4. Use in Code_Aster

In Code_Aster, Ritz methods can be used in transition mainly by the DYNA_TRAN_MODAL [U4.53.21] operator.

A phase of projecting the stiffness and mass matrices on a vector base is carried out by the operators PROJ_MATR_BASE [U4.63.12] and PROJ_VECT_BASE [U4.63.13].

The generalized dynamic problem is then solved in the operator DYNA_TRAN_MODAL by an explicit (EULER or DEVOGELEARE) or implicit (NEWMARK) integration scheme. The characteristics and properties of integration schemes are presented in note [R5.06.04]. For structures for which the hypothesis of BASILE does not apply (non-proportional damping) we will also project the damping matrix which does not become diagonal. The integration of the coupled system is then necessarily done with the implicit (NEWMARK) or explicit (EULER) schema.

Localized nonlinearities are specified directly in the DYNA_TRAN_MODAL operator. We can introduce localized nonlinearities such as shock and friction (see [R5.06.03] Modeling shocks and friction), modal forces that are a function of displacement or speed.

Subsequent static truncation corrections are available in the case of a single excitation (see [R4.05.01] Seismic Response).

The addition of static modes can be done by using the operators MODE_STATIQUE [U4.52.14] and DEFI_BASE_MODALE [U4.64.02] beforehand. When the problem involves non-linearities only explicit diagrams can be used.

For global nonlinearities [éq 3.1-4], it is possible to use the DYNA_NON_LINE [U4.53.01] command with the PROJ_MODAL operand, which allows the internal forces to be calculated according to the physical degrees of freedom at each time step, and then to project the problem onto a modal basis.

An operation to return to the physical base is then necessary to obtain physical quantities such as displacement, speed or acceleration on the structure. This operation is performed by the operator REST_GENE_PHYS [U4.63.31] for the whole structure or RECU_FONCTION, (keyword factor RESU_GENE) [U4.32.01] for tracking a quantity in a node. In the case of global nonlinearities (calculation with DYNA_NON_LINE) we use the REST_COND_TRAN [U4.63.33] operator.

More generally, the Ritz approach can be used in harmonic calculation by the DYNA_LINE_HARM [U4.53.22] command and in power spectral density by the DYNA_ALEA_MODAL [U4.53.23] command.

Finally, dynamic substructuring can be considered as a specific Ritz method [R4.06.02].