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Simulation and Optimisation in Engineering AERO2598
创建日期：2024-05-14 18:02:12
标签： AERO2598
Simulation and Optimisation in Engineering

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Simulation and Optimisation in Engineering
AERO2598
Assignment 2
2.1 Numerical Integration
2.2 Curve Fitting - Polynomial Fitting of Data
School of Engineering
Simulation and Optimisation in Engineering
AERO2598
Assignment 2.1
Numerical Integration
Assessment:
Assignment 2.1 is worth 15%.
Submission:
Location: AERO2598 Canvas Site
Objectives:
Evaluate numerical integrals of functions using a combination of Simpson’s Rules and
Trapezoidal Rule using Matlab program.
Submission Instructions:
 There are two m-files provided to you on Canvas. One is the template for the function file
you are to submit, and the other is a script file that you may use to test your function file
(Please do not submit the script file used for testing your function).
 Submit a single function m-file via Canvas before submission due date.
 Note that the Last Name and student ID are required as a comment line at the beginning of
the m-file, but after the function header (i.e. use the % symbol at the beginning of
commented lines).
 The m-file submitted on Canvas should be named as:
o LastName_StudentID_composite_integration.m (write your function in this one)
 Do not zip any files.
 Failure to submit before the due date will require a reasonable explanation and approval
from the Course Coordinator before any late submission can be accepted.
Page 1
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
Page 2
 RMIT late assignment submission policy will apply (10% deduction of marks for each day
delay).
 Use the Special Consideration process for any submission later than one week.
 Submit only the files related to Assignment 2.1.
Background
Often when dealing with data from experiments or computations, the data will be discrete and may
not have uniform spacing. We could use trapezoidal integration on it but it would be more
beneficial (generally more accurate) to use the appropriate Simpson’s rule in regions where we have
a consecutive interval spacing that is uniform or constant.
As an example, given the following x and y data to integrate, there are regions with uniform spacing
that can use Simpson’s rule.
x y
0.000 0
1.0000 0.8415
2.0000 0.9093
3.0000 0.1411
4.0000 -0.7568
4.5000 -0.9775
5.0000 -0.9589
5.1000 -0.9258
5.2000 -0.8835
5.3000 -0.8323
5.8000 -0.5507
6.1000 -0.1822
Simpson’s 1/3rd rule, with step=1
Simpson’s 1/3rd rule, with step=0.5
Simpson’s 3/8th rule, with step=0.1
Trapezoidal rule, with step=0.5
Trapezoidal rule, with step=0.3
Page 3
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
Task
Write a MATLAB function that will integrate discrete data, using Simpson's rules (1/3rd and 3/8th)
where possible and trapezoidal rule elsewhere.
The following function header MUST be used with LastName and StudentID being replaced
with your last name and student number (sxxxxxxx). The m-file must be saved with the same name
as the function name for the function to work.
function int_y_dx = LastName_StudentID_composite_integration(x,y)
You MUST use the following variables for your function input arguments: x and y, where y
represents a vector of values dependent on the corresponding values in vector x. The user of the
function will input two vectors into your compsite_integration function. Please see the test
script for an example of the input arguments for your function.
You MUST use the variable int_y_dx for your function output. This will represent the numerical
integral of area covered by x and y.
Coding Requirements
1. You may utilise the MATLAB trapz() function as part of your program. But for
Simpson’s 1/3rd and 3/8th rules you should use the basic equation given on page 6. (for
example, You must NOT use the integral() function)
2. You should not make any sub-functions inside your function.
3. If the x and y data given to the function is not the same length, the function MUST use the
error() function to give the user an error message and exit.
4. There must NOT be any user interaction within the function (i.e. no use of input() ) .
5. Your code MUST return the SIGNED area under the graph (i.e., the ACTUAL value of the
integral of y with respect to x, NOT absolute area). NOTE: The signed area above x-axis
(bounded by the x-axis and a curve) is positive, whereas the signed area below x-axis
bounded by a curve is negative. Conversely, an unsigned area is the absolute value of the
integral, meaning that the unsigned area below the x-axis bounded by a curve is positive.
6. To attain maximum accuracy, your function must avoid using trapezoidal rule
wherever possible. For example, if there is a run of 4 equally spaced INTERVALS,
then you should use 2× Simpson 1/3rd rules rather than 1× Simpson 3/8th and 1x
trapezoidal. To help you, the following table indicates which rule must be used for a
given number of consecutive equal intervals (as opposed to points).
