Development of a new formulation for a beam element with displacement DOF using an inverse method

A superconvergent beam element with displacement degrees of freedom only is presented using an inverse method. The inverse method consists of two steps in obtaining element formulation. First, the element formulation in parametric form is constructed by fulfilling geometrical symmetries of the element and rigid body modes requirements. Second, unknown parameters of the element formulation are determined by minimizing discretization errors. These errors are the disparity between the resulting parametric finite element discrete formulation and its corresponding continuous governing equations. The errors were minimized based on the idea that the residual errors in adjacent nodes must be equal in magnitudes but with opposite signs. This causes to a zero bias error and leads to a more accurate model. Here, the best stiffness matrix with highest convergence rate using the inverse method for the element is the same as the stiffness matrix obtained using linear shape functions. However, the developed mass matrix is different from those reported in the literature. The new element formulation is compared with the classical model in the literature through numerical eigen-solution examples. The results exhibit that the new element formulation produces a much faster convergence rate than those reported in the literature.