NUMERICAL PROJECT REPORT PDF

Preview

Summary

Download NUMERICAL PROJECT REPORT PDF

Description

Application of Numerical Mathematics in Civil Engineering Area

COURSE CODE SUBJECT

BFF BFC 25203

CODE COURSE

Numerical Mathematics

NAME FACULTY

Faculty of Civil and Environmental Engineering 1. Mohamad Faisal Irwan bin Mokhtar ( CF200192 ) 2. Mohamad Syahir bin Mustaffa ( CF200126 )

GROUP 3. Nurul Syifa’ binti Nordin ( CF200108 )

MEMBER

4. Nur Adillah binti Omar ( CF200112 ) SECTION NAME OF

8 Dr. Roshayati binti Yahya @ Atan

LECTURER DATE

07 December 2021

MARKS

1.0 INTRODUCTIONS

1.1 BACKGROUND OF STUDY As we know, numerical method use in civil engineering works. We take thermodynamic process as our study to identifiy the method and solve the tabulated data given. In addition, we focused on numerical integration chapter. Numerical integrations is needed when direct integration is impossible. There are severals numerical methods that can be used to approximate integrals such as trapezoidal rule, Simpson’s 1/3 Rule, Simpson’s 3/8 Rule and Gauss-Quadrature.

1.2 PROBLEM STATEMENTS The work produced by a constant temperature, pressure volume thermodynamic process can be computed as 𝑏

𝑊 = ∫ 𝑝 𝑑𝑉 𝑎

Where W is work, p is pressure, and V is volume. Given the data in the table below, using a combination of the trapezoidal rule, Simpson’s 1/3 rule, and Simpson’s 3/8 rule to compute the work W. The combination needs to inclue all these three rules. P (kPa)

328

296

263

258

256

247

180

160

V (m3)

0.5

2

3

4

6

8

10

11

(Hint: the distances between the data points are not equal, and you need find a method to group the segments of the same length together, and then apply the integration rules.)

1.3 FILA TABLE Facts

•

Ideas

In thermodynamic process,

•

Pressure volume

•

Actions

How to compute

thermodynamic

the

can be used to

thermodynamic

constant

determine work in

process?

temperature.

thermodynamic

produced

work

Learning Issues

by

a

work

in

•

Use

trapezoidal

rule,

simpson’s

1 3

rule

simpson’s

and 3 8

rule to

determine the work

1.4 OBJECTIVES a) To determine the volume under the graph or tabulated data. b) Identify an approximate method to be used c) To determine the combination of the three methods

2.0 LITERATURE REVIEW 2.1 Trapezoidal Rule Trapezoidal Rule is a rule that evaluates the area under the curves by dividing the total area into smaller trapezoids rather than using rectangles. This integration works by approximating the region under the graph of a function as a trapezoid, and it calculates the area. This rule takes the average of the left and the right sum. The Trapezoidal Rule does not give accurate value as Simpson’s Rule when the underlying function is smooth. It is because Simpson’s Rule uses the quadratic approximation instead of linear approximation. Both Simpson’s Rule and Trapezoidal Rule give the approximation value, but Simpson’s Rule results in even more accurate approximation value of the integrals. Belows shown the formula for Trapezoidal Rule 𝑏

∫ 𝑓(𝑥)𝑑𝑥 ≈ 𝑎

(𝑏 − 𝑎) [𝑓(𝑎) + 𝑓(𝑏)] 2

𝟏

2.2 Simpson’s 𝟑 Rule For minimum segment in trapezoidal rule is one while in Simpon’s 1/3 rule is two. By using Newton forward difference (up to second order), we obtain an area A1, with n=2 segments under Simpson’s 1/3 rule, given by a basic formula of 2

ℎ

A1= ∫0 𝑓(𝑥)𝑑𝑥 ≈ 3[f0 +4f1 + f2] For better accurancy, the interval [a,b] is divided into n=2m (m, n are integer) subintervals [xi, xi+1] of equal width h=

𝑏−𝑎 𝑛

such that the integral is approximated by composite Simpson’s 𝑏

1 3

rule as

ℎ

∫𝑎 𝑓(𝑥)𝑑𝑥 ≈ 3 [𝑓1 + 4(𝑜𝑑𝑑) + 2(𝑒𝑣𝑒𝑛) + 𝑓last]

𝟑

2.3 Simpson’s 𝟖 Rule Simpson's 3/8 rule, also called Simpson's second rule, is another method for numerical integration proposed by Thomas Simpson. It is based upon a cubic interpolation rather than a quadratic interpolation. Simpson's 3/8 rule is as follows 𝑏

∫𝑎 𝑓(𝑥)𝑑𝑥 ≈

3ℎ 8

(f0 +3f1+3f2+f3) 3

For better accurancy, the integral is approximated by composite Simpson’s 8 rule 𝑏

∫𝑎 𝑓(𝑥)𝑑𝑥 ≈

3ℎ 8

𝑛

( )−1

3 [f0+fn+3∑ 𝑛/3 𝑖=1(f3i-2 + f3i-1) + 2∑𝑖=1

𝑓3i]

