Reliability Index Approach Research Articles

MPP-Based Dimension Reduction Method for Accurate Prediction of the Probability of Failure of a Performance Function

A hybrid reliability analysis method is proposed to yield very accurate failure rate calculation of a performance function when dealing with highly nonlinear electromagnetic systems in the presence of uncertainties. To achieve this goal, the first-order reliability method called reliability index approach is first conducted for searching a most probable failure point (MPP) at a given design. However, its result may have significant errors especially for nonlinear or multi-dimensional performance functions. To overcome the drawback, the univariate dimension reduction method is additionally executed at the obtained MPP, and then the probability of failure of a performance function is recalculated through additively decomposing an n-dimensional function into n 1-D functions. A mathematical example and TEAM workshop problem 22 are provided to demonstrate numerical efficiency and accuracy of the proposed method by comparison with the existing reliability methods.

Enhanced single-loop method for efficient reliability-based design optimization with complex constraints

Reliability-based design optimization (RBDO) has been widely implemented for engineering design optimization when considering the uncertainty. The single loop approaches (SLA) are highly efficient but is prone to converge with inappropriate results for highly nonlinear probabilistic constraints. In this paper, a novel RBDO algorithm is proposed based on single loop approach and the enhanced chaos control method, named as enhanced single-loop method (ESM). The performance of SLA is enhanced using an adaptive inverse reliability method with limited number of iterations. The adaptive step size is computed based on a merit function which is computed using the results of the new and previous iterations. The iterations of the probabilistic constraints of RBDO models are manually controlled in the range from 1 to 10 in ESM. The efficiency and accuracy of the ESM are compared through four nonlinear RBDO problems with complex constraints, including a nonlinear mathematical problem, two engineering problems and a practical complex stiffened panel example with complex buckling constraint for aircraft design. Results illustrate that the proposed ESM is more efficient and robust than the performance measure approach and reliability index approach for RBDO problems.

Mixed uncertainty based analytical target cascading: an approach for hierarchical multilevel optimization under probabilistic and interval mixed uncertainties

Analytical target cascading (ATC) has been extended to the probabilistic formulation to deal with the hierarchical multilevel optimization problem under uncertainty. However, only aleatory uncertainties can be handled by the probabilistic theory. Considering both aleatory and epistemic uncertainties exist in engineering design, a mixed uncertainty based analytical target cascading (MUATC) approach is proposed in this paper. The probability distribution and interval are used to represent the aleatory uncertainties and epistemic uncertainties, respectively. To figure out the mixed uncertainty propagation(MUP) problem, the reliability index approach based MUP (RIA-MUP) and performance measurement approach based MUP (PMA-MUP) are proposed firstly. Based on PMA-MUP, the MUATC method is established. MUATC firstly decouples the mixed uncertainty all-in-one (MUAIO) problem into deterministic optimization problem and mixed uncertainty analysis, and then hierarchically decomposes them into subproblems. All-in-one (AIO) mixed uncertainty analysis and hierarchically mixed uncertainty analysis methods are established to calculate the characteristic of the interrelated responses and shared variables. The accuracy and effectiveness of MUATC are demonstrated with three examples.

Reliability based design optimization for a rock tunnel support system with multiple failure modes using response surface method

Reliability-based design optimization (RBDO) is applied to a rock tunnel support system, considering the ground-support interaction and the parametric uncertainties. Both reliability index approach (RIA) and performance measure approach (PMA) are employed as the RBDO algorithms. The first-order reliability method (FORM) and the response surface method (RSM) are used to perform the reliability analysis. The shotcrete thickness and its installation position are considered as two design variables to be optimized in an illustrative example of a circular rock tunnel. The influence of the two design variables on the reliability results is first investigated with respect to three failure modes, namely, the support capacity criterion, the tunnel convergence criterion and the rockbolt length criterion. Then, the proposed RBDO procedure is used to seek for the design variables of minimum shotcrete thickness and the optimal installation position for the rock tunnel example. The computational efficiency and stability between RIA and PMA are compared. The influences of the rock mass quality, the variation of the shotcrete thickness and the target reliability requirement on the optimized results are also investigated and discussed.

Structural Behavior of Tall Building Raft Foundations in Earthquake Zones

The paper studies the behavior of reinforced concrete raft foundations for multi-story buildings. It also develops a reliability assessment tool for multi- story building raft foundations subjected to earthquake loading. Several multi-story buildings with various configurations, heights, and soil profiles, were subjected to several ACI code combinations of gravity and earthquake loads from different seismic zones. The reliability of the raft foundations of these buildings was assessed using the reliability index approach based on their resistance to the applied loads. Also, the responses of the multi-story buildings under these loading combinations were studied and analyzed in order to draw recommendations and guidelines for the preliminary design of structurally efficient and reliable raft foundations in earthquake zones.

