Shape functions of superconvergent finite element models

In structural dynamics superconvergent element models are obtained by eigen-value convergence analysis, or minimizing the discretization errors leading to maximum convergence rates in their eigen-solutions. The element formulations developed by these inverse strategies are obtained in local coordinates. As no shape functions are employed in their development transforming them to global coordinates is a challenge and prevents their use in practical finite element models. To remove this obstacle a new method is proposed to obtain shape functions for superconvergent element models attained directly from the eigen-value convergence analysis or discretization error analysis. The method employs series of trigonometric functions to obtain shape functions corresponding to the superconvergent element formulations. Using the proposed strategy, the shape functions for superconvergent rod, beam and transverse vibration membrane are obtained. It is shown transformation of the superconvergent element formulation to the global coordinates using the obtained shape functions does not affect the eigen-value convergence rates.