### ENGF0004 Mathematical Modelling and Analysis II

matlab作业 | 代写mathematical | lab – 这是一个关于matlab的题目, 主要考察了关于matlab的内容,是一个比较经典的题目, 涵盖了matlab/ios等程序代做方面, 这是值得参考的lab代写的题目

### Late Summer Assessment 2021/

### Coursework 2

#### Standard Formula Handbook: Assessment section on MMA II Moodle site

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Topics Covered: Topics 5 - 8
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Expected Time on Task: 10 hours
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### Guidelines

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Failure to follow this guidance might result in a penalty of up to 10% on your marks.
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I. Submit a single PDF document with questions in ascending order. This can be produced
for example in Word, LaTeX or mat lab Live Script. Explain in detail your reasoning for
every mathematical step taken.
II. Do not write down your name, or student number, or any information that might help
identify you in any part of the coursework. Do not write your name or student number in
the title of your coursework document file. Do not copy and paste the coursework
questions into your submission simply rewrite information where necessary for the sake
of your argument.
III. Insert relevant graphs or figures, and describe any figures or tables in your document. All
figures must be labelled, with their axes showing relevant parameters and units.
IV. You will need MATLAB coding to solve some questions. Include all code as pasted text
(for the purposes of plagiarism checks) in an Appendix at the end of your document.
Remember to comment on your code, explaining your steps.
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**LONDONS GLOBAL UNIVERSITY**

This coursework counts towards 12.5% of your final ENGF000 4 grades and comprises one

question, referred to as model, worth 100% of your coursework grade.

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On Academic Integrity (Read more about it here)
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Academic integrity means being transparent about our work.
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- Research: You are encouraged to research books and the internet. You can also include and paraphrase any solution steps accessible in the literature and online content if you reference them.
- Acknowledge others: We are happy when you acknowledge someone else’s work. You are encouraged to point out if you found inspiration or part of your answers in a book, article or teaching resource. Read more about how to reference someone else’s work here and how to avoid plagiarism here.
- Do not share and do not copy: We expect students not to share and not to copy assessment solutions or MATLAB code from their peers, even if partially.
- Do not publish ENGF000 4 assessment material: We expect students not to share ENGF000 4 assessment materials on external online forums, including tutoring or "homework" help websites.

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Students found in misconduct can receive a 0 mark in that assessment component and
have a record of misconduct in their UCL student register. In some extreme cases,
academic misconduct will result in the termination of your student status at UCL.
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### Model 1 : Streamlines in Fluid Dynamics [ 5 0%]

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Figure 1. Image of streamlines around an airplane wing.
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Differentials often occur in mathematical modelling of practical problems, for example in fluid dynamics such as those observed in the flow around an airplane in flight. The image in Figure 1 shows the streamlines around a commercial airplane in flight. Streamlines are the paths along which the fluid flows there is no flow across a streamline (the volume of fluid between two streamlines remains constant).

The definition of a streamline function (,)=. in incompressible flow where the velocity at the point (,) has components

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=((,),(,)),
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is

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(,)=
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##### (,)=

##### .

These should be satisfied alongside the continuity equation

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##### + ( ) = 0 ,

where (,,) is the density of the fluid, =(,) is the 2 D velocity vector. In the case of incompressible 2-dimensional flow, the continuity equation reduces to

##### =

##### +

##### = 0.

Question 1 [ 20 marks]

We will begin the analysis by proving the reduced continuity equation for incompressible flow. By definition, incompressible flow means that the total derivative of the density of the fluid with respect to time following the fluid is equal to 0 :

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##### =

##### +

##### +

##### = 0.

The rates of change with respect to time of the spatial dimensions and following the fluid are defined as

```
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##### =,

##### =.

Show that by applying the definition of incompressibility to the continuity equation, it can be used to prove that the divergence of the velocity in incompressible flow is equal to 0.

Question 2 [20 marks]

Consider the case of steady-state incompressible fluid flow in two dimensions, where it is known there are distinct stream lines in the flow, observed experimentally through colouring with a die.

Find the stream function (,) for the incompressible flow that is such that the velocity at the point (,) is

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=(
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##### (^2 +^2 )

##### ,

##### (^2 +^2 )

##### ).

Question 3 [1 0 marks]

Plot the streamlines you found in Question 2 in MATLAB. What type of flow does that streamline function correspond to? Discuss what technique taught in Vector Calculus in MMA 2 does the method applied in Question 2 resemble.

### Model 2: Vision-based monitoring [50%]

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Figure 2. Image of UCL in greyscale.
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Vision-based monitoring measures the displacements of civil infrastructure such as towers and bridges as a way of performing remote observation and make decisions on maintenance actions. This relies on video processing from a single-point displacement measurement, (i.e. the mid-span of a bridge).

However, to keep costs low, in practice such systems often have limited storage and processing capabilities and require that the images taken are compressed. In this model, we will use Singular Value Decomposition to create compressed images from on original, calculate resulting compression rat ios and discuss the loss of accuracy.

Question 1 [10 marks]

Download the file Portico.csv from Moodle and import it into MATLAB. This comprises a 340 340 matrix containing a greyscale image in Figure 2. To view the image, use the MATLAB command imshow(P). A figure window will appear with the image of the UCL Portico.

Question 2 [15 marks]

Perform a singular value decomposition of the image file into U, S and V in MATLAB using the svd function. Discuss the type of matrices U, S and V are.

Question 3 [15 marks]

Create an image from the first 10 principal components =^10 = 1 T and view it in a new figure. This means that you should only use the first 10 elements from each of the three matrices U, S and V to reconstruct the image.

Then create images using the first 20 and 50 components and compare these three images to the original.

Calculate the compression ratio by comparing the number of elements present in the original 340 340 matrix and the elements present in the U, S and V matrices used to reconstruct the 10, 20 and 50 component images.

Question 4 [10 marks]

Now that we know how to create images with different compression ratios, we can examine the effect and resulting loss of useful information. Below are four vision-based monitoring graphs of deflection from original position in a bridge measured from images with different number of elements (= 55 , 155 , 255 , 355 ).

Consider these images and the differences between the information they convey in terms of location and magnitude of the maximum deflection detected. Discuss the trade off between image size and preservation of useful information.

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Figure 3. Graphs produced in vision-based monitoring that show deflection from original position in a bridge with
different number of elements (= 55 , 155 , 255 , 355 ).
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