UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

FINAL EXAMINATION, Thursday, April 27, 2017 DURATION: 2 and Y2hrs

Third Year - Mechanical Engineering

M1E334H1 - Numerical Methods I

Calculator Type: 2

Exam Type: C

Examiner - T. Filleter

Rules and Guidelines:

The examination is 150 minutes, and is comprised of six (6) questions, worth a total of

100 points. You should attempt all questions.

The exam contains 4 pages, including this one.

This exam is closed-book. A one-page (double-sided) aid sheet is allowed. No other

external material is allowed, such as lecture or tutorial notes, quizzes, midterms, final

exams, or books.

You must answer all questions in the separate booklets provided.

Write your name and student number on each booklet used.

Clearly highlight all final answers, and show all intermediate workings. We cannot

give partial marks if you do not.

Question 1 /16 Question 4 /14

Question 2 / 14 Question 5 /20

Question 3 / 14 Question 6 /22

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1. (Short answers ONLY)

(3 points) Name any 3 numerical methods that are derived using the Taylor series.

(7 points) Consider using fixed-point iteration (FPI) to estimate the root of the following equation:

f(x) = lOOe_x - x'°

If you know that the root lies somewhere between x = 1 & x = 2, suggest an equation (in FPI form) that you would use to estimate the root. Clearly justify your choice of equation (you DO NOT need to estimate the root).

(C) (3 points) For fitting a cubic spline to a set of n + 1 data points what is: The number of interior knots? The minimum number of equations in the system of equations to be solved? The "special" form of the matrix that defines this system of equations?

(d) (3 points) For Runge-Kutta methods of solving ordinary differential equations: What is the order of the local truncation error for Heun's method? Is the order of the global error for Heun's method higher or lower than the local error? How does the order of local error for Heun's method compare to Ralston's method?

2. (14 points) Determine the values of a, b, and c to develop a 2nd order accurate backward

approximation for df(x)

dx based on the following relation:

df(x)

dx - af(x1) + bf(x_1) + cf(x_2) -

For this purpose, using the Taylor series expansion, write out the relations for f(x1-1) and

f(x1-2) and substitute back into the above equation. Then compare the left-hand side with the right-hand side to find the values of a, b, and c. Clearly show why the approximation is 2nd order accurate. (j: x1_1 = x - h, x_2 = x1 - 2h).

3. (14 points) First, write the quadratic polynomial version of the Lagrange interpolation method based on its general formula. Next, using the obtained quadratic Lagrange expression, estimate the value of y at x = 3.5 to the best possible accuracy (for the quadratic method). Retain 5 significant figures in your calculations.

X 0 1 2.5 3 4.5 5 6

y 2 5.4375 7.3516 7.5625 8.4453 9.1875 12

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4. (14 points) The following data was collected for the distance traveled versus time by a

rocket:

t(s) 0 25 50 75 100 125

y(km) 0 32 58 78 92 100

Choose a 4th order accurate numerical differentiation method and use it to estimate

the rocket's velocity at t = 75 s. Retain 3 significant figures in your calculations.

Choose a 2nd order accurate numerical differentiation method and use it to estimate

the rocket's acceleration at t = 0 s. Retain 3 significant figures in your calculations.

5. (20 Points) The elastic deflection of a rectangular cantilever beam (i.e. fixed on one end

with a load applied to the opposite end as shown below) is given by:

EI— = F(L - x) dx

where E is the Young's modulus, I is the moment of inertia, y is the deflection of the

beam in the vertical direction, x is the horizontal location along the beam (where x = 0

is the fixed end of the beam), F is the force acting on the unfixed end of the beam, and L

is the length of the beam.

Use Euler's method to estimate the vertical deflection at both the center and the

unfixed end of the steel beam (E = 200 GPa, I = 720 cm4) which has a 2.7 kN load

being exerted to its end. Use a step size of 0.5 m in your calculations. For each step,

summarize your calculations in a table. Consider that y(0) = 0 and y'(0) = 0.

Retain 4 significant figures in your calculations.

Determine the true percent relative error of your estimate at the unfixed end

considering that the deflection at the unfixed end can be calculated by:

FL YL =

A

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6. (22 points) A researcher conducts a tensile test (to failure) of a graphene sheet and measures the following stress vs. strain data:

a(GPa) 0 43.5 77.4 98.1 108.4

c(mm/mm) 0 0.05 0.10 0.15 0.20

If the stress-strain relationship for graphene is known to be non-linear and given by the following relationship:

a = EE + Dc

What type and order of curve fitting to the data would you use to estimate the elastic coefficients E and D? Explain your answer in 1-2 sentences ONLY.

If the researcher finds the following coefficients from curve fitting:

E = 998.5 GPa, D = —2012 GPa

Calculate the toughness (UT ) of the graphene sheet by analytical integration of the

known stress-strain expression between s = 0 and 0.2. Retain 4 significant figures in

your calculations.

Then use the best combination of the trapezoidal AND/OR Simpson's 1/3 AND/OR Simpson's 3/8 rules to obtain an estimate of toughness directly from the stress-strain data. Retain 4 significant figures in your calculations.

Explain (1-2 sentences ONLY) your choice of method/methods used in (c).

How do you expect your results in (b) and (c) to compare? Should they be equal? Why or why not? (1-2 sentences ONLY)

Finite-divided difference formulas:

f' (x1) f(x1)—f(x)

f' (x1 ) f(XL+2)+4f(XL+1)3f(X)

f"(xt) ______________________

- Ii - 2h h2

—f (x +3) + 4f (x 2) - 5f (x 1) + 2f (xi)

h2

f(x1)—f(x1_1) f'(xt)

31(xt)-4f(xj_1)+f(Xj_2) P(Xi) "

— h ' 2/i f (Xi) -

f(x11)—f(x1_1) f ()

, —f(x12)+8f(x +1)-8f(x_1)+f(x1_2)

P(Xi) (xt) - 2/i ' 12/i

f"(xt) f(xt+i)-21(xt)+f(xt_i)

f"(x1) —f(x1+2)+16f(x1)-30f(x1)+16f(x_1)—f(X1_2)

- ' - 12h2

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