From nodal patch schemes to averaged strain elements

In this paper we present a review of the patch averaged assumed strain element technique, recently proposed in [1, 2, 3] for elastic problems and then developed for structural models such as beams and plates structures with isotropic or composite materials [4, 5, 6]. In this approach a patch averaged strain-displacement matrix is constructed for each node of the mesh, yielding a smoothed representation of strain and stress fields. The appeal of this representation of recovered quantities is clear: recovered quantities does not require to be post-processed in order to obtain unique values at the inter-element interfaces. One of the main novelty of the proposed approach lies in the use of the so-called variational consistency condition: the assumed-strain operators are directly derived by this condition. A comparison with similar averaging techniques shows that usually this condition is used only as an ex post facto test to asses the properties of the assume-strain operators. We detail that the finite element approximation converges uniformly to the exact solution also for problematic cases such as the nearly incompressible case or in the limit of thin plates. The present review investigate also some aspects related to the choice of sampling points to use with the construction of the averaged strain matrix. Upon this choice the proposed technique produces a wide set of elements: accuracy, convergence properties and insensitive response in the presence of shape distortion are among the feature of each proposed element.