Reliability Index Approach Research Articles

A simplified reliability-based design method for multiple design variables considering different limit states

A generalized reliability model comprising the objective, constraint, and judgment functions is established for the reliability index approach (RIA), taking parameters’ properties of engineering practice and negative reliability index into consideration. Based on this, the reliability-based design (RBD) problem with multiple design variables is translated into the solution to the nonlinear equations, and a simplified method consisting of a simple variant of the Newton iteration method and the finite difference method (FDM) is proposed. Numerical examples are presented to verify the efficiency of the proposed reliability approach and to determine the incremental step size for FDM. RBD of a simply supported beam is illustrated and the variabilities of design variables are investigated considering the uncertainties in the manufacturing process and practical operations. Results reveal that the variations of the design variables should not be ignored. Moreover, analysis results show that the design value might not intuitively increase with the increase of its coefficient of variation (CoV), and it might not increase with the increasing reliability requirement for problems involving multiple variables. The reasons for this phenomenon are very complicated, and it is a systematic problem. One should be aware of this phenomenon, and specific analysis is required for specific problems.

Damage-reliability approach for fatigue crack propagation in MDPE gas pipe

Fatigue crack propagation tests are carried out on arc-shaped specimens prepared from MDPE gas pipes. Damage zone characterization is achieved using the diamond wafer sectioning technique from partially propagated and prematurely arrested cracks. Damage ahead of the crack-tip is used to assess a damage parameter and reliability based on statistical laws and subsequent use the PHIMECA Software. It is shown that a length and a width associated with a corresponding change in thickness at the fracture surface satisfactorily describe the damage zone size. The 3-parameter Weibull model gives the best reliability behavior and a critical lifetime of 82%. When considering separately both the Dugdale model and experimental damage zone measurements, it is possible to establish the evolution of the reliability index as a function of crack length. It is concluded that the reliability index approach based on damage provides a more consistent representation compared to analytical models.

Supplement to the reliability index approach and its application to tunnel reliability problems

With the growing demands on limit state design in engineering practices, the reliability index approach (RIA) has been widely used because it is simple and convenient. However, the RIA using a norm function cannot distinguish the negative reliability index, though it is based on the same concept with the HLRF method. Focusing on this issue, a finding from the HLRF concerning negative reliability indices is presented, and accordingly the sign of RIA is resolved. And then a supplement to the RIA is formed with a suggested formula of sensitivity factors for both positive and negative cases. Two nonlinear examples illustrating negative reliability problems are presented, which validates that the proposed supplement can ascertain negative cases whereas the conventional RIA could not. Additionally, results of sensitivity factors agree well with the proposed formula. Furthermore, the proposed supplement is applied to address tunnel reliability problems including assessment and design. Results reveal that the conventional RIA cannot get the real sign of reliability indices and could mislead search directions of the RBD, which would lead to dangerous decisions. Fortunately, the proposed supplement can help to make correct decisions and thus it can facilitate the application of RIA in the limit state design.

Improving the evaluation efficiency of failure probability for large and complex systems using an innovative active set strategy

Reliability-based design optimization (RBDO) is a well-known design procedure for solving uncertainties and seeking better design, but evaluation of its failure probabilistic constraints involves an unbearable computational burden. In this study, the innovative active set strategy (ASS) for the reliability index approach is proposed to improve the efficiency and robustness of evaluating probabilistic constraints for RBDO by establishing an ASS and defining an update factor. The innovative ASS is defined using inequality to effectively perform reliability analysis and obtain the feasible most probable failure point in the inner loop. Meanwhile, a defined update factor is applied to eliminate the unnecessary probability constraints and renew the active constraints in the outer loop. The inner loop and outer loop are synchronized to enhance computational efficiency during the whole optimization process. Five commonly used examples are presented, demonstrating that the ASS is a superior efficient and robust method for a large-scale system.

A novel reliability index approach and applied it to cushioning packaging design

This paper is aimed at proposing a new approach to Reliability-based Design Optimization (RBDO) and applying it to cushioning packaging design with a highly nonlinear system. The problem is formulated as an RBDO problem, which is included a minimizing cost function and probabilistic constraints. Here, the thickness of the cushion material is dealt with uncertainty and uncontrolled parameters. The traditional reliability index approach (RIA) has evolved as a powerful tool to solve the RBDO problem; however, due to its convergence problem, the modified reliability index approach (MRIA) is proposed. Although the MRIA method solves the problems of the traditional RIA, it inherits the low efficiency of searching for the most probability point (MPP). Thus, we developed a novel RIA based on MRIA to improve the efficiency and robustness during the RBDO process. The innovation active set strategy is developed in reliability assessment, which is a strict inequality to determine whether the current constraint is active or inactive. An application example is presented, and the results are compared with MRIA to assess cost-effectiveness and efficiency. Results indicate the proposed method is feasible to solve the uncertainty problem of packaging materials in the processing process and is also an efficient RBDO method.

