Numerical Control of the Hourglass Inst Ability

Numerical integration of stiffness matrices in finite element analysis can represent a significant portion of the overall computational effort. To improve computational efficiency, it has become a practice of some analysts to 'underintegrate' the various element stiffnesses. i,e. to employ a numerical quadrature rule of an order less than that required to yield exact values for polynomial integrands defined on regular meshes. While such underintegration can significantly reduce computational effort. the resulting matrices may be rank deficient and solutions to the equations of the resulting discrete system may contain 'spurious modes', i.e, zero energy components which are manifested in the computed solutions as oscillatory patterns called hourglass modes. Procedures exist for eliminating or damping hourglass instabilities. One technique is to add corrective terms to the underintegrated stiffness matrix prior to solving the discrete problem. These so-called a priori stabilization procedures have been studied and developed by Belytschko and Liu and their collaboration (see, for example. References 1-4). Alternatively. an a posteriori stabilization technique has been introduced in which the' glob~ll rank-deficient stiffness matrix is used to compute an approximate solution and the spllrious modes are then eliminated by a projection scheme-a so-called hourglass filter. This idea has been advanced by Jacquotte and Oden.s-s Surprisingly good solutions can often be obtained using the a posteriori liItering technique. Indeed, for one-point integration of QJ(4-node quadrilateral) elements in approximations of the two-dimensional Poisson problems. a filtered solution can be obtained which exhibits the same rate of convergence as the fully integrated solution (for sufficiently regular solutions) at a fraction of the cost. Similar results have been obtained for other types of isoparametric elements. Our mission in the present paper is to review some of the concepts underlying the a posteriori underintegration scheme and to present representative numerical results obtained using this procedure. We also provide the results of numerical experiments designed to assess the performance of this scheme for problems with singularities and for cases in which irregular meshes are used.