Non-probabilistic Reliability Research Articles

Non-probabilistic Integrated Reliability Analysis of Structures with Fuzzy Interval Uncertainties using the Adaptive GPR-RS Method

Due to its weak dependence on the quantity of variable samples, the non-probabilistic reliability analysis method based on the convex set model is applicable to practical problems in structural engineering with inherent uncertainties. However, when dealing with the black-box limit-state function issues in practical complex structural engineering, the traditional quadratic polynomial response surface (QP-RS) method has the problem of insufficient precision in approximating a highly nonlinear function. Meanwhile, fixing the limits of interval variables is trickier in case of scant samples and meager statistical information. To remedy the above deficiencies, this paper introduces a reasonable integrated reliability analysis approach. First, an adaptive Gaussian process regression response surface (GPR-RS) method that dynamically improves the fitted accuracy near the design point of the black-box limit-state function is formulated. Furthermore, the integrated reliability index with consideration of fuzzy interval uncertainties is presented. Three validation and two application examples are employed, which have justified the approach as a more reasonable assessor of practical complex structural reliability with safer results.

Optimal Maintenance Design-Oriented Nonprobabilistic Reliability Methodology for Existing Structures Under Static and Dynamic Mixed Uncertainties

The performances of engineering structures are degrading in service and maintenance interventions can enhance the structural reliability and make sure that the structures are effectively functional. Engineering structures often exhibit obvious time cumulative effects due to multisource time-varying uncertainties including property degradation of materials, changeable environment conditions, and dynamic loading processes, which makes the reliability assessment and maintenance design much harder. In this paper, a new time-varying nonprobabilistic reliability-oriented optimal maintenance design method with the consideration of static and dynamic mixed uncertainties is proposed. The interval model is utilized to quantify the uncertain and time-varying characteristics and the structural reliability can be assessed combining the interval model and the first-passage theory. We try to establish a more general maintenance cost model enlightened by Frangopol's work under the nonprobabilistic system and the maintenance strategy including uncertainty assessment, reliability evaluation, and cost optimization is constructed. Three numerical examples are presented to demonstrate the effectiveness of the developed methodology.

Efficient strategy for reliability-based optimization design of multidisciplinary coupled system with interval parameters

• The uncertainty propagation method in multidisciplinary system based on subinterval theory is presented. • The interval reliability displacement based multidisciplinary design optimization methodology is proposed. • The partial gradient region for interval reliability is converted to full gradient region for reliability displacement. • The convergence process can be accelerated by utilizing the gradient optimization algorithms. Non-probabilistic reliability based multidisciplinary design optimization has been widely acknowledged as an advanced methodology for complex system design when the data is insufficient. In this work, the uncertainty propagation analysis method in multidisciplinary system based on subinterval theory is firstly studied to obtain the uncertain responses. Then, based on the non-probabilistic set theory, the interval reliability based multidisciplinary design optimization model is established. Considering that the gradient information of interval reliability cannot be acquired in the whole design domain, which causes convergence difficulties and prohibitive computation, an interval reliability displacement based multidisciplinary design optimization method is proposed to address the issue. In the proposed method, the interval reliability displacement is introduced to measure the degree of interval reliability. By doing so, not only the connotation of the interval reliability is guaranteed, but more importantly, the partial gradient region for interval reliability is equivalently converted into full gradient region for reliability displacement. Consequently, the gradient information can be acquired under any circumstances and thus the convergence process is highly accelerated by utilizing the gradient optimization algorithms.

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.

Non-Probabilistic Time-Varying Reliability-Based Analysis of Corroded Pipelines Considering the Interaction of Multiple Uncertainty Variables

Reliability analysis of corroded pipelines is critical to the integrity and safe working of pipeline infrastructure. Aiming at less probability information is obtained for corrosion pipeline engineering, and the mechanical properties of pipeline with corrosion defects deteriorate caused by the accumulative effect of corrosion growth. Based on the quasi-static analysis method and non-probability theory, this paper presents a reliability model for assessing corroded pipelines with corrosion growth. In fact, reliability analysis of corroded pipelines needs to consider the interaction of multiple uncertainty variables. By introducing interaction theory, a mathematical model of corrosion defects considering the interaction of variables is put forward. Moreover, this paper develops a non-probabilistic time-varying reliability method for pipeline systems with multiple defects. Thus, several numerical examples are investigated to discuss the effectiveness of the proposed methodology. The results show that a two-dimensional or even three-dimensional ellipsoid model with correlation has more accurate results to evaluate corroded pipelines under the interaction of multiple corroded defects with poor information. Furthermore, a non-probabilistic time-varying reliability model is established according to the time-varying characteristics of the corroded pipeline under the influence of multiple factors. An effective complement to the theory of non-probabilistic reliability analysis of system is investigated. The analysis of the results suggests that interaction of corroded pipeline has a negligible impact on reliability. It also provides a theoretical basis for maintenance and is of great significance for risk- and reliability-informed decisions regarding buried oil and gas pipelines.