Enhanced modified reliability index approach for efficient and robust reliability‐based design optimization

AbstractThe reliability index approach (RIA) is one of the effective tools for solving the reliability‐based design optimization (RBDO) probabilistic model, which models the uncertainties with probability constraints. However, its wide application in engineering is limited due to low efficiency and convergence problems. The RIA‐based modified reliability index approach (MRIA) appears to be very robust and accurate than RIA but yields inefficient for the most probable point (MPP) search with highly nonlinear probabilistic constraints. In this study, an enhanced modified reliability index approach (EMRIA) is developed to improve the efficiency and robustness of searching for MPP and is utilized for RBDO. In the EMRIA, an innovative active set using rigorous inequality is applied to construct the region of exploring for MPP, where the unnecessary probabilistic constraint could be eliminated adaptively during the iterative process. Moreover, the double loop strategy (DLS) is integrated into the EMRIA to strengthen the efficiency and robustness of large‐scale RBDO problems. Two numerical examples demonstrated that the EMRIA is an efficient and robust method for MPP search in comparison with current first‐order reliability methods. Six RBDO problems quoted also indicate that DLS‐based EMRIA has good performance to solve complex RBDO problems.

Active set strategy-based sequential approximate programming for reliability-based design optimization

To improve the evaluation efficiency of failure probability in RBDO models with uncertainty, many RIA-based, PMA-based methods have evolved as a powerful procedure, including the modified reliability index approach (MRIA), PMA two-level, PMA with sequential approximate programming (SAP). However, MRIA may encounter inefficiency and instability when applied to complex concave performance functions, and so does PMA two-level, not for PMA with SAP. The active set strategy-based SAP (ASS-based SAP) for PMA is proposed to accelerate computational efficiency through establishing an active set strategy and a deciding factor. The active set strategy defined by using inequality is to identify the feasible most probable target point (MPTP) in the inner loop. The decision factor integrates the reliability index and the active set strategy to quickly renew the active constraints in the outer loop. The reliability assessment and outer optimization are driven simultaneously, thereby the computational efficiency is strengthened. Numerical examples are compared with other reliability methods to demonstrate the excellent performance of the proposed method in efficiency and robustness. Results also show that the proposed method has the ability to solve complex RBDO problems.

RELIABILITY BASED DESIGN OF GEOTECHNICAL SYSTEMS BASED ON A DECOUPLED RELIABILITY ANALYSIS AND OPTIMIZATION APPROACH

Reliability-based design intends to design an engineering system subject to probabilistic design constraints. This paper presents a reliability design optimization approach of geotechnical engineering problems based on a decoupled reliability analysis and optimization concept. Geotechnical and foundation systems involve considerable uncertainties in soil properties, loading conditions, and inadequate site characterization. A traditional reliability design optimization method, such as the reliability index approach, requires a double-loop optimization structure, since reliability analysis is implemented for constraint function evaluation using a numerical optimization algorithm. The decoupled approach separates the reliability analyses of the constraint functions from the design optimization loop. An adaptive metamodeling method is proposed to create approximate functions of the reliability indices, so that the reliability design optimization is performed using the approximate functions of the reliability constraints. Radial basis functions are implemented to build the metamodels. To progressively improve the accuracy of predicting the reliability indices, an adaptive sampling technique is employed, so that new samples are added using the optimal points found in previous design iterations. Numerical examples are solved and presented, and the proposed design approach works well. It is shown to be a useful alternative for solving reliability-based design of geotechnical and foundation systems.

