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 co nvergence. However, it is not widely known that if a high-order elemen t based on a mapping from a master-element space is distorted by displ acing one of the side nodes slightly from the traditional midside posi tion then the rate of convergence drops one order. The proof for a thr ee-node element in one dimension has been given previously. Here, a nu merical demonstration is presented to show how quickly the rate is los t as the second node is moved from the midpoint. For a six-node triang ular element, a similar convergence study is performed in two dimensio ns 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 suffic iently small element size. To prevent this drop in the rate of converg ence as a side node is displaced, a procedure is given for developing the element stiffness matrix without formulating element basis functio ns. For the six-node triangle, a complete quadratic representation is retained, but at the expense of element compatibility between nodes. T he numerical investigation shows that convergence appears to be retain ed but that the error associated with the incompatibility is greater t han the error obtained with the use of distorted isoparametric element s. The results of this study are particularly appropriate for domains with curved boundaries and for non-linear problems in which node posit ions are updated according to the deformation history.