Stabilization of the zero-energy modes of under-integrated isoparametric finite elements

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The coefficient matrices of isoparametrically distorted finite elements are frequently computed by numerical integration formulas with a lower degree of exactness than required by the degree of the integrand. This may result in a rank deficiency of the element matrices which causes an oscillatory instability in the solution. To avoid this effect the element matrices may be stabilised, i. e. their rank may be raised artificially. A new approach is suggested which approximates exact integration without the computational expense usually connected with it. For this purpose, a new family of integration formulas is introduced which is based on the standard Gauss product formulas extended by derivatives of the integrand. For the stabilization, these derivatives can be approximated very easily. The procedure generates a stabilization matrix which, when added to the under-integrated matrix, produces the correct rank. If e. g. shear locking is a problem, the stabilization matrix may be scaled down. Two- and three-dimensional Lagrange elements in second and fourth order problems and Mindlin plate elements are presented together with the results of computations and numerical experiments.