4. Audit and Enforcement Phase#

4.1. Calculation of the expansion base restricted to the measured degrees of freedom#

First, the mesh of the measurement is projected onto the mesh of the numerical model. The participation of the nodes of the numerical model is then determined for each measurement node via the shape function of the element that contains the measurement node. The correspondence obtained between the nodes is provided in file MESSAGE of the Aster study.

The second treatment consists in calculating the field component (expansion base) at the measurement node according to the measured degrees of freedom.

4.2. Generalized coordinate calculation#

The solution to the minimization equation is given by:

\(\begin{array}{ccc}\eta (0)& \text{=}& {\left[{\stackrel{ˉ}{\Phi }}_{\mathrm{num}}^{T}{\stackrel{ˉ}{\Phi }}_{\mathrm{num}}\right]}^{\text{-}1}{\stackrel{ˉ}{\Phi }}_{\mathrm{num}}^{T}{q}_{\mathrm{exp}}(0)\\ \eta (i)& \text{=}& {\left[{\stackrel{ˉ}{\Phi }}_{\mathrm{num}}^{T}{\stackrel{ˉ}{\Phi }}_{\mathrm{num}}+\alpha (i)\right]}^{\text{-}1}({\stackrel{ˉ}{\Phi }}_{\mathrm{num}}^{T}{q}_{\mathrm{exp}}(i)+\alpha (i){\eta }_{\mathrm{priori}})\end{array}\)

With:

  • \(\eta (i)\): generalized coordinates for the order number i (ti or fi),

  • \({q}_{\mathrm{exp}}(i)\): measure at order number i,

  • \({\stackrel{ˉ}{\Phi }}_{\mathrm{num}}\): expansion base restricted to the degrees of freedom of measurement,

  • \(\alpha (i)\): coefficients allowing to specify the weight assigned to the information a priory to the order number i. These variables or functions are defined by the user in the COEF_PONDER or COEF_PONDER_F operands of the keyword factor RESOLUTION. They are introduced in the form of a list of real numbers or functions and correspond, term by term, to each vector of the expansion base selected.

Depending on the method used, the preceding parameters are divided as follows:

Without regularization: \(\alpha =0\)

Minimum standard (NORM_MIN): \({\eta }_{\mathrm{priori}}=0\)

Tikhonov « relative » (TIK_RELA): \({\eta }_{\mathrm{priori}}={\eta }_{i-1}\)

Note 1:

If a weighting coefficient is negative, the treatment stops in a fatal error.

Note 2:

If all the weighting coefficients are zero for a given order number and the number of measurements is strictly less than the number of base vectors, an alarm message is sent to warn of the risk of a singular matrix (in fact, in this case, there is no uniqueness of the solution) .

At the end of the calculation, the generalized coordinates identified are derived in order to calculate the corresponding speeds and accelerations.

The result of the reversal is a tran_gene, harm_gene, or mode_gene concept.