PHI2 Method Research Articles

The first-order time-variant reliability expansion method

Time-variant reliability problems are frequently encountered in engineering due to factors like material degradation or random loading modeled as random processes. The PHI2 method, which employs the First Order Reliability Method (FORM), is commonly used to solve such problems. However, it requires repeated searches for Most Probable Points (MPPs), making it computationally expensive. To improve efficiency with little sacrifice of accuracy, this study proposes a First Order Time-variant Reliability Expansion (FOTRE) method, which provides an efficient explicit formulation for MPP regarding time, in contrast to the expensive optimization approach of the PHI2 method. It requires only a single accurate search for the so-called “worst MPP” over the whole lifespan and offers the “adaptive accuracy of outcrossing rate”, which avoids the repeated search for MPPs ensuring computational accuracy. The inspiration behind the FOTRE method stems from the observation that the outcrossing rate tends to be small at time points with relatively large reliability indexes compared to the minimum reliability index βmin, which has a negligible impact on the subsequent structural failure probability over the entire lifespan. This innovative approach significantly improves the efficiency of solving time-variant reliability problems without compromising much of the numerical accuracy. The effectiveness and accuracy of the FOTRE method are demonstrated through several numerical examples.

Reliability analysis for air-breathing hypersonic vehicles based on a new dynamic PHI2 method

Purpose This paper aims to present a novel dynamic reliability model that considers the interval mixed uncertainty for the air-breathing hypersonic flight vehicle (AHFV) to guarantee flight safety and structural reliability. Design/methodology/approach Initially, the force condition of the fuselage is analyzed based on the longitudinal elastic model of an AHFV. Subsequently, a new high-efficiency dynamic reliability model is presented to describe the failure probability evolution of the fuselage structure. For the random uncertainty problem with interval distribution parameters, the interval PHI2 method of time-dependent reliability is used to obtain the time-dependent reliability interval of the AHFV. Finally, the key variables that affect the failure probability accumulation are determined, which provide an important reference for ensuring structural reliability and improving the life span of AHFVs. Findings It is demonstrated that the proposed reliability model can obtain more accurate dynamic reliability results for the fuselage, and it is confirmed the key variables that affect the failure probability accumulation. The results also provide an important reference for the reliability analysis of hypersonic vehicles. Originality/value The novelty of this work comes from the first application of the PHI2 method (considering the interval mixed uncertainty) in the AHFV and the development of a new reliability model for the entire body of AHFVs. The proposed analysis scheme is implemented on the dynamic model of the AHFV, which provides a more accurate reference for improving the structural reliability and life span of AHFVs.

A single-loop time-variant reliability evaluation via a decoupling strategy and probability distribution reconstruction

In this paper, a single-loop approach for time-variant reliability evaluation is proposed based on a decoupling strategy and probability distribution reconstruction. The most attractive feature of the proposed method is that the reliability at a specified time instant can be captured by performing time-invariant reliability analysis only once. In this method, the expansion optimal linear estimation is first employed to discretize the loading stochastic process. Then, a decoupling strategy that decouples the loading stochastic process and degradation processes is developed to formulate a single-loop method for time-variant reliability analysis, where an equivalent extreme value limit state function (EEV-LSF) is obtained. To improve the accuracy and robustness, the Box–Cox transformation is applied to get a transformed EEV-LSF. The maximum entropy method with fractional exponential moments is employed to robustly derive the probability distribution of transformed EEV-LSF. Once the probability distribution is captured, the time-variant failure probability can be readily computed. To handle a large number of random variables, a weighted sampling method is applied for moment assessment to ensure an efficient solution. Numerical examples including a complex real-world case are studied to validate the proposed method, where pertinent Monte Carlo simulations and PHI2 method are conducted for comparisons.

Developing a method for time-variant reliability assessment of passive heat removal systems in nuclear power plants

Nowadays, the use of passive safety systems is considered as the most important candidate to improve the safety level of new generation nuclear power plants. However, the functioning parameters of the passive systems vary significantly over time and are extremely tied to uncertainty. Time dependency may be due to the changes in certain and uncertain parameters and the mission of the passive systems. In this paper, a time-variant reliability analysis method that combines the out-crossing approach (PHI2) and the first-order reliability method (FORM) is developed and applied to analyze the passive heat removal system (PHRS) of VVER-1000 NPP as a case study. The thermo-hydraulic model of the PHRS and its related parameters besides the solving process of PHRS parameters in the PHI2 method are illustrated. The cumulative failure probability of the PHRS is calculated in terms of the system functioning parameters. The results of this study demonstrate that the failure probability of passive safety systems is significant in terms of amplitude and its variation over time and must be considered in the design and operation of these systems in the nuclear industry.

