Eigenvalue-based optimization of incompressible media using mixed finite elements with application to isolation devices

An alternative formulation for the eigenvalue optimization of structures made of incompressible material is presented. The proposed methodology is based on the adoption of a ‘‘truly-mixed” variational formulation that descends from the principle of Hellinger–Reissner and passes the inf-sup condition [F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991] even for incompressible materials. This therefore allows to avoid the locking phenomenon, that is a difficulty often found handling rubber material by means of commonly used displacement-based finite elements. The approach implemented in the work consists in the adoption of the discretized form of the ‘‘truly-mixed” formulation that uses the composite element of Johnson and Mercier [C. Johnson, B. Mercier, Some equilibrium finite elements methods for two dimensional elasticity problems, Numer. Math. 30 (1978) 103–116] to address eigenvalue-based optimization problems for rubber-like materials. Firstly classical eigenvalue designs for in-plane vibrating bidimensional structures are studied relying on MMA [K. Svamberg, Method of moving asymptotes – a new method for structural optimization, Int. J. Numer. Methods Engrg. 24 (3) (1987) 359–373] as minimization algorithm. Peculiar attention is paid to numerical troubles that may arise managing such kind of problems, as the appearance of localized modes preventing an adequate convergence pattern and the difficulties in getting pure 0–1 designs typical of plane strain conditions. Suitable mass and stiffness interpolations are presented and tested to overcome these problems. Afterwards, the proposed framework is used to approach the design of rubber-like materials in a minimum compliance setting with the imposition of an upper limit on the first eigenvalue. A bi-phase stiffness interpolation is moreover implemented to deal with steel–rubber design of aseismic bearing devices in civil engineering. Some forthcoming investigations are eventually highlighted including the implementation of stress constraints to improve the presented designs for manufacturing.