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
Page 4
No of consecutive
equal intervals
Quadrature rule
1 1× trapezoidal
2 1× Simpson’s 1/3rd
3 1× Simpson’s 3/8th
4 2 × Simpson’s 1/3rd
5+ Use the rule for 3 first since it is most accurate, then
apply the appropriate rule for however many
intervals are left over.
e.g. for 7 intervals, you would use 1× Simpson 3/8th,
leaving 4 left over, for which you would use 2×
Simpson 1/3rd.
Another example how you code your function is shown in the following figure.
When creating software it is very important to ensure your program satisfies the requirements.
Requirements are often very specific as your software may need to interact with user and/or other
software components in a very specific way.
Please heed the following warning and ensure your program meets all the requirements described
herein before submitting it.
Assessment
 Correctly performs integration as specified 70%
 Correctly checks for equality of successive intervals as specified 20%
 Correctly performs error check on input values 10%
 Appropriate deductions for not meeting requirements -20%
Page 5
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
HINTS
(i) Do not place any code above your function header. This is mixing a script and function
definition and cannot be done. The user will pass the data they wish to use to the function via its
input arguments.
(ii) Do not store something to x or y in your function. If you store values in x or y in your
function, then it will overwrite what the user specified for x and y.
(iii) Do test your code. Maybe your code should return a value close to that of trapz (but maybe not
for the data given). Make up your own data (that you know the answer to the integration) and test
that your function returns the correct results.
(iv) Do the learning exercises in the tutorials. They will teach you the skills necessary to do this
task.
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
Page 6
Integration Formulae
Trapezoidal Rule:
׬ ݂ሺݔሻ݀ݔ ൌ ௛ଶ ሾ݂ሺݔଵሻ ൅
௕
௔ 2݂ሺݔଶሻ+2݂ሺݔଷሻ…൅2݂ሺݔ௡ିଵሻ ൅ 2݂ሺݔ௡ሻ ൅ ݂ሺݔ௡ାଵሻ] 
i.e. repeated units of ݂ሺݔ௜ሻ ൅ ݂ሺݔ௜ାଵሻ overlapping by one term
Simpson’s 1/3rd Rule:
׬ ݂ሺݔሻ݀ݔ ൌ ௛ଷ ሾ݂ሺݔଵሻ ൅
௕
௔ 4݂ሺݔଶሻ+2݂ሺݔଷሻ ൅ 4݂ሺݔସሻ ൅ 2݂ሺݔହሻ….. 2݂ሺݔ௡ିଵሻ+ 4݂ሺݔ௡ሻ ൅
݂ሺݔ௡ାଵሻ] 
i.e. repeated units of ݂ሺݔ௜ሻ ൅ 4݂ሺݔ௜ା!ሻ ൅ ݂ሺݔ௜ାଶሻ overlapping by one term
Can only be used if number of interval ( n) is an even number.
Simpson’s 3/8th Rule:
׬ ݂ሺݔሻ݀ݔ ൌ ଷ௛଼ ሾ݂ሺݔଵሻ ൅
௕
௔ 3݂ሺݔଶሻ+3݂ሺݔଷሻ ൅ 2݂ሺݔସሻ ൅ 3݂ሺݔହሻ ൅ 3݂ሺݔ଺ሻ ൅
2݂ሺݔ଻ሻ…..൅݂ሺݔ௡ାଵሻ] 
i.e. repeated units of ݂ሺݔ௜ሻ ൅ 3݂ሺݔ௜ାଵሻ ൅ 3݂ሺݔ௜ାଶሻ ൅ ݂ሺݔ௜ାଷሻ overlapping by one term
Can only be used if n is a multiple of 3 (can be odd or even).
݄ ൌ ܾ െ ܽ݊
In the above equations the number of segments/intervals is equal to the value of n because
we start at ݔ଴.
݊ݑܾ݉݁ݎ ݋݂ ݅݊ݐ݁ݎݒ݈ܽݏ ൌ ݊
However, usually the number of points used in the interval has the following relationship
with the number of segments/intervals it is broken up into, illustrated in the below figure.
݊ݑܾ݉݁ݎ ݋݂ ݅݊ݐ݁ݎݒ݈ܽݏ ൌ ݊ݑܾ݉݁ݎ ݋݂ ݌݋݅݊ݐݏ െ 1
a = 0.2 b = 1
No. of points = 5
No. of segments = No. of points-1 = 4
School of Engineering
Simulation and Optimisation in Engineering
AERO2598
Assignment 2.2
Curve Fitting - Polynomial Fitting of Data
Assessment:
The MATLAB Assignment 2.2 is worth 10%.
