Truly-mixed finite elements for the optimal design of structures on regular grids

The so–called truly–mixed version of the Hellinger–Reissner variational principle implements a regular stress field as main variable of the elasticity problem, whereas displacements play the role of Lagrangian multipliers. A small number of robust discrete schemes are available in the literature that fulfil the stability condition, mainly through ad hoc composite finite elements or the weak enforcement of the symmetry of the stress tensor. The available implementations are generally tied to an increased computational cost with respect to conventional displacement–based finite elements and their iterative adoption within minimum problems is therefore highly demanding in terms of CPU time. Recently, new families of mixed finite elements have been proposed in the literature to address the analysis of linear elastic bodies adopting regular grids and a limited number of degrees of freedom per element. The lowest order two–dimensional finite element of this new family is adopted to formulate a topology optimization problem where stresses play the role of main variables. The compliance is computed through the evaluation of the complementary energy and the enforcement of stress constraints is straightforward. Preliminary numerical simulations investigate the features of the proposed framework, addressing a well–known benchmark for conventional displacement–based minimum weight formulations.