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A new five-node locking-free quadrilateral element based on smoothed FEM for near-incompressible linear elasticity. (English) Zbl 1352.74461
Summary: The volumetric locking issue is critical in finite element analysis of nearly incompressible problems. In this paper, a new five-node quadrilateral element (Q5) is proposed by enriching the four-node quadrilateral element (Q4) with a centroid node to solve the volumetric locking problem in FEM. The cell-based smoothed FEM is employed with Q5 element (referred as Q5-CS-SC4) to soften the stiffness in order to obtain a better solution. To eliminate pressure oscillation in near-incompressible problems, an edge-based area-weighted smoothing scheme incorporated with the cell-wise divergence-free Q5 element is carried out (referred as Q5-EAW), and an adjustable area-weighted strain smoothing scheme using a parameter \(p\) is proposed to improve the performance of the Q5 element in dealing with incompressible media (referred as Q5-\(p\)EAW). The formulation of Q5-\(p\)EAW is a simple combination of Q5-CS-SC4 and Q5-EAW by an adjustable area weight. It can search the exact strain energy of the problem in near-incompressible cases. We also introduce another node-based strain smoothing technique (Q5-NAW) into the domain-based selective scheme to obtain a Q5-\(p\)EAW/NAW model to solve the pressure oscillation, which gives a much smoother pressure solution than Q5-EAW. Finally, the Q5 element is extended into hexahedral element to develop a nine-node hexahedral (H9) element shape function by enriching the eight-node hexahedral element (Q8) with a centroid node. An H9-Gi/NAW model similar to Q5-\(p\)EAW/NAW is proposed by using H9 element to solve the 3D volumetric locking. A series of benchmark problems are provided to demonstrate that the proposed Q5-\(p\)EAW/NAW for 2D plane strain problems, and H9-Gi/NAW model for 3D cases are locking-free for nearly incompressible problems.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74B05 Classical linear elasticity
Software:
XFEM
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