A Volumetric Locking-free SFEM Based On Bubble Enriched Element

The volumetric locking issue is critical in finite element analysis of nearly incompressible problems (when the Poisson’s ratio tends to0.5). In this article, Some basic ideas and literature on the volumetric locking issue is introduced, especially the smoothed finite element methods and the modal analysis of traditional elements (FEM using triangle element, FEM using quadrilateral element and node-based smoothed FEM using triangle element). The eigen-mode analysis shows that the locking of traditional elements listed above is due to their inability to describe one or several basic deformation modes accurately in the incompressible limit, especially the so-called "hourglass mode". Subsequently with the aid of a meshfree locking-free formulation, a new five-node Q5element is proposed. It is found that this element is capable of representing the basic deformation modes. It is also found that the selective reduced integration used by mostly commercial FEM software can not solve the pressure oscillation caused by volumetric locking, also it can not pass the numerical Inf-Sup test, the main cause may be the discontinuous strain between the element. Therefore, two smoothed integrated scheme using Q5element is proposed:the edge-based area-weighted strain smoothing scheme (EAW) and the node-based area-weighted strain smoothing scheme (NAW), a series of smoothed integrated scheme using Q5element is proposed. Subsequently based on the bubble enriched formulation of Q5element, a series of so-called bubble enriched element and related smoothed integrated scheme is proposed, including the nine-node hexahedral element(H9), the four-node triangle element(Tri4) and the five-node tetrahedral element(Tet5). Several benchmark problems demonstrate the proposed Q5element and its derived integrated scheme is locking-free for nearly incompressible problems. The results shown the Q5-pEAW/NAW and H9-Gi/NVW model can be locking-free in dealing the nearly incompressible problems and it can also pass the numerical Inf-Sup test. The main works are as follow:(1) Inspired by the meshfree locking-free formulation, a new five-node Q5element is proposed. It is found that this element is capable of representing the basic deformation modes, especially the so-called "hourglass mode".(2) Introduced the Q5element into the SFEM, a series of locking-free smoothed integrated scheme is proposed, including the cell-based strain smoothing scheme (Q5-CS-SC4), the edge-based area-weighted strain smoothing scheme (Q5-EAW), the edge-based adjustable area-weighted strain smoothing scheme (Q5-pEAW), the domain-based selective integration using the node-based area-weighted strain smoothing scheme (Q5-pEAW/NAW model). The Q5-CS-SC4can soften the stiffness matrix of Q5element using Gauss integration, it can also inherit the property of Q5element using Gauss integration. The Q5-EAW can also inherit the property of Q5-CS-SC4by using the area-weighted strain smoothing scheme, besides it can also solve the pressure oscillation. The Q5-pEAW can improve the accuracy of the solution by adjusting the numerical stiffness matrix. The Q5-pEAW/NAW model inherit the property of Q5-pEAW and Q5-NAW, it has better accuracy compared to Q5-EAW and it is easier to extend to three-dimensional problems. The geometry nonlinear formulation using Q5-pEAW/NAW model is also carried out.(3) Based on the bubble enriched formulation of Q5element, a series of so-called bubble enriched element is proposed, including the nine-node hexahedral element(H9), the four-node triangle element(Tri4) and the five-node tetrahedral element(Tet5), the related smoothed integrated scheme is also carried out.