Non-probabilistic Reliability Research Articles

Dynamic failure assessment analysis of 15MnTi steel supporting structure with a circumferential surface crack under impact loads

Supporting structures, whose strength have great influence on the safe operation of the whole equipment, are widely used in various ship equipment. In this paper, dynamic failure assessment analysis is carried out to discuss the structural integrity of the cracked supporting structure under impact loads. Considering the dynamic response characteristics of structures, the dynamic fracture toughness determined by the loading rate of dynamic J-integrals is used to calculate the failure assessment point (FAP). Due to the widespread uncertainties in practical engineering, such as crack size and mechanical properties of materials, the failure assessment results will also have uncertainty To analyze the structural integrity affected by uncertain variables, the polygonal convex set (PCS) model is introduced to the failure assessment diagram (FAD) to propose a non-probabilistic failure assessment method. The FAP is extended to a failure assessment region, and structural integrity can be analyzed by the non-probabilistic reliability degree. Compared with the traditional interval model, the non-probabilistic reliability degree calculated by PCS model is closer to the probabilistic reliability degree, which avoids premature judgement of structural failure. For the 15MnTi supporting structure with a circumferential semi-elliptical surface crack under impact loads, the critical depth obtained by non-probabilistic methods decrease by 10% compared with deterministic results, and the failure mechanism is summarized as plastic collapse resulting from the excessive net section stress. In the absence of statistical data on crack size or mechanical properties, the non-probabilistic failure assessment method can be used as an effective alternative to the probabilistic method.

A non-probabilistic reliability topology optimization method based on aggregation function and matrix multiplication considering buckling response constraints

A non-probabilistic reliability topology optimization method based on aggregation function and matrix multiplication considering buckling response constraints

A non-probabilistic reliability analysis method with the fuzzy failure criterion

In engineering practice, the failure boundary of structures is not usually distinct due to some ambiguous information, which brings a tricky challenge to reliability assessment. The fuzzy theory is introduced to solve it with the traditional probability assessment method. For the non-probabilistic reliability system, however, this problem tends to be at a dead end because it lacks some necessary probability information. In this study, the non-probabilistic reliability model with fuzzy failure criterion is established, which involves two different evaluation indices in terms of the limit state surface. Based on this, a corresponding calculation method is proposed to deal with it, and the interval model and the ellipsoid model are shown for instance. In addition, the HRHL-RF method owning strong robustness is utilized to calculate the index of the ellipsoid model. Eventually, an established algorithm frame can achieve the calculation of the unified fuzzy reliability index, and three examples with the classical fuzzy membership functions are computed to show its feasibility and efficiency.

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.

Double-time-scale non-probabilistic reliability-based controller optimization for manipulator considering motion error and wear growth

Motion error and wear growth are two crucial factors that determine the performance of the manipulator system. The two factors are distinct from the sensitivity to time and respectively indicate its control performance and lifespan level. In this paper, a double-time-scale non-probabilistic reliability (DTSNPR)-based optimization method, which considers time-sensitive factor motion error as time-dependent reliability (TDR) and time-insensitive factor clearance wear growth as time-independent reliability (TIR), is proposed to comprehensively evaluate and optimize the controller of the manipulator system in consideration of these multi-scale factors. Meanwhile, for a highly nonlinear response problem of a manipulator system, the adaptive subinterval collocation method (ASICM) which transfers the highly nonlinear uncertainty propagation problem into a sequence of small subintervals, is adopted to obtain the response range more precisely. The proposed DTSNPR-based optimization method is applied to three numerical manipulator systems and ensures each of them under a predefined level of reliability upon both motion error and wear growth. The results also indicate that the proposed ASICM owns an advantage over computational cost and precision, with only 0.4% error and 1% computational cost compared with the Monte-Carlo methods.

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.

Non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the strength constraint parameterized level set method and interval mathematics

In this paper, a study on non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the parameterized level set method (PLSM) is conducted. The P-norm is used to replace the local stress constraint with a single comprehensive stress constraint that approximates the maximum stress to optimize the structure model with minimum volume. In the interval-set model, the stress constraint is transformed into the non-probabilistic reliability-based scheme and the reliability is evaluated by the optimization feature distance (OFD). In the NRBTO scheme, the unknown-but-bounded (UBB) uncertainties of the material and external load of the optimized structure are considered. By interpolating the level set function (LSF) with the multiquadric (MQ) splines, the partial differential equation (PDE) is transformed into an ordinary differential equation (ODE). In addition, the interval parameter vertex method is employed to obtain the stress constraint interval, and the shape derivative and adjoint vector methods are used to obtain the evolution velocity field of LSF. The optimality criterion (OC) method is used to solve the optimization problem. Finally, three examples are given, and the optimization model of stiffness constraint is introduced, which not only verifies the applicability of the topology optimization scheme based on non-probabilistic reliability but also compares the difference between the topology optimization with strength constraint and the topology optimization with stiffness constraint.