RELIABILITY BASED DESIGN OF GEOTECHNICAL SYSTEMS BASED ON A DECOUPLED RELIABILITY ANALYSIS AND OPTIMIZATION APPROACH

Reliability-based design intends to design an engineering system subject to probabilistic design constraints. This paper presents a reliability design optimization approach of geotechnical engineering problems based on a decoupled reliability analysis and optimization concept. Geotechnical and foundation systems involve considerable uncertainties in soil properties, loading conditions, and inadequate site characterization. A traditional reliability design optimization method, such as the reliability index approach, requires a double-loop optimization structure, since reliability analysis is implemented for constraint function evaluation using a numerical optimization algorithm. The decoupled approach separates the reliability analyses of the constraint functions from the design optimization loop. An adaptive metamodeling method is proposed to create approximate functions of the reliability indices, so that the reliability design optimization is performed using the approximate functions of the reliability constraints. Radial basis functions are implemented to build the metamodels. To progressively improve the accuracy of predicting the reliability indices, an adaptive sampling technique is employed, so that new samples are added using the optimal points found in previous design iterations. Numerical examples are solved and presented, and the proposed design approach works well. It is shown to be a useful alternative for solving reliability-based design of geotechnical and foundation systems.

Reliability-based topology optimization of piezoelectric smart structures with voltage uncertainty

Currently, most of the piezoelectric structures are designed under deterministic conditions, where the influence of uncertain factors on the output motion accuracy is ignored. In this work, a probabilistic reliability-based topology optimization method for piezoelectric structure is proposed to deal with the working voltage uncertainty. A nested double-loop optimization algorithm of minimizing the total volume while satisfying the reliability requirement of the displacement performance is established, where the PEMAP-P (piezoelectric material with penalization and polarization) model is used for parameterization of stiffness matrix, piezoelectric coupling matrix, and polarization direction. This strategy consists of an inner loop for reliability analysis and an outer loop for topology optimization. The reliability index approach based on most probable point (MPP) is used for realizing the evaluation of reliability constraint in reliability analysis. The sensitivities of reliability constraint with respect to the random variables and design variables are detailed using the adjoint variable method. Typical examples are performed to illustrate the effectiveness of the proposed RBTO method. A comparison of the optimization results for different reliability indexes, standard deviations of the voltage, spring stiffnesses, and displacement limits are conducted, as well as the deterministic topology optimization results.

An innovation active set strategy reliability optimization method for cushioning design based on dynamic stress–strain curve

The uncertainty of cushioning volume in the process of manufacturing is the main cause of products damaged during transport. Many methods have been presented in protecting the products from damage. However, the uncertainty factors are still difficult to overcome in practical engineering. In order to handle those issues, an innovation active set strategy (IASS) reliability method with probabilistic constraints is proposed. This proposed method is tested through a packaging uncertainty problem, and the results obtained are compared with the modified reliability index approach (MRIA) by evaluating the probability constraints to demonstrate the cost‐effectiveness and efficiency. The results obtained have been indicated that the IASS not only solves the uncertainty problems in packaging engineering but also an efficient and faster reliability analysis method in the reliability‐based design optimization (RBDO) process without sacrificing numerical accuracy and stability. Furthermore, the proposed IASS provides a new design method for cushioning packaging design based on the dynamic stress–strain curve of cushioning material with minimum materials cost. In addition, the application of the RBDO methods to handle the uncertainty problems in packaging engineering has not been studied so far.

Uncertainty quantification of quadratically nonlinear local damage size quantification using nonlinear ultrasonics

Backscattered and forward scattered harmonic waves from local damage modeled as quadratically nonlinear material in a 1D bar are used to solve a deterministic inverse problem to find the local damage size by Ghodake (2020). Due to highly nonlinear (|sin(x)/x|) nature of the proposed forward problem, the constrained nonlinear deterministic optimization problem was solved using non-gradient-based algorithms. The forward model is a function of input frequencies, linear wave velocity, and local damage size. In practice, slight uncertainty of these parameters gives a completely different solution. To address this problem, uncertainty quantification of the modified forward problem is carried out along with solving the inverse problem using reliability-based design optimization (RBDO) and Bayesian Inverse (BI) approaches. In RBDO studies, Quantile Monte Carlo (QMC) approach gives more reliable and accurate results in comparison with approaches such as reliability index approach (RIA), performance measure approach (PMA), sequential optimization and reliability assessment (SORA), and inverse first-order reliability method (FORM) integrated with optimization methods such as interior-point (IP) and sequential programming (SQP). Multiple models are solved using the BI approach. Support vector machines regression (SVR), polynomial chaos expansions (PCE), and Kriging surrogate models are used along with RBDO and BI to reduce computational time.

