Engineering Problem Engineering Problem Solving with ComputersSolving with Computers

MEPP Summer ResidencyAugust 2008

Outline for SessionOutline for SessionAbout meCourse overviewCasesExcel programming and

OptimizationMatlab Primer

IntroIntroJake Blanchard, Professor,

Engineering PhysicsPhD in Nuclear Engineering, UCLA,

1988Research: fusion technology, solid

mechanics, nuclear batteries for MEMS, laser effects in metals

Born and raised in Southern California

married with two kids (11 and 9 yrs)

My Research InterestsMy Research Interests

Fusion TechnologyFission Reactor Fuels

and StructuresNuclear

Microbatteries

Goal of EPSCGoal of EPSCStudents who have completed this

course should have a broad understanding of:◦Solution techniques for several equation

types◦Level of difficulty for various problem

types◦Appropriate tools for solving various

problems◦Advantages and disadvantages of using

computers to solve engineering problems

Goal of This SessionGoal of This SessionI just want to get you started with

the tools we use in my courseMy theory is that some exposure

now will benefit you in the SpringWe will repeat everything we do

today when the course starts, though in more depth

Therefore, don’t worry if you don’t catch on to everything right away

Approach Used in EPSCApproach Used in EPSCProblem-based using case

studiesAssigned and student-identified

projectsNo exams

TextbooksTextbooksWe will use a text I put together

from two full texts: one Excel-based and one Matlab-based ◦You won’t all need this and no

problems will be assigned from it◦Buy it from Rose

Many other background books are available if you need additional support

Numerical TopicsNumerical TopicsLinear and nonlinear equationsQuadratureOrdinary differential equations

◦initial value◦boundary value

OptimizationData AnalysisDatabases

Prerequisites and ResourcesPrerequisites and ResourcesMath skills: linear algebra

(matrices and vectors) and ordinary differential equations (first and second order)

Computer skills: some programming is useful, but not necessary

Used These Before?Used These Before?

FORTRANCC++PascalBasicJavaOthers?

• EES

• MathCAD

• Matlab

• Excel/Quattro

• Mathematica

• Maple

• Others?

ToolsToolsMicrosoft Excel (97, 2000, 2003,

etc.)Matlab from Mathworks

◦there is a student version for ~ $100◦Any version beyond Matlab 4 will work

fine, though a couple things got easier with version 6 and a few commands have changed

◦You can run it as “tethered” software from CAE if you are interested (www.cae.wisc.edu/tethered)

CasesCasesBroad coverage of engineering

disciplines and topicsThe more students can see

relevance of cases, the better

Case Titles and TopicsCase Titles and TopicsMacros in ExcelDatabases

◦Material Properties (choose your own for lesson)

Quadrature◦Automobile suspension

Roots◦natural frequencies of satellite boom

Linear Systems◦fluid flow network

Case Titles and TopicsCase Titles and Topics

Initial Value◦PSII target heating

Boundary Value◦chip cooling

Optimization (Linear Programming/Constraints)◦manufacturing

Case Titles and TopicsCase Titles and TopicsOptimization (Monte Carlo)

◦maximize profit with uncertain inputsCurve Fits

◦radioactive decay

Hands-On SessionHands-On SessionOptimization Problem in Excel

(using the Solver)Introduction to Matlab

First Download FilesFirst Download Fileshttp://www.cae.wisc.edu/~blanchar/

MEPP.html

Optimization of Water Optimization of Water DistributionDistributionConsider two bottling plants:

Mexico City and TepicConsider three customers: one in

Monterrey, one in Mazatlan, and the other in Acapulco

Question is: which plants should ship to which cities?

Plant DataPlant Data

Plant Production cost ($/gal)

Capacity

(Million gal/y)

Distance to

Acapulco (km)

Distance to

Monterrey (km)

Distance to

Mazatlan (km)

Mexico City

0.5 1 411 989 1085

Tepic 0.6 0.5 1064 1004 278

Customer DataCustomer Data

Customer Demand (Million gal/y)

Acapulco 0.32

Monterrey 0.44

Mazatlan 0.16

Best GuessBest GuessUse pencil and paper to make

best guess of optimum distribution

Goal is to meet demand with minimum cost

Selling price is the same in each city

Sheet SetupSheet Setup

What is total cost for your guess at the optimal distribution?

Using the SolverUsing the Solver

Tools/Add-Ins…SolverTools/Solver

Take a look at this link for further information:

http://econltsn.ilrt.bris.ac.uk/cheer/ch9_3/ch9_3p07.htm

The Main Solver WindowThe Main Solver Window

Dialog for Adding ConstraintsDialog for Adding Constraints

Results DialogResults Dialog

Your TaskYour TaskSet up the Solver, with all

appropriate constraints, to optimize this problem

Next ScenarioNext ScenarioWhat if the production cost in

Tepic drops to 0.45 $/gal?

Next ScenarioNext ScenarioWhat is optimum if demand

increases in each city by 50%? Put Tepic back at 0.6 $/gal on the production cost.

