# Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage
## 4 Frequency Response Analysis under Uncertainty
### Stochastic Collocation  
We consider an anisotropic Smolyak-type collocation operator defined with respect to a finite multi-index set (see e.g., [20; 21; 22; 23]). In principle, a simpler formulation would also suffice for the purpose of our numerical experiments. However, we wish to keep the general form of the operator here, since it allows for efficient computation even in high-dimensional parameter spaces.  
Let \(L_{p}\) be the univariate Legendre polynomial of degree \(p\). Denote by \(\{\chi_{k}^{(p)}\}_{k=0}^{p}\) the zeros of \(L_{p+1}\) and by \(\{w_{k}^{(p)}\}_{k=0}^{p}\) the associated Gauss-Legendre quadrature weights. We define the one-dimensional Lagrange interpolation operators \(\mathcal{I}_{p}^{(m)}\) via  
\[\left(\mathcal{I}_{p}^{(m)}v\right)(\xi_{m})=\sum_{k=0}^{p}v\Big{(}\chi_{k}^{ (p)}\Big{)}\,\ell_{k}^{(p)}(\xi_{m}), \tag{32}\]  
where \(\{\ell_{k}^{(p)}\}_{k=0}^{p}\) are the related Lagrange basis polynomials of degree \(p\). Observe that  
\[\int_{-1}^{1}\left(\mathcal{I}_{p}^{(m)}v\right)(\xi_{m})\,\frac{d\xi_{m}}{2} =\sum_{k=0}^{p}v\Big{(}\chi_{k}^{(p)}\Big{)}\,w_{k}^{(p)}.\]  
Now, let \(\mathcal{A}\subset\mathbb{N}_{0}^{M}\) be a finite set of multi-indices. For \(\alpha,\beta\in\mathcal{A}\), we write \(\alpha\leq\beta\) if \(\alpha_{m}\leq\beta_{m}\) for all \(m=1,\ldots,M\). We assume that \(\mathcal{A}\) is monotone In the following, sense:  
\[\exists\alpha\in\mathcal{A}\text{ such that }\beta\leq\alpha\quad\Rightarrow \quad\beta\in\mathcal{A}.\]  
The sparse collocation operator is defined as  
\[\mathcal{I}_{\mathcal{A}}:=\sum_{\alpha\in\mathcal{A}}\bigotimes_{m=1}^{M} \left(\mathcal{I}_{\alpha_{m}}^{(m)}-\mathcal{I}_{\alpha_{m}-1}^{(m)}\right), \tag{33}\]  
where \(\mathcal{I}_{-1}^{(m)}:=0\). This may be rewritten in a computationally more convenient form  
\[\mathcal{I}_{\mathcal{A}}=\sum_{\alpha\in\mathcal{A}}\sum_{\beta\in\mathcal{G }_{\alpha}}(-1)^{\|\alpha-\beta\|_{1}}\bigotimes_{m=1}^{M}\mathcal{I}_{\beta _{m}}^{(m)}, \tag{34}\]  
where \(\mathcal{G}_{\alpha}:=\{\beta\in\mathbb{N}_{0}^{M}\mid\alpha-1\leq\beta\leq\alpha\}\). The collocation points are of the form  
\[\chi_{\gamma}^{(\alpha)}=\left(\chi_{\gamma_{1}}^{(\alpha_{1})},\ldots,\chi_{ \gamma_{M}}^{(\alpha_{M})}\right)\in\Gamma \tag{35}\]  
for some \(\gamma\in\mathbb{N}_{0}^{M}\) such that \(\gamma\leq\alpha\in\mathcal{A}\). Similarly, we define the tensorized quadrature weights \(w_{\gamma}^{(\alpha)}=w_{\gamma_{1}}^{(\alpha_{1})}\cdots w_{\gamma_{M}}^{( \alpha_{M})}\). Statistics, such as the expected value and variance, for the collocated solution may now be computed by applying the quadrature rule  
\[\mathbb{E}_{\Gamma}\!\left[\bigotimes_{m=1}^{M}\mathcal{I}_{\alpha_{m}}^{(m)}v \right]=\sum_{\gamma\leq\alpha}v\left(\chi_{\gamma}^{(\alpha)}\right)w_{ \gamma}^{(\alpha)} \tag{36}\]  
on the terms in (34).  
The accuracy of the collocated approximation is ultimately determined by the smoothness of the solution as well as the choice of the multi-index set \(\mathcal{A}\subset\mathbb{N}_{0}^{M}\). We refer to [21; 23] for a detailed analysis.