Download Chapter 1

Applied Numerical Methods
with MATLAB
Part1
Chapter 1
Mathematical Modeling,
Numerical Methods and
Problem Solving
1-1
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Why Study Numerical Methods?
• As engineers we
solve mathematical
problems in order
to
– Understand
Physical Systems
– Create Designs
– Verify Behavior
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Why Study Numerical Methods?
• Not all problems are easy
to solve analytically
• Not all problems are
POSSIBLE to solve
analytically
• Numerical methods offer
us an alternative way to
model complex systems
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Problem Solving Process
• Define the Problem
• Create a mathematical
model
• Solve either analytically,
numerically or graphically
• Communicate your
results so that they can
be implemented
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• A mathematical model is represented as a functional
relationship of the form
Dependent
Variable
=f
independent
forcing
variables, parameters, functions
• Dependent variable: Characteristic that usually reflects the
state of the system
• Independent variables: Dimensions such as time and space
along which the system’s behavior is being determined
• Parameters: reflect the system’s properties or composition
• Forcing functions: external influences acting upon the system
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Newton’s 2nd law of Motion
• States that “the time rate change of momentum
of a body is equal to the resulting force acting
on it.”
• The model is formulated as
F = m a (Equation 1.2)
F=net force acting on the body (N)
m=mass of the object (kg)
a=its acceleration (m/s2)
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• Formulation of Newton’s 2nd law has several
characteristics that are typical of mathematical
models of the physical world:
– It describes a natural process or system in
mathematical terms
– It represents an idealization and simplification of
reality
– Finally, it yields reproducible results,
consequently, can be used for predictive purposes.
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• Some mathematical models of
physical phenomena may be much
more complex.
• Complex models may not be solved
exactly or require more sophisticated
mathematical techniques than simple
algebra for their solution
– Example, modeling of a
falling bungee jumper:
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Example – A Falling Bungee
Jumper
• The jumper
experiences two
forces
– Force due to
gravity
– Force due to drag
(air resistance)
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F  ma
F
a
m
dv Ftotal

dt
m
F  FD  FU
FD  mg
dv mg  cv

dt
m
FU  cv
2
2
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c 2
dv
g v
m
dt
• This is a differential equation and is written in
terms of the differential rate of change dv/dt of
the variable that we are interested in
predicting.
• If the jumper is initially at rest (v=0 at t=0),
using calculus
v(t ) 
 gcd 
gm
tanh 
t


cd
m


Dependent variable
Forcing function(g)
Independent variable
Parameters (m,cd )
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1-13
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Terminal Velocity
• As you can see from the graph – the
parachutist approaches a terminal
velocity
• At that point, the acceleration is equal
to 0 - ie dv/dt = 0
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dv
c 2
g v 0
dt
m
m
v  g*
c
2
68.1
v  9.8*
 51.67m / s
0.25
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1-17
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Numerical Solution
Euler’s Method
dv
c 2
g v
dt
m
Differential Equation
v(ti 1 )  v(ti )
dv v


dt t
ti 1  ti
Finite Difference
Approach
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What happens as delta t gets smaller?
dv
v
 lim
dt
t
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dv v(ti 1 )  v(ti )
c
2

 g  v(ti )
dt
ti 1  ti
m
Rearrange to give:
c

2
v(ti 1 )  v(ti )   g  v(ti )  *  ti 1  ti 
m


dv
dt
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Euler’s Method
New value = old value + slope * step size
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New value = old value + slope * step size
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New value = old value + slope * step size
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Numerical
Solution
Analytical
solution
What happens when you change the
value of h?
Try it with the example code.
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Conservation Laws and Engineering
• Conservation laws are the most important and
fundamental laws that are used in engineering.
Change = increases – decreases (1.13)
• Change implies changes with time (transient).
If the change is nonexistent (steady-state), Eq.
1.13 becomes
Increases =Decreases
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Fig 1.6
• For steady-state incompressible fluid flow in pipes:
Flow in = Flow out
or
100 + 80 = 120 + Flow4
Flow4 = 60
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Conservation in Engineering
• Mass
Chemical Engineers
• Momentum Civil and Mechanical
• Charge
Electrical
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Summary of Numerical Methods
• The book is divided into five categories of
numerical methods:
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Next time – Chapter 4
• Chapters 2 and 3 are MATLAB review
chapters
• Chapter 4 covers a number of different
concepts related to error measurements
including the Taylor series
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