Received: 26 June 2018 Accepted: 08 July 2018 DOI: 10.1002/pamm.201800114 A probability-box approach on uncertain correlation lengths by stochastic finite element method Mona M. Dannert 1, * , Amelie Fau 1 , Rodolfo M.N. Fleury 1 , Matteo Broggi 2 , Udo Nackenhorst 1 , and Michael Beer 2,3,4 1 Institute of Mechanics and Computational Mechanics, Leibniz Universität Hannover, Appelstraße 9a, 30167 Hannover 2 Institute for Risk and Reliability, Leibniz Universität Hannover, Callinstraße 34, 30167 Hannover 3 School of Engineering & Institute for Risk and Uncertainty, University of Liverpool, United Kingdom 4 Department of Structural Engineering & Shanghai Institute for Disaster Prevention and Relief & International Joint Research Center for Engineering Reliability and Stochastic Mechanics, Tongji University, China In order to regard mixed aleatory and epistemically uncertain random fields within stochastic finite element method, a proba- bility box approach using stochastic collocation method is introduced. The influence of an interval-valued correlation length on the output is investigated. c  2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction In engineering application, uncertainties are inevitable and therefore quite significant within a structural analysis. It is distin- guished into two kinds of uncertainties, the aleatory (stochastic) and the epistemic (lack of knowledge) ones [1]. In reality, there usually exists a mixture of both uncertainties, leading to a need for advanced approaches such as probability box (p-box) approach. Before introducing this approach implemented with a nested collocation algorithm in section 3, a possible occur- rence of this mixed uncertainty within random fields is discussed in the following section. The impact of a mixed uncertain random field on the result of a stochastic finite element simulation is illustrated by an academic example in section 4. 2 Mixed uncertainty within random fields A Gaussian random field X(x, ω), describing a parameter X depending on space x and chance ω, can be discretised by Karhunen-Loève expansion (KLE) [2]. Therefore, it is expanded by a series truncated at order T : X(x, ω) ≃ ˆ X(x, ω) = E(X)+ T  i=1  λ i φ i (x)ξ i (ω), (1) where E(X) is the parameter’s expected value and ξ i are standard normal distributed random variables. Further, λ i and φ i are the eigenvalues and eigenfunctions, respectively, given by the spectral decomposition of the covariance Cov(x 1 ,x 2 ) of the random values at two arbitrary points x 1 and x 2 within the random field. In this work, a Gaussian covariance kernel is assumed: Cov(x 1 ,x 2 ) = Std(X) 2 exp  −  |x 1 − x 2 | l c  2  , (2) where Std(X) is the parameter’s standard deviation and l c is the correlation length. While the aleatory parameters E(X) and Std(X) can be determined by experiments, the correlation length is difficult to identify exactly. It is therefore assumed to be epistemically uncertain and modelled by an interval-valued parameter. 3 Probability-box algorithm One approach capturing mixed uncertain parameters is given by a probability box (p-box) [3]. Instead of stating a unique cumulative distribution F X (x), a p-box [ F X ,F X ] is defined by a left and right bound, F X (x) and F X (x), respectively. This way, the epistemically uncertain part is added to the probabilistic model of the aleatory uncertainty. The algorithm works as follows [4]: In the first step, the epistemic uncertainty, given here as interval-valued correlation length, is discretised in the outer loop. For each correlation length, a KLE of the resulting random field is performed and the occurring problem can be solved by stochastic finite element method within the inner loop. Here, a stochastic collocation method using sparse Gauss-Hermite grids within the Smolyak algorithm [5] is used. The number of collocation points within the grid is influenced by T and the so-called Smolyak level k. ∗ Corresponding author: e-mail mona.dannert@ibnm.uni-hannover.de, phone +49 511 762 3707, fax +49 511 762 19053 PAMM · Proc. Appl. Math. Mech. 2018;18:e201800114. www.gamm-proceedings.com c  2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 of 2 https://doi.org/10.1002/pamm.201800114