An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

The relationship between the reliability probabilities of the component and the system is hard to get. If this relationship can be obtained easily, the reliability of the system can be calculated by using the reliability structures of the components. The common method to express this relationship is using the linear correlation index, which only shows the linear correlation between the components failure rather than the relationship between high nonlinear functions. In order to describe this relationship accurately and calculate the system reliability using the component reliability structures, a Uniform Design (UD)-Saddlepoint Approximation (SA)-based system reliability analysis method is proposed. The system reliability analysis method is decomposed to three simple steps: (1) calculating the weight coefficient which represents the contribution rate of each component to system reliability, (2) approximating Cumulant Generation Function (CGF) of each component, (3) calculating CGF of the system and approximating the system reliability with SA method. The weight coefficient of each component is derived from UD method, and a variable interval selection method is developed to decrease the required number of samples and increase the accuracy of the weight coefficients. First-Order Saddlepoint Approximation (FOSA) method or Mean-Value First-Order Saddlepoint Approximation (MVFOSA) method is used to analyze the CGF of a component performance function. Then the CGF of the system can be obtained by the weighted addition law by combining the CGFs of components performance functions with the weight coefficients. Finally, the system reliability can be approximated by SA method. Four examples are employed to demonstrate that the new method outperforms other methods for system reliability analysis in terms of efficiency and accuracy.

An improved high order moment-based saddlepoint approximation method for reliability analysis

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bounds of system failure probability. The second method is SA based Nataf approximation, in which SA is used to estimate the probability density function (PDF) and cumulative distribution function (CDF) for the approximately linearized performance function of each failure mode. After the PDF of each failure mode and the correlation coefficients among approximately linearized performance functions are estimated, Nataf distribution is employed to approximate the joint PDF of multiple structural system performance functions, and then the system failure probability can be estimated directly by numerical simulation using the joint PDF. The third method is SA based line sampling (LS). The standardization transformation is needed to eliminate the dimensions of variables firstly in this case. Then LS method can express the system failure probability as an arithmetic average of a set of failure probabilities of the linear performance functions, and the probabilities of the linear performance functions can be estimated by the SA in the non-normal variables space. By comparing basic concepts, implementations and results of illustrations, the following conclusions can be drawn: (1) The first method can only obtain the bounds of system failure probability and it is only acceptable for the linear limit state function; (2) the second method can give the estimation of system failure probability, and its error mostly results from the approximation of Nataf distribution for the joint PDF of the structural system performance functions and the linearization of the performance functions; (3) the SA based LS method can obtain the estimator of system failure probability, which converges to the actual value along with the increase of sample size. The SA based LS method considers the influence of nonlinearity of limit state function on the failure probability, and it is acceptable for the structural system both with a single failure mode and with multiple failure modes, therefore it has the widest applicability.

