Shock and Vibration

Volume 2018, Article ID 5817940, 16 pages

https://doi.org/10.1155/2018/5817940

## Cylindrical Shell with Junctions: Uncertainty Quantification of Free Vibration and Frequency Response Analysis

Aalto University, School of Science, Department of Mathematics and Systems Analysis, P.O. Box 11100, FI-00076 Aalto, Finland

Correspondence should be addressed to Harri Hakula; if.otlaa@alukah.irrah

Received 27 April 2018; Revised 29 September 2018; Accepted 16 October 2018; Published 2 December 2018

Guest Editor: Aly Mousaad Aly

Copyright © 2018 Harri Hakula et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Numerical simulation of thin solids remains one of the challenges in computational mechanics. The 3D elasticity problems of shells of revolution are dimensionally reduced in different ways depending on the symmetries of the configurations resulting in corresponding 2D models. In this paper, we solve the multiparametric free vibration of complex shell configurations under uncertainty using stochastic collocation with the *p*-version of finite element method and apply the collocation approach to frequency response analysis. In numerical examples, the sources of uncertainty are related to material parameters and geometry representing manufacturing imperfections. All stochastic collocation results have been verified with Monte Carlo methods.

#### 1. Introduction

Numerical simulation of thin solids remains one of the challenges in computational mechanics. The advent of stochastic finite element methods has led to new possibilities in simulations, where it is possible to replace isotropy assumptions with statistical models of material parameters or incorporate manufacturing imperfections with parametrized computational domains and derive statistical quantities of interest such as expectation and variance of the solution. Here, we discuss two special classes of such uncertainty quantification problems: free vibration of shells of revolution and directly related frequency response analysis. What makes the specific shell configurations interesting from the application point of view is that they include junctions and shell-solid couplings. Free vibrations of cylinder-cone configurations have been studied by many authors before [1–4]. There also exists an exhaustive literature review on shells with junctions [5].

In vibration problems, the sources of uncertainty relate to materials and geometry. In this paper, the stochastic vibration problems are solved using a *nonintrusive* approach, the stochastic collocation [6]. Collocation methods are particularly appealing in problems with parameterized (random) domains. The standard approach is to solve the problem at specified quadrature points and interpolate over the points. If the number of random variables is high, this leads to the *curse of dimensionality* which can be alleviated up to a point with special high-dimensional quadratures, the so-called sparse grids [7–10]. If the uncertainty is only in the material parameters, one can apply the *intrusive* approach such as stochastic Galerkin methods, see [11] and references therein.

Sparse grids are designed to satisfy given requirements for quadrature rules. It is, however, possible to construct similar interpolation operators even if the point set is limited to some subset of the parameter space (partial sparse grids), or, crucially, points are added to the sparse grid. In both cases, the construction is the same [12]. Every point in a sparse grid corresponds to some realization of the random field. If some real measurements become available and the data can be identified as points in the parameter space, they can be incorporated into the interpolation operator.

The two model problems studied here are a wind turbine tower resonance problem and the free vibration of a classical long cylinder-cone junction combined with T-junction via kinematic constraints. The wind turbine tower problem is inspired by an earlier study [13], where the tower is modeled as a solid due to solver limitations. The long cylinder case is in turn inspired by a photograph of a collapsed pipe ([14], p. 82).

We have studied related stochastic shell eigenproblems in the previous work [15], but only for cylinders with constant thickness. In this study, in all cases one, of the random parameters is geometric which changes the character of the computational problems considerably. However, the underlying theoretical properties remain the same. One of the characteristic features of multiparametric eigenvalue problems is that the order of eigenpairs in the spectrum may vary over the parameter space, a phenomenon referred to as *crossing* of the modes.

In this paper, we show how the existing methods can be applied to shell problems that have characteristics that differ from those covered in the literature until now. We do observe crossing of the modes in the wind turbine tower problem which is satisfying since it highlights the fact that this feature is not an artificial mathematical concept but something that is present in standard engineering problems. The stochastic framework for eigenproblems is applied to frequency response analysis in a straightforward manner, yet the resulting method is new. We also demonstrate in connection with frequency response analysis that interpolation operators based on partial sparse grids can provide useful information.

All simulations have been computed using *p*-version of the finite element method [16]. The results derived using collocation on sparse grids have been verified using the Monte Carlo method.

The rest of this paper is organized as follows. In Section 2, the two shell models used in this paper are introduced; in Section 3, the stochastic variant is given with the solution algorithms; Section 3.5 covers the multivariate interpolation construction; the frequency response analysis is briefly outlined in Section 4; the numerical experiments are covered in detail in Section 5; finally, conclusions are drawn in Section 6.

#### 2. Shell Eigenproblem

Assuming a time harmonic displacement field, the free vibration problem for a general shell leads to the following abstract *eigenvalue* problem: find and such thatwhere represents the shell displacement field, while represents the square of the eigenfrequency. In the abstract setting, and are differential operators representing deformation energy and inertia, respectively. In the discrete setting, they refer to corresponding stiffness and mass matrices.

Simulation of thin solids using standard finite elements is difficult, since one (small thickness) dimension dominates the discretization. In the following, we shall derive two variants of problem (1) by employing different dimension reductions for shells of revolution. Instead of working in 3D, the elasticity equations are dimensionally reduced either in thickness or in the case of axisymmetric domains, using a suitable ansatz.

##### 2.1. Shell Geometry

In the following, we let a profile function revolve about the *x*-axis. Naturally, the profile function defines the local radius of the shell. Shells of revolution can formally be characterized as domains in of typewhere *D* is a (mid)surface of revolution, is the unit normal to *D*, and is the thickness of the shell measured in the direction of the normal. An illustration of such a configuration is given in Figure 1.