3.0 METHODOLOGY

Finding topics for project title

Study and state the: • •

Problem statements Objectives

Make some background study and literature review

Identify approximate method and solve the problem

Discuss the method and calculations result

Conclusion

4.0 CALCULATIONS V (m3)

0.5

2

3

4

6

8

10

11

P (kPa)

328

296

263

258

256

247

180

160

V (m3)

0.5

2

P (kPa)

328

296

1st group :

h = interval size = 2 – 0.5 = 1.5m3 using trapezoidal rule of integration : 2

𝑊1 = ∫0.5 𝑃𝑑𝑣 = 𝑊1 =

ℎ [(𝑃0 2

+ 𝑃1 )]

1.5 [328 + 296] 2

𝑊1 = 468 𝑘𝑃𝑎𝑚3 𝑜𝑟 𝐾𝐽

2nd group : V (m3)

2

3

4

P (kPa)

296

263

258

1

n = 2 , since ‘n’ is even we can use simpson’s 3 rule : h= 3 – 2 = 4 – 3 = 1m3 ℎ

4 𝑊2 = ∫2 𝑃𝑑𝑣 = 3 [(𝑃0 + 𝑃2 ) + 2 (𝑒𝑣𝑒𝑛 𝑃′𝑠) + 4(𝑜𝑑𝑑 𝑃′𝑠)] ℎ

= 3 [(𝑃0 + 𝑃2 ) + 2(0) + 4 (𝑃1 )] 1

= 3 [(296 + 258) + 4(263)] 𝑊2 = 535.33 𝐾𝐽

3rd group: V (m3)

4

6

8

10

P (kPa)

258

256

247

180

n=3, since ‘n’ is odd we cannot use simpson’s

1 3

rule, so we must use simpson’s

3 8

rule, since ‘n’ is

multiple of 3. 3ℎ

10

𝑊3 = ∫4 𝑃𝑑𝑣 =

8

=

3ℎ [(𝑃0 8

=

3(2) [(258 + 8

[(𝑃0 + 𝑃3 ) + 2 (𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3 , 𝑃′𝑠) + 3 𝑥 (𝑜𝑡ℎ𝑒𝑟 𝑡ℎ𝑎𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3, 𝑃′𝑠)]

+ 𝑃3 ) + 2(0) + 3𝑥 (𝑃1 + 𝑃2 )] 180) + 3(256 + 247)]

𝑊3 = 1460.25 𝐾𝐽 4TH group: V (m3)

10

11

P (kPa)

180

160

h = 1m3, n = 1 using trapezoidal rule: 11

ℎ

𝑊4 = ∫10 𝑃𝑑𝑣 = 2 [𝑃0 + 𝑃1 ] 1

= 2 [180 + 160] 𝑊4 = 170 𝐾𝐽

total work done = W

= W1 + W2 + W3 + W4 = 468 + 535.33 + 1460.25 + 170 = 2633.58 KJ

5.0 DISCUSSION 1

3

From the result obtained using a combination of trapezoidal rule, Simpson’s 3 rule, and Simpson’s 8 rule of calculation, this is the most suitable method in solving our problem. The total work done that we get from this solution is 2633.58 𝑘𝑃𝑎 which is accurate to the work produced by a constant temperature, pressure volume thermodynamic process. Other than, we also try use another method conducted shows differences in result which differentiate the level of accuracy and correctness of it. Moreover, it is best to use the most accurate result to prevent failure occurring to the accurate to the work produced by a constant temperature, pressure volume thermodynamic process.

6.0 CONCLUSION As a conclusion, the main objective of our work is to determine better numerical integration formula for our problem statement which is compute the pressure volume thermodynamic. Therefore, we apply Trapezoidal rule, Simpson’s 1/3 rule and Simpson’s 3/8 rule to solve various numerical problems and compare the result with their exact solution. From the methods examined in our calculation, we are capable of showing numerical integration for finding the smallest error value by using the methods of Trapezoidal as well as Simpson’s 1/3, Simpson’s 3/8 rules that we have discussed. We have tried to display some examples as well as emphasized the condition for which Simpson’s 1/3 method is the best. Consequently, we see that Simpson’s 1/3 rule gives the smallest error value among the rules as well as formally it is the most effective and appropriate methods among the mentioned rules in the case.

7.0 REFERENCES 1. Numerical Methods Text Book 2. Trapezoid Rule And Simpson’s Rule,©2002, 2008, 2010 Donald Kreider and Dwight Lahr 3. Lecture Class Video https://drive.google.com/drive/folders/1CsbCgYT0ZlXKm3dkPYKVekxQ1mWKvAH9 4. Youtube e-learning https://www.youtube.com/watch?v=7EqRRuh-5Lk 5. Lecture Slide Presentation https://slidetodoc.com/simpsons-13-rd-rule-of-integration-civil-engineering/...