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Sobol’ indices as dimension reduction technique in evolutionary-based reliability assessment

Gonçalo das Neves Carneiro (Faculty of Engineering, University of Porto, Porto, Portugal)
Carlos Conceição António (Faculty of Engineering, University of Porto, Porto, Portugal)

Engineering Computations

ISSN: 0264-4401

Article publication date: 30 August 2019

Issue publication date: 16 January 2020



In the reliability assessment of composite laminate structures with multiple components, the uncertainty space defined around design solutions easily becomes over-dimensioned, and not all of the random variables are relevant. The purpose of this study is to implement the importance analysis theory of Sobol’ to reduce the dimension of the uncertainty space, improving the efficiency toward global convergence of evolutionary-based reliability assessment.


Sobol’ indices are formulated analytically for implicit structural response functions, following the theory of propagation of moments and without violating the fundamental principles presented by Sobol’. An evolutionary algorithm capable of global convergence in reliability assessment is instrumented with the Sobol’ indices. A threshold parameter is introduced to identify the important variables. A set of optimal designs of a multi-laminate composite structure is evaluated.


Importance analysis shows that uncertainty is concentrated in the laminate where the critical stress state is found. Still, it may also be reasonable in other points of the structure. An accurate and controlled reduction of the uncertainty space significantly improves the convergence rate, while maintaining the quality of the reliability assessment.

Practical implications

The theoretical developments assume independent random variables.


Applying Sobol’ indices as an analytical dimension reduction technique is a novelty. The proposed formulation only requires one adjoint system of equilibrium equations to be solved once. Although a local estimate of a global measure, this analytical formulation still holds because, in structural design, uncertainty is concentrated around the mean-values.



The authors gratefully acknowledge the funding by Fundação para a Ciência e Tecnologia (FCT), Portugal, through the funding of the “Associated Laboratory of Energy, Transports and Aeronautics (LAETA)”, UID/EMS/50022/2019.


das Neves Carneiro, G. and António, C.C. (2020), "Sobol’ indices as dimension reduction technique in evolutionary-based reliability assessment", Engineering Computations, Vol. 37 No. 1, pp. 368-398.



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