Efficient strain-gradient mixed elements using shared degrees of freedom for the discretised fields

A displacement-only finite-element formulation of strain-gradient models requires elements with $C^1$ continuous interpolation. Mixed formulations have been proposed to allow the use of more common $C^0$ element shape functions. These mixed formulations are based on the interpolation of two different fields, displacement and some kind of displacement gradient, with the relation between the two fields enforced using either Lagrange multipliers or penalty methods. All elements proposed in the literature for such formulations use a distinct set of degrees of freedom to discretise each field. In this work, we introduce for the first time shared degrees of freedom, that lead to a mixed formulation with a significantly better numerical performance. We describe how this novel mixed formulation can be derived, present individual elements implementing this, and discuss the significance of the results.