Shock and Vibration

Volume 2015 (2015), Article ID 812069, 13 pages

http://dx.doi.org/10.1155/2015/812069

## Hybrid Stochastic Finite Element Method for Mechanical Vibration Problems

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

Received 9 April 2015; Revised 23 June 2015; Accepted 28 June 2015

Academic Editor: Evgeny Petrov

Copyright © 2015 Harri Hakula and Mikael Laaksonen. 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

We present and analyze a new hybrid stochastic finite element method for solving eigenmodes of structures with random geometry and random elastic modulus. The fundamental assumption is that the smallest eigenpair is well defined over the whole stochastic parameter space. The geometric uncertainty is resolved using collocation and random material models using Galerkin method at each collocation point. The response statistics, expectation and variance of the smallest eigenmode, are computed in numerical experiments. The hybrid approach is superior to alternatives in practical cases where the number of random parameters used to describe geometric uncertainty is much smaller than that of the material models.

#### 1. Introduction

In standard engineering models many physical quantities such as material parameters are taken to be constant, even though their statistical nature is well understood. Similarly, assumptions of geometric constants, such as thickness of a structure, are not realistic due to manufacturing imperfections. In a detailed report on a state-of-the-art verification and validation process comparing modern simulations with the set of experiments performed in the Oak Ridge National Laboratory in the early 70s, Szabo and Muntges report discrepancies of over 20% in some quantities of interest [1]. These discrepancies are attributed to machining imperfections not accounted for in the computations. Also, in important nonlinear problems such as buckling of a shell, it is known that variation between manufactured specimens has a profound effect in the actual performance [2]. This suggests that a stochastic dimension should be added to the models.

The modern era of uncertainty quantification starts with the works of Babuska et al. [3, 4] and the ETH-group led by Schwab and Gittelson (e.g., [5]) with provably faster convergence rates than the standard Monte Carlo methods. Although the so-called stochastic finite elements had been studied for a relative long time before (the classic reference is Ghanem and Spanos [6]), their application was thus limited to highly specific cases.

The solution methods can broadly speaking be divided into* intrusive* and* nonintrusive* ones. The same division applies to stochastic eigenvalue problems (SEVPs) as well. SEVPs have attracted a lot of attention recently and various algorithms have been suggested for computing approximate eigenpairs [7–10], specially the power iteration [11] and inverse iteration [12].

In this paper our focus is on effects of material models and manufacturing imperfections of geometric nature. It should be noted that in the context of this paper it is assumed that the problems are positive definite and the eigenpair of interest is the ground state, that is, the one with the smallest eigenvalue which, in theory, can be a double eigenvalue. Our experimental setup is nonsymmetric and, thus, a spectral gap exists and the inverse iteration converges to the desired eigenmode.

In stochastic eigenvalue problems one must address two central issues that do not arise in stochastic source problems: first, the eigenmodes are defined only up to a sign and, second, the eigenmodes must be normalized over the whole parameter space; that is, every realization must be normalized in the same way. Here our quantity of interest is the eigenvalue and, therefore, we do not necessarily have to fix the signs. The normalization is handled by solution of a nonlinear system of equations as in [12].

The main result of this paper is the new hybrid algorithm which combines a nonintrusive outer loop (collocation) with an intrusive inner one (Galerkin). The randomness in geometry is resolved using collocation and that of materials with Galerkin* at every collocation point*. The model problem is an idealized engine bed vibration problem, where a 2D hollow square is clamped from one side (see Figure 1). The inner cavity is assumed to be of fixed shape but random orientation and Young’s modulus is taken to be random in the normal direction into the body. The choice of the geometric uncertainty models the case of sand casting with moulds of fixed shape. The hybrid approach combines the geometric flexibility of collocation with much faster computation times of the Galerkin approach used in the inner loop and, thus, combines the best of both worlds.