Hybrid reliability-based sequential optimization for PID vibratory controller design considering interval and fuzzy mixed uncertainties

Vibration control is an important problem in practical engineering. Optimal control is widely used, but the uncertainties in practical engineering may lead to the failure of the system. To reach a balance of safety and energy-consuming, taking the widely-used proportional-integral-differential control as an example, a framework for hybrid reliability-based sequential optimization for vibratory controller design considering interval and fuzzy mixed uncertainties is proposed in this paper. Firstly, the fuzzy uncertainties are transformed into interval uncertainties according to the cut-level strategy. Then the response interval under a certain membership can be calculated by the subinterval collocation method, which can guarantee the accuracy of the calculated response interval. In this process, to choose proper memberships as cut-levels and improve efficiency, the adaptive fitting method based on the differences of areas is adopted. Next, taking interval and fuzzy uncertainties into consideration, a novel non-probabilistic reliability evaluation is performed based on the area ratio, which is composed of upper and lower boundaries of the minimum value of the extreme state function. Finally, the sequential optimization is adopted to improve the efficiency of optimization, in which the constraint is changed according to the proposed non-probabilistic area-ratio reliability. Two engineering examples are presented to demonstrate the accuracy and practicability of this framework.

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.

Non-Probabilistic Reliability Bounds Method for Series Structural Systems Considering Redundant Failure Modes

Non-probabilistic structural reliability analysis is based on the convex model and more applicable for practical engineering problems with limited samples. Recently, the authors proposed a non-probabilistic reliability bounds method (NRBM) for series structural systems as an effective means for the assessment of systems’ non-probabilistic reliability. A novel non-probabilistic reliability bounds method considering redundant failure modes is proposed in this paper for series structural systems to further improve the efficiency and accuracy of the NRBM. By decomposing the system into several subsystems with two or three failure modes, three identification criteria for redundant failure modes are developed for these subsystems. A bounding formula for the system’s non-probabilistic failure degree is then derived after removing the redundant failure modes. An investigation of three numerical examples indicates that the proposed method has a higher efficiency and at least equivalent accuracy compared to the NRBM.

Credible Bayesian reliability model for structures with interval uncertain parameters

If a non-probabilistic interval model is adopted to describe the uncertainty, the determination of the interval bounds is challenging, which degrades the accuracy and credibility of the reliability analysis. In this study, based on Bayesian theory, a non-probabilistic credible Bayesian reliability model, which can be updated by introducing new sample points, is proposed for three failure modes and developed to model the credibility of non-probabilistic reliability. The relations between the classical probabilistic confidence reliability model and the proposed non-probabilistic credible Bayesian reliability model are investigated and discussed. Two relevant numerical examples are selected for the investigation to demonstrate the effectiveness of the proposed model and the limitations of the classical confidence reliability in the evaluation of structural safety.

Non-Probabilistic Reliability Analysis of Robot Accuracy under Uncertain Joint Clearance

The development of industrial robots in high-precision fields is currently constrained by the reliability of motion. Considering the influence of the joint clearance on the motion reliability of the industrial robot, the kinematic model of the industrial robot is established, the kinematic equation of the robot is deduced, and positive kinematic solutions are performed. The non-probability positioning accuracy reliability measure of a robot end-effector is proposed, based on the non-probability theory and method, combined with the prescribed permission interval and error interval, and different states of reliability can be judged according to the position relationship, the non-probability reliability properties are outlined, and the positioning accuracy reliability assessment model is established. Combined with the joint clearance modeling theory, the simulation of the robot end-effector under the influence of six joint clearances is carried out, and the displacement error interval of the end-effector under the preset motion path is analyzed for the industrial robot motion reliability problem. The motion path is split by time, and the end effector moves to different workspace areas in different time periods. The motion reliability of each segment is analyzed, and it is concluded that the reliability of the end-effector under the influence of uncertain joint clearance parameters changes in different working regions. Based on the above, the research direction of space division and partition parameter calibration is proposed, which lays a foundation for the study of partition non-probabilistic calibration of robot workspace.

Global non-probabilistic reliability sensitivity analysis based on surrogate model

Sensitivity analysis is used to find the key variables which have significant effect on system reliability. For a product in early design stage, it is impossible to collect sufficient samples. Thus, the probabilistic-based reliability sensitivity analysis methods are difficult to use due to the requirement of probability distribution. As an alternative, interval can be used because it only requires few samples. In this study, an effective global non-probabilistic sensitivity analysis based on adaptive Kriging model is proposed. The global accuracy Kriging model is constructed to reduce overall computational cost. Subsequently, the global non-probabilistic sensitivity analysis method is developed. Compared to existing non-probabilistic sensitivity analysis methods, the proposed method is a global non-probabilistic reliability sensitivity analysis method. The proposed method is easy to use and does not require probability distribution of the input variables. The applicability of proposed method is demonstrated via two examples.

Structural Reliability Analysis by Using Non-Probabilistic Multi-Cluster Ellipsoidal Model.

