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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering - Stage 3
(NFQ – Level 8)
Summer 2007
Computing and Numerical Methods
(Time: 3 Hours)
Instructions
Answer FOUR questions.
All questions carry equal marks.
Examiners: Dr. R. Sheehy Mr. P. Clarke Prof. M. Gilchrist
Q1. (a) Show that for steady state heat flow in a thin plate the temperature T(x, y) at any interior
point obeys Laplaces Equation.
y
T
x
T
∂
∂
∂
∂ (4 marks)
(b) Replace Laplaces Equation
y
T
x
T
∂
∂
∂
∂
by a finite difference approximation.
If the boundary values T(x, y) are assigned on all four sides of a square show how a
linear algebraic system results. (6 marks)
(c) Show, using a suitable example, how
(i) Derivative Boundary Conditions and
(ii) Irregular Boundaries
Contribute to the system of linear algebraic equations. (10 marks)
(d) Outline the general structure of a program to implement Liebmann’s Method for a
rectangular plate. Your program should incorporate both Dirichlet and Neumann
boundary conditions. (5 marks)
Q2. The flow through a particular porous media was described by the Poissons Equation
y
h
x
h where h = head
(a) Find the head at grid points of a 1 cm network for a rectangular plate 4cm x 3cm,
if the boundary conditions are top and bottom face = 0 ∂
y
h ,
left face h = 10, and right face (^) = 1 ∂
x
h (14 marks)
(b) Find the velocity at any grid point.
Note: Velocity is related to head by D’Arcy’s Law x
h q (^) x K ∂
K = 5 x 10
(c) Develop the A.D. I. Scheme for the 2D Laplace Equation
2
2
2
2
= ∂
y
T
x
T
and the 2D heat conduction equation.
2
2
2
2
y
T
x
T
k t
T
(6 marks)
Q3. The vertical deflection u(x) of a bar with distributed load P(x) is given by Poisson’s
Equation P( )x
dx
du AE (^2)
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)
(All imperial units)
(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)
Q6. (a) Briefly explain explicit and implicit finite difference methods in the solution of partial
differential equations.
Use an explicit finite difference method to obtain a solution to the heat conduction
equation
x
T
k t
T
in a thin rod of length 10 cms. At t = 0 the temperature of the rod is zero and the
boundary conditions are fixed for all times at T(0) = 100˚C and T(10) = 50˚C.
Note rod is aluminium: k = 0835 cm
2 /s. (10 marks)
(b) Develop the simple implicit method for the rod in part (a).. (6 marks)
(c) Outline the main elements of a program to implement an implicit finite difference method
(e.g. A.D.I) method. (9 marks)