The optimal design of multidisciplinary systems under uncertainty involves three types of iterative analyses- multidisciplinary analyses, reliability analyses, and optimization. If these processes are carried out in a nested manner in reliability based multidisciplinary design optimization (RBMDO), the computational expense can be enormous. Computationally efficient simultaneous analysis and design (SAND) or All-AtOnce methods have been developed earlier for deterministic MDO. Single loop RBDO methods have also been developed to integrate reliability analysis and optimization in a single loop. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints. The formulation allows for solution with a single design optimization loop by using the KKT optimality conditions of the first-order reliability method (FORM) as constraints in the design optimization along with disciplinary consistency constraints. Alternative solution processes which remove the FORM MPP’s from the decision space and many constraints from the problem are developed. The alternative solution methods are more efficient and more robust than using nonlinear programming to solve the single loop formulation. A numerical example is solved with the two proposed solution algorithms. I. Introduction In the design of multidisciplinary systems it is often necessary to make decisions under uncertainty. This requires that multidisciplinary systems modeling be integrated with uncertainty analysis and optimization. A common example of such an engineering design application would be the design of an aircraft wing under uncertain aerodynamic loading conditions, with multiple structural components with uncertain capacities, such that the entire structural system achieves a prescribed level of reliability. The problem has coupled physics, in that the pressure distribution is dependent upon the deflection, and the deflection is dependent on the pressure distribution. Such designs have been performed using multidisciplinary optimization (MDO). Other examples where MDO may be applied include electronics packaging, in which the disciplines of heat transfer and circuit analysis are coupled, the design of new types of aircraft and integrating them into an existing fleet of aircraft, and even the management of interconnected homeland infrastructures. While MDO has matured with the development of different types of algorithms for various types of problems and design organizations, there has been little work to incorporate reliability analysis into MDO. However, system and component reliabilities are often a major concern in the design of multidisciplinary systems. MDO can be computationally expensive due to the need to achieve consistency across multiple disciplinary analysis codes; each disciplinary analysis code may be computationally expensive to run. Further, the optimal design of multidisciplinary systems under uncertainty involves three types of iterative procedures- multidisciplinary analyses, reliability analyses, and optimization. The integration of these three procedures is referred to in this paper as reliability based multidisciplinary design optimization (RBMDO). If these processes are carried out simultaneously, the computational expense can be enormous. Computational savings can often be achieved with simultaneous analysis and design (SAND) or All-At-Once methods, which allow for the solution of the multidisciplinary analyses and the optimization in one single loop. Single loop RBDO methods allow for the integration of reliability analysis and optimization in a single loop. Applying SAND and single-loop RBDO together may allow for large computational savings when applied to problems in RBMDO. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints based on the SAND method for MDO and by use of the KKT conditions of the first order reliability method (FORM). The resulting formulation may be solved as a standard nonlinear programming problem. While the single loop formulation presented in this paper is a solution of the RBMDO problem, robustness and efficiency are improved through two improved algorithms which can remove the MPP’s from the decision space, thereby significantly reducing the optimizer’s workload.

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

Dynamic Model-based Saddle-point Approximation for Reliability and Reliability-based Sensitivity Analysis

Purpose – The purpose of this paper is to study the consideration of uncertainty from analysis modules for aircraft conceptual design by implementing uncertainty-based design optimization methods. Reliability-Based Design Optimization (RBDO), Possibility-Based Design Optimization (PBDO) and Robust Design Optimization (RDO) methods were developed to handle uncertainties of design optimization. The RBDO method is found suitable for uncertain parameters when sufficient information is available. On the other hand, the PBDO method is proposed when uncertain parameters have insufficient information. The RDO method can apply to both cases. The RBDO, PBDO and RDO methods were considered with the Multidisciplinary Design Optimization (MDO) method to generate conservative design results when low fidelity analysis tools are used. Design/methodology/approach – Methods combining MDO with RBDO, PBDO and RDO were developed and have been applied to a numerical analysis and an aircraft conceptual design. This research eva...

Reliability-based design optimization is much more computationally expensive than deterministic design optimization. To alleviate the computational demand, the First Order Reliability Method (FORM) is usually used in reliability-based design. Since FORM requires a nonlinear transformation from non-normal random variables to normal random variables, the nonlinearity of a constraint function may increase. As a result, the transformation may lead to a large error in reliability calculation. In order to improve accuracy, a new reliability-based design method with Saddlepoint Approximation is proposed in this work. The strategy of sequential optimization and reliability assessment is employed where the reliability analysis is decoupled from deterministic optimization. The accurate First Order Saddlepoint method is used for reliability analysis in the original random space without any transformation, and the chance of increasing nonlinearity of a constraint function is therefore eliminated. The overall reliability-based design is conducted in a sequence of cycles of deterministic optimization and reliability analysis. In each cycle, the percentile value of the constraint function corresponding to the required reliability is calculated with the Saddlepoint Approximation at the optimal point of the deterministic optimization. Then the reliability analysis results are used to formulate a new deterministic optimization model for the next cycle. The solution process converges within a few cycles. The demonstrative examples show that the proposed method is more accurate and efficient than the reliability-based design with FORM.