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.

Nonprobabilistic reliability oriented topological optimization for multi-material heat-transfer structures with interval uncertainties

This study presents a nonprobabilistic reliability-based topology optimization (NRBTO) framework that combines a multi-material interpolation model and interval mathematics to achieve an optimal layout design for heat-transfer structures under unknown but bounded (UBB) uncertainties. In terms of the uncertainty quantification (UQ) issue, the interval dimension-wise method (IDWM) based on set collocation theory is first proposed to effectively determine the bounds of nodal temperature responses. For safety reasons, the interval reliability (IR) index corresponding to the thermal constraint is defined, and then a new design policy, i.e., the strategy of nonprobabilistic reliability oriented topological optimization is established. To circumvent problems of large-scale variable updating in a multi-material topology optimization procedure, theoretical deductions of the design sensitivity analysis are further given based on the adjoint-vector criterion and the chain principle. The validity and feasibility of the developed methodology are eventually demonstrated by several application examples.

Dynamic reliability analysis model for structure with both random and interval uncertainties

Aiming at analyzing the safety of the dynamic structure involving both input random variables and the interval ones, a new dynamic reliability analysis model is presented by constructing a second level limit state function. Two steps are involved in the construction of the dynamic reliability model. In the first step, the non-probabilistic reliability index is firstly extended to the dynamic structure, in which the uncertainties of interval inputs can be analyzed by fixing random inputs and time parameter. In the second step, the second level limit state function is constructed by considering the fact that the non-probabilistic reliability index larger than one corresponds to the safe state, in which the uncertainties of random inputs are taken into account. Generally, the actual reliability of dynamic structure with both random and interval inputs is an interval variable, and theoretic analysis illustrates that the proposed reliability is equivalent to the lower bound of the actual reliability, which can provide an efficient way for measuring the safety of dynamic structure. For estimating the proposed reliability, a double-loop optimization algorithm combined with Monte Carlo Simulation as well as the active learning Kriging method is established. Several examples involving a cylindrical pressure vessel, an automobile front axle and a planar 10-bar structure are introduced to illustrate the validity and significance of the established reliability model and the efficiency and accuracy of the proposed solving procedure.

A direct-integration-based structural reliability analysis method using non-probabilistic convex model

In practical structural reliability analysis, there is not only random uncertainty but also fuzzy uncertainty. Aiming at the fuzzy reliability of structure, a novel fuzzy reliability method is proposed based on direct integration method and ellipsoidal convex model. Firstly, the decomposition of fuzzy mathematics principle is used to convert fuzzy reliability model into non-probabilistic reliability model, in which fuzzy variables are converted into interval variables. The upper and lower bounds of interval variables are determined by the possibility distribution function on the membership value. Secondly, multidimensional ellipsoid convex models are constructed to quantify the uncertainty because of the complexity of non-probabilistic reliability. Finally, sigmoid function with adjustable parameter is introduced to direct integration method for approximating the step function, and then direct integration method is used to solve the fuzzy reliability. Numerical examples are investigated to demonstrate the effectiveness of the present method, which provides a feasible way for the structural fuzzy reliability analysis.

An importance learning method for non-probabilistic reliability analysis and optimization

With the time-consuming computations incurred by nested double-loop strategy and multiple performance functions, the enhancement of computational efficiency for the non-probabilistic reliability estimation and optimization is a challenging problem in the assessment of structural safety. In this study, a novel importance learning method (ILM) is proposed on the basis of active learning technique using Kriging metamodel, which builds the Kriging model accurately and efficiently by considering the influence of the most concerned point. To further accelerate the convergence rate of non-probabilistic reliability analysis, a new stopping criterion is constructed to ensure accuracy of the Kriging model. For solving the non-probabilistic reliability-based design optimization (NRBDO) problems with multiple non-probabilistic constraints, a new active learning function is further developed based upon the ILM for dealing with this problem efficiently. The proposed ILM is verified by two non-probabilistic reliability estimation examples and three NRBDO examples. Comparing with the existing active learning methods, the optimal results calculated by the proposed ILM show high performance in terms of efficiency and accuracy.

