Failure Probability Constraint Research Articles

Quantile-based sequential optimization and reliability assessment method under random and interval hybrid uncertainty

Under random and interval hybrid uncertainties, solving hybrid reliability based design optimization (HRBDO) can acquire an optimal balance between structural performance and reliability. Since solving HRBDO includes a triple nested framework involving minimum analysis of performance function (PF), failure probability constraint analysis and design parameter optimization, the computational complexity of HRBDO is high, especially for dealing with complex structures. Therefore, a quantile-based sequential optimization and reliability assessment method (QSORA) is proposed for reducing the computational complexity of HRBDO. In the proposed QSORA for HRBDO, failure probability constraint is firstly transformed into minimum PF (MPF) quantile one corresponding to target failure probability. Then, approximating the difference between PF and its target quantile at current iteration by that at previous one, the failure probability constraint analysis is decoupled from the design parameter optimization. Moreover, by approximating the minimum point of the PF with respect to the interval input in the current iteration by that in the previous one, the minimum analysis of PF is separated from the design parameter optimization. By the separation of minimum analysis and failure probability constraint analysis from the design parameter optimization in the proposed QSORA, the triple nested framework of HRBDO is decoupled sequentially as the deterministic design optimization, the minimum analysis of the PF and the target MPF quantile estimation, and this way of reconstructing the HRBDO from the triple nested framework to three single-loop frameworks can significantly enhance the efficiency of solving HRBDO. Furthermore, the MPF quantile at the current design parameter is estimated by stochastic collocation based statistical moment method, in which the stochastic collocation method is employed to efficiently estimate the MPF moment to approximate the probability density function of MPF. The efficiency and accuracy of the QSORA are validated by four numerical and engineering examples finally.

Safety Lifetime Analysis Using Two-Phase Subset Simulation Combined with Kriging Model

The safety lifetime analysis under the required time-dependent failure probability (R-FP) constraint is essential to ensure structure service safety, but existing methods are unaffordable for solving the safety lifetime under a small R-FP constraint. For this issue, a novel method is proposed by combining a two-phase subset simulation with an adaptive kriging model (TP-SS-AK). The innovations of the TP-SS-AK include three aspects. First, it creatively transforms the safety lifetime under a small R-FP constraint into that under a large conditional R-FP constraint by subset simulation, which can greatly improve efficiency because the large conditional R-FP constraint only needs a small candidate sample pool for searching the safety lifetime. Second, by deducing the misclassification probability of the time-dependent structure state in the first phase, the first new learning strategy is established to train the kriging model of the time-dependent performance function and further determine a reasonable lifetime search interval efficiently. Third, by deriving the misclassification probability of the first failure instant, the second new learning strategy is constructed to efficiently update the kriging model in the second phase and further search the safety lifetime. Because the proposed TP-SS-AK combines the advantages of the two-phase subset simulation and two new learning strategies, it is superior to the existing methods in efficiency, and its superiority is verified by the examples.

Stochastic Response Analysis and Reliability-Based Design Optimization of Nonlinear Electromechanical Energy Harvesters With Fractional Derivative Elements

Abstract A methodology based on the Wiener path integral (WPI) technique is developed for stochastic response determination and reliability-based design optimization of a class of nonlinear electromechanical energy harvesters endowed with fractional derivative elements. In this regard, first, the WPI technique is appropriately adapted and enhanced to account both for the singular diffusion matrix and for the fractional derivative modeling of the capacitance in the coupled electromechanical governing equations. Next, a reliability-based design optimization problem is formulated and solved, in conjunction with the WPI technique, for determining the optimal parameters of the harvester. It is noted that the herein proposed definition of the failure probability constraint is particularly suitable for harvester configurations subject to space limitations. Several numerical examples are included, while comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the satisfactory performance of the methodology.

