# Low-Rank Approximation of Frequency Response Analysis of Perforated Cylinders under Uncertainty
## 2 Preliminaries
### Shell Eigenproblems  
The free vibration problem for a general shell under the assumption of a time harmonic displacement field leads to the following abstract _eigenvalue_ problem: Find \(\mathbf{u}\in\mathbb{R}^{3}\) and \(\omega^{2}\in\mathbb{R}\) such that  
\[\begin{cases}\mathbf{S}\,\mathbf{u}=\omega^{2}\mathbf{M}\,\mathbf{u}\\ +\,\,\text{boundary conditions}.\end{cases} \tag{1}\]  
Above, \(\mathbf{u}=\{u,v,w\}\) represents the shell displacement field, while \(\omega^{2}\) represents the square of the eigenfrequency. In the abstract setting, \(\mathbf{S}\) and \(\mathbf{M}\) are differential operators representing deformation energy and inertia, respectively. In the discrete setting, they refer to corresponding stiffness and mass matrices.  
Reduction in Thickness  
In the case of a shell of revolution with constant thickness \(t\), the free vibration problem for a general shell (1) can be stated as the following eigenvalue problem: Find \(\mathbf{u}(t)\) and \(\omega^{2}(t)\in\mathbb{R}\) such that  
\[\begin{cases}t\,\mathbf{A}_{M}\mathbf{u}(t)+t\,\mathbf{A}_{S}\mathbf{u}(t)+t ^{3}\,\mathbf{A}_{B}\mathbf{u}(t)=\omega^{2}(t)\,\mathbf{M}(t)\,\mathbf{u}(t) \\ +\,\,\text{boundary conditions}.\end{cases} \tag{2}\]  
Again, \(\mathbf{u}(t)\) represents the shell displacement field, while \(\omega^{2}(t)\) represents the square of the eigenfrequency. The differential operators \(\mathbf{A}_{M}\), \(\mathbf{A}_{S}\) and \(\mathbf{A}_{B}\) account for membrane, shear, and bending potential energies, respectively, and are _independent of_\(t\). Finally, \(\mathbf{M}(t)\) is the inertia operator, which in this case can be split into the sum \(\mathbf{M}(t)=t\,M^{l}+t^{3}\,M^{r}\), with \(M^{l}\) (displacements) and \(M^{r}\) (rotations) _independent of_\(t\).  
Figure 1: Trommel screen. Shell of revolution generated by a profile function \(\phi(x)=1\). Transverse deflection fields under angular excitation of wave number \(=20\). Effect of the deterministic parameters \(t\) (thickness) and \(f\) (frequency) is illustrated, (**a**) \(t=1/100\), \(f=5\), (**b**) \(t=1/100\), \(f=40\), (**c**) \(t=1/1000\), \(f=5\), (**d**) \(t=1/1000\), \(f=40\). Notice how, from (**a**) to (**b**), the maximal intensity moves from the area of the large perforations to the small ones, but from (**c**) to (**d**), the same effect does not take place. In (**d**), the solution is locally dominant and the energy is concentrated on the boundary layers.  
The finite element method is based on the variational formulation of problem (2). Accordingly, when the space \(V\) of admissible displacements is introduced, the problem becomes: Find \((\mathbf{u}(t),\omega^{2}(t))\in V\times\mathbb{R}\) such that  
\[\begin{split} t\,a_{m}(\mathbf{u}(t),\mathbf{v})+t\,a_{s}( \mathbf{u}(t),\mathbf{v})+t^{3}\,a_{b}(\mathbf{u}(t),\mathbf{v})=\\ \omega^{2}(t)\,m(t;\mathbf{u}(t),\mathbf{v})\quad\forall\mathbf{v }\in V,\end{split} \tag{3}\]  
where \(a_{m}(\cdot,\cdot)\), \(a_{s}(\cdot,\cdot)\), \(a_{b}(\cdot,\cdot)\) and \(m(t;\cdot,\cdot)\) are the bilinear forms associated with the operators \(\mathbf{A}_{M}\), \(\mathbf{A}_{S}\), \(\mathbf{A}_{B}\) and \(\mathbf{M}(t)\), respectively. Obviously, the space \(V\) and the three bilinear forms depend on the chosen shell model (see, for instance, [24]).  
The energy norm \(|||\cdot|||\) is defined in a natural way in terms of the deformation energy \(\mathcal{E}(\mathbf{u}):=|||\mathbf{u}|||:=\sqrt{\mathbf{A}(\mathbf{u}, \mathbf{u})}\). Similarly, for (squared) bending, membrane, and shear energies, e.g., \(\mathcal{B}(\mathbf{u}):=t^{2}\mathbf{A}_{B}(\mathbf{u},\mathbf{u})\). The shell model used throughout the paper, the Reissner-Naghdi model, is outlined in Appendix A.