LagrangeINT留学生辅导、讲解Computational Project程序、MATLAB编程调试、辅导Data files

Computational Project 3
due: 03/04/2019, 5 p.m.
Last updated February 27, 2019.
Instructions
New format guideline! Submit electronic copy of code, which must produce and
display all requested figures and output, and also submit a hard copy only of answers
to questions not already included directly in your code output. Results output to the
screen must use fprintf and an indicator of which question is being answered, for
example,
fprintf(‘Prob 1(b): best estimate of viscosity is %f \n’, viscosity)
Data files such as stress.txt and the LagrangeINT.m will be located in the TAs’ local
directory when they test your code, so you don’t need to include these in your code.
Problems
1. Consider a fluid with velocity vx in the x-direction changing as a function of y.
The sheer stress in a fluid, τ , is related to the viscosity μ and the strain rate, γ,
where γ =vxy . For pseudo plastic fluids, e.g., shampoo, these are related as
τ = μγn
For Newtonian fluids, these are related asτ = μγ
For Bingham plastics (e.g., toothpaste), there is a threshold, τy, to overcome before
Newtonian fluid properties are exhibited. In this case the stress-strain relationship isτ = τy + μγ
You have performed experiments at a variety of strains and measured the resulting
stresses on an unclassified fluid sample. The data are in the strain.txt and
stress.txt files. Using this data set, write code that uses linear regression to fit the
data for each of the three potential models described above. You may solve the regression
equations yourself or use MATLAB statistical commands, such as regstats.
For each of the three models, your code should generate a scatter plot of the stress
versus strain data along with the regression line. In a separate plot, also show the
1residuals (ri
’s) as a function of the strain rates for each fit. Use the subplot command
to put all of these plots into a single figure. Clearly label your plots (titles,
axis labels, legends).
Based on further analysis of your results, include a writeup that answers the following
questions:
(a) to which class of fluids does this sample most likely belong? Justify your answer.
(b) what is your best estimate of the sample’s viscosity?
22. In this problem we will explore the fitting of data points yi with polynomial functions
of order p at various values of xi
, where
yi = a0 + a1xi + a2x2i + a3x3i + . . . + apxpi + ri,
and ri is the residual error. This relationship can be summarized more concisely as
y = Xa + r. The following steps explore how different approaches to finding the
coefficients a lead to better or worse fits to the data.
(a) Download data for x (temperature.txt) and y (density.txt) for a liquid metal.
Plot the density [g/cm3
] as a function of temperature [K].
(b) Calculate the coefficients of a 5th order regression polynomial using the polyfit
function. Then, using polyval, evaluate this polynomial as x goes from 396 to
512 in intervals of size 0.5. Add a plot of this polynomial to your figure as a
solid black line.
(c) Calculate the coefficients of a 5th order regression polynomial by directly solving
a = (XT X)1XT y.
Then, using polyval, evaluate this polynomial as x goes from 396 to 512 in
intervals of size 0.5. Add a plot of this polynomial to your figure as a solid blue
line. For a calculated this way, you may need to flip a in order to use polyval.
(d) Repeat (b), but for p=24. Add a plot of this polynomial to your figure as a
dashed black line.
(e) Repeat (c), but for p=24. Add a plot of this polynomial to your figure as a dashed blue line.
(f) Use the LagrangeINT.m function that comes with your textbook to evaluate a
24th order interpolating polynomial as x goes from 396 to 512 in intervals of
size 0.5. Add this polynomial to your plot as a solid green line.
(g) Add a legend to your plot, labeling all lines. Use the axis command to restrict
the y-axis range of your plot so that the different lines and datasets are clearly
discernible.
(h) Which numerical method of finding the regression polynomial works better for
small values of p? For large values of p?
(i) Using the best polynomial you found, estimate the metal’s density at 440 K.
(j) BONUS: For polynomial fits of order p = 2 to p = 24, calculate the sum of
squared residual error (E =Pni=1 r2i) using both the method in part (b) and
part (c). Plot E for each as a function of p on the same plot. The range of
values of E spans several orders of magnitude, so use a logscale for the y-axis
with the semilogy command.
3Table 1:
x y
0 0
1 -6.6
2 9.1
3 6.4
4 -7.6
3. Given the following data,
calculate the equations for an ‘unnatural’ quadratic spline. This type of spline imposes
as an additional constraint that the second derivative of the spline is zero for
the last spline (between x = 3 and x = 4), instead of for the first spline.
(a) Write down the set of equations you need to solve to find all of the spline
coefficients.
(b) Find the polynomial coefficients of each spline function.
(c) Plot the data and the interpolating spline functions between the data points.
Tip: here is some code to plot a quadratic polynomial with coefficients a(1),
a(2), and a(3) over the range from x(1) to x(2),
syms xs
f = a(1) + a(2) * xs + a(3) * xs^2
ezplot(f,[x(1) x(2)])
(d) Add to your plot a cubic spline calculated using MATLAB’s spline command.