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#

Adaptive Analysis

The generic h-adaptive procedure consists in the appropriate combination of
the following components

- finite element solver for the problem under consideration,
- reliable posteriori error estimator providing a measure of the local error in a given
norm to locate the regions that should be subjected to (de)refinement,
- refinement strategy translating the output from the
error analysis into the desired distribution of mesh density,
- mesh generator capable of (de)refinement according to the
supplied mesh density distribution,
- mapping operator which projects the current state of the model
from the old mesh to the new (adapted) one.

Initially, the problem is discretized using the mesh generator, optionally
taking into account a-priori error assessment (typically based on the
user experience). This discretization is then analyzed using the
finite element solver. After each step of the nonlinear solution, the
error estimation is invoked. If the global relative error
exceeds the prescribed value the refinement strategy
is applied to create mesh density distribution map which is afterwards
used by the mesh generator to produce the next, appropriately
(de)refined mesh. The finite element solver is then restarted from the
previous step, when the relative error was still below the defined
threshold. The solution (primary unknowns as well as the state
variables) from this step is mapped on the new mesh, the
equilibrium and consistency are recovered and
the analysis continues as long as the error remains within the permissible
range. This process is repeated until the analysis is completed.
Clearly, the efficiency of such a methodology depends not only on the
efficiency of its individual components but also on the way how it
is automated to minimize the user intervention.

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*Daniel Rypl *

2005-12-03