CVEN302-502Computer Applications in

Engineering and Construction

Modeling is the development of a mathematicalrepresentation of a physical/biological/chemical/economic/etc. system

Putting our understanding of a system/problem into math

Numerical methods are one means by whichmathematical models are solved

Example:

Falling ParachutistF=ma =Fdown +Fup

=mg-cv (gravity minus air resistance)Where does mg come from?

Observations.Where does -cv come from?

More observations!

Now we have fundamental physical laws, so we combine those with observations to modelsystem.

A lot of what you will do is “canned” but needto know how to make use of observations.How have computers changed problem solving in engineering?

Allow us to focus more on the correct description of the problem at hand, rather than worry about how to solve it.

Example: Finite elements and structural analysisSimple truss - force balance Complex truss

Instead of limiting our analysis to simple cases, numerics allows us to work on realistic cases.

What are the fundamental laws we use in modeling?

Conservation of mass - i.e. traffic flow estimation

Conservation of momentum -i.e. force balance in structures

Conservation of energy - i.e. redox chemistry in water treatment plant

Issues to be considered in modeling and numeric methods

1.Nonlinear vs. Linear2.Large vs. Small systems3.Nonideal vs. Ideal4.Sensitivity analysis5.Design

Back to our example: the falling parachutist

F=ma=mg-cv

mcvmg

dtdv

cvmgdtdvm

Analytic solution (from calculus)

tmcecgmtv /1

Numerical solution

discretize original equation

iiiii

iii

ii

ii

ii

tttvmcgtvtv

tvmcg

tttvtv

tttvtv

tv

dtdv

11

1

1

1

1

Finally, combining analytical and numerical techniques

Catenary cable (power lines)From force balances, displacement can be modeled by a differential equation

2

2

2

1

dxdy

Tw

dxyd

a

0

2

4

6

8

10

12

-6 -4 -2 0 2 4 6 8 10 12

Ta

W=ws

Tb

Forces acting on catenary

Can solve by integration

wTyx

Tw

wTy a

a

a

0cosh

Where

xx eex 21cosh

This equation is not analytically solvable for w or Ta

Say we are given w, y0 and the value of y at an x. Can solve for Ta using numerical methods

wTyx

Tw

wTy a

a

a

0cosh

Becomes

Try different values of Ta until lhs is 0

0cosh 0a a

a

T Tw x y y xw T w

14

We will use Matlab as the computer language of choice for this course

• Anything you can do in Fortran or C you can do in Matlab

• Easier debugging system

• Built-in graphics

• Many, many functions already exist

• Excellent help capabilities

Matlab Truss example – nice animation

In short, you will use numeric methods throughout your career

- even if you don’t write programs

- even if you go into management

If we didn’t have numerical methods, we might as well be...