ME3291 – NUMERICAL METHODS IN ENGINEERING
(Semester 2 : AY2013/2014)
Time Allowed : 2 Hours
INSTRUCTIONS TO STUDENTS:
1.
Please write your Student Number only. Do not write your name.
2.
This assessment paper contains FOUR (4) questions and comprises FOUR (4) printed pages.
3.
Students are required to answer ALL FOUR (4) questions.
4.
Students should write the answers for each question on a new page.
5.
This is a CLOSED-BOOK ASSESSMENT with authorized materials. Students are allowed to bring two A4 size sheets of notes/formulae written on both sides.
6.
All questions carry equal marks.
PAGE 2
ME3291
QUESTION 1
The heat conduction equation in 1D is given by
T/ t = b
2
T/ x2.
Here T is the temperature and b is the thermal conductivity.
You are interested to use the DuFort & Frankel discretization scheme to obtain the finite difference equation of the governing equation because you have heard of its inherent stable properties. The DuFort & Frankel scheme is given as:
(Tpq+1 - Tpq-1)/(2 t) = (b / ( x)2) [Tp+1q – (Tpq-1 + Tpq+1) + Tp-1q]. where Tpq = T (p x, q t) is the finite difference representation.
You are interested to use the von Neumann (Fourier) stability analysis to determine if it is inherently stable or otherwise. If otherwise, then you show the criterion for the limit of stability. You may assume that Tpq = q ei ph where is the amplification factor, is a particular spatial
Fourier mode, i is the complex number (-1)0.5, and h x in the analysis.
(a)
Determine the (quadratic) equation for
(10 marks)
(b)
Hence or otherwise, determine the possible range of r ( t/( x)2) so that there is no source or sink in the governing heat conduction equation.
≤ 1.0 since
(15 marks)
PAGE 3
ME3291
QUESTION 2
The one-dimensional wave equation is given as
T/ t + T/ x = 1.0,
(1)
with the initial condition
T (x, 0) = 10 + 10x
0
x
,
T (0, t) = 10 + 10t
0
t
.
and boundary condition
Here x is the spatial coordinate and t is the time coordinate.
(a)
Find and sketch the characteristics of the system on a t-x plot. Hence using the method of characteristic, obtain the (analytical) solution for T(t,x).
Use the (analytical) solution and calculate for T(1, 1)
(10 marks)
(b)
You are given three different types of finite difference representation for the governing wave equation. These are given as follows.
(i)
Euler-backward for time and Euler-forward for space discretizations w.r.t Ti,j, i.e.
( Ti,j - Ti,,j-1)/ t
(ii)
+
(Ti+1,,j - Ti,,j)/ x
= 0;
Euler-backward for time and Euler-backward for space discretizations w.r.t Ti,j,
i.e.
( Ti,j - Ti,,j-1)/ t
+
(Ti,,j - Ti-1,,j)/ x
= 0;
(iii) The Lax-Wendroff discretisation of the general governing equation a T/ t + b T/ x = c is given as:
Ti+1, j+1 = Ti,j + ((b t - a x) /(b t + a x))(Ti+1,j –Ti,j+1) + (2c t x)/ (b t + a x) where a, b and c are constants.
Here Ti,j pertains to T(i x, j t). By taking x= t=1.0, calculate T(1,1) using the three forms of finite difference representations given by (i), (ii) and (iii).
(10 marks)
(c)
By comparing your finite difference solutions with the analytical solution given by the method of characteristics, briefly explain the reason(s) for difference or similarity of the finite difference solution w.r.t the analytical solution.
(5 marks)
PAGE 4
ME3291
QUESTION 3
Five springs are connected by the arrangement as shown in the figure below. The spring stiffness coefficients are indicated as (i) (i=1,2,3,4,5) respectively. The node 1 is fixed. The displacement extension at node 2 is given at 10. In question (c) and (d), assume (i) =1
(i=1,2,3,4,5).
(a)
Number the element and nodes clearly.
(2 marks)
(b)
Determine the global stiffness matrix.
(10 marks)
(c)
Compute the displacement of each spring.
(8 marks)
(d)
Compute the acting forces of each spring.
(5 marks)
Figure 1
QUESTION 4
For each of the following differential equations and stated boundary conditions, obtain a oneterm approximation solution using Galerkin’s method of weighted residuals and the specified trial function.
(a)
