Mixed Finite Element Analysis and Topology Optimization of Viscoelastic Devices

In the first part of this paper, a truly-mixed approach of the Hellinger-Reissner type for the analysis of viscoelastic beams and continua is presented. Stresses represent the main variables of the formulation whereas velocities (not displacements) play the role of Lagrange multipliers. Ad-hoc finite elements are then introduced for the relevant mixed discrete problem that are discussed in some detail. In the second part of the paper, a topology optimization approach is introduced for the optimal design of viscoelastic devices with respect to eigenvalue-dependent objective functions. The approach represents a first proposal in the literature that is nearly entirely limited to the case of elastic structures. Even though the procedure is general, in this paper the numerical examples concerning topology optimization are limited to the beam case leaving applications to two dimensional systems for future papers.