CVEN302502. 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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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
Falling Parachutist
F=ma
=Fdown +Fup
=mgcv (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=mgcv
dv
m
=
mg

cv
dt
dv
mg

cv
=
dt
m
Analytic solution (from calculus)
gm
(
)
(
)
(
)

/
c
m
t
v
t
=
1

e
c
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
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
Matlab Truss example – nice animation
14
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...