Non-probabilistic Reliability Index Research Articles

Aero-Structural Comprehensive Nonprobabilistic Reliability-Based Optimization Method for High Aspect Ratio Wings

An aero-structural comprehensive reliability-based optimization method is proposed to construct the optimization framework of designing the high aspect ratio wing. There are three parts included in this method. 1) The ideal cruising shape is obtained by optimizing the geometrical parameters in the cruising conditions, where only the aerodynamic shape optimization is involved and the structural deformation is not considered. 2) The jig-shape optimization method is presented, where the aerodynamic performance of the wing structure with deformation is improved by adjusting the twist angle along the span direction to make sure that the shape with the aeroelastic deformation approaches the ideal cruising shape. 3) A nonprobabilistic reliability-based structural optimization is conducted by updating the thickness of composite layer, where the dispersion of mechanical properties of composite materials is considered and a nonprobabilistic reliability index according to strength and stiffness constraints is established. The global stiffness of the wing structure is changed after jig-shape optimization and structural optimization, and the jig-shape and structural scheme are modified alternately. The previously mentioned process needs to be repeated until the changes in the jig-shape and structural scheme in the iterations can be ignored. Finally, the effectiveness of this method has been verified through the numerical example followed by some conclusions.

The reliability index estimation of truss rods with interval uncertainty of the initial data

The article presents an approach to evaluation the reliability index of steel truss bars with the uncertainty of random variables expressed in the presence of information only about the bounds of variability. Different methods of estimating the bounds of v
 ariability for random variables are presented. The new approach is also developed using the provisions of the theory of possibility and the Dvoretzky–Kiefer–Wolfowitz inequality (DKW). The reliability index allows to compare various design solutions by the safety criterion, identify structural elements with the highest failure probability for monitoring the technical state and to quantify the increase in the safety level with strengthening of structural elements. The Monte Carlo statistical simulation data reflect the analogy of the non-probabilistic reliability index in the considered approach with the non-failure probability of the truss bar.

Non-Probabilistic Reliability Analysis of Slopes Based on a Multidimensional Parallelepiped Model

Aiming at the problem that the non-probabilistic reliability analysis method of slope engineering, which is based on an interval model, cannot consider the cross-correlation of geotechnical parameters, a non-probabilistic reliability analysis method of slopes based on a multidimensional parallelepiped model is proposed. This method can effectively alleviate the problem of difficult data survey in the field of geotechnical engineering. Using the limited sample data of soil parameters, the multidimensional parallelepiped model is constructed. The performance function of the slope is constructed based on Latin hypercube sampling and the quadratic response surface method. Then, the limit state equation of the slope can be standardized using the multidimensional parallelepiped model. The non-probabilistic reliability indexes of the slope are calculated based on the global optimal solution to judge the stability state of the slope. The example analysis verifies the feasibility of the proposed method. The results show that the correlation of shear strength parameters of soil has a great influence on the non-probabilistic reliability indexes of the slope. When the correlation coefficients of the shear strength parameters are between −1.0 and 0, the smaller the correlation coefficient is, the greater the non-probabilistic reliability index of the slope is; when the correlation coefficients of the shear strength parameters are between 0 and 1.0, the non-probabilistic reliability index of the slope does not change with the correlation coefficient. The non-probabilistic reliability indexes of the slope obtained using the multidimensional parallelepiped model are between the results obtained using an ellipsoid model and those obtained using an interval model, which are validated by Monte Carlo method and relatively more reasonable. In the absence of a large number of geotechnical sample data, this method provides a new way for slope stability analysis and expands the application field of calculation methods based on non-probabilistic theory.