An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.

An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses

Reliability-based design optimization (RBDO) algorithms have been developed to solve design optimization problems with existence of uncertainties. Traditionally, the original random design space is transformed to the standard normal design space, where the reliability index can be measured in a standardized unit. In the standard normal design space, the modified reliability index approach (MRIA) measured the minimum distance from the design point to the failure region to represent the reliability index; on the other hand, the performance measure approach (PMA) performed inverse reliability analysis to evaluate the target function performance in a distance of reliability index away from the design point. MRIA was able to provide stable and accurate reliability analysis while PMA showed greater efficiency and was widely used in various engineering applications. However, the existing methods cannot properly perform reliability analysis in the standard normal design space if the transformation to the standard normal space does not exist or is difficult to determine. To this end, a new algorithm, ensemble of Gaussian reliability analyses (EoGRA), was developed to estimate the failure probability using Gaussian-based kernel density estimation (KDE) in the original design space. The probabilistic constraints were formulated based on each kernel reliability analysis for the optimization processes. This paper proposed an efficient way to estimate the constraint gradient and linearly approximate the probabilistic constraints with fewer function evaluations (FEs). Some numerical examples with various random distributions are studied to investigate the numerical performances of the proposed method. The results showed that EoGRA is capable of finding correct solutions in some problems that cannot be solved by traditional methods. Furthermore, experiments of image processing with arbitrarily distributed photo pixels are performed. The lighting of image pixels is maximized subject to the acceptable limit. Our implementation showed that the accuracy of the estimation of normal distribution is poor while the proposed method is capable of finding the optimal solution with acceptable accuracy.

Reliability-Based Design Optimization Combining the Reliability Index Approach and Performance Measure Approach

Reliability-Based Design Optimization Combining the Reliability Index Approach and Performance Measure Approach

Step length adjustment iterative algorithm for inverse reliability analysis

Performance Measure Approach (PMA) is an alternative way for evaluation of probabilistic constraints in reliability-based design optimization other than traditional Reliability Index Approach (RIA). In PMA, the probabilistic performance measure (PPM) is obtained through locating the minimum performance target point (MPTP) with the specified target reliability index in standard normal space, which is also called inverse reliability analysis. The advanced mean-value (AMV) method is well suitable for locating MPTP due to its simplicity and efficiency. However, AMV may have difficult to converge for highly nonlinear performance function. In this paper a step length adjustment (SLA) iterative algorithm, which introduces a "new" step length to control the convergence of the sequence, is proposed. This step length is new because the line search process for step length selection is not needed and it may be constant during the whole iteration process or decrease successively several times using a self-adjust strategy. It is proved that the AMV method is a special case of the SLA algorithm when the step length tends to infinity and the reason why AMV diverges is illustrated. SLA is as simple as AMV and does not need the prior knowledge of convexity or concavity of the performance function as other modified algorithms do. Numerical results of several highly nonlinear performance functions including an engineering application indicate that SLA is effective and robust.

Investigation of reliability analysis algorithms for effective reliability‐based optimal design of electromagnetic devices

To extend the reliability-based optimal design to engineering electromagnetic problems, the essential prerequisite is to develop a numerically efficient method of reliability calculation. From a practical point of view, this study presents a comparative study on different algorithms of reliability analysis. For reliability analysis of electromagnetic problems, the author proposed one sensitivity-assisted Monte Carlo simulation (SA-MCS) method in the previous research. To seek a higher accuracy of reliability analysis, in this study, the second-order derivatives are incorporated into the SA-MCS method. About calculation of second-order derivatives (also called sensitivity analysis in the finite element analysis), a hybrid direct differentiation–adjoint variable method is applied. Finally, analysis performances of the suggested SA-MCS and traditional methods such as the reliability index approach and the performance measure approach are investigated through applications to analytic examples and a superconducting magnetic energy storage system.