Optimization Approaches for the Numerical Design of Structures Under Consideration of Polymorphic Uncertain Data

Abstract In this paper, optimization approaches for the numerical design of structures are presented. The uncertainty of structural parameters is considered by means of random variables and interval design parameters within the optimization. A new space subdividing technique is introduced to substitute time-consuming failure probability constraint evaluations. In order to solve optimization problems with polymorphic uncertain parameters, surrogate objectives are formulated and solved by means of a particle swarm optimization (PSO) approach in combination with Monte Carlo simulation and interval analysis. Two computational schemes are presented and verified by a benchmark example. Finally, an application is shown, where the reinforcement layout of a reinforced concrete bridge structure is optimized by minimizing the crack widths at the reinforcement layers in order to improve the durability of the structure. A nonlinear finite element (FE) model is used to compute the uncertain crack patterns and the load bearing capacity. The stochastic objective function and the failure probability constraint are approximated by neural network based surrogate models.

Truss structure optimization for two design variable elements using Genetic Algorithm with stress and failure probability constraints

This paper presents the results of a study on trusses which are need to satisfy optimal conditions, i.e. the lowest Cost possible cost with maximal performance. The trusses considered were statically indeterminate steel structures with a multi-system loading. The cost is represented by the material volume of the structure and the maximal performance is reflected by the high working stresses reflect the maximum performance within allowable stress limits. The material strength was modeled as a random variable with a Log Normal distribution. The structures are also required to meet a failure probability of Pf =10-3, which may occur locally within the elements as well as globally on the structure as a whole. The complexity of optimization problems depends in general on the number of the considered variables. The larger the number of variables considered, the more complicated becomes the solution process. Therefore, cases of single variable elements such as multi variables ones were considered in this study. Optimization problems are usually solved by applying iterative procedures, frequently resorting to mathematical programming. In these procedures the process usually converges to unreliable solutions; it even may completely bogged down with no solution. To circumvent this problem, iteration was carried out by applying Genetic Algorithms where the process proceeds in a stochastic manner.

System reliability-based design optimization and risk-based optimization: a benchmark example considering progressive collapse

ABSTRACTSystem Reliability-Based Design Optimization (RBDO) considers a single system failure probability constraint. The formulation allows different failure modes to compete with each other. Very few methods and benchmark examples involving optimal design considering system behaviour with progressive failure can be found in the literature. In this article, the conventional System RBDO formulation is compared with a risk-based formulation. A benchmark example involving progressive failure of hyper-static truss is addressed. It is demonstrated that typical System RBDO formulations always lead to isostatic structures, since the formulation does not offer any incentive for the permanence of hyper-static members. In the risk-based formulation, failure costs are differentiated with respect to primary member failure, in hyper-static structures (existence of warning before eventual collapse), and failure of isostatic members (no warning). In this formulation, optimal designs also include hyper-static structures. Results presented herein are relevant in the modern context of robust design considering progressive collapse.

Link importance incorporated failure probability measuring solution for multicast light-trees in elastic optical networks

The light-tree-based optical multicasting (LT-OM) scheme provides a spectrum- and energy-efficient method to accommodate emerging multicast services. Some studies focus on the survivability technologies for LTs against a fixed number of link failures, such as single-link failure. However, a few studies involve failure probability constraints when building LTs. It is worth noting that each link of an LT plays different important roles under failure scenarios. When calculating the failure probability of an LT, the importance of its every link should be considered. We design a link importance incorporated failure probability measuring solution (LIFPMS) for multicast LTs under independent failure model and shared risk link group failure model. Based on the LIFPMS, we put forward the minimum failure probability (MFP) problem for the LT-OM scheme. Heuristic approaches are developed to address the MFP problem in elastic optical networks. Numerical results show that the LIFPMS provides an accurate metric for calculating the failure probability of multicast LTs and enhances the reliability of the LT-OM scheme while accommodating multicast services.

Maximum probable life time analysis under the required time-dependent failure probability constraint and its meta-model estimation