Topology optimization design of improved response surface method for time-variant reliability

This paper proposes an effective and efficient method for topology optimization with a time-variant reliability constraint. A decoupling strategy is used to decompose the time-variant reliability topology optimization into a time-variant analysis and time-invariant deterministic topology optimization, thus improving the efficiency of the optimization calculation. The reliability analysis will use the PHI2 method and an improved response surface method to estimate the time-variant reliability level and satisfy the reliability constraint. Continuously updated design points derived from a time-varying reliability analysis are used for topology optimization using the BESO method. Finally, some numerical examples of the time-variant reliability topology optimization problems are provided to illustrate the efficiency of the proposed method and its ability to determine the optimal solution.

Application of adaptive surrogate models in time-variant fatigue reliability assessment of welded joints with surface cracks

The time-variant fatigue reliability of welded joints subjected to stochastic loading is assessed based on the PHI2 method, which takes the time-variant reliability problem as a two-component parallel system reliability problem. The complex fatigue process that takes place at the weld toe is modeled as a semi-elliptical surface crack growth process. Surrogate models, representing surface crack sizes at a time instant, are applied to solve the reliability problem involving time-consuming fatigue crack growth analyses. Polynomial regression models and Kriging interpolation models are both employed. Their corresponding adaptive procedures are also adopted to improve the reliability results. Some adjustments of the active refinement algorithm based on the Kriging model are proposed to make it more suitable for the present study. The time-variant fatigue reliability assessment of a T-plate welded joint is carried out. Without using the adaptive procedures, the Kriging models can better represent the crack sizes compared with the polynomial models. The outcrossing rate formulated by the PHI2 method is sensitive to the reliability index of the first component of the two-component parallel system. The effectiveness of the adaptive procedures, which improve the reliability results, is demonstrated.

Time-variant fatigue reliability assessment of welded joints based on the PHI2 and response surface methods

A time-variant fatigue reliability assessment model for welded joints subjected to stochastic loading is presented. The PHI2 method, which allows one to solve time-variant problems using time-invariant methods, provides the framework of the model. A sophisticated fatigue crack growth model that is capable of taking the residual stress effect into account is employed to compute crack sizes at different times under stochastic loading. The crack sizes are explicitly represented by response surface models. Limit state functions are formulated based on combined fracture criteria, which consider both brittle and ductile fracture, and the response surface models. The time-variant fatigue reliability of a T-plate welded joint with an edge crack located at the weld toe is assessed. It appears that results from the time-invariant fatigue reliability assessment may be too optimistic. The weld induced residual stress effects are considered based on the stress intensity factor due to residual stress. Its effects on the crack size and time-variant fatigue reliability are significant and cannot be ignored.

Extending first-passage method to reliability sensitivity analysis of motion mechanisms

Identifying the parameters that substantially affect the time-dependent reliability is critical for reliability-based design of motion mechanism. The time-dependent local reliability sensitivity and global reliability sensitivity are the two effective techniques for this type of analysis. This work extends the first-passage method and PHI2 method, which are commonly used for estimating the time-dependent reliability, for efficiently estimating the time-dependent local reliability sensitivity and global reliability sensitivity indices of the motion mechanism. Both the local reliability sensitivity and global reliability sensitivity indices are analytically derived based on the Poisson assumption–based first-passage method and the first-order Taylor’s expansion of the motion error function. Compared with the current envelope function method for estimating the time-dependent local reliability sensitivity and global reliability sensitivity indices, the developed method does not need to estimate the second-order derivatives of motion error function, thus is more applicable. The accuracy and effectiveness of the proposed method are demonstrated by a numerical example and a satellite antenna, the direction of which is controlled by a four-bar function generator mechanism.

The effect of copulas on time-variant reliability involving time-continuous stochastic processes

In structural reliability the dependence structure between random variables is almost exclusively modeled by Gauss (normal or Gaussian) copula; however, this implicit assumption is typically not corroborated. This paper is focusing on time-variant reliability problems with continuous stochastic processes, which are collection of dependent random variables and to our knowledge are not modeled by other than Gauss copula in structural reliability. Therefore, the aim of this contribution is to qualitatively and quantitatively analyze the impact of this copula assumption on failure probability. Three illustrative examples are studied considering bivariate Gauss, t, rotated Clayton, Gumbel, and rotated Gumbel copulas. Time-variant actions are modeled as stationary, ergodic, continuous stochastic processes, and the PHI2 method is adopted for the analyses. The calculations show that the copula function has significant effect on failure probability. In the studied examples, application of Gauss copula can four times underestimate or even 10 times overestimate failure probabilities obtained by other copulas. For normal structures agreement on copula type is recommended, while for safety critical ones inference of copula type from observations is advocated. If data are scare, multiple copula functions and model averaging could be used to explore this uncertainty.

