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Mechanical Engineering Exam: Computing & Numerical Analysis for BEng (Hons), Exams of Mathematical Methods for Numerical Analysis and Optimization

Baddi University of Emerging Sciences and TechnologiesMathematical Methods for Numerical Analysis and Optimization

The instructions and questions for an exam in the computing & numerical analysis module of a mechanical engineering degree. The exam covers topics such as numerical methods for obtaining roots of an equation, numerical integration, and differential equations. Students are required to answer any four questions, each carrying equal marks.

Typology: Exams

2012/2013

Uploaded on 04/13/2013

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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 2
(EMECH_8_Y2)
Summer 2008
Computing & Numerical Analysis
(Time: 3 Hours)
Instructions
Answer any FOUR questions.
All questions carry equal marks.
Examiners: Dr. R. Sheehy
Prof. M. Gilchrist
Mr. P. Clarke
Q1. (a) Explain the terms convergence and stability as applied to numerical methods for
obtaining roots of an equation and show that 1)(
'<xg is a necessary condition for
convergence in the case of linear iteration. (8 marks)
(b) The displacement of a structure is defined by tCosey t210 5.0
= for damped oscillation.
Estimate the time required for the displacement to decrease to 4. (5 marks)
(c) Write a program that searches for real roots in an interval [a, b] using step size h.
Your program calls a subroutine which uses either Bisection or Newton-Rapshon Method
to zone to the root. (8 marks)
(d) Illustrate, using a suitable example, an equation with multiple roots. Outline any
numerical method for obtaining multiple roots. (4 marks)
Q2. (a) Briefly describe the rationale behind:
(i) Newton Cotes Integration formulae
(ii) Gauss Quadrature
Derive the two point Gauss Quadrature formula. (8 marks)
(b) Outline the general structure of a program for Numerical Integration using either Gauss
Quadrature or a Newton Cotes formulae (e.g. Simpson ⅓ Rule). (8 marks)
pf3

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Cork Institute of Technology

Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 2

(EMECH_8_Y2)

Summer 2008

Computing & Numerical Analysis

(Time: 3 Hours)

Instructions Answer any FOUR questions. All questions carry equal marks.

Examiners: Dr. R. Sheehy Prof. M. Gilchrist Mr. P. Clarke

Q1. (a) Explain the terms convergence and stability as applied to numerical methods for

obtaining roots of an equation and show that g '^ ( x )< 1 is a necessary condition for convergence in the case of linear iteration. (8 marks)

(b) The displacement of a structure is defined by (^) y = 10 e 0.^5 t^ Cos 2 t for damped oscillation. Estimate the time required for the displacement to decrease to 4. (5 marks)

(c) Write a program that searches for real roots in an interval [a, b] using step size h. Your program calls a subroutine which uses either Bisection or Newton-Rapshon Method to zone to the root. (8 marks)

(d) Illustrate, using a suitable example, an equation with multiple roots. Outline any numerical method for obtaining multiple roots. (4 marks)

Q2. (a) Briefly describe the rationale behind:

(i) Newton Cotes Integration formulae (ii) Gauss Quadrature Derive the two point Gauss Quadrature formula. (8 marks) (b) Outline the general structure of a program for Numerical Integration using either Gauss

Quadrature or a Newton Cotes formulae (e.g. Simpson ⅓ Rule). (8 marks)

(c) Use two point Gauss Quadrature to evaluate the integral of f ( x )= Sinx between the limits x = 0 and x = π. (5 marks) (d) Explain Rômberg Integration. (4 marks)

Q3. (a) Use central difference formulae to estimate the first and second derivative of

f ( x )= e^ x at x = 0. 5 using step size h = 0.1. (7 marks) (b) Use Richardsons extrapolation to obtain an O( h^4 ) estimate of the first derivative at x = 0· (9 marks) (c) Show that the Differential Operator D is related to the Difference Operator ∆

by:- D = 1 h ln( 1 +∆).

Hence, show that the n th^ derivative

DN f (^) i ≈ (^) h^1 N^ ∆ Nfi (9 Marks)

Q4. (a) Briefly describe the terms: (i) Interpolation (ii) Extrapolation. (6 marks) (b) State the formula for Newton's Interpolating Polynomial fn (x) of degree^ n. Derive this formula for the case n = 2 (Quadratic Interpolation). (6 marks) (c) The points (1,0), (4,1.386), (6,1.791) lie on the curve f(x) = ln x. Fit a 2nd^ order Newton's Interpolating polynomial to the data and hence calculate ln 2. (6 marks) (d) Use the additional data point (5,1.609) to estimate the error for the 2nd^ order polynomial in (c). (7 marks)