Objectives:
Generate curves by fitting data using polynomials
Submission Instructions:
 There are two m-files provided to you on Canvas. One is the template for the function file
you are to submit and the other is a script file that you should use to test your function file
(please do not submit the script file used for testing your function).
 Submit a single function m-file via Canvas before submission due date specified above.
 Note that the Last Name and Student ID are required as a comment line at the beginning of
the m-file, but after the function header.
 The m-file submitted on Canvas should be named as:
o LastName_StudentID_polynomial_fitting.m (write your function in this one)
 Do not zip any files.
 Failure to submit before the deadline will require a reasonable explanation and approval
from the Course Coordinator before any later submission can be accepted.
 RMIT late assignment submission policy will apply (10% deduction of marks for each day
delay).
 Use the Special Consideration process for any submission later than one week.
 Submit only the files related to the Assignment 2.2.
Background
When dealing with data from experiments or computations, the data will be discrete and not have
uniform spacing. Fitting the data within a polynomial can help to predict outputs and to deal with
the data in further calculations.
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
The aim of fitting your discrete data to a polynomial is to be able to predict a value of the dependent
variable in terms of the independent variable for those values which do not have corresponding
discrete data. Depending on the relationship of the variables, polynomials of different orders should
be used to find a good fit between the polynomial and the real data. In the figure shown below, real
measurements of liquid water density were taken in increments of 10 °C and several polynomials
were calculated using polyfit()to see the correlation between them and the discrete data.
After a polynomial is obtained, we can try to interpolate or extrapolate y-values for any x-value. Be
careful if you are extrapolating outside the boundaries of your discrete data, the polynomial could
fit well inside your data, but maybe it does not represent well the reality outside them. What is the
density of liquid water (pressurized) for temperatures greater that 100°C? In the figure above, we
can try to extrapolate them, but we do not know if the physical phenomena will be still fitting our
polynomials.
Task
Your task is to write a function which takes two vectors, x (independent variable) and y (dependent
variable), and fits polynomials from order 1 (linear relation) up to order 10 (or the maximum order
possible for the data size). The function should plot the x and y data with markers (no line) and plot
each fitted polynomial (with lines), all on the same plot axis. In a separate figure (or in separate
plots within one figure using subplot), plot the interpolated/extrapolated y value for a given x-point
defined by the user.
Page 2
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
Page 3
Coding Requirements
1. Use the MATLAB built-in function polyfit.
2. You should not make any sub-functions inside the function.
3. Ensure the x and y data given to the function have the same length. If it is not the case, the
function MUST use the error() function to give the user an error message and exit.
4. There must NOT be any user interaction within the function.
5. To fit a function with a polynomial of an order (also called grade or degree) “n” there
must be at least “n+1” points. Your code MUST detect if the number of points is not
sufficient enough and show an error message before trying to calculate wrong polynomials.
6. The user will enter the x and y vectors to the function from the script, and it will also specify
a point “x2” which the function will use to extrapolate its value y(x2) using all the
polynomials calculated.
7. You should create extra x-points in your function to make the polynomial line plots
smoother using more points.
8. If the given data fits exactly a polynomial of an order lower than 10, your function MUST
NOT plot the polynomials of a greater order. You could use a tolerance to test this.
Hint: In order to fulfill requirement number 8, you should compare your coefficients with a certain
tolerance (e.g 10e-6) if you calculate your polynomial, and from certain coefficient all the higher
order coefficients (beyond it) are lower than the tolerance, you may assume that it is a perfect fit for
a polynomial, excluding those higher order coefficients (i.e. they are equal to zero).
Page 4
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
Examples
Example 1
In this first example, the data fits exactly an order 2 polynomial, therefore the code will only plot
two polynomials (order 1 and 2)
The user will produce a call to the function in the workplace such as:
In this case, the function output will produce two plots. The first one is the polynomial fits with the
original data. As the polynomial of order 2 fits exactly the function, only 2 polynomials have to be
plotted.
Page 5
AERO2598 Simulation and Optimisation in Engineering: Assignment 2
The second plot output contains the predictions at position x2 (x-axis is polynomial order, y-axis is
the predicted y value). You have to plot only the number of predictions of the plotted polynomials.
As in this case the original data was a perfect x^2 relation, we expect all polynomials of this order
or greater to perfectly fit the data and extrapolate the same value. In this subplot you can choose
between plotting the guesses of all the polynomials or plot only those guesses for the polynomials
shown in the previous figure.
Example 2.
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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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