1. science
2. mathematics
3. algebra

Incorporating the development of a graphical
user interface in courses teaching numerical
methods for engineers
P. B. Kosasih (corresponding author) and A. K. Tieu
School of Mechanical, Material and Mechatronics Engineering, Faculty of Engineering,
University of Wollongong, NSW 2522, Australia
Abstract This paper describes how students’ learning interests can be stimulated and their
understanding of topics in numerical methods can be improved by incorporating the development of
graphical user interface (GUI) in a numerical methods course or related courses. The students’ interest
and understanding are raised because they are able to implement the theory to solve practical
problems and see how the methods can be used in practical situations with the help of the GUI that
they design and create. To illustrate how development of GUI can be incorporated into numerical
methods courses, two illustrative examples are given: solution of systems of ordinary differential
equations related to vibratory motion of masses, springs and dampers system; and solution of an elliptic
partial differential equation (i.e. Poisson equation). The tool of choice in the authors’ class was
MATLAB. The two examples have been tested on final-year undergraduate and postgraduate
engineering courses. The authors also discuss delivery strategy of GUI-related materials in their
numerical course.
Keywords graphical user interface (GUI); numerical methods; ordinary differential equation; partial
differential equation
Introduction
Mathematical modellings have played a significant role in many advances in engineering, science and technology during the past century. This is primarily due to our
ability to develop accurate mathematical models of physical processes and to solve
them either analytically or numerically. Most of the equations describing the physical world are differential equations, which are either ordinary differential equations
(ODEs) or partial differential equations. Hence graduate engineers would be better
equipped if they could master basic skills in solving differential equations.
There are two ways to solve differential equations. The analytical solution is mathematically demanding and only a handful of students with sound mathematical skill
might attempt to do it this way. A more convenient and familiar way is to apply
numerical methods to attain approximate solutions with acceptable accuracy.
Numerical methods courses are taught as part of undergraduate and postgraduate
engineering degrees. Ideally, numerical method courses could be clearly linked to
other courses in which students are taught how to derive mathematical models of
physical systems. However, in teaching numerical methods to engineering students,
it is often the case that the teaching is concentrated on techniques/methods to solve
given differential equations only. Thus the link between the physical problem and
the mathematical model that is supposed to represent the problem is lost in the
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Incorporating a GUI in numerical methods
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subtlety of the solution methods, and consequently students will quickly lose interest in learning the methods. This is because they could easily perceive that they are
being trained to arrive at approximate solutions of mathematical problems rather
than physical problems, which is often the case in solving differential equations.
Therefore it is important that the links between the physical problem and the mathematical model be carefully maintained in the teaching of numerical methods. It is
the aim of this paper to show that these links can be maintained throughout the
course via exercises in the building of a graphical user interface (GUI).
In standard numerical courses students are normally introduced to the mathematics of finite-difference, finite-element or finite-volume techniques and solution of
systems of linear equations. They are taught how the methods work and made aware
of the accuracy of each method so that they can control the accuracy of the approximate solutions associated with each method. During the courses they are required
to construct programs implementing the methods of solving given mathematical
problems and finally analyse the results in terms of their accuracy and physical correctness. Using a GUI, checking physical correctness can be made more accessible
and fun.
This benefit of incorporating a GUI in the standard course is illustrated as follows.
Students will normally be able to solve a given differential equation specific to an
engineering problem. When different scenarios, for example different sets of boundary conditions, are to be imposed, the program is normally re-run, which will involve
recompilation of the program. By giving students, in addition, the skills to build a
GUI, they will be able to build software and use it to explore all possible scenarios
of the problem interactively. In fact, the students can build a research tool. With the
visibility and interactivity associated with the software, students are able to relate
the physical problem to the mathematical and numerical methods with ease. Another
factor that we think also helps boost the students’ interest and confidence is that they
get the chance to create numerical software. This is a tangible outcome of the course.
