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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering - Stage 3
(NFQ – Level 8)
Summer 2006
Computing and Numerical Methods
(Time: 3 Hours)
Instructions Answer FOUR questions.
Examiners: Mr. R. Sheehy Mr. J. E. Hegarty Prof. M. Gilchrist
Q1. (a) Obtain the temperature distribution at 1 cm grid points of a (4 x 5) cm rectangular plate. Poissons Equation
2 Q Q^5 ,k^16
2 2
2 ∂∂ xT +∂∂ yT =− k = = ⋅
holds for all interior points and the boundary conditions are: Left and right edges held at constant temperature of 20º C. Upper and lower edges lose heat such that ∂∂ Ty (^) = − 15 ° C / cm ( upper ), ∂∂ Ty = 15 ° C / cm ( lower )
(Two iterations using a suitable relaxation factor suffices) (16 marks) (b) Find the flux at any interior grid point. Note: plate aluminium k' = · 49. (4 marks) (c) Develop the A.D. I. Scheme for Poissons Equation in (a). (5 marks)
Q2. (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 Leibmann’s Method for a rectangular plate. Your program should incorporate both Dirichlet and derivative boundary conditions. ( marks)
Q3. (a) Use Galerkin’s Method to develop element equations for the boundary value problem:-
dx f x whereT(o,t)=T^1 andT(10,t)=T^2
d T = − (8 marks)
(b) A rod of length 9 cms has boundary temperatures T(o,t) = 20 and T(9,t) = 200 with heat source f(x) = 15. Obtain a 3 element solution. (9 marks) Show both the finite element and analytical solution on a graph. Briefly explain the concept of convergence outlining the main sources of error in a finite element solution. (c) Outline the structure of a program to model the steady state distribution of temperature in a rod using the finite element method. (8 marks)
Q6. (a) Illustrate using suitable examples both Initial Value and Boundary Value Problems. The
steady-state heat balance for a rod is 2 1 ( ) 0
2 ddx T + h Ta − T =. Obtain an analytical solution
for a 10m rod with h^1 = .01 T (^) a = 20 , T(O) = 40 , T (^) (10) = 200. (3 Marks) (b) Use the Shooting Method to obtain a numerical solution to the problem in part (a). (7 Marks) (c) Use a Finite Difference Method with ∆X = 2 to solve the same problem as in part (a). (7 Marks) (d) Outline the structure of a program to implement the shooting method for a linear 2nd order differential equation. (8 Marks)
