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Locking phenomena in classic finite element analysis are a well-known problem and many solutions have been developed over the years to reduce or suppress their undesirable effects. The most common types of locking are shear locking, that can occur with slender elements, and volumetric locking, that can occur when the material is quasi-incompressible (with a Poisson’s ratio close to 0.5, for instance). The goal of this study is to assess the performance of finite elements created using the FE² technique. This technique consists in creating a sub-mesh on each element and was developed to help model materials with complex behaviours. In this study, however, the FE² technique is used in the elastic domain with homogeneous isotropic materials. 
To conduct this study, a finite element program was created using MATLAB to solve two dimensional elastic problems. This program was used to create the equivalent stiffness matrices of the FE² elements, and then to test these elements.
Finally, the elements created were tested for each type of locking mentioned above. The test on shear locking (a cantilever beam subject to bending) revealed that even though the FE² method showed some improvement compared to classic fully integrated elements, shear locking still appeared and was non-negligible. To test the appearance of volumetric locking, a cylinder subjected to internal pressure was studied for different values of the Poisson’s ratio. The results on volumetric locking, on the other hand, are very promising as the new elements developed showed little to no locking. The results on volumetric locking were also compared to results obtained with industrial elements, and showed better or equivalent performances.

Finite Element Method --- FE² --- Locking --- Ingénierie, informatique & technologie > Ingénierie civile