Implemented in Code_Aster
=============================




Practical implementation
----------------------

The operator CALC_ERC_DYN [:ref:`U4.53.41 <U4.53.41>`] was developed to solve the problem of finding optimal fields :math:`(u,v,w)` associated with the constrained optimization problem described in the previous section.

Concretely, for a list of given frequencies, it solves the system of linear equations :math:`Al=b` detailed above, which makes it possible to obtain the fields :math:`u` and :math:`u-v` sought. It should be noted that field :math:`u-w` is not proposed as the output of the operator because it is trivially the result of a proportional relationship with :math:`u-v`.

This operator also makes it possible to evaluate, if desired, the value of the functional :math:`{e}_{\omega }^{2}` in the :math:`(u,v,w)\text{}` solution triple. This is particularly interesting when you want to use the :math:`{e}_{\omega }^{2}` function in a realignment or identification problem.

The operator uses five mandatory keywords:


* the keywords MATR_RIGI and MATR_MASSE correspond respectively to the real assembled stiffness and mass matrices of the model studied;


* the keyword MATR_NORME: this keyword allows you to fill in the generalized assembled matrix corresponding to the positive definite symmetric matrix :math:`{G}_{r}` serving as the norm of errors in the observation space.


* the keyword MATR_PROJ: this keyword is used to give the projection matrix associated with the :math:`H` operator. It is a geometric correspondence between the mesh associated with the mass and stiffness matrices on the one hand, and the mesh associated with the :math:`\widehat{u}` observations on the other hand. The projection matrix requested here must be the result of a calculation with the operator PROJ_CHAMP [:external:ref:`U4.72.05 <U4.72.05>`] with the method 'COLLOCATION'.


* the MESURE keyword: this keyword allows you to fill in the fields that will be used as an :math:`\widehat{{u}_{i}}` observation. This concept should contain as many order numbers as the number of frequencies that will be studied when the operator is called (operands FREQ/LIST_FREQ). Moreover, the dimension of the fields contained in this concept must be consistent with the dimension of the standard matrix entered under the operand MATR_NORME. In addition, the fields contained must not include Lagrange-type degrees of freedom. To do this, it is advisable to condition this concept using the operator OBSERVATION [:external:ref:`U4.90.03 <U4.90.03>`] beforehand;


* the CHAMP_MESURE keyword to define the type of field contained in the measurement;


* the FREQ/LIST_FREQ keyword to define the frequencies at which the problem will be solved;


* the keywords ALPHAet GAMMA corresponding to the weighting parameters of the functional comparable to a regularization term (:math:`\alpha`) and to the error weighting parameter :math:`(u-v)` and :math:`(u-w)` (:math:`\gamma`);


* the EVAL_FONCT keyword to determine if the functional value for the optimal field triplet will be evaluated and stored in the result;


* the SOLVEURpermettant keyword to choose the type of solver that will be used when solving the problem. In the current release, the solvers available are MUMPS (default) and LDLT.