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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?


Unitoverview


errors

Roundofferror

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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?


Unitoverview


errors

Roundofferror

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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?


Unitoverview


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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?


Unitoverview


errors

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Abraham F. F. et.al. PNAS;2002;99:5783-5787

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Computer model can predict(e.g., weather forecast)

Outline

Whatis

engineering

compu:ng?

Whydothisobject?

How?


Unitoverview


errors

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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?


Unitoverview


errors

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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?


Unitoverview


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?


Unitoverview


errors

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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?


Unitoverview


errors

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How? Software/Programming

To program, we will use a computer packagecalled MATLAB

Outline

Whatis

engineering

compu:ng?

Whydothisobject?

How?


Unitoverview


errors

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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?


Unitoverview


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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?


Unitoverview


errors

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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?


Unitoverview


errors

Roundofferror

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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?


Unitoverview


errors

Roundofferror

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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?


Unitoverview


errors

Roundofferror

Trunca:onerror

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Outline

Whatis

engineering

compu:ng?

Whydothisobject?

How?


Unitoverview


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?


Unitoverview


errors

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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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


errors

Roundofferror

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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?


Unitoverview


errors

Roundofferror

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Unit Overview

Computer laboratory contents

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Outline

Whatis

engineering

compu:ng?

Whydothisobject?

How?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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?


Unitoverview


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.

mec3456-2013-lecture01

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