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DepartmentofMechanicalEngineeringMonashUniversity
MEC3456 Engineering Computational Analysis
Lecture 01 Introduction
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
My detailsDr. Zhe LiuRoom 219, Building 82 (New Horizons)Phone 990 53627
EMAIL:[email protected]
Available consultation hours:
Monday: 2pm-5pm
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What is Engineering computing?
In general, engineers arerequired to relate physicalphenomena to mathematicalmodels.
An example is to design theracing car to obtain the
optimal aerodynamicperformance.
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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What is Engineering computing?
A mathematical model is broadly defined as a formulationor equation that expresses the essential feature of aphysical system.
Often it is difficult to get an analytical solution. A computer model is thus required to solve the
mathematical models, often, repetitively.
Solving simultaneous equations Solving ordinary or partial differential equations integration
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Dependent
variables= f
independent
variables
, parameters,
forcing
function
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Why do this project?
Why would an engineer use a computer to solveproblems?
Problem may be difficult to test in practice High precision required Economic concerns Fast results required Many models to test
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
ComputerScien:stslearnhowtoprogram
Theyhavenotlearntabout
ThermodynamicsSolidMechanicsFluidMechanicsKinema:cs
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Why do this subject?
Computer can model where experiments cantventure
(e.g., crack propagation and dislocation motion)
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Abraham F. F. et.al. PNAS;2002;99:5783-5787
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Why do this subject?
Computer model can predict(e.g., weather forecast)
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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Why do this subject?
Today, 97 of the top 100 industrial companiesinvest in engineering simulation as a key strategyto win in a globally competitive environment.
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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How? Hardware
Manufacturer --- IBM Location --- Lawrence Livermore National Laboratory, California, US Funded by --- National Nuclear Security Administration as part
of the Advanced Simulation and Computing Program (ASC)
Processors and speed ---A total of 1,572,864 cores;16.32 petaflop/s Power consumption --- 7.890 MW.
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Theworldfastestcomputer:Sequoia http://www.top500.org/
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How? Hardware
Victorian Life SciencesComputation Initiative
Processors and speed:65,536 cores; 65 TB of RAM;
690,197 GFlop/s
Power consumption: 328.75 kW Application:
Life science and medical scienceresearch
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
FastestcomputerinAustralia:AVOCA 31th on the top 500 list
In 2013, ANU will have a newsupercomputer manufactured byJapanese ICT and Fujitsu performingat 1.2TFlop/s.
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How? Software/Programming
Computer program are merelya set ofinstructions that direct the computer toperform certain tasks.
Simple information representation Advanced information representation Input/output Mathematical formulas Logic representation Modular programming
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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How? Software/Programming
To program, we will use a computer packagecalled MATLAB
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
MATLAisspecificallydesignedforscien:ficandengineeringresearch
ItisveryaccurateatmodelingItisveryeasytovisualizeyourdata
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Unit Overview
In this unit, you will learn how to use computerto do the followings
Finding roots of functions/interpolation Numerical Integrate and Numerical Differentiation Solving O.D.E.s
Initial-value problems Boundary-value problems
Solving P.D.E.s 1D. Problem 2D. Problem
Fourier analysis
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
We will focus on
Techniques to solve problemsAccuracy of the technique Stability of the technique Efficiency of the technique
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Unit Overview
At the end of this unit, you will be able to Problem solving Programming A critical approach to a complex problems
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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Unit Overview: Ideas and Attitudes
IdeasComputational solutions:
Having errors May never get an answer May take a long time to obtain May be complex
Attitudes Confidence in writing/using/developing codes
Be critical of numerical results they may be wrong
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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Unit Overview
Lectures
Interpolation and root finding (3 lectures) Open and close methods Newton & Lagrange interpolation Cubic spline Higher dimensional interpolation
Numerical integration Newton-Cotes technique Richardson/Romberg technique Gauss quadrature
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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Unit Overview
Lectures
Solution of ODE (8 lectures) Runge-Kutta method Multi-step method Implicit method Initial and boundary value problems Eigenvalue problem
Solution of PDE (11 lectures) Finite difference method Finite element method
Fourier analysis (2 lectures)
Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
TextbooksOne prescribed textbook
Several other textbooks will also be used in this course
NumericalMethodsfor
Engineers,
Chapra&Canale,6thedi:on
McGraw-Hill.
Largenumberofengineering-
relatedmethodsdetailed.
Virtuallynoprogramming,but
substan:altheory.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
AppliedNumericalMethodswith
MatlabforEngineersandScien:sts,
StevenC.Chapra,McGrawHill,2005.
Thistextisgoodbecauseitmarries
bothMATLAprogrammingandtheapplica:onofNumericalMethods.
Recommended texts and readings
NumericalMethodsinEngineering
withMatlab,
Kiusalaas,CambridgeUniversity,2005.
Thistextisalsogoodbecauseit
marriesbothMATLAprogramming
andtheapplica:onofNumerical
Methods.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
Recommended texts and readingsNumericalMethods,
J.FairesandR.Burden
3rdEd.ThomsonBrooks/Cole,2003
Anexcellenttextbook.Goesintoalotofdetail,someprogramming-usesthe
"Maple"programminglanguage.
NumericalRecipesinC,
Flannery,Ve_erling,Press&Teukolsky.
(alsoavailableinPascal,C++andFortran.)
Explana:ons&codeforawiderangeof
numericalalgorithms.Advanced
techniques.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
How do I deliver this subject Lectures(in class, three hours per week, recorded
videos uploaded online after lectures)
Tutorial questionsrelated to computer lab practices Computer labs(2 hours per week) --- instruction
videos posted online before computer labs.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
Assessment and feedback Final exam paper and in class quiz (70%)
Four in-class quiz (2%) will cover the main topics: root-findingand interpolation, numerical integration, ODE, and PDE.