In this paper, two examples where a GUI can be incorporated in the teaching of
numerical methods are illustrated. The two examples are: solution of a system of
second-order ODEs related to vibratory motion of masses, springs and dampers
system; and solution of an elliptic partial differential equation (i.e. Poisson equation). The first example involves solution of a system of first-order ODE using
Runge–Kutta methods. The second example illustrates solution of partial differential equations using finite-difference method. In both cases students must develop
programs for solving the equations and create a GUI, and finally use the software
to analyse different conditions.
Several OOPs (object-orientated programs) can be used to build a GUI, such VisualBasic, VisualC++, VisualFortran and MATLAB. However, the authors adopt
MATLAB to introduce this concept as it incorporates a specific tool with which to
do so.
The addition of a GUI to a typical numerical course should not burden the already
packed course contents. The typical structure of the course contents and delivery in
teaching numerical methods is shown in Fig. 1. There are four stages in a standard
numerical methods course:
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P. B. Kosasih and A. K. Tieu
Develop
Mathematical Model
OTHER COURSE
NUMERICAL COURSE
1. Derivation of
Numerical Methods
2. Develop Algorithm
3. Develop Program
5. Build GUI
4. Presentation &
Analyses
Fig. 1
(1)
(2)
(3)
(4)
Path for teaching a numerical methods course.
derivation of the numerical methods;
development of algorithm;
implementation in a program;
presentation and analyses of the numerical results.
In addition to these standard approaches we introduce a fifth element, namely
building a GUI in parallel with program implementation. We discuss, after the presentation of two examples, how this fifth element is brought into the class with
minimum disruption to the other four main elements.
Example 1. Motion of spring-mass-damper system
The first example relates to solution system of first-order ODEs. The problem at
hand is motion simulation of three masses, three springs and two dampers, as shown
in Fig. 2 [1]. The motions or displacements of the masses are given by the Newton
law, yielding a system of three second-order ODEs:
y1′ = [−c1y1′ − k1y1 + c1y2′ + k1y2 + F1(t)]/m1
(1)
y 2′ = [c1y1′ + k1y1 − c1y2′ − (k1 + k2)y2 + k2 y3]/m2
(2)
′
3
′
3 3
y = [k2y2 − c y − (k2 + k3)y3]/m3
The system is excited by the forcing function, given by:
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(3)
Incorporating a GUI in numerical methods
Fig. 2
F1 = At
=0
49
Spring-mass-damper system.
0≤t≤T
t >T
(4)
It is subjected to the following initial conditions:
· (0) ·
· (0)
·
· (0)
(0) ·
y1(0) = y1(0), y2(0) = y (0)
2 , y3(0) = y3 , y1(0) = y1 , y2(0) = y2 and y3(0) = y3
Students are first taught methods to solve the boundary-value first-order ODE.
Several methods were introduced including: Euler; improved/modified Euler; and
Runge–Kutta methods [2]. Once they have mastered how to solve a single first-order
ODE, application of these methods to solve systems of first-order ODEs is then
taught. For ODEs of higher order (>2), the equation will be transformed into system
of first-order ODEs (i.e. second-order ODEs into two first-order ODEs and thirdorder ODEs into three first-order ODEs, and so on). The task is to solve the equations for a given set of data and boundary conditions: k1 = k2 = k3 = 1 kgm/s2, m1 =
m2 = m3 = 0.5 kg, c1 = c3 = 0.1 kg/s. Initial conditions are y1(0) = y·1(0) − y2(0) = y·2(0)
= y3(0) = y·3(0) = 0. And F1(t) = 0.1 t N for 0 ≤ t ≤ 1 s, F1(t) = 0 N for t > 1 s. Students
are asked to plot the motion of each mass for the first 30 seconds.
Once this task is completed, they then develop the GUI software for the problem.
The software must enable users to investigate the motion of the masses subjected to
different conditions. In this example the software is designed to allow users to
change the system conditions, such as the masses, the spring constants and damping
coefficients of the dampers. To do this, all the variables that might affect the solution must be identified. There are 14 values that can be changed, namely: m1, m2,
(0) (0) · (0) · (0) · (0)
m3, k1, k2, k3, c1, c3, t0, tn, T, A, initial condition vectors [y(0)
1 y2 y3 y1 y2 y3 ]
and the number of time intervals. These parameters can all be entered in the GUI
(Fig. 3).
