3. Reminders

3.1. Optimization problem associated with an energy functional

This operator solves the following problem:

Find the triple of eligible fields \((u,v,w)\text{}\) that minimize functionality:

\({e}_{\omega }^{2}(u,v,w)=\frac{\gamma }{2}{(u-v)}^{T}[K](u-v)+\frac{1-\gamma }{2}{(u-w)}^{T}{\omega }^{2}[M](u-w)+\frac{1-\alpha }{\alpha }{(\mathit{Hu}-\widehat{u})}^{T}\text{}[\mathit{Gr}](\mathit{Hu}-\widehat{u})\)

under duress:

\(\left[K\right]v-{\omega }^{2}\left[M\right]\text{w=}0\)

Where:

\(K\) represents a real stiffness matrix

\(M\) represents a mass matrix

\(H\) represents an observation matrix

\(\mathit{Gr}\) represents a symmetric positive definite matrix serving as the norm of errors in the observation space

\(\omega =2\pi f\): pulsation of excitement

\(\widehat{u}\) observation of movements at the pulse \(\omega\)

\(\gamma\) parameter for weighting errors \((u-v)\) and \((u-w)\)

\(\alpha\) functional weighting parameter similar to a regularization term

3.2. Problem solving equations

Obtaining the triplet \((u,v,w)\text{}\) associated with the problem under stress above leads, at each frequency considered, to the resolution of the following system of linear equations:

\(Al=b\)

with, for each natural pulsation \({\omega }_{i}\):

\({A}_{i}=\left(\begin{array}{cc}\gamma \left(\text{K+γ}/\left(1-\gamma \right){\omega }_{i}^{2}M\right)& -\gamma \left(K-{\omega }_{i}^{2}M\right)\\ -\gamma \left(K-{\omega }_{i}^{2}M\right)& \left(-\text{2α}/\left(1-\alpha \right)\right){H}^{T}{G}_{r}H\end{array}\right)\); and \({b}_{i}=\left(\begin{array}{c}{0}_{n}\\ \left(-\text{2α}/\left(1-\alpha \right)\right){H}^{T}{G}_{r}{\widehat{u}}_{i}\end{array}\right)\)

For more details on the wording, refer to document [R4.10.07].