A dynamic Gaussian process surrogate model based on the grasshopper optimization algorithm for non-probabilistic reliability analysis of complex structures

In practical complex engineering structures, the performance function (PF) for reliability analysis is commonly implicit and highly nonlinear. Commonly used surrogate models are appropriate for structural reliability analysis with an implicit PF. However, these methods require hundreds of PF values, which are time-consuming to obtain by adopting numerical analysis, such as finite element analysis (FEA). Therefore, since non-probabilistic reliability analysis does not require a large number of samples, it has great development potential. In this paper, a dynamic Gaussian process (GP) surrogate model based on the grasshopper optimization algorithm (DGP-GOA) is proposed for the non-probabilistic reliability analysis. First, with the help of the scale factor of the convex set model, the non-probabilistic reliability analysis problem is transformed into an unconstrained optimization problem. Second, the DGP-GOA fits the PF by constructing a GP surrogate model with a small dataset. Third, the GOA is used to search for the global optimal solution to obtain a non-probabilistic reliability index. Then, a dynamic retraining strategy is proposed to improve the fitting accuracy and efficiency. The results demonstrate that the proposed method is highly applicable to the non-probabilistic structural reliability analysis of complex engineering structures.

Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory

Aimed at the problem of fuzzy uncertainty of geotechnical parameters in slope stability analysis, a non-probabilistic reliability analysis method for slopes based on fuzzy set theory is proposed. Geotechnical parameters are described as fuzzy numbers, which are transformed into interval numbers at different cut set levels by taking fuzzy sets. The corresponding non-probabilistic reliability indexes and failure degrees of the slope are calculated by the non-probabilistic reliability analysis method based on the ellipsoidal model, and then the overall failure degree of the slope is obtained by weighted average to judge the stability state of the slope. The feasibility of the method was verified by a case analysis. The results show that the type and shape parameters of the fuzzy affiliation function of geotechnical parameters have a great influence on the non-probabilistic reliability of the slope. The slope failure degrees obtained from trapezoidal fuzzy numbers were larger, the slope failure degrees obtained from triangular fuzzy numbers and normal fuzzy numbers were medium, and the slope failure degrees obtained from lognormal fuzzy numbers were smaller. When considering soil parameters as triangular fuzzy numbers, normal fuzzy numbers, or lognormal fuzzy numbers, with the reduction of the shape parameters, the non-probabilistic reliability indexes of the slope increased while the failure degrees decreased. Additionally, adopting the overall failure degree to evaluate the stability of the slope can effectively solve the problem where the calculation results are too conservative (if the non-probabilistic reliability index is greater than 1) to judge the stability state of the slope in the traditional non-probabilistic reliability method.

Uncertainty Propagation for the Structures with Fuzzy Variables and Uncertain-but-Bounded Variables.

Various uncertain factors exist in the practical systems. Random variables, uncertain-but-bounded variables and fuzzy variables are commonly employed to measure these uncertain factors. Random variables are usually employed to define uncertain factors with sufficient samples to accurately estimate probability density functions (PDFs). Uncertain-but-bounded variables are usually employed to define uncertain factors with limited samples that cannot accurately estimate PDFs but can precisely decide variation ranges of uncertain factors. Fuzzy variables can commonly be employed to define uncertain factors with epistemic uncertainty relevant to human knowledge and expert experience. This paper focuses on the practical systems subjected to epistemic uncertainty measured by fuzzy variables and uncertainty with limited samples measured by uncertain-but-bounded variables. The uncertainty propagation of the systems with fuzzy variables described by a membership function and uncertain-but-bounded variables defined by a multi-ellipsoid convex set is investigated. The combination of the membership levels method for fuzzy variables and the non-probabilistic reliability index for uncertain-but-bounded variables is employed to solve the uncertainty propagation. Uncertainty propagation is sued to calculate the membership function of the non-probabilistic reliability index, which is defined by a nested optimization problem at each membership level when all fuzzy variables degenerate into intervals. Finally, three methods are employed to seek the membership function of the non-probabilistic reliability index. Various examples are utilized to demonstrate the applicability of the model and the efficiency of the proposed method.

Structural Interval Reliability Algorithm Based on Bernstein Polynomials and燛vidence Theory

Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors. Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid. The uncertain parameters mainly exist in the form of intervals. This method requires a lot of calculation and is often difficult to achieve efficiently. In order to solve this problem, this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty. Based on the non-probabilistic reliability index method, the extreme value of the limit state function is obtained using the properties of Bernstein polynomials, thus avoiding the need for a lot of sampling to solve the reliability analysis problem. The method is applied to numerical examples and engineering applications such as experiments, and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method, especially for some reliability problems with higher nonlinearity. Moreover, this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.

