next up previous contents
Next: Using the LATIN method Up: Discretization with the X-FEM Previous: Discretization with the X-FEM   Contents

Variational Formulation and Time-Stepping

We consider the solution on the time interval , partitioned into time steps as . Considering the thermal conductivities to be isotropic and constant, we proceed as described in section 4 and write (24) at time step as:

and we consider the general Trapezoidal rule:

(29)

Substituting this equation into the above yields, and after proceeding to the variational formulation, we have the following expression:


where we have assumed that .

We now focus on the terms in the above which arise on the freezing front . These terms consist of a pair of heat fluxes:


which are inherently associated with the motion of the interface through (26).

We note that we need to apply a time-step algorithm to the equation describing the motion of the interface. To this purpose, we rewrite equation (26) at time step as:

(32)

And now let's ignore conventional wisdom, and use a different algorithm , the Forward Euler algorithm ( which is the case of the general Trapezoidal rule algorithm when ), yields:

(33)

which can be rewritten as:

(34)


next up previous contents
Next: Using the LATIN method Up: Discretization with the X-FEM Previous: Discretization with the X-FEM   Contents
Renaud Merle 2000-08-26