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8/22/2011 http://numericalmethods.eng.usf.edu 1

Introduction to Scientific

Computing

Major: All Engineering Majors

Authors: Autar Kaw, Luke Snyder

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

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Introduction

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My advice

If you dont let a teacher know at what level

you are by asking a question, or revealing

your ignorance you will not learn or grow. You cant pretend for long, for you will

eventually be found out. Admission of

ignorance is often the first step in our

education. Steven CoveySeven Habits of Highly Effective People

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Why use Numerical Methods?

To solve problems that cannot be solved exactly

due

x u

g

2

2

2

1

T

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Why use Numerical Methods?

To solve problems that are intractable!

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Steps in Solving an

Engineering Problem

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How do we solve an engineering

problem?

Problem Description

Mathematical Model

Solution of Mathematical Model

Using the Solution

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Example of Solving an

Engineering Problem

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Bascule Bridge THG

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Trunnion

Hub

Girder

Bascule Bridge THG

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Trunnion-Hub-Girder

Assembly Procedure

Step1. Trunnion immersed in dry-ice/alcohol

Step2. Trunnion warm-up in hub

Step3. Trunnion-Hub immersed in

dry-ice/alcohol

Step4. Trunnion-Hub warm-up into girder

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Why did it get stuck?

Magnitude of contraction needed in the trunnionwas 0.015 or more. Did it contract enough?

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Video of Assembly Process

Unplugged Version VH1 Version

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Consultant calculations

TDD (vv!( E

FTo18880108 !!(Finin o//1047.6 6v!E

0.01504"

)188)(1047.6)(363.12( 6

!

v!(

D

"363.12!D

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Is the formula used correct?

TDD (!( E

T(oF) (in/in/oF)

-340 2.45

-300 3.07

-220 4.08-160 4.72

-80 5.43

0 6.00

40 6.24

80 6.47

TDD (vv!( E

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The Correct ModelWould Account for

Varying Thermal Expansion Coefficient

dTTDDc

a

T

T

)(!( E

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Can You Roughly Estimate the

Contraction?

dTTDDc

a

T

T

)(!( E

Ta=80oF; Tc=-108

oF; D=12.363dTTDDc

a

T

T

)(!( E

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Can You Find a Better Estimate for

the Contraction?

dTTDD

c

a

T

T

)(!( E

Ta = 80oF

Tc = -108oF

D= 12.363"

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Estimating Contraction

Accurately

dTTDD

c

a

T

T

)(!( E

Change in diameter((D) by cooling it in dryice/alcohol is given by

0150.6101946.6102278.1325

vv! TTE

Ta = 80oF

Tc = -108oF

D= 12.363"

"0137.0!(D

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So what is the solution to the

problem?One solution is to immerse the trunnion in liquid nitrogen

which has a boiling point of -321oF as opposed to the

dry-ice/alcohol temperature of -108oF.

"0244.0!(D

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Revisiting steps to solve a problem

1) Problem Statement: Trunnion got stuck inthe hub.

2) Modeling: Developed a new model

3) Solution: 1) Used trapezoidal rule OR b)Used regression and integration.

4) Implementation: Cool the trunnion in liquidnitrogen.

dTTDD

c

a

T

T

)(!( E

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THE END

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Introduction to Numerical Methods

Mathematical Procedures

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Mathematical Procedures Nonlinear Equations

Differentiation

Simultaneous Linear Equations

Curve Fitting

Interpolation

Regression

Integration

Ordinary Differential Equations

Other Advanced Mathematical Procedures:

Partial Differential Equations

Optimization

Fast Fourier Transforms

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Nonlinear EquationsHow much of the floating ball is under water?

010993.3165.0 423 !v xx

Diameter=0.11m

Specific Gravity=0.6

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Nonlinear EquationsHow much of the floating ball is under the water?

010993.3165.0)(423

!v!

xxxf

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Differentiation

What is the accelerationat t=7 seconds?

dt

dva !t.

tv(t) 89

50001016

1016ln2200

4

4

v

v!

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Differentiation

Time (s) 5 8 12

Vel (m/s) 106 177 600

What is the acceleration at t=7 seconds?

dt

dva !

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Simultaneous Linear EquationsFind the velocity profile, given

,)( 2 cbtattv !Three simultaneous linear equations

106525 ! cba

125 ee t

177864 ! cba60012144 ! cba

Time (s) 5 8 12

Vel (m/s) 106 177 600

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InterpolationWhat is the velocity of the rocket at t=7 seconds?

Time (s) 5 8 12

Vel (m/s) 106 177 600

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RegressionThermal expansion coefficient data for cast steel

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Regression (cont)

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Integration

!(fluid

room

T

T

dTDD E

Finding the diametric contraction in a steel shaft when

dipped in liquid nitrogen.

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Ordinary Differential EquationsHow long does it take a trunnion to cool down?

),( ahAdt

dmc UU

U! roomUU !)0(

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Additional Resources

For all resources on this topic such as digital audiovisual

lectures, primers, textbook chapters, multiple-choice tests,

worksheets in MATLAB, MATHEMATICA, MathCad and

MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/introduction_nu

merical.html

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THE END

http://numericalmethods.eng.usf.edu