Mixed methods for viscoelastodynamics and topology optimization

Abstract

A truly-mixed approach for the analysis of viscoelastic structures and continua is presented. An additive decomposition of the stress state into a viscoelastic part and a purely elastic one is introduced along with an Hellinger-Reissner variational principle wherein the stress represents the main variable of the formulation whereas the kinematic descriptor (that in the case at hand is the velocity field) acts as Lagrange multiplier. The resulting problem is a Differential Algebraic Equation (DAE) because of the need to introduce static Lagrange multipliers to comply with the Cauchy boundary condition on the stress. The associated eigenvalue problem is known in the literature as constrained eigenvalue problem and poses several difficulties for its solution that are addressed in the paper. The second part of the paper proposes a topology optimization approach for the rationale design of viscoelastic structures and continua. Details concerning density interpolation, compliance problems and eigenvalue-based objectives are given. Worked numerical examples are presented concerning both the dynamic analysis of viscoelastic structures and their topology optimization.