# Low-Rank Approximation of Frequency Response Analysis of Perforated Cylinders under Uncertainty
## 1 Introduction  
Cylindrical shell structures are common in engineering. Within computational mechanics, they belong to the class of thin solids, the class which arguably remains computationally one of the most challenging ones. Here, the focus is on perforated cylinders--more specifically, trommel screens and their frequency response. Perforated cylinders are used, for instance, in mass dryers and as screening devices, and are naturally subjected to large loads, which are often of impact type or otherwise concentrated [1, 2]. Ideally, one would like to be able to analyse perforated structures as homogenised, where the effect of the perforation pattern is transferred to material parameters, the so-called effective material parameters [3, 4, 5]. The structural models used in shell analysis are dimensionally reduced, where the thickness is treated as a (dimensionless) parameter. Somewhat surprisingly, there is an intricate interplay between the perforation patterns and the thickness, which sets limits to homogenisation both under static loading and in dynamical eigenmode setting; see [6, 7]. In this study, no attempts at homogenisation are made, and all simulations are carried out in exact geometry. Frequency response analysis of thin solids is a multiparametric problem. In the structures considered here, uncertainty is included in the material properties; hence, the parameters can be divided into the stochastic parameters defining the material properties ([PERSON]'s modulus), and the deterministic parameters, i.e., the dimensionless thickness and the frequency considered. From the point of view of the analysis, no changes are necessary even if the perforation patterns were randomly perturbed as well. Many of these issueshave been addressed in a previous study [8]. Modern stochastic analysis of multiparametric problems leads to a large number of simulations. Low-rank approximation can be a highly effective way of reducing the computational complexity. The Krylov subspace approach pioneered by [PERSON] is the one considered here [9]. Naturally, it becomes necessary to establish limits within the parameter space when such approximations are applicable in the problem class of interest. Frequency response analysis is of interest in many engineering fields, and hence the body of literature is not only vast but also scattered over a large number of journals. On structural health monitoring alone, two review articles cover nearly 300 publications [10; 11]. In this paper, the underlying mathematical models are taken to be linear, and the materials isotropic in order to simplify the discussion. As examples of more complicated configurations in related problems, [PERSON] et al. consider vibration composite beams under random loads [12] and [PERSON] et al. corrugated cylinders and nonlinear buckling [13]. A very recent paper by [PERSON] et al. [14] provides references to papers on complex vibration problems and related uncertainty quantification. A well-known paper by [PERSON] et al. [15] highlights specific problematic issues in the uncertainty quantification of nonlinear systems. In the context of collocation methods such as those used here, the fundamental question is whether the interpolation property of the system is preserved. It is reasonable to expect that, within linear elasticity, this is never violated. The two foundational papers in introducing the Krylov subspace method in this context are by [PERSON] et al. [16] and [PERSON] [17]. Three more widely cited papers building on the connection with the proportionally damped systems are [PERSON] and [PERSON] [18], [PERSON] et al. [19], and [PERSON] and [PERSON] [20], where also material uncertainties are included in a similar way as here. [PERSON] et al. have continued with a series of papers [21; 22]. Moreover, [PERSON] et al. [23] have studied this approach in the context of topology optimisation, where also the computational complexity can increase significantly.