The software makes it possible for students to come up with the solution pertaining to the system. For example, if it is decided to increase one of the damper
coefficients, then using the software they can investigate the effect of increasing the
damper coefficient quickly and interactively. Fig. 4 shows the masses’ motion when
c1 is increased to 0.5 kg/s. Fig. 5 shows the masses’ motion when c3 is increased to
0.5 kg/s. Hence a student can quickly arrive at a solution using the software. In addition to this benefit we found that students seem to be encouraged to explore the
effects of changing other parameters with the software. As a result their understanding of the behaviour of vibration systems is also increased.
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P. B. Kosasih and A. K. Tieu
Fig. 3
Initial window of the GUI for the spring-mass-damper system.
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Incorporating a GUI in numerical methods
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Fig. 4
The masses’ motion with c1 = 0.5 kg/s.
Fig. 5
The masses’ motion with c3 = 0.5 kg/s.
Example 2. Poisson equation
The second example illustrates a two-dimensional field problem governed by an
elliptic partial differential equation (Poisson equation). The equation appears in
many engineering problems, including ones on two-dimensional heat conduction,
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P. B. Kosasih and A. K. Tieu
hydrodynamic lubrication (Reynolds equation) and irrotational flow of an ideal fluid.
The Poisson equation may be written as:
∂  ∂T  ∂  ∂T 
k2
k1
+
= f ( x, y)
∂x  ∂x  ∂y  ∂Y 
(5)
The equation may be subjected to three types boundary conditions: Dirichlet;
Neumann; or Robin. In this example we consider only the case where all the boundary conditions are of Dirichlet type, defined by:
T(x,y0) = f1(x,y), T(x,yn) = f2(x,y) for x0 ≤ x ≤ xn
T(x0,y) = f3(x,y), T(xn,y) = f4(x,y) for y0 ≤ y ≤ yn
To solve the problem, students are first taught how to solve a system of algebraic
linear equations using techniques such Jacobi iteration or Gauss–Seidel [2] discretization techniques using finite difference. Then they are taught to design the GUI.
In the design stage they must identify all the variables and/or functions that might
affect the solution of the equation. For the example, these variables are k1, k2, x0, xn,
y0, yn, n (number of intervals in x) and m (number of intervals in y). The functions
are f(x,y), f1(x,y), f2(x,y), f3(x,y) and f4(x,y). Thus there are eight variables and five
functions that can be changed. The GUI must be designed to allow user to write/enter
the functions and variables in clearly identifiable cells. The 13 cells used in this
example are clearly labeled (see Fig. 6) and strategically placed around the screen.
The position and the labelling must be clear so that user can identify them with ease.
The initial screen setting of the GUI is shown in Fig. 6. Here, f(x,y) = x + y, f1(x,y)
= x, f2(x,y) = 1, f3(x,y) = y, f4(x,y) = 1, and k1 and k2 are constants and set to one. The
numbers of intervals in x and in y are set to 5.
The software allows users to quickly investigate the problem at hand for different boundary conditions and source functions, f(x,y). Fig. 7 shows the screen for
f(x,y) = 0, f1(x,y) = sin(pi * x), f2(x,y) = 0, f3(x,y) = sin(pi * y), f4(x,y) = 0, and k1 and
k2 are constants and set to one. The numbers of intervals in x and in y are set to 15.
Delivery strategy
The GUI element in the numerical course described in the paper was not yet part of
a syllabus or general policy to increase the use of new technologies within mathematics/numerical methods courses. It is, rather, a recommended addition to it. Hence
the introduction of GUI development to the class needs to be done in such a way
that students do not become distracted from the main syllabus. Teachers intending
to incorporate GUI development into such classes must give careful consideration
to how and when to do so, in order not to confuse students with the underlying
numerical methods theory.
