctice-Hall Inc., NJ, 1999, ISBN: 0130138517
S. Nakamura, Numerical Analysis and Graphic Visualization with MATLAB®, 2/e, Prentice Hall, 2002, ISBN:01306548921
A. Gilat and V. Subramaniam, Numerical Methods for Engineers and Scientists, John Wiley & Sons, Inc., 2008, ISBN: 9780471734406
C. Moler, Numerical Computing with MATLAB®, Electronic edition: The MathWorks, Inc., Natick, MA, 2004, http://www.mathworks.com/moler. Print edition: SIAM, Philadelphia, 2004.http://ec-securehost.com/SIAM/ot87.html
J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB®, 3rd ed, Upper Saddle River, NJ: Prentice Hall, 2004, ISBN: 0130652482
J. Kiusalaas, Numerical Methods in Engineering with MATLAB® , Cambridge University Press, 2005, ISBN: 0521852889
17Selis Önel, PhD
Grade Percentages
Quizes 15%
Homeworks 10%
Midterm I 30%
Final (Class exam+Take Home) (25%+25%) 50%
Total 105%
18Selis Önel, PhD
Homeworks
1. Problems to be solved by hand
Improve understanding of numerical methods
Carry out few steps of numerical methods by hand
2. Problems to be programmed in MATLAB®
Improve programming skills
Learn MATLAB® toolboxes
3. Problems in math, science and engineering and
mathematical modeling
Identify engineering problems that need to be solved by
using numerical methods
19Selis Önel, PhD
Midterms
Midterm I : Between March 16/18
Midterm II: Between April 20/22
Finals week: May 24-31
Midterm questions may be reviewed in an
additional tutorial hour if requested
20Selis Önel, PhD
What are your expectations?
Please write in a paragraph or list
1. Programming languages you are familiar with
2. Your familiarity with MATLAB®
3. Your expectations from this course, the instructor and the assistant
4. How you think this course will help you in your education and career
21Selis Önel, PhD
MATLAB Feautures
ComputationLanguage
Graphics
Programming
GUI Toolboxes
MATLAB
Reproduced from: S. Lyshevski, Engineering and Scientific Computations Using Matlab, 2003, p.2
22Selis Önel, PhD
MATLAB is …
A high level computer language
For scientific computing and data visualization
Built around an interactive programming environment
Where MATLAB programs
Can be tested and debugged quickly
Do not need to be compiled, linked and executed after
each correction
Can be developed much faster than FORTRAN or C
programs
BUT
You need to have MATLAB installed on your computer
23Selis Önel, PhD
MATLAB® …
is the short for "matrix laboratory“
was designed by Cleve Moler in late 1970s to give his students access to LINPACK and EISPACK without having to learn Fortran
(C. Moler: chairman of the computer science dept. at the Univ. of New Mexico at that time, now chairman at The Mathworks)
was enjoyed by the applied mathematics community
was rewritten by Cleve Moler, Steve Bangert, and Jack Little in C and set the foundation for The MathWorks in 1984
(rewritten libraries were known as JACKPAC)
24Selis Önel, PhD
MATLAB® …
was first adopted by control design engineers, Little's
specialty, but quickly spread to different fields
is also used in education (linear algebra and numerical
analysis) and is popular amongst scientists involved with
image processing
offers toolboxes for solving mathematical problems in many
popular fields
25Selis Önel, PhD
Links for
Information on MATLAB
http://www.mathworks.com/products/matlab/
http://www.cyclismo.org/tutorial/matlab/
http://kgptech.blogspot.com/2005/07/matlab.h
tml
http://www.mathworks.com/matlabcentral/