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This document is for Coventry University students for their own use in completing their assessed work for this
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7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 1
Faculty of Engineering, Environment and Computing
7015MAA
Mathematical Modelling in Aerospace Engineering
Assignment Brief

Module Title Mathematical Modelling in Aerospace Engineering	Individual	Cohort SEP-JAN	Module Code 7015 MAA
Coursework Title CW1 Resit / Deferral: Data-Driven Modelling	Hand out date (dd/mm/yyyy) Aula: 28/01/2022
Lecturer Mauro Sebastián INNOCENTE	Due date and time Aula: 04/04/2022 18:00 hrs.
Estimated Time 15 hrs. Page Limit: 9 (excluding cover page, table of contents and references).	Coursework type Written	Credit value assessed 5 (33.33% of the module)
File types: 1 pdf file (report) 1 zip file (all codes) Mark and Feedback date (dd/mm/yyyy): 18/04/2022 Mark and Feedback method: Written via Aula

AssignmentTutorOnline

Module Learning Outcomes Assessed
1. Understand how the scientific method works and the process of constructing a mathematical
model.
2. Be aware of different modelling techniques, including their strengths and weaknesses and when
to use them.
3. Develop suitable models to extract the underlying pattern of behaviour of a system, sub-system
or component from physical measurements, observed data, or generated data.
4. Differentiate between mechanistic modelling and data-driven modelling, including grey-box and
black-box models.
6. Critically evaluate the validity of the developed models in terms of their ability to represent the
phenomena under study as well as to clearly articulate their inherent simplifying assumptions
and limitations.
7. Evaluate, communicate and interpret the performance of a given system, sub-system or
component based on the insight gained through modelling.

Task and Mark distribution
1. Report presentation 10%
2. Question 01: Constructing a Mathematical Model. 10%
3. Question 02: Taylor Polynomial for Data-Driven Modelling. 10%
4. Problem 01: Heat Transfer Data-Driven Modelling. 35%
5. Problem 02: Spring Stiffness Identification (Load-Controlled). 35%

General Notes
1) Please notify your registry course support team and module leader for disability support.
2) Any student requiring an extension or deferral should follow the university process as outlined
here.
3) The University cannot take responsibility for any coursework lost or corrupted on disks, laptops
or personal computer. Students should therefore regularly back-up any work and are advised to
save it on the University system.
4) If there are technical or performance issues that prevent students submitting coursework
through the online coursework submission system on the day of a coursework deadline, an
appropriate extension to the coursework submission deadline will be agreed. This extension will
normally be 24 hours or the next working day if the deadline falls on a Friday or over the
weekend period. This will be communicated via your Module Leader.
5) You are encouraged to check the originality of your work by using the draft Turnitin links in Aula.
6) Collusion between students (where sections of your work are similar to the work submitted by
other students in this or previous module cohorts) is taken extremely seriously and will be reported
to the academic conduct panel. This applies to both courseworks and exam answers.
7) A marked difference between your writing style, knowledge and skill level demonstrated in class
discussion, any test conditions, and that demonstrated in a coursework assignment may result in
you having to undertake a Viva Voce in order to prove the coursework assignment is entirely your
own work.
8) If you make use of the services of a proof-reader in your work, you must keep your original version
and make it available as a demonstration of your written efforts.
9) You must not submit work for assessment that you have already submitted (partially or in full),
either for your current course or for another qualification of this university, unless this is
specifically provided for in your assignment brief or specific course or module information. Where
earlier work by you is citable, i.e. it has already been published/submitted, you must reference it
clearly. Identical pieces of work submitted concurrently will also be considered to be self
plagiarism.

Submission Notes
1) Written reports are to be submitted in pdf format via Aula.
2) Programming codes and/or supporting files to be submitted via Aula as one zip-file.
3) Paper size: A4. Maximum size of report is 9 pages, excluding cover page, table of contents, and
reference list.
4) Font type & size: Times, 12pt.
5) Single line spacing and body text justified.
6) Paragraph spacing: 6pt (before and after paragraph).

