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Local multigrid methods provide a technique to solve problems which locally require high accuracy (cracks, discontinuous boundary conditions, entrant corners etc.) in acceptable computational times and memory space. These methods have several advantages such as fast resolution on each sub-grid, work on local fine meshes only...
We propose here to use the Local Defect Correction (LDC) multigrid method, proposed by Hackbush [3], to solve Solid Mechanics Problems. From an initial coarse mesh, this method focuses on recursively adding local sub-grids in areas where higher accuracy is required. We keep adding levels of refinement until the finest grid has the desired accuracy. Then prolongation and restriction operators are used to link the different level of grids. It should be noted that this method is generic : solver, refinement ratio , mesh type, model etc. could be different (or not) between the several levels of refinement.
In order to automatically detect the zones to be refined, we propose to
couple the LDC multigrid method with the Zienkiewicz-Zhu a
posteriori error estimator [5]. Based on the fact that
the stress field calculated by the finite element method is discontinuous
between elements, the Zienkiewicz-Zhu a posteriori error estimator
constructs a smoothed stress field. Then the error estimator is evaluated
by the difference between the finite element stress solution and the
smoothed stress fields. A strategy of coupling the LDC multigrid method
with the Zienkiewicz-Zhu a posteriori error was already introduced
in [1,2]. The authors proposed to refine the
elements for which the indicator is greater than 
% of the
maximum error. This strategy is easy to implement but requires the
knowledge of the (which may depend of the problem) and the
maximal number of sub-grids (because this indicator never stops). We
propose here an automatic procedure of refinement working directly on the
elements for which the stress error is superior than a threshold
(relative error) given by the user. In this way, we avoid the dependence
on a arbitrary coefficient.
We apply this method on an industrial test case that is simulation of the
Pellet-Cladding mechanical Interaction (PCI) in Pressurized Water
Reactors (PWR) [4]. Precise simulations of this problem
require cells of 1 
m for a structure of 1 cm. The LDC multigrid
method could be efficient to overcome this issue. We present results
obtained on the 2D linear elasticity test cases with discontinuous
boundary condition. Whatever the thresholds set by user, the proposed
coupled method gives really satisfactory results. Memory space and CPU
time saving will be also pointed out.