ENGF0004 Mathematical Modelling and Analysis II 2022
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ENGF0004
Mathematical Modelling and Analysis II
2022
Question 1: HREM (50 marks)
Figure 1. Imageofamurinebrain,obtainedusing HREM(high-resolutionepiscopicmicroscopy). Source:C. Walsh,et.al, 2020.
High-Resolution Episcopic Microscopy (HREM) is a novel optical microscopy technique that allows the high resolution imaging of the histology of tissues (Figure 1). Detailed morphology of blood vessel networks and tissue organisation of organs or tumours can be extracted to create microscale models that mimic their function.
As a recently developed technique, work is ongoing in calibrating the imaging signal obtained from the microscope. HREM analyses tissues which have been treated to improve their translucence and set in resin to provide rigidity to the sample. A cross section of the sample is then imaged, cut and removed, and the process is repeated.
By its nature, this technique is sensitive to refraction and reflection deep within the sample, introducing noise in the image. Quantifying the point spread function which characterises this phenomenon is essential to optimising the use of this microscope.
a) [15 marks] The true intensity distribution (henceforth referred to as the true image) of the sample f(x, y, z) is convolved with the point spread function g (x, y, z) when the image is taken, generating the image obtained from the microscope , h (x, y, z) .
Demonstrate how and explain why Fourier transforms can be used to analyse this problem more easily.
What steps can you take as a scientist
o to determine the unknown point spread function 6 (x, 人, z)?
o once 6 (x, 人, z) is known, to find the true image J(x, 人, z)?
b) [15 marks] The point spread function 6 (x, 人, z) characterising HREM imaging is given by
6 (x, 人, z) = Na− 2Dx(2)八 − 2Dz(2) ,
where N, Dx人 and Dz are parameters which can be found by data fitting (assumed to be ‘known’).
The Fourier integral for multidimensional functions is given by
9(n, a, 从) = ∭ 6 (x, 人, z)a−2业!nx−2业!a人−2业!从zpxp人pz,
where ! is the imaginary unit and 9(n, a, 从) is the Fourier transform of 6 (x, 人, z) . Find the Fourier transform of 6 (x, 人, z).
c) [20 marks] During the preparation of the sample for this microscope, a blade sometimes chips the resin rather than slice it cleanly. A potential solution would be to use a laser to cut the sample. However, this would mean that the sample is heated which might distort the image.
The temperature L(x, 人) in the sample from a laser cutting can be modelled using the heat equation
eL = 才 e 2 L
e? ex2 ,
where the initial temperature is L(x, 0) = 70a−3x . To counteract the heating, the bottom of the sample is kept at zero degrees Celsius, i.e. L(T, ?) = 0 . There is no heat transfer from the top of the sample (i.e. x = 0) after the laser instantly cuts it off:
= 0.
Find the temperature distribution through the sample L(x, ?).
Question 2: Hidden Figures (50 marks)
Figure2. KatherineJohnson,ontheleft, workingasacomputeratNASA. Ontheright, MaryJackson,who
workedasacomputerat NASAbeforebecomingthefirst African Americanwomanengineerat NASAin
1958. Source: NASA
Before electronic computers became ubiquitous to our daily lives, they were human – a computer was referred to as the person employed to make computations. This occupation offered little glamour but was essential to many technological advances in the 20th century, when 98% of computer positions were held by women.
The movie “Hidden Figures” sheds light on the role of women, and women of colour especially , in one of humanity’s greatest achievements – space exploration and the Moon landing. Complex orbital calculations were possible due to the work of women like Katherine Johnson, Mary Jackson and Dorothy Johnson Vaughan (Figure 2).
One of the techniques used to calculate re-entry and landing points for manned spacecraft missions was Euler’s method.
a) [5 marks] Given the time derivative of the function y(t) y ′ (t) = f(t)
and the initial condition
y(t0 ) = y0 ,
write the Taylor’s series expansion of y(t + Δt) and derive Euler’s method from first principles (Euler’s method ignores quadratic and higher order terms in the approximation):
y(tn + Δt) = y(tn ) + hf(tn ),
where h = Δt .
b) [10 marks] A more accurate numerical approximation can be found by including the second derivative term to formulate Euler’s modified method:
y(tn + Δt) = y(tn ) + (f(tn ) + f(tn + Δt)).
Derive Euler’s modified method by extending the derivation in part a).
c) [12 marks] The magnitude of the tangential velocity v(t) of a space capsule during re-entry into Earth’s atmosphere is given by the following equation:
= − − v 2 (t)ACD − ṁ,
where is the acceleration due to gravity as a function of the height ℎ of the space capsule
above the surface, R is the radius of Earth, G is the gravitational constant, p is the density of the atmosphere, A is the cross-sectional area of the capsule, CD is the capsule’s drag coefficient, m = m0 − ṁ t is the time-dependent mass of the capsule due to fuel being
consumed at a constant rate ṁ = = F , and c is the magnitude of the velocity of exhaust
gases burning from the capsule’s thrusters to bring it into the atmosphere.
Use any tool at your disposal to find the velocity for the first two time-steps, using Euler’s method with a time step-size of 0.1 seconds and the values for the parameters specified in Table 1.
Table 1. Parameter values.
Parameter Value (units) Description |
||
0 |
902 kg |
Initial mass of the capsule |
0 |
150 km |
Initial distance from Earth on re-entry |
0 |
7 844 m/s |
Initial velocity |
G |
6.67 × 10 −11 Nm2 /kg2 |
Gravitational constant |
M |
5.97 × 1024 kg |
Mass of Earth |
R |
6 371 km |
Earth’s radius |
p |
0.086 kg/m3 |
Density of atmosphere |
A |
2.8 m2 |
Area of capsule exposed to drag |
CD |
1 |
Drag coefficient |
c |
5528 m/s |
Velocity magnitude of thruster gases |
F |
0.03 kg/s |
Rate of burning thruster fuel |
d) [13 marks] A major objective of these calculations is to plan and predict a landing zone for the space capsule upon splashdown. It might be of interest to determine the orbital position of the space capsule in its orbit around Earth at a certain point in time at which braking is applied by engaging the thrusters and re-entry is initiated. This would give the initial position required to find the landing zone.
