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CivilComp Proceedings
ISSN 17593433 CCP: 86
PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 89
Mixed FiniteElements for Eigenvalue Optimization of Incompressible Media M. Bruggi and C. Cinquini
Department of Structural Mechanics, University of Pavia, Italy M. Bruggi, C. Cinquini, "Mixed FiniteElements for Eigenvalue Optimization of Incompressible Media", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 89, 2007. doi:10.4203/ccp.86.89
Keywords: eigenvalue optimization, incompressible media, mixed finite elements.
Summary
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 trulymixed
variational formulation that may be derived from the principle of
HellingerReissner. 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 socalled trulymixed, passes the
infsup condition [1] along with its discretization based
on the composite stress element of Johnson Mercier [4]. Therefore this
trulymixed setting overcomes the locking
phenomenon that often prevents displacementbased finite elements
form a correct analysis of rubberlike material.
The approach presented in this work consists of the adoption of this trulymixed discretized form to solve eigenvalue optimization problems for rubberlike 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 wellknown 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 01 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 01 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. References
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