7) Margins: 2.5 cm on the top and 2 cm elsewhere.
8) Equations MUST be sequentially numbered between parentheses.
9) Tables and figures MUST be sequentially numbered, placing the labels below them followed by a
period and explanatory text (caption).
10) Font size in labels and captions: 10pt.
11) References and citations as well as numbering and cross-references of equations, tables and
figures MUST be automatic (as opposed to manual).
12) Page numbers MUST be included in the report.
13) If citations and references are provided, use any standard reference style (e.g. IEEE 2006 or
Harvard) generated by any reference manager of your choice (e.g. Word built-in reference
manager, EndNote, JabRef, Zotero).

Important definitions
• Problem Formulation
It consists of the translation of the problem statement from plain words into mathematical
language. The problem formulation defines the mathematical model, stemming from an
interpretation of the real-world problem and from the setting of a number of assumptions.
Typically, a formulation is given in terms of algebraic equations; of algebraic functions with
unknown coefficients (data-driven modelling); or in terms of difference or differential equations
and initial and/or boundary conditions.
• Explanations and Solutions
Solutions refer to the problem formulated. Thus, a correct solution may be found for either a
correctly or an incorrectly formulated problem. Typically, solutions consist of coefficients’ values;
closed-forms that satisfy difference or differential equations; or values of discretised fields for
numerical problems (a model could be fit afterwards). Solutions may also entail quantities derived
from them (i.e. velocity at a given location derived from obtained position field; or heat flux at a
given location once temperature field has been obtained). Detailed explanations and justifications
of what has been done and why throughout the problem-solving procedure –between formulation
and results’ presentation– will also be assessed.
Note: If the problem is solved using a numerical method (e.g. Runge-Kutta, Finite Differences,
etc.), then the submitted code will also be assessed here. Make sure it is neat and commented.
• Results Presentation
This does not refer to the professional look of the report, which is assessed as Report Presentation
(see above). Instead, results’ presentation here refers to the relevance, clarity and completeness
of the results presented (e.g. equations, data, tables, figures, etc.) to describe and help interpret
the solution(s) obtained for the problem at hand. They must be easily accessible and
understandable for their analysis and interpretation.
• Results’ Interpretation (Discussion and Conclusions):
This refers to the student exhibiting clear understanding of the meaning of the results obtained
and what they entail, instead of merely enumerating them. This may also include using the model
developed to study the phenomena in question and draw some conclusions. While doing this, it is
quite common to find that results do not make sense and therefore mistakes be identified. The
importance of this component is evidenced by the high weight assigned to it in the marking rubric
below. Different grades will be awarded according to the depth of this analysis. Do not confuse
depth with length. Be clear and concise.

MARKING RUBRIC FOR REPORT PRESENTATION
First 70-100	– Guidelines for the report presentation were followed. – Outstanding, well-structured and professionally looking report.
Upper Second 60-69	– Guidelines for the report presentation were followed for the most part. – Well-structured and professionally looking report.
Lower Second 50-59	– Guidelines for the presentation of the report were partially followed. – Acceptably well-structured report lacking some presentation quality.
Third 40-49	– Guidelines for the presentation of the report were largely ignored. – Rather poorly structured and/or unprofessionally looking report. – Nonetheless, results obtained can still be extracted with minimal effort.
Fail 0-39	– Guidelines for the presentation of the report were largely ignored. – Low quality report, which may be of low standards, poorly structured, poorly presented, and/or incomplete. – Extracting and understanding of results obtained are not straightforward.

MARKING RUBRIC FOR QUESTIONS
First 70-100	– Question answered clearly, concisely, and correctly. – Answer demonstrates a high level of understanding of the topic and of the question. – Evidence of critical thinking and deep knowledge.
Upper Second 60-69	– Answers are mostly correct and understanding of fundamentals is evident. – Level of accuracy and/or details could be improved. – Good applied knowledge.
Lower Second 50-59	– Answer is partly correct. – Answer is mostly descriptive but includes some level of discussion. – Fundamentals are understood, despite some lack of accuracy and detail.
Third 40-49	– Question answered but without detail, precision or discussion. Alternatively, question answered extensively but without coherence (as if combining extracts from different sources). – Some errors and/or inconsistencies may be present. – Descriptive answers showing some level of understanding of the topic and of the question.
Fail 0-39	– Question answered incorrectly, superficially, and/or largely incompletely. – Descriptive answers only. A lack of understanding of the topic and of the question is evident. – If question is unanswered, grade will be set to zero.