As the subject of the following discussion, a numerical course taught to 27 students from various educational backgrounds (many were international students who
had never studied numerical methods before attending this course) by the authors in
autumn 2004 will be used. The course was ENGG952 Engineering Computing and
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Incorporating a GUI in numerical methods
Fig. 6
Fig. 7
53
Initial window of the GUI for solving the Poisson equation.
Window of the GUI for different boundary conditions and source functions.
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P. B. Kosasih and A. K. Tieu
was offered as part of a masters degree. The course covered topics such as: simultaneous linear algebraic equations; matrix eigenvalue problem; ODE initial-value
problems; ODE boundary-value problems; and partial differential equations. The
course contents were delivered over 13 weeks, and in each these were two hours of
lecture and two hours of laboratory session. Each topic was presented following the
logical path shown in Fig. 1. The GUI-related materials, which usually are about
designing an appealing and user-friendly interface suited to the problem at hand,
were discussed once stage 3 was completed and when students were about to tackle
their programming assignments. A GUI element was always attached to the assignments associated with each topic. The presentation of GUI-related materials could
be delivered during either lectures or laboratory sessions. In either case the authors
found that for students to get the most out of their GUI experience it was best for
us (the teachers) to merely introduce to them the main GUI commands (in
MATLAB) – uimenu, uicontrol, get, set, and axes – and basic principles of GUI
design layout. Students then creatively designed their GUI-based presentation and
analysis tools. Students’ creativities are significant factors in determining the quality
of their GUI. As students have different creativity levels, we awarded marks associated with the GUI element only according to its functional correctness and accuracy. This did not seem to affect their pursuit of an attractive and functional
presentation. It was clear that the class was motivated not only by the assignment’s
mark but also the satisfaction brought about by having created well functioning presentation and numerical analysis tools. When time permitted, we also asked some
students to voluntarily present their work to the class and they responded willingly
and proudly.
While we will not claim that the strategy discussed above is the best, it nevertheless is quite clear that the students were greatly motivated and satisfied with what
they learned in the course, as reflected in the survey conducted at the end of the
course. The teaching survey at the University of Wollongong consists of two components: teacher evaluation and student questionnaire. Teacher evaluation measured
teacher attributes such as: teacher’s preparedness (5.9/6), clarity of presentation
(5.6/6), organization (5.7/6), assignment marking and feedback (5.8/6), and so on.
In the questionnaire students provided comments in their own words. The survey
did not, however, ask specific questions about the GUI materials. Overall, students’
performance improved, based on the high proportions of students achieving distinction (≥75) or above, as shown in Fig. 8.
The authors believe that GUI materials could be readily attached to other teaching materials delivered via web-based and e-learning tools and this will be tried in
the future classes.
Conclusion
The paper introduces the development of a GUI as part of the teaching of numerical methods courses to undergraduate and beginning postgraduate students. It can
be easily integrated into any existing numerical methods courses, as illustrated in
the two examples presented. By incorporating this element, the students’ appreciaInternational Journal of Mechanical Engineering Education 35/1
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Incorporating a GUI in numerical methods
55
Distribution of student grades
No. of students
10
8
6
4
2
0
Fig. 8
HD
D
C
P
F
Distribution of students’ grades. HD = high distinction; D = distinction;
C = credit; P = pass; F = fail.
tion of the subject materials is increased. The teaching survey at the end of the course
clearly indicated that students were greatly motivated and their enthusiasm boosted
when they found themselves able to apply the numerical methods and create software implementing the methods. The authors on request can provide the source
codes of the two examples.
Acknowledgement
The first author (PBK) would like to express his gratitude to the school of Mechanical, Materials and Mechatronics Engineering, Faculty of Engineering, at the University of Wollongong, for allowing him to further develop the Engineering
Computing Course.
References
[1] S. Nakamura, Numerical Analysis and Graphic Visualization with MATLAB (Prentice-Hall, New
Jersey, 1996).
[2] L. V. Fausett, Applied Numerical Analysis Using MATLAB (Prentice-Hall, New Jersey, 1999).
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