Adding a “Run” ButtonAdding a “Run” ButtonExcel has a built-in macro

language called Visual Basic for Applications (VBA)

It can be used to create user interfaces

We can use it to add a button for running the Solver

Using MacrosUsing MacrosMacros are written in a Basic-like

language called Visual Basic for Applications

Excel comes with a separate macro editor

To create or edit a macro, go to Tools/Macro/Visual Basic Editor

To add a new module go to Insert/Module

The MacroThe Macro

Sub runsolver() SolverSolveEnd Sub

Creating the ButtonCreating the ButtonGo to View/Toolbars/FormsFrom this Toolbar, click on the

button (row 2, column 2) and then trace out a button on a spreadsheet

Assign the “runsolver” macro to the button

Now click the buttonNote that you may have to add a

“Reference” to the Solver under Tools in the VBA Editor

User-Defined FunctionsUser-Defined FunctionsWe can also create user-defined

functions in VBAThese can be called from cells,

just like any built-in function

Creating a FunctionCreating a Function

Suppose we want to create an Excel function that takes a temperature in Celsius and converts to Fahrenheit

We would type the following in a module:

Function ctof(temp) ctof = 9 / 5 * temp + 32End Function

Using the functionUsing the functionThen you can go to the

spreadsheet and type =ctof(100)Or, you can put the value of

“100” into cell A1 and then type =ctof(A1) into some other cell

In fact, this function can be used just as any built-in Excel function can be used

ExerciseExerciseCopy this function and create

ftoc for converting from fahrenheit to celsius

The Macro LanguageThe Macro LanguageOperators: +, -, *, /, ^, ModComparison: =, <, >, <=, >=,

<>Logical Operators: And, Eqv,

Imp, Not, Or, XorIntrinsic Functions: Abs, Cos,

Sin, Tan, Atn (arc tangent), Exp, Log (natural), Sgn, Sqr (square root), Rnd (random number)

Flow ControlFlow Control

If condition Then statementsElse statementsEnd If

If x=0 Then f=1Else f=sin(x)/xEnd If

Flow ControlFlow Control

For counter=start To end statementsNext

For i=1 To 100 sum=sum+iNext

Flow ControlFlow Control

Do Until condition statementsLoop

i=1x=1Do Until i=50 x=x*i i=i+1Loop

Flow ControlFlow Control

Do While condition statementsLoop

i=1x=1Do While i<50 x=x*i i=i+1Loop

ExampleExampleWrite an Excel function that

calculates the factorial of some number Z

Factorial is Z*(Z-1)*(Z-2)*…*3*2*1

My solutionMy solution

Function fact(Z) x = 1 ans = 1 Do Until x = Z ans = ans * x x = x + 1 Loop fact = ansEnd Function

Another SolutionAnother Solution

Function fact(Z) ans = 1 For i = 1 To Z ans = ans * i Next fact = ansEnd Function

ExerciseExerciseWrite an Excel function that

calculates the sum of the first N cubes and test it

That is: 13+23+33+…+N3

What is the result for N=20?

MatlabMatlabMatlab began as a linear algebra

packageIt’s grown to be a general

purpose equation solverStrengths: robust routines,

excellent performanceWeaknesses: more difficult user

interface

Matlab DemosMatlab DemosType “demo”View:

◦Visualization/3D plots, vibration movie

◦Miscellaneous/Bending Truss

Matlab GUIsMatlab GUIs

Matlab has a tool for creating Graphical User Interfaces

You can start it up by typing guide at the command prompt

See an example by typing nextdemogui to run the file we downloaded earlier

Note that you may need to set the path by setting the Current Directory at the top of the command window

Matlab ToolkitsMatlab ToolkitsSimulink: dynamic systems

simulatorStateflow: event-driven systemsDSP, Signal Processing, Image

ProcessingControl, OptimizationPDE, Financial, Mappingand many more

Starting Out with MatlabStarting Out with MatlabStart Matlab - you’ll see the

“Command Window”Type the following:

A=[2 4; 1 3]

B=inv(A)

A*B

Plotting with MatlabPlotting with Matlab

list=0:0.1:20;y=sin(list);plot(list,y)

Using Scripts (m-files)Using Scripts (m-files)

Start up the editor (File/Open)Open test.mChange sin to cos in the scriptSaveGo back to command window and

execute by typing test

The scriptThe scriptclear allecho onA=[2 4; 1 3]B=inv(A)A*B%%Hit any key to continue

pauselist=0:0.1:20;y=sin(list);plot(list,y)

An ExampleAn ExampleSuppose we’re looking for a $100k,

30-year mortgage. What interest rate do I need to keep the payments below $700 per month?

Solve for i

0)1(

1)1(700100000

360

360

ii

i

ApproachApproachCreate user-defined functionPlot the functionFind point where function is 0

Create the functionCreate the function

function s=f(i)p=100000;n=360;a=700;s=p-a*((1+i).^n-1)./(i.*(1+i).^n);

Plot the FunctionPlot the FunctionFirst save file as f.mNow enter the following:

i=0.001:0.0001:0.01;y=f(i);plot(i,y)

ResultResultZero-crossing is around i=0.006Annual interest rate is 12*i, or

about 7%Try more accurate solution

12*fzero('f',0.006)

This gives about 7.5%

ExerciseExerciseWhat is the interest rate to keep

the payments below $700 for a 15 year mortgage?

Summary and ConclusionsSummary and ConclusionsGoal: broad understanding of

◦Solution techniques◦Level of difficulty◦Appropriate tools◦Advantages and disadvantages

Any comments, requests, or suggestions?

See You Next Spring!See You Next Spring!