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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 3
Bachelor of Engineering in Mechanical Engineering – Stage 3
(NFQ – Level 8)
Summer 2005
Computing & Numerical Methods
(Time: 3 Hours)
Instructions Answer FOUR questions All questions carry equal marks.
Examiners: Dr. R. Sheehy Prof. J. Monaghan Mr. J. E. Hegarty
Q1. (a) Briefly explain, using suitable examples, the concepts of: (i) Stability (ii) Convergence (iii) Sources of Error as applied to the numerical solution of ordinary differential equations. (6 marks)
(b) Briefly describe: (i) RK methods (ii) Multi-step (iii) Predictor Corrector methods (iv) Adaptive step size control and give an example in each case. (4 marks) List the distinguishing properties of Runge-Kutta methods.
(c) Outline the structure of a program to implement the 4th^ order RK Method. (5 marks)
(d) Using any multi-step method obtain a Numerical solution to
dxdy^ =^3 x^2^ y in the interval [0, 1] with y (0) = 1 and h = 0..
Compare your results with the analytical solution. (10 marks)
Q2. The vertical deflection u(x) of a bar with distributed load P(x) is given by Poisson’s
Equation AE dxdu 2 P( )x
2
Use the finite element method to model its deflection. (a) Develop element equations using Galerkins Method. (12 marks) (b) Obtain a 3 element solution for a bar of length L = 9, A = .1, E = 1.5 x 10^7 , P(x) = 50 and M = O, rigidly fixed at both ends. (6 marks) (c) Show both finite element and analytical solution on a graph. (4 marks) (d) Briefly explain the concept of convergence outline the main sources of error in a finite element solution. (3 marks)
Q3. (a) Illustrate using a suitable example both an Initial Value and Boundary Value Problem. (3 marks) (b) The deflection y(x) of a light cantilever of length L subject to a uniform distributed load W is characterised by the initial value problem
2EI 2 2
2 ddx y = W ( L − x ) with y = o (^) dxdy (^) = o at x =o
Obtain an analytical solution and show the deflection y (L) = WL 4 (7 Marks) 8EI
(c) Using Eulers Method, obtain a numerical solution for a beam of length L=5, EI = 1.5 x 107 , W = 50 ( marks)
(d) Use a Finite Difference method with 4 interior nodes to solve the same problem. Compare your results in (c) and (d) with the analytical solution at X=L. (7 marks)
Q4. (a) Briefly explain explicit and implicit finite difference methods in the solution of partial differential equations. (6 marks) (b) Use both explicit and implicit finite difference methods to obtain a solution to the heat conduction equation
2
2 x k^ T t
T
Q6. (a) Using finite difference techniques show how Laplaces Equation reduces to a linear algebraic system of equations. (5 marks)
(b) Show how: (i) derivative boundary conditions, and (ii) irregular boundaries contribute the system of linear algebraic equations. (10 marks)
(c) Outline the general structure of a program to implement Leibmann’s Method for a rectangular plate. Your program should incorporate both Dirichlet and derivative boundary conditions. ( marks)
(d) Outline the structure of a program to model the steady state distribution of temperature in a rod using the finite element method. (5 marks)
