## 9 Uncertainty Quantification for the Vibroacoustic Computational Model
#### 9.1.3 Stochastic Modeling of Random Matrix \([\mathbf{M}^{S}]\)
* Constructing the family \(\{[\mathbf{\hat{N}}^{R}(\omega)]\}_{\omega\geq 0}\) of random matrices using the equation \[[\mathbf{\hat{N}}^{R}(\omega)]=\frac{2}{\pi}\mathrm{p.v}\int_{0}^{+\infty}\frac {\omega^{\prime\;2}}{\omega^{2}-\omega^{\prime\;2}}\left[\mathbf{D}^{S}( \omega^{\prime})\right]d\omega^{\prime}\,,\] (81) (in which p.v means the Cauchy principal value that is defined in Equation (83)) or equivalently, using the two following equations that are useful for computation:\[\text{for}\quad\omega=0\quad,\quad[\widehat{\mathbf{N}}^{R}(0)] = -\frac{2}{\pi}\int_{0}^{+\infty}[\mathbf{D}^{S}(\omega)]\,d\omega\,, \tag{82}\] \[\text{for}\quad\omega>0\quad,\quad[\widehat{\mathbf{N}}^{R}(\omega)] = \frac{2}{\pi}\,\text{p.v.}\int_{0}^{+\infty}\frac{u^{2}}{1-u^{2}} \,\omega\,[\mathbf{D}^{S}(\omega u)]\,du\,,\] (83) \[= \frac{2}{\pi}\,\lim_{\eta\to 0}\{\int_{0}^{1-\eta}+\int_{1+\eta}^{+ \infty}\}\,.\]
* Constructing the random matrix \([\mathbf{K}^{S}(0)]\) such that \[[\mathbf{K}^{S}(0)]=[L_{K^{S}(0)}]^{T}\,[\mathbf{G}_{K^{S}(0)}]\,[L_{K^{S}(0) }]\,,\] (84) in which the deterministic matrix \([L_{K^{S}(0)}]\) is such that \[E\{[\mathbf{D}^{S}(\omega)]\}=[D^{S}(\omega)]\quad,\quad E\{[\mathbf{K}^{S}( \omega)]\}=[K^{S}(\omega)]\,, \tag{77}\]0 (85) The random matrix \([\mathbf{G}_{K^{S}(0)}]\) is a \((N_{S}\times N_{S})\) random matrix belonging to ensemble \(\text{SG}^{+}_{\varepsilon}\) defined in Section 9.1.2 whose probability distribution and generator of independent realizations depend only on dimension \(N_{S}\) and on the dispersion parameter \(\delta_{K^{S}(0)}\) that allows the level of uncertainties to be controlled. Note that random matrix \([\mathbf{G}_{K^{S}(0)}]\) is independent of random matrix \([\mathbf{G}_{D^{S}}]\). * Computing the random matrix \([\mathbf{D}^{+}]\) such that \[E\{[\mathbf{D}^{S}(\omega)]\}=[D^{S}(\omega)]\quad,\quad E\{[\mathbf{K}^{S}( \omega)]\}=[K^{S}(\omega)]\,, \tag{77}\]1 (86)
* Defining the random matrix \([\mathbf{K}^{S}_{0}]\) such that \[E\{[\mathbf{D}^{S}(\omega)]\}=[D^{S}(\omega)]\quad,\quad E\{[\mathbf{K}^{S}( \omega)]\}=[K^{S}(\omega)]\,, \tag{77}\]2 (87)
* Constructing the random matrix \([\mathbf{K}^{S}(\omega)]\) such that \[E\{[\mathbf{D}^{S}(\omega)]\}=[D^{S}(\omega)]\quad,\quad E\{[\mathbf{K}^{S}( \omega)]\}=[K^{S}(\omega)]\,, \tag{77}\]3 (88) It must be verified that \(\omega\mapsto[\mathbf{K}^{S}(\omega)]\) is effectively positive definite. In [101], the following sufficient condition is proven in order for \([\mathbf{K}^{S}(\omega)]\) to be a positive-definite random matrix for all \(\omega\geq 0\): if for all real vector \(\mathbf{y}=(y_{1},\ldots,y_{N_{S}})\), the random function \(\omega\mapsto\mathbf{y}^{T}\,[\mathbf{D}^{S}(\omega)]\,\mathbf{y}\) is decreasing on \([0\,,+\infty]\), then for all \(\omega\geq 0\), \([\mathbf{K}^{S}(\omega)]\) is a positive-definite random matrix. * For \(\omega<0\), the family \(\{[\mathbf{D}^{S}(\omega)],[\mathbf{K}^{S}(\omega)]\}_{\omega<0}\) is deduced from the family \(\{[\mathbf{D}^{S}(\omega)],[\mathbf{K}^{S}(\omega)]\}_{\omega\geq 0}\) by using Equation (78). * A numerical procedure for computing the integrals in Cauchy principal value can be found in [103].