Keywords: eigenvalue optimization, incompressible media, mixed finite elements.

The eigenvalue optimization of structures made of incompressible material is approached by means of an alternative formulation. The core of the proposed methodology is the adoption of a truly-mixed variational formulation that may be derived from the principle of Hellinger-Reissner. In general one may write two different formulations, i.e. a first formulation where continuous displacements are main variables coupled with discontinuous stresses and the dual one where regular stresses are the main variables and discontinuous displacements play the role of Lagrangian multipliers. This last formulation, the so-called truly-mixed, passes the inf-sup condition [1] along with its discretization based on the composite stress element of Johnson Mercier [4]. Therefore this truly-mixed setting overcomes the locking phenomenon that often prevents displacement-based finite elements form a correct analysis of rubber-like material.

The approach presented in this work consists of the adoption of this truly-mixed discretized form to solve eigenvalue optimization problems for rubber-like materials. The problems considered belong to the family of the maximization of the first eigenvalue or maximization of the weighted sum of the first eigenvalues to improve the overall stiffness of the structure, see [3].

When dealing with eigenvalue optimization of incompressible media, two peculiar numerical aspects have to be considered. The first one is the well-known problem arising from localized modes, analyzed in [5]. The second one is concerned with the incompressibility property of the material that may prevent a convergence to a pure 0-1 design under the plane strain condition, as illustrated for minimum compliance problems in [6,2].

For the first problem an alternative mass interpolation is proposed in order to eliminate the existence of low density regions with a very high ratio between penalization of mass and stiffness. For the second one, numerical remedies proposed in minimum compliance problems in [2] are tested in order to obtain pure 0-1 optimal designs.

The numerical section focuses on examples that provide evidence arising from these two numerical troubles. Optimal topologies for plane strain design of incompressible media are therefore presented as the result of a minimization process performed by means of MMA. Some forthcoming investigations are eventually highlighted including the study of a larger number of examples to eventually find topological differences among designs under plane strain and plane stress conditions for structures made of incompressible materials.

1

Brezzi, F., Fortin, M., 1991. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York.

2

Bruggi M. and Venini P., "Topology optimization of incompressible media using mixed finite elements", Comput. Methods Appl. Mech. Eng., in press, 2007. doi:10.1016/j.cma.2007.02.013

3

Kosaka I., Swan C.C., "A symmetry reduction method for continuum structural topology optimization", Comp.& Struct., 70:47-61, 1999. doi:10.1016/S0045-7949(98)00158-8

4

Johnson. C., Mercier B., "Some equilibrium finite elements methods for two dimensional elasticity problems", Numer. Math., 30, 103-116, 1978. doi:10.1007/BF01403910

5

Pederson N.L., "Maximization of eigenvalue using topology optimization", Struct. Mult.Opt., 20, 2-11, 2000. doi:10.1007/s001580050130

6

Sigmund O. and Clausen P.M., "Topology optimization using a mixed formulation: An alternative way to solve pressure load problems", Comput. Methods Appl. Mech. Eng., 196 (13-16):1874-1889, 2007. doi:10.1016/j.cma.2006.09.021

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