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The genesis of an undergraduate course: Reasoning, processes and problems

Matthew Simpson
Centre for Water Research, Department of Environmental Engineering
The University of Western Australia
Finite Element Theory is a widespread technique used by engineers in the solution of differential systems to quantitatively describe several processes of interest. The applicability of Finite Element Theory is unique to environmental engineering problems that are often characterised by irregularly shaped and heterogeneous systems. Problems of this nature are naturally handled by the Finite Element approach. Although the use of Finite Elements is commonplace in the professional life of an environmental engineer, the subject is almost devoid from the Australian undergraduate experience. As a result of this observation, a complete set of teaching resources has been compiled in the area of Finite Element Theory (the current version being freely available from the author). The course is targeted at an audience of senior undergraduate, or beginner graduate students of environmental engineering or related disciplines. The selection of the course content was completely open, in part because of the absence of any text that is particularly focused upon this work for the intended audience. The various issues encountered and the reasoning for the choices made in the selection of the course content and preferred teaching format are proposed and evaluated in light of further experience gained through the development of the course, along with feedback being collected from the first student body being exposed to the work.

Finite Element Method for environmental engineering: Reasoning

The typical undergraduate program aimed at training environmental engineers usually incorporates some introduction to the numerical approach for the solution of differential equations. This training occurs, usually at a superficial level, the aim being to give the student a familiarity with the reasons for solving differential systems and also in the basics of the numerical approach to doing so. Most certainly the focus in this training is firmly grounded in the Finite Difference Method (FDM) as opposed to the Finite Element Method (FEM).

While the FDM provides a similar approach to the solution of differential systems, the FEM has several inherent advantages, which are particularly important when dealing with environmental engineering problems. The problem with focusing exclusively upon the FDM is that several industry standard mathematical models use the FEM, and so a student with no FEM experience is disadvantaged with respect to this.

While it is clear that there is a need for the introduction of the FEM at the undergraduate level, there is a lack of commitment by Australian universities to provide these resources. A survey of subject descriptions offered at Australian universities indicated that is was only the Department of Civil, Surveying and Environmental Engineering at the University of Newcastle which offered a complete subject devoted to the instruction of the FEM to undergraduates. Unfortunately, this teaching has a civil engineering background and so incorporates problems that are fundamentally different to those considered by environmental engineers. Another plausible reason for the reluctance to teach FEM is the lack of a suitable text that simplifies the approach to a digestible, yet practical level. Several texts at an advance level are available (for example, Pinder and Gray, 1977). However, for beginners there needs to be a clear separation between that information which is necessary and superficial, otherwise these texts are overwhelming. There are also several texts available that introduce the concepts at a very simplistic level but do not incorporate the essential step of transforming the theory into computer algorithms, thereby making the theory inapplicable (for example, Chapra and Canale, 2001).

Coupled with the need for a simple explanation of the FEM essentials, it is imperative that the problems discussed with in the instruction of the FEM, relates to the background of the students. This allows for some intuitive reasoning on the student behalf and avoids problems concerning jargon in the applied material.

Finite Element Method for environmental engineering: Processes

The two main processes in the preparation of the teaching material were the decision concerning the course content along with the method of delivery of the information.


The content of the course follows a basic outline shown in Table 1.

Table 1: Course syllabus and timeline

TopicAssigned problemsWeek
FEM Introduction-1
1D Linear Elements-2
1D Confined FlowConfined Groundwater Flow3-4
1D Advection DispersionSolute Transport4
1D Unconfined FlowUnconfined Groundwater Flow5
1D Richards' EquationColumn Infiltration6-7
2D Elements-8
Quadratic Elements-9

The basic concept of the course outline is to separate the learning into three phases. The first two weeks is to provide an understanding of what the FEM encompasses, to introduce the assessment schedule and to communicate the requirements of the course. The next 4 weeks contain the main structure of the information where 4 different types of physical problems are posed and then solved using the FEM. This middle 4 weeks is also when the assigned problems are to be completed; the details of the assignments and their purpose are subsequently discussed. The final three weeks provides a chance to explore some further topics in the FEM that are not incorporated into the problem solving section of the course.

As expected, the bulk of the course preparation time was invested in the design of the assigned problem set (McKeachie, 1999). This is the most important feature of the course as the assigned problems are supposed to be the point where the students are able to independently integrate the theory and concepts to produce a meaningful outcome. The background of the problem set comes from the area of subsurface hydrology. The reason for this choice was twofold, firstly, it is the author's area of research interest and therefore the background information was readily obtainable and compiled by the author. Secondly, and most importantly, the area of subsurface hydrology poses an ideal basis for any numerical methods course to obtain it's problem set from as the subject matter covers several simple and physically intuitive example of several equation types. For example, the 4 problems in the syllabus cover the solution of a linear parabolic equation, a linear hyperbolic equation, as well as two non-linear equations, all of which require different treatment in some particular way. These problems are also effective as they are based upon real physical problems and therefore invite the students intuition to be as much a part of the solution as the theory. The fact that the course material is suited to a senior undergraduate student of environmental engineering, it is likely that the student will have already been exposed to (or being concurrently exposed to) the concepts borrowed from the subsurface hydrology literature. If the student body is not familiar with the subject matter then it is possible to change the physical bases of the problem set and continue with the course.

