Scientific Computing - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Introduction to Scientific Computing

My advice

• If you don’t let a teacher know at what level

you are by asking a question, or revealing your

ignorance you will not learn or grow.

• You can’t pretend for long, for you will

eventually be found out. Admission of

ignorance is often the first step in our

education.

• Steven Covey—Seven Habits of Highly Effective People

Why use Numerical Methods?

• To solve problems that are intractable!

How do we solve an engineering problem?

Problem Description

Mathematical Model

Solution of Mathematical Model

Using the Solution

Bascule Bridge THG

Trunnion

Hub

Girder

Bascule Bridge THG

Problem

After Cooling, the Trunnion Got Stuck in

Hub

Why did it get stuck?

Magnitude of contraction needed in the trunnion

was 0.015” or more. Did it contract enough?

Consultant calculations

∆ D = D × α × ∆ T

D = 12. 363 "

T F

o ∆ = − 108 − 80 = − 188

in in F

o

6. 47 10 / /

− 6 α = ×

∆ D = × − −

Is the formula used correct?

∆ D = D α∆ T

T(oF) α (μin/in/ oF) -340 2. -300 3. -220 4. -160 4. -80 5. 0 6. 40 6. 80 6.

∆ D = D ×α ×∆ T

Can You Roughly Estimate the Contraction?

∆ D = DTT ∫ ca α( T ) dT

D D T dT

c

a

T

T

∆ = ∫α ( ) Ta=80o^ F; Tc=-108 o^ F; D=12.363”

Can You Find a Better Estimate for the

Contraction?

D D T dT

c

a

T

T

∆ = ∫α( )

T a = 80 o^ F T c = -108 o^ F D = 12.363"

So what is the solution to the problem?

One solution is to immerse the trunnion in liquid nitrogen which has a boiling point of -321 o^ F as opposed to the dry- ice/alcohol temperature of -108 o^ F.

∆ D = − 0. 0244 "

Revisiting steps to solve a problem

1. Problem Statement: Trunnion got stuck in the 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 liquid nitrogen.

D D T dT

c

a

T

T

∆ = ∫α( )