ENGR3703-Computational Methods in Engineering 1
Fall 2020
Project
Due Date: 12/10/2020 by 11:59pm
Please submit the python files (with .py extension) together with your
project document.
Please upload all your documents to D2L-assignment folder
Background: Simpson’s 3/8 Method
In Simpson’s 3/8 Method, a cubic (third-order) polynomial is used to approximate
the f(x).
Equation 1
where
Equation 2
P(x) is the third order Newton’s Interpolating polynomial. Using this polynomial
and the definitions of a1 to a4 (found in your textbook and lecture notes) in Eq. 1
to perform the integration and subsequent (and somewhat lengthy algebraic
manipulation) one finds:
Equation 3
This technique as shown integrates only over four points over the interval from a
to b. The accuracy of the integral estimate in Eq. 3 can be improved by
decreasing the spacing of the points. This means one might need repeated
application of Eq. 3 over the entire range from a to b. An example of this is
shown in Figure When the entire interval from a to b is broken into a number of
sub-intervals, N, that is divisible by three, the entire integral can be shown to be:
Equation 4
∫
a
b
f ( x)dx=∫
a
b
P( x)dx
P(x)=a1+a2
(x−x1
)+a3
(x−x1
)(x−x2
)+a4
(x−x1
)(x−x2
)(x−x3
)
∫
a
b
f ( x)dx=∫
a
b
P( x)dx=
3h
8
[f (a)+3 f ( x2
)+3 f ( x3
)+f (b)]
∫
a
b
f (x)dx=
3h
8 [
f (a)+3 ∑
i=2,5,8 ,…
N −1
(f (xi
)+f ( xi+1
))+2 ∑
j=4,7,10
N−2
f ( xj
)+f (b)]
Part 1. Writing a Python code for Simpson’s 3/8 Method
Write a Python function simpson_3_8, that takes end points of the interval (a,b)
and number of sub-intervals, N, as parameters, and uses the function, f, to
calculate Eq. .
Your function should check if the number of sub-intervals is divisible by 3. If
number is not divisible by 3, your function should print out an error message and
exit.
Part 2. Application of Curve Fitting and Simpson’s 3/8 Methods
Background. The calculation of total force due to a distributed force is an
important engineering calculation:
If you have an object that is H meters tall then the total force, F, due to a
distributed force (f(z) ~ force per unit length dz) and its line of action can be found
(see Figure 2).
Figure 1
internship
The total force due to the distributed force is
The location of the line of action, d, of F is the effective location of the force F.
For example, if
f (z)=200z[N /m]
with z ranging from 0 to 15 m.
F=∫
0
15 m
200 zdz=(100 z
2
)0
15 m
=100 (152−0)=2.25 x104N
The line of action would be:
d=
∫
0
H
zf (z)dz
∫
0
H
f (z)dz
=
1
2.25 x104
N
∫
0
15m
200 z
2
dz=
1
2.25x 104
N
200
3
(153−0)=10m
Figure 2
F=∫
0
H
f (z)dz
Equation 5
d=
∫
0
H
zf (z)dz
∫
0
H
f (z)dz
Equation 6
When f(x) is a simple function like in the example above, solving Eqs. 5 and 6
can be done analytically (by hand). However, as in Fig. 2, it is more likely that the
functional relationship is complicated. For this situation, numerical methods
provide the only alternative for determining the integrals.
In this part of the project you need to find the total wind force exerted on the side
of a skyscraper and its line of action. The force is distributed and varies with
height. You only know the force per unit height, f(z) at certain discrete z values –
every 30 meters starting at ground level. These values are given in Table 1.
You will find the force and line of action three different ways:
a. Based on the data in Table 1, determine the coefficients of a third order
Newton’s interpolating polynomial that represents f(z). Once you have the
polynomial integrate it as in Eqs. 5 and 6 to determine the total force and
its line of action, F and d ..
a. VERY IMPORTANT – Only do this for the first three segments (first
four points!!). This way you only have a third order polynomial.
b. ALSO VERY IMPORTANT – to compare this to other integration
techniques you will need to use those techniques (parts b and c) for
the first three segments also.
b. Determine values of m and b such that the data in Table 1 are fit by an
equation of the form:
f (z)=b z
m
If you are curious of why we might expect this force to be of this form see
the last page of this document.
Once, b and m are determined use f(z) to find F and d . Note you will need
to do this from z = 0 to 270 m and from z = 0 to 90 m to compare the
integrals to part (a).
Figure 3
c. Use your python code for Simpson’s 3/8 rule to compute both F and d .
a. Note you will need to do this from z = 0 to 270 m and from z = 0 to
90 m to compare the integrals to part (a).
d. Compare the results for F and d from each technique of calculation…
when we say compare we mean numerically and to analyze how
significant the differences are and why there are differences (or not). This
section of your project is of equal importance to the calculation portions.
This should be a complete analysis of what you did and why there are
differences if any and if there are not significant differences why that is the
case.
a. Again – also compare integrals from z = 0 to 90 to the results of
part (a).
Table 1. Force on side of a skyscraper.
Height (z in m) Force/height (f(z) in N/m)
0 0
30 340
60 1200
90 1600
120 2700
150 3100
180 3200
210 3500
240 3800
270 4000
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