Teaching format

The success or otherwise of this course undoubtedly depends, in part upon the method of delivery to the student body. While there can be no one specific method of delivery to suit all situations there has been a considerable amount of thought invested as to how best deliver the information. One of the striking things about the appearance of the notes is that the content is indeed quite mathematical, this has two almost opposing effects. Firstly, since the material is composed of mathematical symbolism (calculus notations, integral expressions etc.), then the material is not easily digestible by simply reading the notes, and indeed the only way to make headway in learning the FEM is to write out the formulations pencil and paper style. Secondly, in the latter stages of each section, the formulation is composed of matrix algebra equations, which are smothered in superscript and subscript notation, which pose a difficulty in terms of note taking. Therefore, while it is necessary to be rigorous in note taking, it is unreasonable to expect that the transcription of the script from lectern, to board, to notebook shall occur smoothly. One option to deal with this dilemma is to use the board, in part to deliver the material so that the student is also imparted with the appropriate degree of workmanship required by the subject. This can be supplemented by posting the major results upon an easily accessible web site for perusal and printing. This combination was used in the delivery of the material to a class of postgraduate students at Centre for Water Research, 2001, with some success (see notes at http://www.cwr.uwa.edu.au/~simpson/index.htm).

The other major component of the course apart from learning about the fundamentals of the FEM is to produce some working algorithms through the use of a formal procedural computing language. The solutions to the assigned problems are included in the teaching package written in FORTRAN 77. The use of these algorithms, as a part of the material delivery is possible, and would depend upon the computing prowess of the student. It is envisaged that the senior undergraduate student should be familiar with one procedural language, and that some aspects of the FEM fundamentals would be better communicated by distributing sections of the code as handouts during the instruction. This may be particularly relevant when instructing the matrix assembly procedure in week 2.

The course content should be deliverable with the use of two lecture sessions per week. Since the proposed work in some weeks is more rigorous than in others, it would be possible to supplement the slower weeks in terms of the theory, with problem solving sessions or assistance with the programming requirements.


The structure of the proposed course lends its self naturally to traditional assessment methods. The one aspect that is crucial is the suggested assigned problems (or variations upon the suggested problems). Not surprisingly, the key to understanding how to make FEM work is in actually doing it, a student who reads through the lecture notes may become comfortable with the notations, symbolism and concepts. However, these concepts are not verified until a problem is solved from beginning to end with a computer algorithm. One of the benefits of the planned assessment regime is its flexibility. The difficulty of the assessment hinges upon how the first assignment is posed to the student body. If given with no computational aid, then the first assignment problem would be very demanding indeed, however it is envisaged that either part of (or perhaps the entire) computing solution could be distributed to the student body to aid in their familiarisation with the work. Once the first assignment is complete, the second assignment is a simple extension of the first and so on through to the last computing assignment.

If there is a need for additional assessment, there exists the opportunity for the use of a long style "take home" problem to be assessed, very much in the style of the programming assignments. Additionally, a final exam type paper could also be easily set as there are several small exercises buried within the lecture notes that are easily transformed into examination style questions. This is probably most easily achieved using the content of the final two weeks lectures where several of the simple concepts are repeated to reaffirm the simple 1D linear work addressed in the initial section of the course.

Finite Element Method for environmental engineering: Problems

While there are undoubtedly several points in this teaching plan where problems could arise, some thought has been put into being aware of these problems as they may arise and thinking about the best way to address them.


The obvious problem of the content being unfamiliar with the student could be a problem depending upon the audience and their background. If the audience was completely composed of environmental engineering students, then the concepts should be familiar, however this may not always be the case. Fortunately most of the problems in the problem set and the lecture notes can be re-formulated in terms of an equivalent heat flow problem. For example, the content leading up to the first assigned problem can be identically replaced by the heat equation which should be familiar to anyone who has taken a Calculus 1 (or equivalent) course. The heat analogue can be extended into the non-linear equation section, however the physical intuition associated with the simple heat equation or subsurface hydrology problems are lost with this transformation.


The main issue with the proposed assessment format would occur if there were any problems with the completion of the assigned problems. The importance of the student in being able to independently solve the FEM problems via the algorithms can not be stressed enough. This should be considered when the course is planned into the degree program. The subject should only be taught after the student has attained a competent background in programming with some procedural language. If this were to be a problem, then it would be possible to instruct the course, however the instruction of the theory alone without the backup of the assigned work would almost certainly fail to convey the basis of a working knowledge of the FEM.


The importance of the FEM in the instruction of undergraduate environmental engineering students has been clearly presented. The current inequality between the lack of FEM instruction and the proliferation of FEM modelling in the professional life of environmental engineering indicates that there is a clear and present need for thought about effective instruction of the FEM. The teaching package presented here represents a simple, yet effective method for the delivery of the FEM concepts to environmental engineers. The lecture material was planned to be firmly grounded in simple, yet physically meaningful problems that communicate a real essence of FEM for environmental engineers. While the program is in its genesis, there is room for flexibility in the design and delivery of the material. It is envisaged that the program shall be able to address the inequality in regards to the lack of FEM instruction, and help to address the clear need for its widespread instruction.


Chapra, S. C. and Canale, R. P., Numerical Methods for Engineers with Programming and Software Applications, 4th ed., McGraw-Hill, Toronto, 2001.

McKeachie, W. J., Teaching Tips, Strategies, Research and Theory for College and University Teachers, 10th Edition, Houghton Mifflin Co., Boston, 1999.

Pinder, G. F. and Gray, W. G., Finite Element Simulation in Surface and Subsurface Hydrology, Academic Press New York, 1977.

Author: Matthew Simpson, Graduate Student, Centre for Water Research, The University of Western Australia. simpson@cwr.uwa.edu.au

Please cite as: Simpson, M. (2002). The genesis of an undergraduate course: Reasoning, processes and problems. In Focusing on the Student. Proceedings of the 11th Annual Teaching Learning Forum, 5-6 February 2002. Perth: Edith Cowan University. http://lsn.curtin.edu.au/tlf/tlf2002/simpson.html

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