CVEN302-502

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CVEN302-502. Computer Applications in Engineering and Construction. Modeling is the development of a mathematical representation of a physical/biological/chemical/ economic/etc. system Putting our understanding of a system/problem into math Numerical methods are one means by which

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CVEN302-502

Computer Applications in

Engineering and Construction

Modeling is the development of a mathematical

representation of a physical/biological/chemical/

economic/etc. system

Putting our understanding of a system/problem

into math

Numerical methods are one means by which

mathematical models are solved

Example:

Falling Parachutist

F=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 model

system.

A lot of what you will do is “canned” but need

to 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 analysis

Complex truss

Simple truss - force balance

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. Linear

2.Large vs. Small systems

3.Nonideal vs. Ideal

4.Sensitivity analysis

5.Design

Back to our example: the falling parachutist

F=ma=mg-cv

dv

m

=

mg

-

cv

dt

dv

mg

-

cv

=

dt

m

Analytic solution (from calculus)

gm

(

)

(

)

(

)

-

/

c

m

t

v

t

=

1

-

e

c

Numerical solution

discretize original equation

(

)

(

)

dv

D

v

v

t

-

v

t

i

+

1

i

@

=

dt

D

t

t

-

t

i

+

1

i

(

)

(

)

v

t

-

v

t

c

(

)

i

+

1

i

=

g

-

v

t

i

t

-

t

m

i

+

1

i

c

é

ù

(

(

)

(

(

)

)

)

v

t

=

v

t

+

g

-

v

t

t

-

t

ê

ú

i

+

1

i

i

i

=

1

i

m

ë

û

Finally, combining analytical and numerical techniques

Catenary cable (power lines)

From force balances, displacement can be modeled by a differential equation

2

2

d

y

w

dy

æ

ö

=

1

+

ç

÷

2

dx

T

dx

è

ø

a

Forces acting on catenary

12

Tb

10

8

6

W=ws

Ta

4

2

0

-6

-4

-2

0

2

4

6

8

10

12

Can solve by integration

æ

ö

T

T

w

ç

÷

a

a

y

=

x

+

y

-

cosh

ç

÷

0

w

T

w

è

ø

a

Where

1

(

)

x

x

-

cosh

x

=

e

+

e

2

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

æ

ö

T

T

w

ç

÷

a

a

y

=

x

+

y

-

cosh

ç

÷

0

w

T

w

è

ø

a

Becomes

Try different values of Ta until lhs is 0

• 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

14

- 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...