The design of any engineering component requires robust analysis using numerical methods like the Finite Element Method. Of paramount importance is to develop convergent formulations that can achieve accurate estimates for the solution at cheaper computational costs.

We investigate a method for improving the accuracy of the stress predicted from models using the mean-strain finite elements recently proposed by Krysl and collaborators [IJNME 2016, 2017]. In state-of-the-art finite element programs, the stress values at the integration points are commonly post-processed to obtain nodal stresses. The mean stresses are element-wise constant, and hence the nodal values obtained from the mean stresses tend to be less accurate. The proposed method post-processes the uniform stress in each element in combination with a linearly-varying stabilization stress field to compute more accurate nodal stresses. Selected examples are presented to demonstrate improvements achievable with the proposed methodology for hexahedral and quadratic tetrahedral mean-strain finite elements.

The nodally integrated formulations exhibit spuriousness in dynamic analyses (such as in modal analysis). Previously proposed methods involved a heuristic stabilization factor, which may not work for a large range of problems, and a uniform stabilization was used over all the finite elements in the mesh. The method proposed herein makes use of energy-sampling stabilization. The stabilization factor depends on the shape of the element and appears in the definition of the properties of a stabilization material. The stabilization factor is non-uniform over the mesh, and can be computed to alleviate shear locking, which directly depends on the aspect ratios of

the finite elements. The nodal stabilization factor is then computed by volumetric averaging of the element-based stabilization factors. Energy-sampling stabilized nodally integrated elements (ESNICE) tetrahedral and hexahedral are proposed. We demonstrate on examples that the proposed procedure effectively removes spurious (unphysical) modes both at lower and at higher ends of the frequency spectrum. The examples shown demonstrate the reliability of energy-sampling in stabilizing the nodally integrated formulations in vibration problems, just sufficient to eliminate spuriousness while imparting minimal excessive stiffness to the structure. We also show by the numerical inf-sup test that the formulation is coercive and locking-free.