'Right things come in threes' SUMMARY A formulation for the plane 4-node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second-order Taylor series in the physical co-ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices.For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix.The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non-linear analysis. There, an application is given for the calculation of inelastic problems in physically non-linear elasticity.The element is efficient to implement and it is frame invariant. Locking effects and zero-energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.

SUMMARYThe standard plane enode element is written as the summation of a constant gradient matrix, usually obtained from underintegration, and a stabilization matrix. The split is based on a Taylor series expansion of element basis functions. In the incompressible limit, the 'locking'-effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher-order term in the Taylor series.The incompressibility condition is formulated in a weak sense so that the element displacement field is divergence-free when integrated over the element volume. The resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first-order terms of the Taylor series. The corresponding eigenvalue enforces incompressibility implicitly by means of a penalty-constraint. Analytical expressions for that constant-dilatation eigenpair are derived for arbitrary element geometries. It is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner.For several classical and recent elements, the eigensystems are analysed numerically. It is shown that most of the formulations reflect the incompressible constraint identically. In the incompressible limit, the numerical accuracies of the e!ements are compared.

SUMMARYThe numerical simulation of processes undergoing finite deformations requires robust elements. For a broad range of applications these elements should have a good performance in bending dominated situations as well as in the case of incompressibility. The element should be insensitive against mesh distortions which frequently occurs during finite deformations. Furthermore, due to efficiency reasons a good coarse mesh accuracy in required in non-linear analysis. The QS6 element, developed in this paper, tries to fulfil the above-mentioned requirements. The performance is depicted by means of numerical examples.

SUMMARYThe standard bilinear displacement field of the plane linear elastic rectangular 4-node quadrilateral element is enhanced by incompatible modes. The resulting gradient operators are separated into constant and linear parts corresponding to underintegration and stabilization of the element stiffness matrix. Minimization of potential energy is used to generate exact analytical expressions for the hourglass stabilization of the rectangle. The stabilized element is shown to coincide with the element obtained by the mixed assumed strain method.

SUMMARYLinear equality multipoint constraint conditions define a vector space which is used to construct an orthogonal projection operator. Another orthogonal projection operator follows from the complement to the constrained vector space. A procedure is developed in which these operators are used in a systematic fashion to solve a set of algebraic equations subject to a set of multipoint constraints. In addition to elementary examples to illustrate the procedure, a problem associated with an incompressible, isotropic, elastic material is solved using four-node quadrilateral finite elements. The method of orthogonal projections provides the insight to show that, with the application of the incompressibility constraint, convergence is obtained for any arbitrary value of Poisson's ratio chosen sufficiently far from the value of one-half, which is the value normally associated with incompressibility. The bending performance of the incompressible four-node quadrilateral can be adjusted with artificial values of Poisson's ratio. The result is that calculations are performed with full numerical quadrature of the element, and mesh locking, hourglass instabilities and subsequent modifications to the element stiffness matrix are avoided.