The orbital position at time tp can be determined through Equation (1):
2tp
T
where T is the orbital period of the capsule and u is an angle known as the eccentricity anomaly (rad). Equation (1) needs to be solved using numerical methods in this situation.
Simplify Equation (1) by employing a first order Maclaurin Series approximation for sin u and hence specify up to what maximum angle u (according to the remainder term in Taylor’s theorem) this approximation can be applied to keep the maximum error obtained for the value of u below 0.02 rad.
e) [10 marks] Our model presently assumes the Earth is a perfect sphere and thus , its gravitational field is uniform. This is not true in reality. Table 2 lists the gravitational accelerations at different major cities within 40 degrees latitude from the equator and beyond
40 degrees latitude of the equator.
Perform analytically a hypothesis test to determine if the two groups are statistically different at an appropriate level of significance .
Table 2. Gravitational acceleration at different major cities.
Below 40 degrees latitude Above 40 degrees latitude |
|||
City |
Gravitational acceleration (m/2 ) |
City |
Gravitational acceleration (m/2 ) |
Atlanta |
9.7951 |
Chicago |
9.8027 |
Austin |
9.7927 |
Albany |
9.803 |
Baltimore |
9.801 |
Anchorage |
9.819 |
Charleston |
9.7953 |
Boston |
9.8038 |
Charlotte |
9.797 |
Cleveland |
9.8021 |
Dallas |
9.795 |
Lincoln |
9.801 |
Denver |
9.796 |
Newark |
9.802 |
Question 3: Gravity (50 marks)
B
A
Earth s orbit around
the Sun
Figure3. GraphicalrepresentationofEarthorbitingthe Sunduetotheforceofgravity.
Newton’s law of gravity describes the gravitational force between two bodies. If a mass M is positioned at the origin, and a mass m is at a position vector r = (xi + yj + zk) which is a distance r = |r| away, then the gravitational force between them is defined by
F = −
where is a unit vector in the direction of r , G is the gravitational constant, and the force acts towards mass M at the origin.
a) [5 marks] Define the i,j, k components of this force as a function of x, y and z .
b) [13 marks] Find the curl of the force field F . Hence, deduce what the state of Earth’s movement must have been at its creation knowing it now spins around its geographical axis.
c) [7 marks] With the help of some mathematical derivation(s), briefly discuss why your findings
for the curl imply the existence of a gravitational potential function p such that F = ∇p.
Use this to explain in your own words why the gravitational vector field is conservative.
d) [15 marks] Find the gravitational potential function p .
Hence, find the work done by the gravitational field when the Earth moves from point A on its elliptical orbit the furthest away from the Sun (1.52 × 108 km away) to a point B closest to the Sun (1.47 × 108 km away). Use the values for the mass of the Earth m = 5.97 × 1024 kg, the mass of the Sun M = 1.99 × 1030 kg and G = 6.67 × 10 −11 Nm2 /kg2 (Figure 3).
e) [10 marks] The force balance governing the motion of objects in space becomes complicated in systems of three or more bodies, causing instability in orbits over the long term . This explains why even though most asteroids in our solar system move in circular orbits around Mars and Jupiter, they can be disturbed from their orbit, sending them on a path towards Earth. Due to the random nature of such events, predicting them in advance is difficult, but historical information can be used to estimate their occurrence.
The frequency of a meteorite approximately 7m in diameter (a small impact event) striking Earth has been estimated by models and historical data to be once every 4.6 years. Over the past 30 years, 15 impacts with such events have been recorded.
Find the probability of observing 15 such meteorite events over 30 years using the expected frequency estimated in the model. Hence, comment on the accuracy of the model by referencing the recent historical record .
Question 4: Wildlife Conservation (50 marks)
Figure4. Amale Borneanorangutan, arepresentativeofthecriticallyendangeredspecieswhosenumbers havedeclinedby 50%overthelast 60years. Source: Wikipedia
You are working on a proposal for wildlife conservation project for the World Wildlife Fund aiming to gradually increase the population of the Bornean orangutan.
To do this mathematically, a matrix model which describes the population and its growth is developed based on existing data. The model assumes the population is separated into three age groups, each spanning 12 years: children, young adults and adults.
Children: Represented by ci where i represents the number of 12-year cycles that have passed since the initial state in the model. Children are born from young adults at a rate Ry and from adults at a rate Ra . They survive to grow into young adults with a success rate Sc . Children cannot reproduce and at the end of the 12-year period they have either died or grown into young adults.
Young adults: Represented by yi , they are the children that survived into young adulthood. They grow into adults with a success rate of Sy (this means those unsuccessful die in this 12-year period).
Adults: Represented by ai , are the young adults who survived into adulthood. No adults survive beyond a 12-year cycle to the next.
The population of each age group after each twelve-year cycle can then be represented by the following set of equations
ci = Ryyi−1 + R aai−1
yi = S cci−1
ai = Sy yi−1
where ci−1, yi−1 and ai−1 are the numbers of children, young adults and adults of Bornean orangutan at the start of the modelled period and the numbers for the following 12-year cycle are given as ci , yi and ai .
The population of orangutan has been studied at a conservation centre and measurements were taken for the number of individuals in each age group over 36 years (Table 3).
Table 3. Population numbers of Bornean orangutan in each age group in the conservation centre over a 36-year
period.
2022-08-08