Resampling method for reliability-based design optimization based on thermodynamic integration and parallel tempering

In this paper, a fully decoupled simulation method is proposed for reliability-based design optimization (RBDO) based on thermodynamic integration and parallel tempering (TIPT). We show that the failure probability function and its gradient can be obtained simultaneously with once generalized reliability analysis, and thus the RBDO problem is converted to the traditional optimization problem efficiently. Firstly, the design parameters are deemed as uniformly distributed random variables, and an auxiliary probability density function (PDF) of random design variables is constructed to cover its whole parameter space. Then, based on thermodynamic integration, the estimation of failure probability is converted to a series of simple integration problems with smooth integrand, and they are estimated by running multiple Markov chains using the so-called parallel tempering method. Finally, importance sampling (IS) is used to estimate the failure probability function and its gradient, and the IS samples are obtained by resampling from the existing Markov chains without extra computation. The proposed method is tested with severa benchmarks, and the results show that it provides robust solution for problems with various nonlinear constraints compared to other popular methods, include double-loop Monte Carlo simulation (MCS), Quantile MCS, sequential optimization and reliability assessment, performance measure approach and reliability index approach.

Efficient optimum safety factor approach for system reliability-based design optimization with application to composite yarns

Introduction. The integration of reliability and optimization concepts seeks to design structures that should be both economic and reliable. This model is called Reliability-Based Design Optimization (RBDO). In fact, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational cost and weak convergence stability. Materials andMethods. Several methods have been developed to overcome these difficulties. The methods called Reliability Index Approach (RIA) and Performance Measure Approach (PMA) are two alternative methods. RIA describes the probabilistic constraint as a reliability index while PMA was proposed by converting the probability measure to a performance measure. An Optimum Safety Factor (OSF) method is proposed to compute safety factors satisfying a required reliability level without demanding additional computing cost for the reliability evaluation. The OSF equations are formulated considering RIA and PMA and extended to multiple failure case.Research Results. Several linear and nonlinear distribution laws are applied to composite yarns studies and then extended to multiple failure modes. It has been shown that the idea of the OSF method is to avoid the reliability constraint evaluation with a particular optimization process.Discussion and Conclusions. The simplified implementation framework of the OSF strategy consists of decoupling the optimization and the reliability analyses. It provides designers with efficient solutions that should be economic satisfying a required reliability level. It is demonstrated that the RBDO compared to OSF has several advantages: small number of optimization variables, good convergence stability, small computing time, satisfaction of the required reliability levels.

Global optimal reliability index of implicit composite laminate structures by evolutionary algorithms

With uncertainty, reliability assessment is fundamental in structural optimization, because optimization itself is often against safety. To avoid Monte Carlo methods, the Reliability Index Approach (RIA) approximates the structural failure probability and is formulated as a minimization problem, usually solved with fast gradient-methods, at the expense of local convergence, or even divergence, particularly for highly dimensional problems and implicit physical models. In this paper, a new procedure for global convergence of the RIA, with practical efficiency, is presented. Two novel evolutionary operators and a mixed real-binary genotype, suitable to hybridize any Evolutionary Algorithm with elitist strategy, are developed. As an example, a shell laminate structure is presented and the results validated, showing good convergence and efficiency. The proposed method is expected to set the basis for further developments on the design optimization of more complex structures with multiple failure criteria.

Reliability-based Robust Design Optimization with the Reliability Index Approach applied to composite laminate structures

In this paper a new approach to the Reliability-based Robust Design Optimization (RBRDO) of angle-ply composite laminate structures is presented. The problem is defined as a bi-objective Robust Design Optimization (RDO) problem, with deterministic and probabilistic constraints. Design optimization is considered as the minimization problem of the weight (optimality) and the determinant of the variance-covariance matrix of the response functionals of the system (robustness). Reliability assessment is executed by the Reliability Index Approach (RIA), as an inner cycle. The key concept of this methodology is the exclusive use of evolutionary algorithms with elitist strategy, allowing for global convergence in the two optimization cycles. The RIA problem was reformulated and a set of new evolutionary operators were developed. A numerical example is presented and the results compared with other RBRDO and RDO formulations. Results show that deterministic and probabilistic structural integrity measures are quite sensitive to uncertainty, for multi-component composite structures.

