An alternative formulation for the eigenvalue optimization of structures made of rubber- like material is presented. The proposed methodology adopts the truly-mixed variational formulation that descends from the principle of Hellinger-Reissner. This formulation has regular stresses as main variables and discontinues displacements that play the role of lagrangian multipliers. The use of this discretization passes the inf-sup condition [1] even for incompressible materials and therefore allows to overcome the arising of the locking phenomenon, difficulty often found handling rubber material in displacement-based finite element contexts. The approach presented in the work consists in the adoption of a mixed-FEM bidimensional discretization that uses the composite element of Johnson and Mercier [2] as to the discretization of the stress field. Also this discrete formulation, as the continuous one, meets automatically the Babuska-Brezzi conditions of the problem even in presence of incompressible materials. Within this mixed framework, the work addresses classical eigenvalue optimization problems, with the aim of maximizing the first eigenfrequency or a weighted sum of the first eigenvalues, given the domain and a volume constraint [7] , [3]. For all the problems used to test the methodology, MMA [4] is chosen as minimization algorithm, within an optimization framework relying on SIMP method. Peculiar attention is devoted to the problem of the arising of localized modes and to the difficulties in getting pure 0-1 designs in plane strain conditions.
Mixed finite-elements for eigenvalue optimization of incompressible media
BRUGGI, MATTEO;
2007-01-01
Abstract
An alternative formulation for the eigenvalue optimization of structures made of rubber- like material is presented. The proposed methodology adopts the truly-mixed variational formulation that descends from the principle of Hellinger-Reissner. This formulation has regular stresses as main variables and discontinues displacements that play the role of lagrangian multipliers. The use of this discretization passes the inf-sup condition [1] even for incompressible materials and therefore allows to overcome the arising of the locking phenomenon, difficulty often found handling rubber material in displacement-based finite element contexts. The approach presented in the work consists in the adoption of a mixed-FEM bidimensional discretization that uses the composite element of Johnson and Mercier [2] as to the discretization of the stress field. Also this discrete formulation, as the continuous one, meets automatically the Babuska-Brezzi conditions of the problem even in presence of incompressible materials. Within this mixed framework, the work addresses classical eigenvalue optimization problems, with the aim of maximizing the first eigenfrequency or a weighted sum of the first eigenvalues, given the domain and a volume constraint [7] , [3]. For all the problems used to test the methodology, MMA [4] is chosen as minimization algorithm, within an optimization framework relying on SIMP method. Peculiar attention is devoted to the problem of the arising of localized modes and to the difficulties in getting pure 0-1 designs in plane strain conditions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