Reliability and flexural behavior of triangular and T-reinforced concrete beams

The paper studied the behavior of reinforced concrete triangular and T-beams. Three reinforced concrete beams were tested experimentally and analyzed analyti- cally using the finite element method. Their reliability was also assessed using the reliability index approach. The results showed that the finite element vertical displace- ments compared well with those obtained experimentally. They also showed that the vertical displacements obtained using the finite element method were larger than those obtained experimentally. This is a strong indication that the finite element results were conservative and reliable. The results showed that the triangular beams exhibited higher ductility at failure than did the T-beam. The plastic deformations at failure of the triangular beams were higher than that of the T-beam. This is a strong indication of the higher ductility of the triangular beams compared to the T-beam. Triangular beams exhibited smaller cracks than did T-beams for equal areas of steel and concrete. The design moment strengths Mc computed using the American Concrete Institute (ACI) design formulation were safe and close to those computed using experimental results. The experimental results validated the reliability analysis results, which stated that the triangular beams are more reliable than T-beams for equal areas of steel and concrete.

Probabilistic optimization of the main cable and bridge deck of long-span suspension bridges under flutter constraint

The reliability based design optimization of a long-span suspension bridge is performed considering probabilistic flutter constraint accounting for the stochastic nature of parameters such as the extreme wind velocity and experimentally obtained flutter derivatives. The objective of this study is to seek the minimum volumes of the main cables and bridge girder by varying the main cable area and girder plate thicknesses while satisfying the required safety level under flutter. Three methods of Reliability Index Approach, Performance Measure Approach and Sequential Optimization and Reliability Assessment are utilized to solve this problem. The main code to perform the entire RBDO process is written in Matlab®, which is linked to Abaqus finite element models to obtain structural responses and FLAS code developed by our research group to compute aeroelastic responses of the bridge. The three RBDO methods were applied to the Messina Bridge project to examine their numerical performance and computational efficiency.

Rainfall characteristics for anisotropic conductivity of unsaturated soil slopes

The purpose of this study was to investigate the effects of the anisotropic ratio on the stability of slopes using the reliability index approach. A numerical analysis of the relationship between the three rainfall patterns, advanced, normal and delayed, and the anisotropic ratios was designed. This study also considered three different soil properties (sand, silt, and clay) to simulate rain infiltration. In this study, probability analysis was used to evaluate the stability of unsaturated soil slopes. The finite element computer program Geo-Studio was used to simulate the process of rainwater infiltrating a slope. The pore-water pressure results evaluated from seepage analysis (SEEP/W) were imported into the slope stability program (SLOPE/W). Results for the anisotropic ratio of hydraulic conductivity indicate that when the anisotropic ratios become higher, the reduction in the reliability index is insignificant. In addition, the simulation results indicated that when saturated hydraulic conductivity (ks) was less than rainfall intensities (I), the percentage probability of the occurrence of a landslide was larger than when ks was greater than I. Finally, in the cases of anisotropic ks, stability of the high ratio soil slopes was not found to be sensitive to the reliability index variation during the simulation period. Moreover, when ks was greater than I, slope stability decreased earlier than was the case in the opposite situation.

A Multifaceted Approach for Safety Design and Probabilistic Optimization

Safety design and probabilistic optimization are fields that are widely subject to uncertainty, thus making traditional deterministic methods highly unreliable for these two fields. Popular design optimizations methods widely used for safety design and probabilistic optimization are the performance measure approach (PMA) and the performance measure approach (RIA). In a problem where the analysis is performed from an infeasible design space, a modified reliability index approach (MRIA) is employed to address some inefficiency of the traditional RIA to be able to find the optimal solutions. The PMA uses an inverse reliability analysis, which is more computationally efficient at finding the most probable design points but has been reported to have numerical instabilities on some cases. In this paper, three benchmark examples were thoroughly studied with various initial points to examine the stability and efficiency of MRIA and PMA. A hybrid reliability approach (HRA) was then presented after determining a selection factor from the optimum conditions. The proposed HRA aims to determine which of the two optimization methods would be more appropriate during the optimization processes.

Reliability based design optimization of long-span bridges considering flutter

Three reliability based design optimization methods, namely reliability index approach, performance measure approach and sequential optimization and reliability assessment are applied to a long-span suspension bridge under probabilistic flutter constraint. Uncertainties in extreme wind velocity as well as flutter derivatives obtained in wind tunnel are considered. The RBDO problem presented in this research seeks to minimize the bridge girder weight by varying the thicknesses of the girder plate while satisfying the structural reliability level under flutter. In order to solve this problem, the three RBDO methods mentioned above were programmed in Matlab code, which calls Abaqus finite element models to obtain structural responses and FLAS code, developed by our research group, to calculate flutter velocity. The proposed Messina Bridge was used as an application example of these methods. Prior to solve the RBDO problem, reliability analyses were performed in order to obtain the safety level of the original design. The results obtained by different RBDO methods are then compared for their accuracy and computational efficiency.

