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

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

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

Content

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 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.

Assessment

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

Content

Assessment

Conclusions

References

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.