CONSTRUCTION OF STIFFNESS MATRICES TO MAINTAIN THE CONVERGENCE RATE OF DISTORTED FINITE-ELEMENTS

High-order elements are an option for providing an enhanced rate of convergence. However, it is not widely known that if a high-order element based on a mapping from a master-element space is distorted by displacing one of the side nodes slightly from the traditional midside position then the rate of convergence drops one order. The proof for a three-node element in one dimension has been given previously. Here, a numerical demonstration is presented to show how quickly the rate is lost as the second node is moved from the midpoint. For a six-node triangular element, a similar convergence study is performed in two dimensions in which one of the side nodes is moved off centre along a straight line joining the vertices of the triangle. Again, a loss in the order of convergence is shown although the loss is only apparent for sufficiently small element size. To prevent this drop in the rate of convergence as a side node is displaced, a procedure is given for developing the element stiffness matrix without formulating element basis functions. For the six-node triangle, a complete quadratic representation is retained, but at the expense of element compatibility between nodes. The numerical investigation shows that convergence appears to be retained but that the error associated with the incompatibility is greater than the error obtained with the use of distorted isoparametric elements. The results of this study are particularly appropriate for domains with curved boundaries and for non-linear problems in which node positions are updated according to the deformation history.