Time-dependent reliability (failure probability) aims at measuring the probability of the normal (abnormal) operation for structure/mechanism within the given time interval. To analyze the maximum probable life time under a required time-dependent failure probability (TDFP) constraint, an inverse process corresponding to the time-dependent reliability is proposed by taking the randomness of the input variables into consideration. The proposed inverse process employs the monotonicity between the TDFP and the upper boundary of the given time interval which reflects the life time, and an adaptive single-loop sampling meta-model for the time-dependent limit state function is presented to estimate the TDFP at the given time interval flexibly. Since the TDFP is generally monotonic to the upper boundary of the given time interval, thus by adjusting the probable upper and lower boundaries of the time interval in which the corresponding TDFPs include the required TDFP constraint, the proposed approach can always search the maximum probable life time at the required TDFP by the dichotomy. By introducing the time variable as an input which is the same level as the input random variables and constructing the adaptive single-loop sampling meta-model for the time-dependent limit state function in a longer time interval with the TDFP bigger than the required TDFP, the TDFP in any subintervals of the time interval involved in the constructed meta-model can be estimated as a byproduct of the constructed meta-model without any additional actual limit state evaluations. Then the efficiency for analyzing the maximum probable life time is improved by the dichotomy and the unified meta-model of the time-dependent limit state function. Two examples are employed to illustrate the accuracy and the efficiency of the proposed approach.

Spectrum-Aware Survivable Strategies With Failure Probability Constraints Under Static Traffic in Flexible Bandwidth Optical Networks

This paper addresses the problem of spectrum-aware survivable strategies with failure probability constraints under static traffic in flexible bandwidth optical networks. The joint failure probability between primary and backup paths must be below the maximum tolerable joint failure probability for each traffic demand. We develop integer linear program (ILP) models for dedicated-path protection and shared-path protection in order to minimize the total number of frequency slots consumed, and we also propose a spectrum-aware dedicated protection (SADP) algorithm and a spectrum-aware shared protection (SASP) algorithm. Simulation results show that the ILP model solutions consume minimum number of frequency slots, but lead to higher average joint failure probability compared to the SADP and SASP algorithms. Moreover, both the SADP and SASP algorithms achieve a better performance in terms of total number of frequency slots consumed as compared to a conventional dedicated-path protection algorithm and a conventional shared-path protection algorithm, respectively, but lead to higher average joint failure probability.

Spectrum-Efficient Multi-Channel Design for Coexisting IEEE 802.15.4 Networks: A Stochastic Geometry Approach

For networks with random topologies (e.g., wireless ad-hoc and sensor networks) and dynamically varying channel gains, choosing the long term operating parameters that optimize the network performance metrics is very challenging. In this paper, we use stochastic geometry analysis to develop a novel framework to design spectrum-efficient multi-channel random wireless networks based on the IEEE 802.15.4 standard. The proposed framework maximizes both spatial and time domain frequency utilization under channel gain uncertainties to minimize the number of frequency channels required to accommodate a certain population of coexisting IEEE 802.15.4 networks. The performance metrics are the outage probability and the self admission failure probability. We relax the single channel assumption that has been used traditionally in the stochastic geometry analysis. We show that the intensity of the admitted networks does not increase linearly with the number of channels and the rate of increase of the intensity of the admitted networks decreases with the number of channels. By using graph theory, we obtain the minimum required number of channels to accommodate a certain intensity of coexisting networks under a self admission failure probability constraint. To this end, we design a superframe structure for the coexisting IEEE 802.15.4 networks and a method for time-domain interference alignment.

Use of a failure probability constraint to suggest an initial dose in a phase I cancer clinical trial

The primary objective of a Phase I cancer clinical trial is to determine the maximum tolerated dose of a drug. The “failure probability” was proposed and used as a constraint to help identify a suitable initial dose range. The maximum tolerated dose was then determined based on a 3 + 3 cohort-based escalation scheme. Multiple simulations were conducted, and the method was evaluated according to the required sample size and accuracy and precision of maximum tolerated dose estimate. The results indicated that the median of the initial dose range suggested using a failure probability is a suitable initial dose regardless of the dose escalation sequence used for a cancer Phase I study. This initial dose required a smaller sample size and resulted in less bias of the estimated maximum tolerated dose compared with a commonly used initial dose, that is, 10% of the lethal dose. We tested our approach using real dose and toxicity outcome data from two published Phase I studies. These results indicate that adding a failure probability constraint into the calculation of the initial dose range will improve the efficiency of Phase I cancer trials.