An efficient PMA-based reliability analysis technique using radial basis function

Purpose – The performance measure approach (PMA) is widely adopted for reliability analysis and reliability-based design optimization because of its robustness and efficiency compared to reliability index approach. However, it has been reported that PMA involves repeat evaluations of probabilistic constraints therefore it is prohibitively expensive for many large-scale applications. In order to overcome these disadvantages, the purpose of this paper is to propose an efficient PMA-based reliability analysis technique using radial basis function (RBF). Design/methodology/approach – The RBF is adopted to approximate the implicit limit state functions in combination with latin hypercube sampling (LHS) strategy. The advanced mean value method is applied to obtain the most probable point (MPP) with the prescribed target reliability and corresponding probabilistic performance measure to improve analysis accuracy. A sequential framework is proposed to relocate the sampling center to the obtained MPP and reconstru...

Composite First-Order Reliability Method for Efficient Reliability-Based Optimization of Electromagnetic Design Problems

This paper proposes a composite first-order reliability analysis method to effectively perform the reliability-based optimization of electromagnetic (EM) design problems. The proposed method utilizes both of two different ways, reliability index approach (RIA) and performance measure approach (PMA), for reliability analysis of probabilistic constraints. However, each performs a distinct function: the first merely checks the feasibility status of probabilistic constrains, and the second evaluates the failure probability of only active constraints selected from the feasibility identification. Such a unique scheme can substantially enhance computational efficiency during optimization process. The proposed method is applied to two EM design problems, and its numerical efficiency and accuracy are examined by comparison with existing methods.

Efficient strategies for reliability‐based design optimisation of a superconducting magnetic energy storage system‐based on reliability index approach

This study presents an enriched reliability index approach for reliability-based design optimisation of electromagnetic devices in the presence of variability in design variables. To substantially improve computational efficiency, two new optimisation strategies consisting of a constraint indicator condition and a successive optimum search technique are incorporated into the existing method based on the reliability index approach. The constraint indicator decides whether reliability analysis is needed for each intermediate design during the reliability-based optimisation process. Employing the successive optimum search technique, a deterministic optimum is first sought from an initial design and then a probabilistic optimum is obtained by launching at the deterministic optimum. The proposed method is tested with a coil design problem for a superconducting magnetic energy storage system, and is thoroughly investigated by comparing the results with the conventional method based on the reliability index approach and the Monte Carlo simulation in terms of computational efficiency and accuracy. Finally, the results show that the method leads to a significantly decreasing computational cost.

Efficient algorithm for probability-based design optimisation of complex structures and related issues

The probabilistic structural design optimisation problem can be carried out using either the conventional reliability index approach (RIA) or the performance measure approach (PMA). The direct approach for either RIA or PMA is to form a two-level optimisation by simply connecting algorithms of probabilistic calculation and design optimisation, which is computationally prohibitive for complex structures. The authors used the sequential approximate programming (SAP) strategy for RIA and extended to PMA. Examples show that SAP for both RIA and PMA reduce the total number of function evaluations notably, whereas PMA with SAP is more efficient and less dependent on probabilistic distributions and thus serves as a promising approach for complex structures. This study introduces the two SAP approaches in a unified way and shows that in these approaches, Taylor linear approximation is applied to both design variables and random variables. Two kinds of variables are treated in the same way from the mathematical viewpoint.

목표 성능치 기반의 확률구속조건 평가 기법을 이용한 전자기 장치의 신뢰도 기반 최적설계

본 논문에서는 전자기 관련 제품의 효율적인 신뢰도 기반 최적설계를 위해 확률구속조건을 평가하는 기법으로 해의 안정성과 효율성이 우수한 목표 성능치법을 제시 하였다. 목표 성능치법을 적용한 신뢰도 기반 최적설계의 효율성 검증을 위하여 스피커 모델과 초전도 자기에너지 저장장치 모델에 대한 최적설계를 수행하였고, 이를 기존 신뢰도 지수법을 적용한 최적설계 결과와 비교하였다. 또한 몬테카를로 수치모사기법을 이용하여 도출된 최적해의 신뢰도를 재 계산 후 비교함으로써 제안된 기법의 신뢰도 평가 결과의 정밀도를 검증하였다. This paper introduces an effective methodology for reliability-based design optimization of electromagnetic products, where a performance measure approach is adopted to accurately assess probabilistic constrains. Two design problems consisting of a loudspeaker and a superconducting magnetic energy storage system are considered. The efficiency of the proposed method in evaluating the failure probability of performances during the optimization process are compared with the existing method based on the reliability index approach. Moreover, in term of the accuracy of probability failure values, optimized design results are examined with reference values obtained from the Monte Carlo simulation.