First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems by a fractional moments-based mixture distribution approach

C. Ding1, C. Dang1, M. Valdebenito2, M. Faes3, M. Broggi1, M. Beer1

1 Institute for Risk and Reliability, Leibniz University Hannover, Callinstr. 34, Hannover 30167, Germany

2  Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, 2562340 Viña del Mar, Chile

Chair for Reliability Engineering, TU Dortmund University, Leonhard-Euler-Str. 5, Dortmund 44227, Germany

Journal: Mechanical Systems and Signal Processing, Volume 185, 15 February 2023, 109775


First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems is a significant task to be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a novel approach, termed ‘fractional moments-based mixture distribution’, to address such challenge. This approach is implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional moment and mixture distribution. In our context, the fractional moment itself is by definition a high-dimensional integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS). In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments. From the knowledge of low-order fractional moments, the EVD of interest is then expected to be reconstructed. Based on introducing an extended inverse Gaussian distribution and a log extended skew-normal distribution, one flexible mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the first-passage probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance of the proposed method is verified by three examples consisting of two test examples and one engineering problem.


The effect of random field parameter uncertainty on the response variability of composite structures

George Stefanoua,∗, Dimitrios Savvasa, Panagiotis Gavallasa, Iason Papaioannoub

a Institute of Structural Analysis & Dynamics of Structures, Department of Civil Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

b Engineering Risk Analysis Group, Technische Univ. München, 80290 München, Germany

Journal: Composites Part C: Open Access, Volume 9,  October 2022, 100324, 109775


The accurate quantification of the random spatial variation of material properties at different scales is crucial
for the systematic propagation of uncertainties through engineering models. In a previous work, the spatial
variability of the apparent material properties of two-phase composites has been quantified in a Bayesian
framework. This framework enables a consistent modeling of the statistical uncertainty in the parameters of
the respective mesoscale random fields and also allows selecting the most plausible correlation model among
different models belonging to the Matérn family. In this work, the most plausible random field model is
employed in the context of uncertainty propagation of composite structures. Sample functions of the mesoscale
random fields are generated using a covariance decomposition approach and the response variability of various
composite structures is computed through Monte Carlo simulation. Parametric investigations are conducted to
highlight the effect of the identified parameter uncertainty on structural response variability.



Increasing design flexibility by manually adapting the solution space for crashworthiness

P. Ascia1, V. A. Lange2, F. Duddeck1

1 TUM School of Engineering and Design, Technical University of Munich, Arcisstr. 21, 80333, Munich, Germany

2 Research and Innovation Center, BMW Group, Knorrstr. 147, 80788, Munich, Germany

Journal: Journal of Mathematics in IndustryVolume 13, article number 9, (2023)


The solution space methodology, as presented in 2013, was meant to guide developers at the very beginning of the development process of a new mechanically crashworthy car. Several attempts were already made to use this methodology at later development stages. However, they all encountered problems related to its very strict and demanding corridors, thus constricting the design parameters. To allow more flexibility, two different approaches were proposed to relax the initial strict conditions. The first introduced temporal dependencies to widen the corridors. The second locally changed the corridors to adapt to the needs of the development, introducing dependencies between components. We, on the contrary, propose a new method to increase flexibility without introducing any kind of dependencies. We manage this by computing the intervals of solution space under user-defined conditions, hence selecting a custom set of independent corridors that fits the data gathered during development; i.e.: force-deformation curves that can be measured during a drop-tower test simulation. This new methodology of the adaptive solution space allows designers to edit the corridors, in order to have more flexibility for fulfilling high-level requirements when independently designing new components.


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