โ Scribed by F. Molenkamp; D. J. Kidger; I. M. Smith
No coin nor oath required. For personal study only.
In this note the accuracy of the elastic standard 4-node plane-strain finite element is investigated by comparing numerically calculated eigenvalues of the element stiffness matrix to those according to higherorder algebraic approximations. From this study it may be concluded that the 'hourglass' eigenmodes are the only inaccurate ones. For selective reduced numerical integration the hourglass eigenvalues may be expected to be of the order of 1/2 to 2/3 of the analytical values. 'hourglass' instability for reduced integration.
To improve the accuracy of the numerical stiffness, the so-called selective reduced numerical integration' may be applied. In general, in selective reduced integration the volumetric strain is described by a polynomial shape function with one order lower than the one of the deviatoric strain. However, the actual improvement of the accuracy of the numerical stiffness can only be determined by comparing the numerical stiffness to the one according to the analytical solution.
The analytical stiffness could be determined if for any prescribed boundary displacement both the resulting stress and strain distributions could be calculated. However the analytical stiffness may also be approximated by means of computer algebra if higher-order polynomials are used for the expression of the shape function.
In this note the finite-element stiffness according to the full, reduced and selective numerical integration schemes of the 4-node standard 2-dimensional elastic finite element are compared to several approximations with increasing accuracy of the algebraic stiffness. From this comparison conclusions are obtained concerning the accuracy of the various numerical schemes. *Professor. 'Lecturer. 'Professor.
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