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WEAK FORM AND TIME STEPPING

The usual approach to implement a time-integration scheme with finite elements is to discretize in time after developing the Galerkin approximation. In our case, the approximation evolves in time along with a local feature of interest. So an initial point of departure is to reverse the order of these steps as follows.

We consider the solution on the time interval , partitioned into time steps as . Considering the thermal conductivities to be isotropic and constant, we write (1) at time step as:

   on (2)

and we consider a classical time-stepping scheme, namely generalized Trapezoidal rule :

(3)

Substituting this equation into the above yields, eliminating and after rearranging some terms:

   

We now multiply by a kinematically admissible weight function , and integrate over the domain at time step . This gives:


We now follow the standard step and use integration by parts to shift a derivative from the trial function to the weight function. This yield to the following expression:


After solving this system, we know the temperature at instant and we would like to know its time derivative . Basicly and where and are the sets of the basis functions at time steps and . The equation which links these variables in the classical time-stepping scheme is:

(6)

The way to calculate is the same way used in space, that means that we make an projection of the equation (6) onto the set . We multiply the above expression by and integrate on . More precisely:

(7)

After substituting and bye the chosen weight functions and approximations and solving this system, we know the field and we can continue to march in time.


next up previous contents
Next: DISCRETIZATION WITH THE X-FEM Up: Training Report Modeling Phase Previous: PROBLEM FORMULATION   Contents
Renaud Merle 2000-08-26