Research paper
Reliability-based design: Artificial neural networks and double-loop reliability-based optimization approaches

https://doi.org/10.1016/j.advengsoft.2017.06.013Get rights and content

Highlights

  • Two approaches for solving a reliability-based design problem are proposed.

  • Artificial neural network based inverse reliability method has been implemented.

  • Double-loop reliability-based optimization method has been implemented.

  • Verification using examples of structural design was carried out.

Abstract

Two advanced optimization approaches to solving a reliability-based design problem are presented. The first approach is based on the utilization of an artificial neural network and a small-sample simulation technique. The second approach considers an inverse reliability task as a reliability-based optimization task using a double-loop optimization method based on small-sample simulation. Both techniques utilize Latin hypercube sampling with correlation control. The efficiency of both approaches is tested using three numerical examples of structural design – a cantilever beam, a reinforced concrete slab and a post-tensioned composite bridge. The advantages and disadvantages of the approaches are discussed.

Introduction

Tremendous progress has been made in the areas of both reliability and optimization during the last two decades. Reliability-based optimization (RBO), reliability-based design (RBD) and reliability-based design optimization (RBDO) – these terms appear in the literature and represent a combined strategy where we have to deal with the repeated evaluation of an objective function (optimization) and repetitive evaluations of a limit state function (reliability). The concept itself appeared quite early in reliability engineering; see e.g. [1], [2], [3], [4], [5]. From those first pioneering works, the concept progressed from reliability-based to risk-based optimization approaches, e.g. [6], emphasizing robustness in structural optimization, e.g. [7], [8]. Despite these achievements in the fields of both optimization and reliability, the computational effort required is still enormous for practical problems and we need efficient methods that are easy to apply.

When performing either reliability assessment or engineering design, it is certainly essential to take uncertainties into account using advanced fully probabilistic analysis. Reliability assessment requires forward reliability methods for reliability estimation. On the other hand, engineering design requires an inverse reliability approach in order to determine the design parameters needed to achieve the desired target reliabilities that represent the desired level of reliability in the limit state design of structures. The achievement of such reliabilities is generally not an easy or straightforward task.

Some sophisticated approaches to the determination of design parameters (material properties, geometry, etc.) related to particular limit states have been proposed under the name “inverse reliability methods”, e.g. a reliability contour method [9], [10], an iterative algorithm based on the modified Hasofer-Lind-Rackwitz-Fiessler scheme used in reliability analysis [11], the use of a Newton–Raphson iterative algorithm to find multiple design parameters [12], [13], the decomposition technique [14] and various implementations of artificial neural networks (ANNs) with other soft-computing techniques [15], [16], [17]. The use of ANNs in [1], [18] was motivated by the approximate concepts inherent in reliability analysis and the time-consuming repeated analyses required by Monte Carlo type simulation for large-scale structural systems.

The two advanced methods proposed in this paper attempt to overcome the shortcomings of existing inverse reliability methods and are both transparent and relatively easy to apply. Existing inverse reliability methods are generally limited to simple problems and cannot be applied to computationally time consuming problems (such as large finite element computational models). This was the main motivation for the development and software implementation of the techniques presented in this paper. The first method utilizes an ANN too, but in a different way: computational time is reduced by using a small-sample simulation technique called Latin hypercube sampling in an ANN-based inverse problem previously proposed by Novák and Lehký in [19], [20].

The second method is the double-loop RBO approach. Classical deterministic optimization usually leads to solutions that lie at the boundary of the admissible domain, and that are consequently rather sensitive to uncertainty in the design parameters. In contrast, RBO aims at designing the system in a robust way by minimizing an objective function under reliability constraints. It provides the means for determining the optimal solution for a certain objective function, while ensuring that there is only a predefined small probability that a structure fails. RBO methods thus have to mix optimization algorithms together with reliability calculations. The approach known as “double-loop” consists in nesting the computation of the failure probability with respect to the current design within the optimization loop. A FORM-based double-loop approach has been proposed by Dubourg in [21], [22]. The authors of the present paper have developed a double-loop reliability-based optimization approach based on small-sample simulation and the first order reliability method (FORM) [23].

The efficiency of both approaches is tested using three numerical examples of structural design – a cantilever beam, a reinforced concrete slab and a post-tensioned composite bridge. This current paper is based upon Lehký et al. [24], but includes a more detailed theoretical explanation of the small-sample double-loop reliability-based optimization method, including an Aimed Multilevel Sampling strategy for the reduction of sampling space. In addition, an application to a real bridge structure is also included for demonstration purposes and there is a discussion of the practical usability of both approaches.

Section snippets

Reliability problem formulation

The aim of classical (forward) reliability analysis is the estimation of unreliability using a reliability indicator called the theoretical failure probability, defined as:pf=P(Z0),where Z = g(X) is a variable called safety margin, which is a function of a random vector, X = {X1, X2, …, XNvar}T, where Nvar is the number of random variables. Random vector X follows a joint probability distribution function (PDF) fX(x); in general, its marginal variables can be statistically correlated. The

Cantilever beam

The first example demonstrates an application of the method in civil engineering structural design. A cantilever beam with a rectangular cross-section of width b and depth h is subjected to two point loads: one at the free end of the beam, and another at a distance L1 from the fixed support, as shown in Fig. 7. Failure is considered to occur when the deflection at the free end of the beam exceeds a maximum wmax = L/100, where L is the span of the beam. The deflection at the free end due to the

Conclusions

Finding a more realistic optimum structural design in the presence of uncertainties is not an easy or straightforward task. The main reason for this is the increasing computational effort required when dealing with optimization and reliability concepts. The paper presents two alternative approaches to solving reliability-based design problems. Both approaches utilize small-sample simulation LHS to reduce computational effort. They provide very good results, as is indicated in the numerical

Acknowledgements

This work has been supported by project FIRBO No. 15-07730S, awarded by the Czech Science Foundation (GACR), and project No. LO1408 ‘‘AdMaS UP – Advanced Materials, Structures and Technologies”, supported by the Ministry of Education, Youth and Sports of the Czech Republic under ‘‘National Sustainability Programme I”.

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