Minimized Spectrum Resource Consumption With Rescaled Failure Probability Constraint in Flexible Bandwidth Optical Networks

We investigate the problem of minimizing spectrum resource consumption for a set of connection requests in a flexible bandwidth optical network under the constraint that the failure probability of each connection must be below a specified threshold. To optimize the total frequency slots consumed, three schemes are proposed and evaluated: the rescaled failure-probability-aware algorithm (RFPA), the traffic cognition algorithm with rescaled failure probability (RFPTC), and an integer linear programming (ILP) model. We also introduce two traditional Dijkstra’s algorithms with load balancing and spectrum assignment by first fit and traffic cognition (FF_DB and TC_DB) to compare with the proposed three schemes. For static traffic in a small network, the total frequency slots consumed by the RFPA and RFPTC algorithms will approach that of the optimal ILP solution as K increases, as well as the average hops (AH). Furthermore, the average rescaled failure probabilities (RFPs) of the RFPA and RFPTC algorithms are much better than those of the ILP solutions. In addition, similarly, the results of the heuristic algorithms, FF_DB, TC_DB, RFPA, and RFPTC, in the large network have the same characteristics as in the small network. For dynamic traffic in a large network, the RFPTC algorithm reduces blocking probability and makes the best use of spectrum resources compared with the other schemes, which also reflects that discontinuous spectrum fragmentation is greatly reduced by the traffic cognition method. However, both RFPTC and RFPA result in much higher average RFP but perform fewer AH compared to TC_DB and FF_DB.

A comparison of deterministic, reliability-based and risk-based structural optimization under uncertainty

In this paper, the effects of uncertainty and expected costs of failure on optimum structural design are investigated, by comparing three distinct formulations of structural optimization problems. Deterministic Design Optimization (DDO) allows one the find the shape or configuration of a structure that is optimum in terms of mechanics, but the formulation grossly neglects parameter uncertainty and its effects on structural safety. Reliability-based Design Optimization (RBDO) has emerged as an alternative to properly model the safety-under-uncertainty part of the problem. With RBDO, one can ensure that a minimum (and measurable) level of safety is achieved by the optimum structure. However, results are dependent on the failure probabilities used as constraints in the analysis. Risk optimization (RO) increases the scope of the problem by addressing the compromising goals of economy and safety. This is accomplished by quantifying the monetary consequences of failure, as well as the costs associated with construction, operation and maintenance. RO yields the optimum topology and the optimum point of balance between economy and safety. Results are compared for some example problems. The broader RO solution is found first, and optimum results are used as constraints in DDO and RBDO. Results show that even when optimum safety coefficients are used as constraints in DDO, the formulation leads to configurations which respect these design constraints, reduce manufacturing costs but increase total expected costs (including expected costs of failure). When (optimum) system failure probability is used as a constraint in RBDO, this solution also reduces manufacturing costs but by increasing total expected costs. This happens when the costs associated with different failure modes are distinct. Hence, a general equivalence between the formulations cannot be established. Optimum structural design considering expected costs of failure cannot be controlled solely by safety factors nor by failure probability constraints, but will depend on actual structural configuration.

Extensions of Stochastic Optimization Results to Problems with System Failure Probability Functions

We derive an implementable algorithm for solving nonlinear stochastic optimization problems with failure probability constraints using sample average approximations. The paper extends prior results dealing with a failure probability expressed by a single measure to the case of failure probability expressed in terms of multiple performance measures. We also present a new formula for the failure probability gradient. A numerical example addressing the optimal design of a reinforced concrete highway bridge illustrates the algorithm.

Reliability optimization using probabilistic sufficiency factor and correction response surface

Reliability-based design optimization for low failure probability often requires millions of function analyses. Response surface approximation of the response functions (analysis response surface(ARS)) is often used to reduce the cost of failure probability calculations. Failure probabilities obtained from numerical sampling schemes are noisy and unsuitable for gradient-based optimization. To overcome this, response surfaces have been fitted to the failure probability of the designs (design response surface (DRS)) as a function of the design variables and used in optimization. Two shortcomings of the approach are that (i) the ARS fitting is extremely expensive for a large number of variables, especially for the high accuracy required to obtain very accurate reliability estimates, and (ii) DRS introduces fitting errors which affect the tails of the distributions which are significant for low failure probabilities. An approach to obtaining high-accuracy reliability estimates using the probabilistic sufficiency factor and correction response surface is investigated in this article. The method is demonstrated using a thin-walled box beam structure designed for minimum weight with failure probability constraints. The design is subjected to buckling, strength, and displacement constraints. Two methods of correcting low-fidelity analyses are compared for accuracy and efficiency. It is shown that correction to the response function is more accurate than the correction fitted to the probabilistic sufficiency factor.