Computer lab attendance and reports (30%) Attendance accounts 2%, and lab reports contribute 28% Laboratories will start in the 2nd weekof semester The first laboratory is a MATLAB tutorial, and will not be
assessed.
Every two weeks, you submit lab reports in pair by Friday 5pm(week 3, 5, 7, 10, and 12), e.g., lab report 1 and 2 submitted onweek 3, lab report 3 and 4 submitted on week 5, and so on.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
Online lecture notesand other readings Lecture notes will be online (Moodle) one week before the
courses
Also, several articles and links to topics of interestarranged by lecture topic.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
Tutorial Questions and Solutions Several practical questions of math derivation and hand
calculations for each topic.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
Students must attend computer laboratory session assigned viaAllocate+ system. Please contact me ASAP to report any clashes.
Lab questions will be available on Moodle. An instruction video will be provided before the lab session. You are suggested to read the laboratory handout and watch the
instruction video prior to arriving at the laboratory class.
Computer laboratories
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
Computer laboratory contents
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Unit Overview
Feedbacks Lecture contents
You are welcome to ask questions during/after thelectures.
Every week, office hours on Monday from 2pm to 5pm(Room 219, New Horizons Building)
Moodle online forum. Computer labs: programming and lab reports
You are welcome to ask me and our demonstrators duringthe lab sessions.
Comments will be provided in you marked lab reports.Any disagreements related to report marks will be
discussed during the lab session.
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DepartmentofMechanicalEngineeringMonashUniversity
Lecture 01 computational error
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Computational Error
Although the numerical technique yields estimatethat close to the exact analytical solution, there wasa discrepancy or error.
There are two types of computational errors in ournumerical models.
Round-off error (induced by hardware) Truncation error (induced by algorithm)
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Round-off Error
How do computer store numbers?
Computer handles two types of numerical data Integer (fixed point numbers)
E.g., 0, -21, 973,
Real number (floating point numbers)E.g., 0.921, 0.2491
10
-1
IEEE binary floating point arithmetic standard 754-1985 The manner of a real number expressed and stored in computer
memory (b=2 for binary number system)
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Round-off Error
How computer store numbers?
Typically, real number is stored 32 bit (single precision) in acomputer. This is only enough to maintain 7-8 significantfigures within between 10-40 to 1040
The typical relative error is called machine accuracy (m) The machine accuracy of a 32bit computer is about:
m ~ 310-8
Round-off error is accumulative. Every time a floating pointarithmetic operation is performed an error will be introducedof size m.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Round-off Error
How round-off error develop?
These errors mostly add up randomly. AfterNoperations theerror is typically approximately
Total error =N1/2m
However, it is possible that these errors could all be of thesame sign and then the error estimate is:
Total error =Nm
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Round-off Error
How round-off error develop?
Some arithmetic operations will result in significant loss ofaccuracy.
For example, the adding a small number 0.0010 to a largenumber 4000
The subtraction two nearly equal floating point numbers
0.4000 i104
0.0000001 i104
0.4000001 i104
Forahypothe:calcomputerwith
4-digitman:ssa,thefinalresultis
0.4000104
x =b b
2 4ac
2a
Thiswillproducebig
errorsifb2>>4ac
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Truncation Error
Truncation errors are those that results from using anapproximation in place of an exact mathematical formula.
Truncation error is due to the algorithm used. Thinktruncation in terms of the removal of higher-order
terms in a Taylors series expansion.
In this course, we will keep using a discrete method toapproximate a continuous problem (e.g., numericaldifferentiation)
In the limit of using more and more terms, or points, orintervals, the approximation should become increasinglyaccurate.
f(xi+1) = f(xi )+ f '(xi )(xi+1 xi ) + f ''(xi )(xi+1 xi )2+ ...
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Stability of the Solution Technique
Another important issue isSTABILITY.
This is associated with the interaction between the round-off error and the numerical algorithm.
Any error that exists at some stage of the solution process ismagnified (usually exponentially) so that after a certainstage the error swamps the solution and the approximationis meaningless.
We will later look at ways to determine the stabilityproperties of a given method.
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Examples
A round-off error
Consider the simple program snippet in MATLAB:
pi =3.141592653589793238462643;
MATLAB will only store the underlined digits
fprintf('%30.25f\n',pi)
with output:
3.1415926535897931000000000
The digits underlined are incorrect.
More digits than
are stored in
pi
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Examples
A truncation error
Consider the Taylor series expansion of ln(1+x)
y(x) = ln(1+ x) =(1)
i+1
ix
i
i
= x 1
2
x2+1
3
x3
1
4
x4+1
5
x51
6
x6+ ...
y(x)
y(0.5) = 0.5
y(0.5) = ln(1+ 0.5) = 0.405465108
TheTRUNCATIONerroristrue-approxima:on
=0.4054651080.5 =-0.094534892(23%)
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Outline
Whatis
engineering
compu:ng?
Whydothisobject?
How?
Hardware SoBware/programming
Unitoverview
LectureTextbookAssessmentComputa:onal
errors
Roundofferror
Trunca:onerror
Summary
In this lecture, we have learned What is engineering computational analysis? computer
models to solve mathematical equations numerically.
Why do we need this object? It can provide informationthat cant be provided by experiments and theory.
How to do it? the topic of this unit! Numerical errors
Round-off error (induced by hardware) Truncation error (induced by algorithm)
In the next lecture, I will give you a brief introduction ofMATLAB.