6. Use in Code_Aster

6.1. The necessary data

The characteristics of the beams (section \(S\), inertia tensor \(I\), torsional constant \({J}_{z}\)) are entered directly under the keyword factor POUTRE of the command AFFE_CARA_ELEM.

The characteristics of the homogenized coefficients and of the reference cell are entered under the keyword factor POUTRE_FLUI of the AFFE_CARA_ELEM command. For simple keywords, the correspondence is as follows:

B_T: \({b}_{11}\)

B_N: \({b}_{22}\)

B_TN: \({b}_{12}\)

A_ FLUI: \({Y}_{F}\)

A_ CELL: \(Y={Y}_{F}+{Y}_{S}\)

COEF_ECHELLE: \(\mu\)

The characteristics of the materials are specified in order DEFI_MATERIAU. For tubes, the keyword factor ELAS is used to indicate Young’s modulus (E: \(E\)), Poisson’s ratio (NU: \(\nu\)), and density (RHO: \({\rho }_{S}\)). For the fluid, the keyword factor FLUIDE is used to indicate the density of the fluid (RHO: \({\rho }_{F}\)).

6.2. Orientation of the axes of the beams

The generators of the cylindrical cells are necessarily parallel to the axis of the beams and the bases of the cells perpendicular to this same axis. When developing the mesh, it is necessary to ensure that the order of the nodes (local numbering) of each cylindrical mesh is correct: the nodes of the lower base then the nodes of the upper base (or vice versa). The direction of the axis of the beams is specified under the keyword factor ORIENTATION of the AFFE_CARA_ELEM command.

The following hypothesis was made: the reference frame is the same as the main inertia reference frame of the characteristic tube representing the homogenized medium. This means that in equations [éq 3.1-3], the term \({I}_{\mathrm{xy}}\) is zero.

6.3. Modal calculation

The finite element developed makes it possible to characterize the vibratory behavior of a network of beams bathed by a fluid. It is interesting to determine the vibration frequencies of such an air and water network.

To perform a modal calculation in air (\({\rho }_{F}=0\)), you must block all the degrees of freedom corresponding to the fluid displacement potential \(\Phi\), otherwise the stiffness matrix (and even the shifted matrix of the modal problem) is non-invertible [R5.01.01].

To perform a modal calculation in water (\({\rho }_{F}\ne 0\)), you must use CALC_MODES in the command with OPTION =” CENTRE “. The shifted matrix \((\tilde{K}-\sigma \tilde{M})\) is then invertible if \(\sigma\) is not an eigenvalue or if \(\sigma\) is different from zero.