Convexity-oriented reliability-based topology optimization (CRBTO) in the time domain using the equivalent static loads method

Owing to the wide existence of multi-source uncertainties in practical engineering, the reliability-based design optimization for structural dynamics is particularly important and meaningful. However, compared to the size and shape optimization, it has rarely been considered in topology optimization with the lack of reasonable uncertainty cognition and the difficulty in the sensitivity solution for reliability constraints calculated from dynamic characteristics. In this research, a new design strategy, which combines the equivalent static loads (ESLs) algorithm and the set-theoretical convex method, is proposed for dynamic response topology optimization under convex reliability constraints. Time-varying boundary rules of the layout response are determined using the state-space transformation and the set perturbation theory. As the safety measurement, a new non-probabilistic reliability index (related to the function of dynamic compliance near the peaks) is defined under the hypervolume-ratio principle and its design sensitivity is further analyzed by solving central finite difference (CFD) equations. Three typical optimization cases, including a rectangular plate, an equivalent control surface and a win rib, are presented to demonstrate the validity and effectiveness of the developed methodology.

Hybrid active learning method for non-probabilistic reliability analysis with multi-super-ellipsoidal model

Non-probabilistic reliability analysis is of great importance in both reliability measure and reliability based design, the efficiency and precision of non-probabilistic reliability analysis, as well as uncertainty quantification, have attracted great attention currently. In this study, considering the coexistence of correlated and independent uncertain-but-bounded variables in engineering applications, a multi-super-ellipsoidal model is used to quantify the uncertainties. Furthermore, inspired by both “norm” based reliability index and “volume-ratio” based reliability index, a hybrid non-probabilistic reliability index is derived to measure the reliability extent more accurately and intuitively. To improve the accuracy and efficiency of solving the hybrid non-probabilistic reliability index, an effective Kriging based hybrid active learning method (HALM) is further developed. Finally, four examples are used to verify the effectiveness and robustness of the proposed HALM. The results show that the selection of multi-super-ellipsoidal model has a certain effect on the estimation of reliability index. Compared with the sampling-based analysis method, HALM presents better performance in terms of the trade-of between in computational efficiency and accuracy.

Interval uncertainty‐based multidisciplinary reliability analysis method for information‐poor complex system

AbstractDue to integrated structures and multiple functions, complex systems, such as large‐scale equipment and aerospace vehicle, faced prominent reliability problems. However, in real‐world applications, collecting sufficient reliability data is costly and time‐consuming. To overcome this difficulty, the information‐poor variables are modeled with interval models and the corresponding reliability analysis method for complex system is studied in this paper. First, we establish a multidisciplinary nonprobabilistic reliability model based on interval analysis, which is an optimization framework with an objective of nonprobabilistic reliability index and two constraints of interdisciplinary consistency equation and limit state equation. Second, two types of algorithms for the above model are studied. Based on typical methods of multidisciplinary design optimization (MDO), the direct algorithms including RA‐MDF and RA‐IDF are developed. Then, a decoupled reliability analysis algorithm is proposed to realize parallel multidisciplinary reliability analysis, in which two methods of multidisciplinary interval uncertainty analysis are formulated to decouple the coupling relationship between disciplines. Finally, three examples are employed to illustrate the validity and efficiency of the proposed methods.

Invariance problem in structural non-probabilistic reliability index

Invariance problem in structural non-probabilistic reliability index

An efficient method for calculating system non-probabilistic reliability index

Collecting enough samples is difficult in real applications. Several interval-based non-probabilistic reliability methods have been reported. The key of these methods is to estimate system non-probabilistic reliability index. In this paper, a new method is proposed to calculate system non-probabilistic reliability index. Kriging model is used to replace time-consuming simulations, and the efficient global optimization is used to determine the new training samples. A refinement learning function is proposed to determine the best component (or performance function) during the iterative process. The proposed refinement learning function has considered two important factors: (1) the contributions of components to system nonprobabilistic reliability index, and (2) the accuracy of the Kriging model at current iteration. Two stopping criteria are given to terminate the algorithm. The system non-probabilistic index is finally calculated based on the Kriging model and Monte Carlo simulation. Two numerical examples are given to show the applicability of the proposed method.