Uncertainties are normally unavoidable in engineering practice, which should be taken into account in the structural design and optimization so as to reduce the relevant risks. Yet, the probabilistic models of the uncertainties are often unavailable in the problems due to the lack of samples, and the precision of the conventional non-probabilistic models are not satisfactory when the samples are of multi-cluster distribution. In view of this, an improved method by using a non-probabilistic multi-cluster ellipsoidal model (multi-CEM) for the critical structural reliability analysis is proposed in this paper, which describes the samples in a more accurate and compact way and helps to acquire more satisfactory reliability analysis results. Firstly, a Gaussian mixture model (GMM) is built for the multi-cluster samples with performing expectation maximization (EM) algorithm, based on which the multi-CEM can be constructed. In the structural reliability analysis, two cases, respectively, considering whether the components of the multi-CEM are intersected or not are researched in detail. The non-probabilistic reliability (NPR) indexes for each component of the multi-CEM are computed using the Hasofer–Lind–Rackwitz–Fiessler (HL-RF) algorithm, and then the multidimensional volume ratios of the safe domain to the whole uncertainty domain are computed based on these indexes, indicating the structural NPR. In the end, two numerical examples and a practical application are conducted and analyzed to testify the effectiveness of the method.

A Grasshopper Optimization Algorithm-Based Response Surface Method for Non-Probabilistic Structural Reliability Analysis with an Implicit Performance Function

Non-probabilistic reliability analysis has great developmental potential in the field of structural reliability analysis, as it is often difficult to obtain enough samples to construct an accurate probability distribution function of random variables based on probabilistic theory. In practical engineering cases, the performance function (PF) is commonly implicit. Monte Carlo simulation (MCS) is commonly used for structural reliability analysis with implicit PFs. However, MCS requires the calculation of thousands of PF values. Such calculation could be time-consuming when the structural systems are complicated, and numerical analysis procedures such as the finite element method have to be adopted to obtain the PF values. To address this issue, this paper presents a grasshopper optimization algorithm-based response surface method (RSM). First, the method employs a quadratic polynomial to approximate the implicit PF with a small set of the actual values of the implicit PF. Second, the grasshopper optimization algorithm (GOA) is used to search for the global optimal solution of the scaling factor of the convex set since the problem of solving the reliability index is transformed into an unconstrained optimal problem. During the search process in the GOA, a dynamic response surface updating strategy is used to improve the approximate accuracy near the current optimal point to improve the computing efficiency. Two mathematical examples and two engineering structure examples that use the proposed method are given to verify its feasibility. The results compare favorably with those of MCS. The proposed method can be non-invasively combined with finite element analysis software to solve non-probabilistic reliability analysis problems of structures with implicit PF with high efficiency and high accuracy.

Structural reliability with credibility based on the non-probabilistic set-theoretic analysis

A structural reliability analysis method based on the non-probabilistic credible set model is proposed in this paper. Compared with the probabilistic method, the developed method overcomes the high requirement for the sample size of structural uncertain parameters. Moreover, it surpasses the traditional non-probabilistic reliability analysis method by relating the reliability index to its corresponding credibility. Via introducing the non-probabilistic credible set uncertainty quantification, the distribution range of the required credibility for structural uncertain parameters can be determined by scant samples. Accordingly, the improved uncertainty propagation analysis methods based on the non-probabilistic credible set uncertainty quantification are established to obtain the bounds of structural response by transforming into local coordinate system. The non-probabilistic credible reliability index is constructed and three advanced efficient approaches are developed to calculate that. Moreover, the procedure of structural non-probabilistic credible reliability analysis method is built by combining the above content. Besides, the influence of credibility on non-probabilistic credible reliability is discussed. A cantilever beam example, an aeroengine turbine disk case and a pressure vessel case are utilized to verify the feasibility of applying the proposed approach to various engineering problems.

An Efficient Sequential Strategy for Non-probabilistic Reliability-based Topology Optimization (NRBTO) of Continuum Structures with Stress Constraints

In the study, the sequential strategy is proposed for non-probabilistic reliability-based topology optimization (NRBTO) of continuum structures with stress constraints. Firstly, the qp-relaxation criterion is involved to circumvent the stress singularity. Then the deterministic topology optimization formulation by Stress-Influence-Function (SIF) approach is described. Uncertainty quantification analysis is conducted to determine the feasible bounds of the elemental stresses under unknown-but-bounded uncertainties. For safety reasons, the non-probabilistic reliability concept is introduced in the optimization formulation. The sequential strategy is proposed to handle the NRBTO problem and thus the original nested process of topology optimization and reliability analysis was decoupled, in which the update of design variables and the shift of the strength feature in the stress influence function are conducted alternately in the optimization process. The Performance Measure Approach (PMA) is applied to calculate the shift value of the strength feature and a shift value function is presented to avoid the numerical oscillations caused by the large variation of strength feature. Three numerical examples are eventually given to illustrate the applicability and validity of the developed sequential strategy for NRBTO with stress constraints.