Finite element stabilization matrices-a unification approach

In this paper, we show that the stabilization vector γ can be obtained naturally by taking the partial derivatives with respect to the natural coordinates. Hence, the components of the strains and stresses can be expressed in terms of a set of orthogonal coordinates. With these definitions of stresses and strains, a general form of the finite element stabilization matrices is then constructed for underintegrated, irregular shape and anisotropic elements. The explicit expressions of the stabilization matrices for the 4-, 9- and 8-node Laplace and continuum elements and the 4-node plate element are then derived from the general form. These are sufficiently general for the developments herein. The generalization of the stabilization-matrices concept to nonlinear problems which usually require the numerical integration of a fairly general class of nonlinear, finite-deformation constitutive equations is also presented. The computer implementation aspects and numerical evaluation of these stabilized elements are also considered. Numerical tests confirm the stability and accuracy characteristics of the resulting elements. In particular, the numerical experiments reported here also show that the rates of convergence in the L₂-norm and in the energy norm agree well with the expected convergence rates.