MARKING RUBRIC FOR PROBLEMS
WEIGHTS GRADES	PROBLEM FORMULATION 30%	EXPLANATIONS AND SOLUTION 25%	RESULTS PRESENTATION 15%	RESULTS INTERPRETATION (DISCUSSION) 30%
First 70-100	– Mathematical formulation clearly presented in the report. – The student shows clear evidence of understanding the problem at hand. – Problem correctly translated from plain words to mathematical formulation (main features captured). – Clear explanations of how the formulation has been derived. – Main simplifying assumptions stated.	– Clear, detailed, well structured, consistent, concise and yet in depth step-by-step explanation of what has been done, how, and why. – If numerical method used to solve the problem, the submitted code runs when executed, and it is written neatly and well commented. – Entirely appropriate methods used to solve the formulated problem. – Correct solutions obtained for the formulated problem.	– Results obtained are the ones requested and can be easily found, read and understood. – High-quality plots presented in a professional manner. – Entirely appropriate plots to convey results obtained.	– Interpretation of results is sound, mostly correct and consistent. – Elaborate and articulate interpretation of results which goes beyond the obvious. – Model developed used to study the phenomena of interest, drawing some conclusions.
Upper Second 60-69	– Mathematical formulation clearly presented in the report. – The student shows some evidence of understanding the problem at hand. – The mathematical formulation of the problem is mostly correct, with minor mistakes or incomplete. – If mistakes were made, the formulation still makes engineering sense. – Some simplifying assumptions stated.	– Clear explanations of what has been done, how, and why, despite some lack of depth. – If numerical method used to solve the problem, the submitted code runs when executed, and it is written neatly and with some comments. – Adequate methods used to solve the formulated problem. – Solutions for the formulated problem may or may not be correct. If incorrect, this is due to minor mistakes in an otherwise sound problem-solving procedure.	– Results obtained are the ones requested but are not highlighted and not so easy-to-identify in the report. – Good quality plots presented in a professional manner. – Adequate plots to convey results obtained, though some convenient one(s) may be missing.	– Interpretation of results is mostly correct and consistent. – Evidence of understanding the problem at hand, the solution obtained, and the implications of the latter to some extent. – Solution may be incorrect, still making engineering sense. – Solution may be evidently incorrect, which the student identifies and discusses in a fair attempt to identify the reason why.

Lower Second 50-59	– The mathematical formulation is not clearly presented, but relevant equations and analysis indicate the student is on the right track. – Alternatively, the mathematical formulation appears mostly correct but it is not explicitly written. – Alternatively, the mathematical formulation is incorrect, nonetheless still making sense within an engineering context. – The student shows some difficulty in understanding the problem at hand.	– Problem-solving procedure not clearly presented. – The student’s train of thought and process followed to tackle the formulated problem is hard to follow. – If numerical method used to solve the problem, the submitted code runs when executed, though it is hard to read and lacks explanatory comments. – Adequate methods used to solve the formulated problem. – Solutions may be incorrect due to mistakes in the use or implementation of the chosen problem solving technique. – If solutions are correct, there is no justification or clear path as to how they were obtained.	– Results obtained are hard to identify in the report. Some of the requested ones may be missing. – Plots of rather low quality and hard-to read. – The choice of plots to convey the results obtained may not be entirely adequate or insufficient.	– Interpretation of results is mostly correct, yet not very deep. – Evidence of understanding the problem at hand, the solution obtained, and the implications of the latter to a lesser extent. – Some relatively important implications of the solution obtained may be missing. – Solution may be incorrect, still making engineering sense. – Solution may be evidently incorrect, which the student identifies and discusses in a fair attempt to identify the reason why.
Third 40-49	– The mathematical formulation is not clearly presented or incorrect, though still making some sense within an engineering context. – The student shows strong difficulties in understanding what the problem consists of. – Lack of clarity in important concepts.	– Problem-solving procedure not clearly presented and discussed in little to no detail. – The problem-solving procedure followed by the student is unclear and/or inconsistent. – If numerical method used to solve the problem, the submitted code runs when executed. – Adequate methods used to solve the formulated problem. – Solutions may be incorrect due to mistakes in the use or implementation of the chosen problem solving technique. – If solutions are correct, there is no justification or clear path as to how they were obtained.	– Some of the requested results are very hard to identify in the report, or simply missing. – Low quality plots which are hard to interpret. – Some important plots to convey the results obtained are missing, and/or the choice of plots is inadequate.	– Interpretation of results is rather poor in quality and/or quantity, showing weaknesses. It may be inaccurate and/or incomplete. – Lack of clarity in understanding the problem and/or the implications of the results obtained. – Some important implications of the solution obtained may be missing. – Incorrect solutions make little sense within an engineering context and hence should have been identified by the student during this exercise.