An interval updating model for composite structures optimization

In practical engineering, the uncertainties commonly exist in the design process of composite structures due to the dispersion of composite materials. This paper proposes an uncertain optimization method based on interval model updating technique and non-probabilistic reliability (NPR) theory. A novel interval model updating method is firstly established by modifying the deterministic constraint condition parameters, which retains the quantification results of the material parameters and has a convinced updating model. In the design section, a new two-step optimization process for composite structures, which combines NPR method and progressive failure theory, is presented based on the updated simulation model. The traditional composite structural design method (safety factor based method) is also introduced to be compared. The NPR based optimization can optimize the nominal value and deviation of the structural response simultaneously, and will achieve a more significant weight reduction effect. A complete process including the uncertainty quantification, interval model updating, NPR based optimization and experimental verification for a composite stiffened plate under the compressive and shear load conditions were performed to verify the effectiveness of the proposed methodology.

A nonprobabilistic time-variant reliability-based optimization approach to the reliable active controller design of structural vibration considering convex uncertainties

It is observed that the controller, as the “brain” of the active control system, plays a vital role in the vibration-rejecting aspect, and thereby, techniques of the optimal controller design are attracting increasing attention in dynamical engineering recently. However, due to the existence of uncertainty effects originating from modeling parameters and environmental conditions, current design means based on the classical optimal control theory are no longer feasible. In view of this, a new nonprobabilistic time-variant reliability-based optimization (NTRBO) strategy considering convex uncertainties is presented in this study for the reliable controller design of structural vibration. Boundary rules and autocorrelation characteristics of dynamical responses under the linear quadratic regulator control are first determined using the state-space transformation and the convex process theory. Inspired by the out-crossing idea and the safety factor notation, a new piecewise function of the time-variant reliability index is defined, and the nonprobabilistic reliability constraint is then established. Weighing matrices in the Riccati equation, as the keys to construct the controller, are indeed optimized by solving the developed NTRBO model. The usage, validity, and efficiency of the presented reliable control methodology are eventually demonstrated by several example applications.

Approach of non-probabilistic reliability topology optimization using evidence theory

It is of great importance to consider uncertainty due to insufficient data or imprecise information in topology optimization design. In this paper, evidence theory is presented to handle the imprecise data situation. The plausibility measure based on evidence theory is introduced to overcome the difficulty of constructing the precise probabilistic constraint. Furthermore, the topology problem in this evidence-based optimization design is solved by combined strategy of the differential evolution and superior topology technique. In order to overcome the difficulty of intensive computational cost in calculating plausibility measure, an improved method of evolutionary optimization design integrating with imprecise extremum idea is proposed. Two truss examples are given to demonstrate the proposed approach. The research indicates that deterministic results may be the failure solutions under epistemic uncertainty. Although evidence-based optimum topology designs are more conservative than deterministic results in aspects of weight and optimal structural topology layout, it gains a more robust design under epistemic uncertainty.

Nonprobabilistic Reliability Evaluation for In-Service Gravity Dam Undergoing Structural Reinforcement

Influenced by many factors such as material performance attenuation, structural defect repair, and reinforcement, the structural reliability of a gravity dam changes in different periods, which are called dynamic effects. Traditional methods are inclined to calculate and analyze the structural probability reliability from the random or fuzzy uncertain characteristics of parameters. However, in practice, it is difficult to obtain the probability distribution of uncertain parameters as well as the performance function, which is too nonlinear to be expressed. In addition, gravity dam failures are low probability events. Thus, a reliability calculation model and method for the gravity dam system considering dynamic effects under the practice of reinforcement are studied by integrating the theory of interval mathematics and nonprobability reliability. First, with the help of the monitoring data, and physical model and numerical simulation results of the gravity dam before and after the reinforcement, we establish a nonprobabilistic reliability (N-PR) calculation model for the gravity dam element and system based on the bounds of uncertain parameters and propose an inversion method for the bounds of uncertain parameters. Subsequently, by combining the response surface method with the central composite design technique, an N-PR index calculation method for gravity dams considering dynamic effects is developed. It turns out that the given model and method can effectively pave the way to avoid falling into the limitations of classical probabilistic reliability analysis of which the uncertain parameters require to be randomly changed and the results are highly sensitive to the parameters. In addition, it can better adapt to the characteristic of gravity dam performance function, which is too nonlinear to be expressed explicitly. When employing this method to analyze the reliability of gravity dam services, we are able to evaluate and predict the changing process and the trend of it more objectively, and have a closer look at the comprehensive influence and contribution of reinforcement to dam service reliability.

