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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