Fault Tree Analysis With Interval Uncertainty: A Case Study of the Aircraft Flap Mechanism

This article proposes an uncertainty analysis method for evaluating the reliability of a system and calculating the relative importance of the system inputs when the probabilities of basic events are characterized by intervals. The reliability requirement or the safety criterion is first considered in the system reliability assessment. A nonprobabilistic reliability index, called R-index, is defined to evaluate the reliability of the system. Two importance measure indices, called location effect and size effect, are introduced for estimating the influence of the probability uncertainties of basic events on the R-index, thus identifying influential and noninfluential basic events and comparing their relative importance. Then, the proposed method is applied to an illustrative fault tree example and the fault tree analysis of unilateral asymmetric movement of an aircraft flap mechanism, thereby explaining the significance of this method. The results show that the proposed method provides a viable tool for system reliability and importance measure analysis when probability intervals are involved.

A general assessment index for non-probabilistic reliability of structures with bounded field and parametric uncertainties

This paper focuses on the reliability assessment of structures exhibiting both bounded field uncertainties and uncertain-but-bounded parameters in a non-probabilistic framework. Based on the bounded field model for spatially varying uncertainties and the multi-ellipsoid convex set description for uncertain parameters, the mathematical definition of a new reliability index is proposed to quantify the safety measures of a structure. The problem is defined as finding the maximal allowable range of uncertainties within the safety margin of the performance function in the normalized uncertain space. A gradient-based mathematical programming algorithm is used to seek the most probable failure realization of uncertainties and to obtain the non-probabilistic reliability index. The proposed index provides a general and unified formation of structural non-probabilistic reliability with uncertain-but-bounded field or parameters. Numerical examples, considering the reliability assessment of structural performance in stiffness, eigenfrequency, and buckling problems, are also given to illustrate the validity and applicability of the proposed method.

A Bayesian Updating Method for Non-Probabilistic Reliability Assessment of Structures with Performance Test Data

For structures that only the predicted bounds of uncertainties are available, this study proposes a Bayesian method to logically evaluate the nonprobabilistic reliability of structures based on multi-ellipsoid convex model and performance test data. According to the given interval ranges of uncertainties, we determine the initial characteristic parameters of a multi-ellipsoid convex set. Moreover, to update the plausibility of characteristic parameters, a Bayesian network for the information fusion of prior uncertainty knowledge and subsequent performance test data is constructed. Then, an updated multi-ellipsoid set with the maximum likelihood of the performance test data can be achieved. The credible non-probabilistic reliability index is calculated based on the Kriging-based surrogate model of the performance function. Several numerical examples are presented to validate the proposed Bayesian updating method.

A novel approach of reliability-based topology optimization for continuum structures under interval uncertainties

Purpose The purpose of this paper is to propose a novel non-probabilistic reliability-based topology optimization (NRBTO) method for continuum structural design under interval uncertainties of load and material parameters based on the technology of 3D printing or additive manufacturing. Design/methodology/approach First, the uncertainty quantification analysis is accomplished by interval Taylor extension to determine boundary rules of concerned displacement responses. Based on the interval interference theory, a novel reliability index, named as the optimization feature distance, is then introduced to construct non-probabilistic reliability constraints. To circumvent convergence difficulties in solving large-scale variable optimization problems, the gradient-based method of moving asymptotes is also used, in which the sensitivity expressions of the present reliability measurements with respect to design variables are deduced by combination of the adjoint vector scheme and interval mathematics. Findings The main findings of this paper should lie in that new non-probabilistic reliability index, i.e. the optimization feature distance which is defined and further incorporated in continuum topology optimization issues. Besides, a novel concurrent design strategy under consideration of macro-micro integration is presented by using the developed RBTO methodology. Originality/value Uncertainty propagation analysis based on the interval Taylor extension method is conducted. Novel reliability index of the optimization feature distance is defined. Expressions of the adjoint vectors between interval bounds of displacement responses and the relative density are deduced. New NRBTO method subjected to continuum structures is developed and further solved by MMA algorithms.