Interval non-probabilistic reliability of surrounding jointed rockmass considering microseismic loads in mining tunnels

The frequent microseismicity affects the stability and reliability of surrounding rock and the safety of supporting structures in deep mining tunnels. The limit state equations of rock blocks under microseismic loads were developed to indicate the stability conditions and sliding characteristics. The scarce field data result in the uncertainty of the physico-mechanical parameters of tunnel surrounding rock. According to the insufficient statistical information for the rockmass, a method of the interval non-probabilistic reliability was proposed to analyze the stability of surrounding jointed rockmass. The method considered both microseismic loads and the uncertainty of rockmass parameters. The solving approach of interval non-probabilistic reliability was optimized, and it can be available when the field data is scarce. To verify the proposed method, the interval non-probabilistic reliability was used to evaluate the stability of the mining tunnel rockmass in the Yongshaba mine (China). The calculated interval non-probabilistic reliability was compared with the safety factor and random reliability. Results show that the interval non-probabilistic reliability model is in agreement with practical situations. It is proved that the proposed method of interval non-probabilistic reliability, considering both the uncertainty of the rockmass parameters and the microseismic loads, is a beneficial complement to the traditional analysis methods of safety factor and random reliability.

A non-probabilistic reliability-based topology optimization (NRBTO) method of continuum structures with convex uncertainties

This paper develops a non-probabilistic reliability-based topology optimization (NRBTO) framework for continuum structures under multi-dimensional convex uncertainties. Combined with the solid isotropic material with penalization (SIMP) model and the set-theoretical convex method, the uncertainty quantification (UQ) analysis is firstly conducted to obtain mathematical approximations and boundary laws of considered displacement responses. By normalization treatment of the limit-state function, a new quantified measure of the non-probabilistic reliability is then defined and further deduced by the principle of the hyper-volume ratio. For circumventing optimization difficulties arising from large-scale design variables, the adjoint vector scheme for sensitivity analysis of the reliability index with respect to design variables are discussed as well. Numerical applications eventually illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

Non-probabilistic reliability analysis and design optimization for valve-port plate pair of seawater hydraulic pump for underwater apparatus

Seawater hydraulic axial piston pumps are widely used in underwater apparatus systems in recent years. Reliability of the valve-port plate pair is vital to the performance of seawater hydraulic axial piston pumps. As the distribution information of the material combinations and working conditions of seawater hydraulic axial piston pumps are insufficient, the probability reliability analysis methods are not available for the valve-port plate pair. This paper develops a non-probabilistic reliability convex model for the valve-port plate pair. Firstly, the rotation of the pump cylinder is partitioned into six stages. Based on this partition, the dynamic models of contact pressure and pv value (the product of contact specific pressure and linear velocity) of the valve-port plate pair are derived by accounting for the pre-compression angle. Secondly, the marginal interval for the limiting [pv] value (It is the usual criterion to evaluate the critical operating conditions under which the material fails) of the material combination was obtained through adhesive wear tests. Thirdly, a limit-state function is then established based on the stress-strength interference model. Finally, the analysis and sea test results indicate that the reliability of the valve-port plate pair has significantly improved by adding auxiliary support on the port plate.

New target performance approach for a super parametric convex model of non-probabilistic reliability-based design optimization

Non-probabilistic reliability-based design optimization (NRBDO) is an essential tool to deal with an optimization model with uncertain-but-bounded variables. A major disadvantage of NRBDO is that it conventionally suffers from excessively high computational costs. In the present study, a new target performance approach (TPA) is proposed to eliminate high computational cost incurred by complicated equality constraints in non-probabilistic reliability analysis in which the target performance constraint is applied to substitute for the non-probabilistic reliability index constraint. To further enhance the optimization efficiency, a new NRBDO algorithm is developed based on the proposed TPA in which the target performance iterative method (TPIM) is established to search for the target concern point while a convex approximate method is used to further improve the convergence. A mathematical example, three numerical examples, and two complex engineering examples, i.e., an axially compressed stiffened shell in a launch vehicle and a tower crane in civil engineering, are utilized to demonstrate the effectiveness of the proposed method in comparison with that of other existing methods. The results indicate that the proposed method is promising in terms of complicated engineering problems in the absence of prior knowledge of uncertain parameters.

Reliability assessment of fiber-reinforced composite laminates with correlated elastic mechanical parameters

This paper is aimed at developing a novel reliability assessment approach for fiber-reinforced composite laminates with correlated elastic mechanical parameters. Firstly, based on non-statistical grey mathematical theory, uncertainty information including nominal values, standard variances, and uncertainty matrix can be evaluated for elastic mechanical parameters of composites. Here, uncertain parameters are dealt with as correlated interval variables. Furthermore, the first-order Taylor interval expansion method is developed for interval estimation of structural responses. Here, the correlations between any two parameters are taken into account. Finally, by virtue of obtained intervals for composite laminates, the non-probabilistic reliability measurement is adopted for reliability evaluation. Besides, in order to account for the feasibility and validity of the proposed method, two numerical examples are given including an uncertain buckling problem and a first-ply failure analysis with temperature and moisture effects for composites. The results show that the proposed method can be applied efficiently in the process of reliability evaluation for composites. It can be concluded that it is appropriate to take correlations of uncertain elastic parameters into account for reliability assessment of composites, which can give birth to a more reasonable and better result.