Description of the problem

The objective was to use the X-FEM in a simple case of the heat equation where we knew the exact solution. The particular case I have chosen is to calculate the temperature in a rod composed of one phase, where the exact solution is actually the sum of two terms, both depending on time : one which can be represented by a classical Finite Element Method, the second constituing of a moving term with a very local behavior to illustrate the effectiveness of the X-FEM for this kind of problem:

where and are the abscissa and the time, and the position of the front. The function is given by:

The appearance of this solution is the sum of a dissipative term on time (the sinus), and a moving term (the spike) as shown on Fig 2:

Figure 2: The exact solution.

In this case the forcing term on the right hand side of (2) is given by :

I use the X-FEM with the classical piecewise shape functions and the enrichment is given by (see also Fig. 3 and Fig. 4) :

Figure 3: The classical shape function.

Figure 4: The enrichment function g.

We note that, using this method, the mass and stiffness matrices need to be rebuilt at every time step. But basicly, it is just some terms which are changing. In our case, we enriched only two nodes: the one which is immediately before the front, and the one which is immediately after the front (see Fig. 5).

Figure 5: The enriched nodes.

The code corresponding to this case was coded with MATLAB. First I had to learn MATLAB and its environment. The structure of the code is very easy, but there is quite a bit of syntax to learn.