Adaptive conjugate single-loop method for efficient reliability-based design and topology optimization

The single-loop approach (SLA) for reliability-based design optimization (RBDO) is one of the most efficient schemes for optimization problems with linear and weak nonlinear probabilistic constraints. However, it may produce unstable results or increase computational efforts when using the ordinary search direction to determine the optimum design in RBDO problems with highly nonlinear probabilistic constraints. The conjugate gradient (CG) is a promising sensitivity vector for locating the most probable point (MPP) of highly nonlinear concave performance functions. However, the MPP computation using the CG may require a high computational burden for convex constraints To overcome the drawbacks of the SLA the adaptive conjugate single-loop approach (AC-SLA) is proposed for RBDO problems with a large variety of nonlinear constraints. The sensitivity vector of the probabilistic constraints is adaptively computed using the CG vector with a dynamical conjugate scalar factor (DCF). The DCF is adjusted within the range from 0 to 2 using two adaptive coefficients, which are adapted based on the new and previous points. Moreover, the Lyapunov exponents are developed as a general tool for detecting the robustness of different MPP approximation algorithms. The method is also applied to solve a reliability-based topology optimization (RBTO) problem. The ability of the AC-SLA in six RBDO benchmark problems, one applicable to an RBDO aircraft engineering problem and one for RBTO problem, is compared in terms of both robustness and efficiency using the SLA, performance measure approach (PMA), reliability index approach (RIA), and sequential optimization and reliability assessment (SORA). The numerical experiments demonstrate that the convergence capability of the proposed AC-SLA is significantly superior to the PMA, RIA, and SORA. The computational effort of the AC-SLA is significantly reduced, with stable results.

The importance of correlation among flutter derivatives for the reliability based optimum design of suspension bridges

The design of long-span bridges is constrained by the uncertainty in the evaluation of flutter velocity. Among all the elements that take part in the flutter assessment, the uncertainty level in experimentally obtained flutter derivatives has the most impact. It is therefore important in the evaluation of flutter velocity to assess the uncertainty which is associated with the adopted experimental method for flutter derivatives. By using a method of coupled motion only to identify eight flutter derivations simultaneously, it is also essential to consider correlations among the points that define the full set of flutter derivatives since they are not independent from one another. In this research, an experimental campaign was carried out to obtain the statistical information of flutter derivatives and to assemble the correlation matrix. Several cases of reliability analyses were performed to illustrate the importance of considering correlation among random variables as well as the significance of uncertainty level in flutter derivatives on bridge flutter failure. Moreover, a study of Reliability Based Design Optimization (RBDO) was carried out to see the influence of correlations among flutter derivatives on the optimum designs. The RBDO of a suspension bridge was performed under probabilistic flutter constraint using Reliability Index Approach (RIA) method, and this methodology was applied to the Great Belt East Bridge.

Enhanced sequential approximate programming using second order reliability method for accurate and efficient structural reliability-based design optimization

• The sequential approximate programming (SAP) is enhanced using SORM for RBDO. • The proposed enhanced SAP-based SORM has the same efficiency as well as the FORM-based RBDO. • ESAP is applied several nonlinear, applicable RBDO problems and reliability-based topology optimization. • Results demonstrated that the ESAP shows the superiority of performance compared to RIA, PMA, SAP, and SORA. Second-order reliability method (SORM) can provide sufficient accuracy for evaluating the probabilistic constraints in reliability-based design optimization (RBDO). However, the application of SORM in RBDO significantly increases the computational burden, as it is necessary to calculate the second-order sensitivities of the performance function. In order to achieve equal efficiency to that of the first-order reliability method-based RBDO approach, enhanced sequential approximate programming (ESAP) is proposed by implementing the SORM-based RBDO method. Based on the diagonal quadratic approximation method, the Hessian matrix is calculated without generating additional computational costs for providing the design sensitivity analysis of probabilistic constraints within the same iterations. Furthermore, ESAP is applied to the reliability-based topology optimization domain, and five numerical benchmark RBDO problems with two complex engineering examples are studied. The proposed ESAP is compared with other RBDO methods, including the reliability index approach, performance measure approach, sequential optimization and reliability assessment method, and SAP, and the results demonstrate the superiority of the proposed ESAP.

A New Reliability Analysis Method Combining Adaptive Kriging With Weight Index Monte Carlo Simulation

In order to extend the application of reliability-based design optimization to engineering electromagnetic problems, the main contribution of this paper is to propose an efficient reliability analysis method: adaptive Kriging-assisted weight index Monte Carlo Simulation (MCS). The Kriging model is constructed to substitute for expensive finite-element analysis of engineering problem during optimization and reliability analysis. To improve the interpolation accuracy of Kriging, a learning function is adopted in adaptive sampling process. A weight index form of MCS method is proposed to enhance the efficiency of conventional MCS method. Finally, taking the result of conventional MCS as a reference, proposed reliability analysis method is compared with reliability index approach and sensitivity-assisted MCS methods.