Numerical Method & Optimization-Engineering-Nov2019 - Grad Plus

Numerical Method & Optimization-Engineering-Nov2019

T.E. (Mechanical/Auto. Engg./Sandwich/Automobile)
NUMERICAL METHODS AND OPTIMIZATION
(2015 Pattern) (Semester – II)

Time : 2½ Hours
Max. Marks : 70
Instructions to the candidates:
1) Solve Q.1 or Q.2, Q.3 or Q.4, Q.5 or Q.6., Q.7 or Q.8, Q.9 or Q.10. and Q.11 or Q.12.
2) Neat diagrams must be drawn whenever necessary.
3) Figures to the right side indicate full marks.
4) Use of scientific calculator is allowed.
5) Assume suitable data if necessary.


Q1) Evaluate the error in the volume of a tank V\;=\frac\pi4d^{2l} = at d = 1.5 m and I = 2.5 if error in the measurement of diameter is ± 0.010 m and length is ± 0.020 m. [6M]

Q2) A chip-tool interface temperature model is expressed as below : [6M]

T = 314 × V 0.42 × f 1.5

Where, T-Average chip-tool interface temperature (°C), V-cutting speed (m/min) and f – feed (mm/rev).
Find maximum feed (f) in mm/rev to which the temperature of the tool will not increase above 900 °C for cutting speed of 340 m/min. Take initial guesses as [0, 0.5]. Solve for 5 iteration.

Q3) Use the Jacobi method to approximate the solution of the following system of linear equations [6M]

5x1 – 2x2 + 3x3 = –1
–3 x1 + 9x2 + x3 = 2
2x1 – x2 – 7x3 = 3
Continue the iterations until two successive approximations are identical when rounded to two significant digits. Take initial approximation as x1 = 0, x2 = 0, x3 = 0.

OR

Q4) Solve following set of equations using Thomas Algorithm [6M]

2.04 X1 – X2 = 48.8
-X1 + 2.04 X2 + X3 = 0.8
-X2 + 2.04 X3 = 0.8

Q5) a) Write a note on following with example [4M]

i) Slack Variable
ii) Surplus Variable

b) Write a note on simulated annealing. [4M]

OR

Q6) a) Solve following LP problem using graphical method : [5M]
Minimize Z = 5X1 + 6X2
Subject to 2X1 + 5X2 ≥ 1500;
3X1 + X2 ≥ 1200
Where X1, X2 ≥ 0.

b) Write a short note on Genetic Algorithm. [3M]

Q7) a) Draw a flowchart for Runge-Kutta 4th order method. [6M]

b) Solve, \frac{\partial u}{\partial t}\;=\;\frac{\partial^2u}{\partial x^2} for the following condition. [12M]
At x = 0 and x = 0.5, u = 1 for all values of t. At t = 0, u = 2x + 1 for 0 < x < 0.5. Take increment in x as 0.1 and increment in t as 0.01. Find all values of u for t = 0 to t = 0.03.

OR

Q8) a) Given that dy/dx = yz, dz/dx = xy, y(0) = 1, z(0) = 1, h = 0.1 [8M]
Use R-K 2nd order method to find value of y and z at x = 0.1

b) The temperature inside a slab of thickness 16 cm is given by [10M]
\frac{dT}{dx}\;=\;-\frac qAx\frac1{0.5\;\ast(1+0.01\ast T)} (T is in deg.C deg.C)

Find the temperature of other surface by taking step size = 4cm, if heat flux (q/A) is 1000 W/m2 and temperature at one surface, 500 deg.C. Use R-K 4th order method.

Q9) a) A set of x values and respective y values are given below. Using appropriate interpolation method, find the value of y at x = 11.5.[8M]

x25101215
y4568759098

b) The values of Nusselt numbers (Nu) and Reynold numbers (Re) found experimentally are given below. If the relation between Nu and Re is of the type Nu = a.Reb, find the values of a and b for the given values of Nu and Re. [8M]

200024002800320036004000
13.010213.509114.078914.419215.129716.7535

OR

Q10) a) A set of x values and respective y values are given below. Using Lagrange inverse interpolation method, find the value of x at y = 0.42 [8M]

x1020304050
y0.11050.19850.27270.41010.123

b) Fit a quadratic equation of the form y = a0 + a1 x + a2 x2 for a set of given values : [8M]

x25811520
y0.28412.863112.08223.261211.67251.2792

Q11) a) The Velocity v (m/min) of moped which start from rests is given at fixed interval of time t(min) as follows : [10M]

t (min)02468101214161820
v(m/min)010182529322011520
Estimate approximately the distance covered in 20 minutes. Use Simpson’s 1/3 and Trapezoidal rule.

b) Use Gauss-Legendre three-point formula to evaluate I\;=\int_{-1}^1e^xdx [6M]

OR

Q12) a) Draw a combined flowchart for Simpson’s 1/3 rule and Simpson’s 3/8 rule. [8M]

b) Evaluate I\;=\int_0^1\left[\int_0^1e^{(x+y)}dx\right]dy [8M]
using trapezoidal rule, Take strip size for x and y axis as 0.5.

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