Fail 1-39	– The mathematical formulation is not clearly presented or incorrect, making little to no sense within an engineering context. – The student does not evidence sufficient understanding of what the problem at hand is about.	– The problem-solving procedure is not discussed in sufficient details. Major Inconsistencies can be identified. – The methods used to solve the formulated problem are not appropriate or are not presented in any detail. – If numerical method used to solve the problem, the submitted code does not run when executed or no code was submitted. – Solutions are incorrect due to choosing an inadequate problem solving method, or due to misuse / incorrect implementation of the chosen method. – If solutions are correct, there is no justification or clear path as to how they were obtained.	– Some of the requested results are either very hard to find or plainly missing. – Low quality plots, which are hard to interpret. – Some important plots to convey the results obtained are missing, and/or the choice of plots is inadequate.	– Interpretation of results is poor in quality and/or quantity, showing strong weaknesses. – Interpretation of results largely inaccurate and/or incomplete. – Lack of clarity in understanding the problem and the implications of the results obtained. – Incorrect solutions make no sense within an engineering context, which the student fails to identify. – If no results’ interpretation or discussion is offered, this section will be graded as zero.
0	Late submission. NOTE: By regulation, one minute past the deadline is considered late submission.

Example of Coursework Grade Calculation
Partial markings are manually entered in the white cells.
Problem
Formulation
Explanations
and Solutions
Results’
Presentation
Results’
Interpretation
(Discussion &
Conclusions)
0.30 0.25 0.15 0.30
75 0.10 7.50
Q1. Constructing a Mathematical Model 80 0.10 8.00
Q2. Taylor Polynomial for Data-Driven Modelling 65 0.10 6.50
P1. Heat Transfer Data-Driven Modelling 85 72 75 20 61 0.35 21.26
P2. Spring Stiffness Identification (Load-Controlled) 70 70 70 0 49 0.35 17.15
60
Report Presentation
Prob. Quest.
GRADE
ITEMS ASSESSED for CW01 – RESIT
(SEP-JAN 2021/22) Marks Weights Weighted Mark
Weights
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 8
In order to further clarify the important definitions in page 3, which apply to the marking rubric, a
simple worked example is presented below to show what is expected from the students.
Example of How to Solve the Two Problems in this Coursework
Problem statement
Given the pairs shown in the table below, obtain the polynomial of lowest degree that interpolates
all points exactly.

