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This paper investigates 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 values of stress. The mean stresses are element-wise constant, and hence the nodal values obtained from the mean stresses tend to be of lower accuracy. The proposed method post-processes the uniform stress in each element in combination with a linearly-varying stabilization stress field to produce a more accurate representation of the nodal stresses. Selected examples are presented to demonstrate improvements achievable with the proposed methodology for hexahedral and quadratic tetrahedral mean-strain finite elements.

A method for stabilizing the mean-strain hexahedron was described by Krysl (in IJNME 2014). The technique relied on a sampling of the stabilization energy using the mean-strain quadrature and the full Gaussian integration rule, which was shown to guarantee consistency and stability. The stabilization energy was assumed to be generated by a modified constitutive matrix based on the spectral decomposition. The stabilization required user-selected values of the stabilization parameters. In the present work we eliminate the arbitrariness of the stabilization parameters. We formulate the technique more precisely as an assumed-strain method, and we express the stabilization energy in terms of input parameters of the real material. Finally, we fix the value of the stabilization parameters in a quasi-optimal manner by linking the stabilization to the bending behavior of the hexahedral element. For simplicity the developments are limited to linear elasticity, but with an arbitrarily anisotropic elasticity matrix. The accuracy and convergence characteristics of the present formulations compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements as implemented in Abaqus and with a number of successful hexahedral and shell elements and we demonstrate that the present element performs very well when used with large aspect ratios for thin structures such as plates or shells.

Nodally integrated elements exhibit spurious modes in dynamic analyses (such as in modal analysis). Previously published methods involved a heuristic stabilization factor, which may not work for a large range of problems, and a uniform amount of stabilization was used over all the finite elements in the mesh. The method proposed here 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 finite elements in vibration problems, just sufficient to eliminate the spurious modes 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.

The nodally integrated plate element formulation is an assumed strain finite element technique for shear deformable plates that enforce weakly the balance and the kinematic equations. The strain-displacement operators are derived via nodal integration satisfying a priori the kinematic weighted residual statement. The present work analyzes the NIPE technique, including the new element, by testing thoughtfully the sensitivity of the elements to severe geometry distortions. We propose an improvement that confers robustness to all element shapes developed by the NIPE formulation. We present also a new nine-node NIP element configuration. The new nine-node element uses bi-quadratic interpolations of the transverse displacement and rotations and is computed by means of a nine-node quadrature rule. A brief review of the improved triangular and quadrangular NIPEs is reported for elastic plate analyses. A few challenging benchmarks carried out on extreme distorted meshes illustrate the performance of the introduced NIP-Q9 element. We detail that the new NIPE formulation confers insensitivity to extreme distortions for the quadratic quadrilateral element and allows to solve for severe distortion with the NIPE family.