Gradient degrees of freedom and finite-element topology, mobility, and performance

Using position gradients as independent nodal coordinates in rotation-free finite-element (FE) formulations has created new fundamental issues related to the definition of the boundary conditions, FE performance and convergence, interpretation of results, and FE solution accuracy. In case of an unconstrained one-element mesh, different choices of the nodal coordinates that replace the interpolating-polynomial coefficients lead to the same solution regardless of the applied forces. This is not the case when geometric and stress boundary conditions are applied to different coordinate types because the mesh stiffness in response to loads depends on the constrained coordinates. This paper discusses the new fundamental problem of the FE topology , defined by independent displacement and stress constraints that restrict FE mobility . New longitudinal-vibration boundary conditions that cannot be defined using analytical and conventional FE formulations are introduced. Interpretation of the gradient (strain and stress) constraints is discussed and their effect on the FE topology is quantified using a frequency comparison. Effect of using inconsistent constitutive relationships on the frequency prediction is also demonstrated. It is shown that the error in the fundamental natural frequency is very small and within the range 0.01 %–0.06 % when using models consistent with the analytical models.