xi	yi
0	5
1	3
2	5
3	23

Table 1. Data points
Problem Formulation
The polynomial of lowest degree that interpolates four points exactly is given by a third-degree
polynomial. Therefore, the interpolation function is as follows:
𝑓(𝑥) = 𝑎0 + 𝑎1 ⋅ 𝑥 + 𝑎2 ⋅ 𝑥2 + 𝑎3 ⋅ 𝑥3 (1)
where the coefficients in red are the unknowns whose values need to be found so that the function
passes through all the data points.
Explanations and Solution
By evaluating the polynomial in Eq. (1) for each of the four data-points, 𝑓(𝑥𝑖) = 𝑦𝑖, a system of
four linear equations with four unknowns is obtained:
{
𝑎0 + 𝑎1 ⋅ 0 + 𝑎2 ⋅ 02 + 𝑎3 ⋅ 03 = 5
𝑎0 + 𝑎1 ⋅ 1 + 𝑎2 ⋅ 12 + 𝑎3 ⋅ 13 = 3
𝑎0 + 𝑎1 ⋅ 2 + 𝑎2 ⋅ 22 + 𝑎3 ⋅ 23 = 5
𝑎0 + 𝑎1 ⋅ 3 + 𝑎2 ⋅ 32 + 𝑎3 ⋅ 33 = 23
Operating:
{𝑎 𝑎 𝑎 𝑎0 0 0 0 + + + + 0 1 2 3 ⋅ ⋅ ⋅ ⋅ 𝑎 𝑎 𝑎 𝑎1 1 1 1 + + + + 0 1 4 9 ⋅ ⋅ ⋅ ⋅ 𝑎 𝑎 𝑎 𝑎2 2 2 2 + + + + 0 1 8 27⋅ ⋅ ⋅ 𝑎 𝑎 𝑎⋅ 3 3 3 𝑎3= = ==5 3 523
In matrix form:
𝐀 ⋅ 𝐱 = 𝐛
𝐀 = (1 1 1 1 0 1 2 3 0 1 4 9 27 0 1 8 ) ; 𝐛 = (23 5 3 5 ) ; 𝐱 = (𝑎 𝑎 𝑎𝑎10 2 3)
Solving the system (you are allowed to use commercial software to solve systems of equations):
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 9
𝒙 = (𝑎𝑎 𝑎𝑎20 1 3=== =-05 24)
Results Presentation
The polynomial function that interpolates the four data points exactly is given by:
𝑔(𝑥) = 2 ⋅ 𝑥3 – 4 ⋅ 𝑥2 + 5 (2)
The colour-coded data-points are shown in Table 2.

x	y
0	5
1	3
2	5
3	23

Table 2. Colour-coded data points.
A plot of the interpolant in Eq. (2) passing through the colour-coded data points is shown in Fig. 1.
Fig. 1. Third degree polynomial interpolant fitting the four data-points provided exactly.
Interpretation of Results
(Example of easy and obvious interpretation)
Clearly, the interpolant passes through the data-points, and therefore it can be inferred that it has
been obtained accurately. As a sanity check, it has been evaluated at the points to confirm this.
Thus, the trend of the data has been captured with a cubic polynomial.
(Example of interpretation beyond the obvious)
In addition to the above, by observing a plot of the data-points alone, it is not evident that the
interpolant should exhibit such steep and positive slopes to the left of the data range. It is possible
that this is not the actual trend of the phenomenon underlying this data but a side effect of choosing
a 3rd degree polynomial (as required for exact interpolation). Therefore, it may be safe to use this
interpolant within the interpolated data (i.e. 𝑥 ∈ [0,3]) but not to extrapolate outside that range.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 10
Assignment Brief Introduction
This is an individual coursework which is aimed at assessing the student’s ability to understand a
given engineering or scientific problem formulated in plain words, choose or develop an appropriate
mathematical formulation to describe it, and then solve such formulation.
The mathematical formulations of the problems in this coursework are given by parametric functions,
whilst their solutions consist of the values of the parameters which maximise the fitting to the
available data.
Problems are purposely kept simple to prevent the student from getting lost in the complexity of a
problem-solving technique and instead focus on the whole process of understanding the problem,
formulating it mathematically, and finally attempting to find a solution and interpreting the results.
The student is expected to identify a model’s strengths and limitations, use it to study the physical
phenomena of interest (being modelled), and prepare a professional report.
Please, read the submission details, marking rubrics, and important definitions thoroughly. It is
a common recurrence that students forget about interpreting the results, therefore losing 30% of the
marks even if they had solved the problem correctly. Thus, for each of the two problems, it is
recommended that the student use the following sub-headings (or others along these lines):
• Mathematical Formulation
• Solution
• Results Presentation
• Results Interpretation (or Discussion and Conclusions)
1. Q1: Constructing a Mathematical Model
In your own words,
a) Discuss the process of constructing a Mathematical Model of a real system or phenomenon.
Use a diagram (of your own) to help you describe the process.
b) Clearly differentiate between Mathematical Formulation and Solution (see definitions in page
3 and solved example).
c) Define verification and validation.
2. Q2: Taylor Polynomial for Data-Driven Modelling
a) Mathematically derive the Taylor Polynomial and explain the process.
b) Explain its terms.
c) Explain how it can be used for data-driven modelling.
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 11
3. P1: Heat Transfer Data-Driven Modelling
You are required to obtain the temperature profile of a bar experimentally. However, your budget is
limited and can only carry out 3 experiments. Setting up the bar horizontally for the experiment, its
left end is at x = 0 meters whilst its right end is at x = 8 meters. The temperature is thus measured at
three locations, as shown in Table 1.

