Development of super-convergent plane stress element formulation using an inverse approach

Abstract

New formulation for the plane stress element with super-convergent properties is presented using an inverse method. The element formulation is developed in parametric form satisfying geometrical symmetries of the element and producing rigid body and constant strain modes requirements. The remaining higher order modes of the element are assigned by minimizing the difference between the resultant parametric finite element discrete formulation and the corresponding continuous governing equations, i.e. the discretization errors. Classically minimization of discretization errors is performed by starting from the lowest order error terms and setting them equal to zero. In this paper, it is shown the effect of these errors may also be minimized by allowing the residual errors in adjacent nodes to be equal in magnitudes but with opposite signs. This causes zero bias error in the associated eigen-problem and leads to an element model with super convergent eigen-solution properties. The proposed method of minimizing the errors is applied to the plane stress element formulation and it is proven a more accurate model is achieved over the classical method of error minimization. The new element formulation result super-convergent eigen-solution and in a numerical example these convergence properties are compared to the reported models in the literature.