Time-variant assessment for adhesively bonded assemblies with non-linear behaviour in naval applications

Adhesive bonding is becoming a widely used assembling process in the naval field; it can allow great productivity, simplify design constraints and can lead, in general, to a significant weight reduction. Unfortunately, this technology is not completely mastered; parameters like the cure process, size of defects, joint adhesive thickness variations, size of parts to be assembled and non-linear behaviour of the adhesive joint, have to be considered to assess the risks and predict the service life. Moreover, for naval applications adapted state functions have to be proposed in order to take into account this variability of material properties, the time dependency of parameters like damage, solicitations or loads. Under these assumptions, time-independent reliability is insufficient to allow a pertinent-related failure probability calculation. The solution of such a problem can be given by the use of time-dependent reliability tools. This passes through coupling structural reliability software evaluation with a non-linear finite element code. This choice requires robust mechanical models able to describe the behaviour of naval structure with an acceptable numerical cost in order to perform the reliability study. This paper presents a time variant reliability approach based on the PHI2 method. Cohesive-zone models are used to model the adhesive. This choice allows limiting the numerical cost of simulations and permits us to model the non-linear behaviour of adhesive. It also presents a good compromise between numerical costs and quality of results. The damage at the interface can be modelled and it has been proven that using cohesive zone models is well suited for bonded joints as the possible degradation zone is well defined. In order to determine the cohesive zone model parameters, tests were performed. The most relevant results are presented. Different numerical results showing the feasibility of this approach for adhesively bonded assemblies are discussed and the case of a representative part of a naval structure has been studied in this paper.

Analytical derivation of the outcrossing rate in time-variant reliability problems

The usual approach to time-variant reliability problems is based on the computation of the outcrossing rate through the limit state surface and its time integration. The so-called PHI2 method allows one to compute the outcrossing rate by solving a two-component parallel system reliability problem using the First Order Reliability Method (FORM). Following this approach, the present paper provides new analytical expressions of the outcrossing rate and their implementation. The corresponding improvement of the PHI2 method in terms of accuracy is shown. The method is first validated using a simple time-variant reliability problem, for which an analytical expression of the associated outcrossing rate exists. Then, it is applied to evaluate the reliability of a corroded steel beam submitted to a midspan random load.

A time-variant reliability approach applied to corroded naval structures with non-linear behaviour

As corrosion effects can lead to catastrophic consequences in naval structures, methods to take it into account are a source of concern. Its variability in time and space, in addition to the consideration of random and temporal character of material behaviours, environmental conditions and loads, requires adapted strategies from mechanical and reliability points of view. In this paper, we propose a 2D non-linear finite element mechanical approach modelling local decreases of the safe thickness of the structure, avoiding successive re-meshing. A time-variant reliability analysis of a corroded plate submitted to a stochastic load is carried out to validate the propounded strategy. The reliability analyses are posttreated in a time-variant way by the PHI2 method, whose two existing formulations are compared in terms of stability and convergence speed.

A time-variant reliability approach applied to corroded naval structures with non-linear behaviour

As corrosion effects can lead to catastrophic consequences in naval structures, methods to take it into account are a source of concern. Its variability in time and space, in addition to the consideration of random and temporal character of material behaviours, environmental conditions and loads, requires adapted strategies from mechanical and reliability points of view. In this paper, we propose a 2D non-linear finite element mechanical approach modelling local decreases of the safe thickness of the structure, avoiding successive re-meshing. A time-variant reliability analysis of a corroded plate submitted to a stochastic load is carried out to validate the propounded strategy. The reliability analyses are posttreated in a time-variant way by the PHI2 method, whose two existing formulations are compared in terms of stability and convergence speed.

Time‐variant Reliability of Nonlinear Structures: Application to a Representative Part of a Plate Floor

AbstractThis paper is concerned with the feasibility study of time‐variant reliability analysis in the case of nonlinear structure behaviour. It is a very important point for designers in order to take into account the real behaviour of structures, which is often nonlinear, especially for naval structures (corrosion, welded joints, fatigue cracks, etc.). As an application, the time‐variant reliability of a part of a plate floor represented by a perforated plate with a visco‐plastic behaviour under a stochastic load is studied. The second point is to analyse the numerical cost of such a complex computation in order to obtain accurate numerical results. The mechanical model is defined using the finite‐element software CAST3M®. The stochastic load applied to the perforated plate is represented by the so‐called series Expansion Optimal Linear Estimation method developed by Li and Der Kiureghian. The time‐variant reliability analysis is computed by the PHI2 method, presented by Andrieu‐Renaud et al. Copyright © 2006 John Wiley & Sons, Ltd.

The PHI2 method: a way to compute time-variant reliability

Time-variant reliability problems appear in the engineering practice when (a) the material properties of the structure deteriorate in time or (b) random loading modelled as random processes is involved. This paper presents a method called PHI2 which is based on the outcrossing approach and allows to solve such problems using classical time-invariant reliability tools such as FORM/SORM methods. The PHI2 method is first presented. Then it is benchmarked with the well-established ‘asymptotic methods’ [Stochast. Process. Appl. 13 (1988) 195; J. Offshore Mech. Arctic Engng 113 (1991) 241; Probab. Engng Mech. 10 (1995) 53; J. Struct. Engng 25 (1998) 1] on three examples dealing with scalar or vector processes and linear or non-linear limit state functions. The PHI2 method appears more accurate in all cases. As an application example, the method is finally applied on a case representing a mechanical system (a beam) placed in an environment that can have exceptional configuration.

On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process

On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process