x [m]	T [°C]
2.00	182.5
4.70	272.5
6.50	210.0

Table 1. Temperatures measured at three locations of a bar.
You suspect that the temperature profile is quadratic and are requested to obtain:
a) The temperature profile along the bar.
b) The maximum temperature along the bar.
c) Extra budget comes along, and you are able to measure the temperature at the right end of the
bar, which returns 81.5 °C. Assuming this measurement is accurate, what is the error of the
original model at this location?
You MUST implement your own code to solve the regression or interpolation problem. You are
encouraged to use off-the-shelf (commercial) tools to verify your results.
NOTE: Calling an external optimiser or a built-in function to solve a system of equations within your
implementation will still be considered “your own code”.
Bear in mind that you are expected to formulate the problem, solve it, explain the problem-solving
procedure, and finally interpret/discuss the results obtained. The quality of the results’ presentation
and that of the report overall will also be assessed (see marking rubrics).
This document is for Coventry University students for their own use in completing their assessed work for this
module and should not be passed to third parties or posted on any website. Any infringements of this rule
should be reported to [email protected].
7015MAA, CW01 Resit / Deferral SEP-JAN 2021/2022 Page | 12
4. P2: Spring Stiffness Identification (Load-Controlled)
Springs store energy when stretched or compressed by a force and release stored energy when the
force is removed. Within their elastic limit, massless springs can be modelled as obeying Hooke’s
law. Hence the force required to vary the length of the spring from l0 to lf = l0 + x is given by:

𝑊 = 𝑘 ⋅ 𝑥,

(1)

where W is the acting force [N]; x is the deflection [m]; and k [N/m] is the spring constant/stiffness.
Let us have a massless spring fixed at one end to the ceiling and holding an object at the other end,
as shown in Fig. 1 (right). The object’s self-weight W [N] stretches the spring thus producing a
deflection x = lf ‒ l0 [m]. The equilibrium length l0 corresponding to W = 0 N is shown on the left.
Fig. 1. Object hanging from massless spring fixed to ceiling.
Suppose that you are provided with a spring whose unknown stiffness (k) you wish to find. You
proceed to carry out some experiments obtaining the following data:

W [N]	x [cm]
0	-0.08
10	0.30
20	1.10
30	1.50
40	1.90
50	2.60
60	2.80
70	3.60
80	3.90
90	4.40
100	5.20
110	5.60
120	6.10

Table 2. Experimental data to calculate spring stiffness.
You are asked to:
1- Estimate the spring stiffness k(x).
2- Evaluate the spring stiffness at x = 6 cm.
3- Evaluate the mechanical potential energy (U) of the spring at x = 6 cm.
You MUST implement your own code to solve this data-driven modelling problem. You are
encouraged to use off-the-shelf (commercial) tools to verify your results such as Matlab’s built-in
functions and Toolboxes (‘polyfit’, ‘csape’, ‘spline’, ‘Curve Fitting Toolbox’, etc.).
NOTE: Calling an external optimiser or a built-in function to solve a system of equations or
optimisation problem within your implementation will still be considered ‘your own code’.
Bear in mind that you are expected to formulate the problem, solve it, explain the problem-solving
procedure, and finally interpret/discuss the results obtained. The quality of the results’ presentation
and that of the report overall will also be assessed (see marking rubrics).

l0 k [N/m] lf

k [N/m]
W

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