Over the last 40 years, finite element analysis has become the standard method for analysing complex problems that involve systems of partial differential equations
Over the last 40 years, finite element analysis (FEA) has become the standard method for analysing complex problems that involve systems of partial differential equations with arbitrary boundary conditions.
By breaking the problem into small components (elements) and building them up into complex shapes with mathematical constraints at the element intersections, scientists and engineers are able to analyse not only the net result on the complex shape, but also how the properties under analysis vary over distance or time.
By breaking the problem into many hundreds or thousands of simpler elements, and using sufficiently powerful computers, one can analyse the stress distribution in complex engineering components under various loading conditions, the temperature distribution in electronic components or pressure distributions in flowing fluids, to name just a few applications. In general, the greater the number of elements, the more accurate the model, and thus, the greater the computing power required for the analysis.
Until recently, FEA required very powerful computers and highly skilled programmers to build the models, run the analysis and interpret the results.
Now, thanks to the rapid reduction in the cost of computation, and the development of easy-to-use processing tools, the power of this technique is being transferred from the specialist to the people that need the analysis.
However, this has created a problem because FEA is tending to be used as a 'black box' by those who have little knowledge of the underlying techniques, increasing the likelihood of using the results blindly and possibly erroneously. As the use of FEA increases, so does the demand for knowledge of its techniques.
Maple is being used increasingly to present the principles of FEA to students at many universities, with interactive exercises to help reinforce the students' understanding.
One teacher who has had great success with this is Professor Artur Portela at the New University of Lisbon, Portugal.
He runs an introductory course on FEA with excellent results.
"The students love the symbolics and graphics.
They can customise, modify and program new graphic routines to visualise results," says Portela.
"This is not usual with traditional FEA codes, since they are usually problem oriented".
Maple's symbolic capability allows Portela to present and manipulate the extensive mathematics in a way that is straightforward and meaningful to his students.
He likes the way he can solve a general class of problems with Maple's symbolics and get a particular solution with Maple's evaluation tools.
"In FEA, for instance, I compute the stiffness matrix of a single symbolic element," Portela told us, "Then, I simply use eval or subs to obtain the stiffness matrix of all the elements of the mesh.
This is an elegant and clean process which can be extended to many processes in FEA".
Portela is planning to run the same course for non-university students over the internet and in London, UK, in the near future.
Maple is supplied and supported in the UK and Ireland by Adept Scientific.