A novel experimental data-driven exponential convex model for reliability assessment with uncertain-but-bounded parameters

The convex model is commonly applied to quantify the uncertain-but-bounded parameters. However, the typical interval and ellipsoid models may lead to inaccurate approximation for existing experimental data, which may incur either too risk or conservative for safety assessment. To this end, this study aims to create a novel data-driven exponential convex model to achieve accurate approximation for experiment data, in which the dimension reduction minimum volume method plays the key role. Furthermore, a novel relaxed exponential nominal value method (RENVM) is developed to evaluate the corresponding non-probabilistic reliability index robustly and efficiently, and the sensitivities are also derived based on the straight forward perturbation method to guarantee its efficiency. Through numerical and experimental studies, the accuracy and validity of the proposed data-driven exponential convex model are validated compared to the interval and ellipsoid models, and the robustness and efficiency of the proposed RENVM are also demonstrated for solving both linear and nonlinear problems.

A nonprobabilistic reliability–based topology optimization method of compliant mechanisms with interval uncertainties

SummaryOn account of the inevitable multisource uncertainty factors in compliant mechanisms, which seriously affect the accuracy of output motion, a nonprobabilistic reliability–based topology optimization (NRBTO) framework for compliant mechanisms with interval uncertainties is introduced. Combined with the solid isotropic material with penalization (SIMP) model and the set‐theoretical interval method, the uncertainty quantification analysis is conducted to obtain mathematical approximations and boundary laws of considered mean compliance. By normalization treatment of the limit‐state function, a new quantified measure of the nonprobabilistic reliability is then defined. The compliance‐based NRBTO design method ensures the output motion realizing its target value accurately considering the uncertainty factors. The sensitivities of the nonprobabilistic reliability index with respect to design variables are calculated by the adjoint vector method. Two engineering examples are eventually presented to illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

Truss layout design under nonprobabilistic reliability‐based topology optimization framework with interval uncertainties

SummaryA nonprobabilistic reliability‐based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown‐but‐bounded uncertainties) is proposed in this paper. The cross‐sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single‐loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.

Improved Nonprobabilistic Global Optimal Solution Method and Its Application in Bridge Reliability Assessment

Utilizing the improved one-dimensional optimization algorithm conveniently solved the nonprobabilistic reliability index, however, only searching the part of probable failure points. Utilizing the global optimal solution method produced heavy computation, although it was capable of searching all probable failure points. This paper, based on these two methods, proposed an improved nonprobabilistic global optimal solution method. The presented method possessed the advantages of searching all probable failure points and generating less computation, by means of which the values of interval variables were determined based on the monotonicity of performance function to the corresponding variables. Without losing any probable failure points, this method was contributive to reducing root value point equations, lowering the computational complexity, and improving the computational efficiency. The effectiveness and feasibility of the presented method were verified by two examples. The proposed method was also introduced to build the nonprobabilistic reliability evaluation process of existing bridges. Taking the Longganhu bridge for example, the nonprobabilistic reliability index of it was calculated using the presented nonprobabilistic reliability evaluation process. The computed nonprobabilistic reliability index η = 0.737 < 1 indicated that the Longganhu bridge was unreliable and needed to be reinforced. Reinforcement measures were carried out for a hollow slab and bridge deck of the Longganhu bridge, respectively. The reinforced bridge was reevaluated as η = 2.159 > 1. The results showed that the bridge was reliable after reinforcement. The study illustrated the application and feasibility of the improved nonprobabilistic global optimal solution method and nonprobabilistic reliability evaluation process in reliability assessment and reinforcement of existing bridges.