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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:
and we consider a classical time-stepping scheme, namely generalized Trapezoidal
rule :
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:
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:
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: DISCRETIZATION WITH THE X-FEM
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Renaud Merle
2000-08-26