Optimal Design with Probabilistic Objective and Constraints

Significant challenges are associated with solving optimal structural design problems involving the failure probability in the objective and constraint functions. In this paper, we develop gradient-based optimization algorithms for estimating the solution of three classes of such problems in the case of continuous design variables. Our approach is based on a sequence of approximating design problems, which is constructed and then solved by a semiinfinite optimization algorithm. The construction consists of two steps: First, the failure probability terms in the objective function are replaced by auxiliary variables resulting in a simplified objective function. The auxiliary variables are determined automatically by the optimization algorithm. Second, the failure probability constraints are replaced by a parametrized first-order approximation. The parameter values are determined in an adaptive manner based on separate estimations of the failure probability. Any computational reliability method, including first-order reliability and second-order reliability methods and Monte Carlo simulation, can be used for this purpose. After repeatedly solving the approximating problem, an approximate solution of the original design problem is found, which satisfies the failure probability constraints at a precision level corresponding to the selected reliability method. The approach is illustrated by a series of examples involving optimal design and maintenance planning of a reinforced concrete bridge girder.

Maintenance Planning for Deteriorating Concrete Bridges

In an effort to prioritize the repair, rehabilitation, and replacement of existing bridges, bridge management programs have been developed. In recent years, reliability-based optimum-maintenance planning strategies applicable to bridges have been proposed. Optimum inspection and repair times can be identified using these strategies. In this study, the effect of repair on time-variant failure probability is illustrated for several repair/replacement strategies. Time-variant reliability computations are performed using a combined technique of adaptive importance sampling and numerical integration. Several repair strategies are investigated for a typical bridge in Colorado. For this bridge, it is shown that for a single repair performed about halfway through the design life, the postrepair strength must be at least 80% of the original strength for the repair to have a significant influence on the system failure proability of the bridge during its remaining service life. Also, for maintenance using shotcrete, multiple repairs appear to have little influence on the lifetime system failure probability. Also presented are optimal maintenance strategies in which optimal inspection/repair times are identified based on minimum expected cost within prescribed reliability constraints. It is shown that the optimum repair time is dependent on the failure cost of the bridge and that the discount rate influences the optimum repair time and life-cycle cost. As the discount rate increases, the optimum repair time increases until it reaches an upper bound that is dependent on the system failure probability constraint. The results can be applied to the prediction of optimal lifetime maintenance planning strategies for concrete bridges under corrosion.

Inspection Planning of Fatigue Deteriorating Structures Using Genetic Algorithm

Genetic Algorithm (GA) is applied to obtain optimal inspection plan for fatigue deteriorating structures. The optimization problem is defined so as to minimize inspection cost in the life-time of the structure under the constraint of failure probability. Optimization parameters are the inspection timing and the inspection quality. The inspection timing is selected from the discrete intervals such as one year, two years, three years, etc. The inspection quality is selected from the followings ; no inspection, normal inspection, sampling inspection or precise inspection.The applicability of the proposed GA approach is demonstrated through the numerical calculations assuming a single member set and a structure consisting of several member sets. Special elite selection at reproduction and multiple point crossover method are applied to increase the efficiency of optimization. The influence of population size, probability of mutation and probability of crossover on GA are also investigated. Markov Chain Model is employed to calculate fatigue reliability as well as inspection effect. Through the numerical calculations, it is made clear that GA approach is a very useful tool at the optimization of inspection planning.

Optimum Design of 2-Dimensional Continuum Considering Reliability.

In design environment, most of variables and parameters such as loads, strength, sizes and material constants have some randomness and/or in many cases designers can't know exact values of those variables. In such a case, the concept of reliability shoud be included in the design object instead of conventional safety factor. This paper proposes a problem of optimum shape design for 2 dimensional continuum, taking the reliability concept into consideration. When sizes, strengths and outer forces vary according to the normal distribution, such a shape is determined that minimizes the average volume under the constraint of failure probability. In the body with the obtained shape, the probability of failure is uniform.

Optimum structural design with failure probability constraints.

Optimum structural design with failure probability constraints.