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A meshfree-enriched finite element method for compressible and near-incompressible elasticity. (English) Zbl 1242.74174
Summary: In this paper, a two-dimensional displacement-based meshfree-enriched FEM (ME-FEM) is presented for the linear analysis of compressible and near-incompressible planar elasticity. The ME-FEM element is established by injecting a first-order convex meshfree approximation into a low-order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker-delta property on the element boundaries. The gradient matrix of ME-FEM element satisfies the integration constraint for nodal integration and the resultant ME-FEM formulation is shown to pass the constant stress test for the compressible media. The ME-FEM interpolation is an element-wise meshfree interpolation and is proven to be discrete divergence-free in the incompressible limit. To prevent possible pressure oscillation in the near-incompressible problems, an area-weighted strain smoothing scheme incorporated with the divergence-free ME-FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu-Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near-incompressible problems.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity
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