Uncertainty analysis, which assesses the impact of the uncertainty of input random variables on performance functions, is an important and indispensable component in engineering design under uncertainty. In this paper, a simulation method based on the Saddlepoint Approximation (SPA) is proposed to estimate accurately and efficiently the distribution of a response variable. The proposed method combines both simulation and analytical techniques and involves three main steps: (1) sampling on input random variables, (2) approximating the cumulant generating function (cgf) of the response variable with its first four cumulants and (3) estimating the cumulant distribution function (cdf) and probability density function (pdf) of the response variable using the SPA. This method provides more computationally efficient solutions than the general Monte Carlo Simulation (MCS) while maintaining high accuracy. The effectiveness of the proposed method is illustrated with a mathematical example and two engineering analysis problems.

This study is concerned with uncertainty analysis of MELCOR simulation of a hypothetical severe accident initiated by station blackout (SBO) in a Swedish pressurized water reactor (PWR). 9 input parameters are chosen, and 12 safety-related output parameters are selected as the figures of merit (FOMs). In random sampling (RS) method, 800 MELCOR cases are run to produce empirical cumulative distribution functions (CDFs) and empirical 95th percentiles of FOMs. Given this sufficient sample size, uncertainty analyses through statistical analysis, can be performed to obtain the first two statistical moments and 95/95 estimates from the first order Wilks’ method. However, RS method including Wilks’ method turns out to be time consuming and computationally expensive since many MELCOR cases require iterative tuning of MELCOR input to restart and finish calculations. To overcome this issue encountered in RS method, in the present study three deterministic sampling (DS) methods are applied to uncertainty analyses, and a coupled approach by combining DS methods with a coverage factor 1.65 is proposed. Comparable results show that DS methods can generally capture the first two statistical moments quickly, and acceptably accurate 95th percentiles can be calculated by the coupled approach when the output parameters can be described as normal distributions. Besides, estimated 95th percentiles from the coupled approach are covered by estimates (boxplots) from the first order Wilks’ method. Hence, it can be concluded that the coupled approach has its potential to work as an alternative in engineering to RS method including Wilks’ method for numerically equivalent estimates of 95/95 tolerance limit with an acceptable accuracy and a significantly less computational cost in MELCOR severe accident simulation.

Uncertainties, including aleatory and epistemic uncertainties, always exist in multidisciplinary system. Due to the discontinuous nature of epistemic uncertainty and the complex coupled relation among subsystems, the computational efficiency of reliability-based multidisciplinary design optimization (RBMDO) with mixed aleatory and epistemic uncertainties is extremely low. A novel RBMDO procedure is presented in this paper based on combined probability theory and evidence theory (ET) to deal with hybrid-uncertainties and improve the computational efficiency. Firstly, based on Bayes method, a novel method to define the probability density function of the aleatory variables is proposed. Secondly, the conventional equivalent normal method (J-C method) is modified to reliability analysis with hybrid-uncertainties. Finally, a novel RBMDO procedure is suggested by integrating the modified J-C method into the frame of sequence optimization and reliability analysis (SORA). Numerical examples and engineering example are applied to demonstrate the performance of the proposed method. The examples show the excellence of the RBMDO method both in computational efficiency and accuracy. The proposed method provides a practical and effective reliability design method for multidisciplinary system.

<p><span>Measurement uncertainty analysis based on combining the state-of-knowledge distributions requires evaluation of the probability density function (PDF), the cumulative distribution function (CDF), and/or the quantile function (QF) of a random variable reasonably associated with the measurand. This can be derived from the characteristic function (CF), which is defined as a Fourier transform of its probability distribution function. Working with CFs provides an alternative and frequently much simpler route than working directly with PDFs and/or CDFs. In particular, derivation of the CF of a weighted sum of independent random variables is a simple and trivial task. However, the analytical derivation of the PDF and/or CDF by using the inverse Fourier transform is available only in special cases. Thus, in most practical situations, a numerical derivation of the PDF/CDF from the CF is an indispensable tool. In metrological applications, such approach can be used to form the probability distribution for the output quantity of a measurement model of additive, linear or generalized linear form. In this paper we propose new original algorithmic implementations of methods for numerical inversion of the characteristic function which are especially suitable for typical metrological applications. The suggested numerical approaches are based on the Gil-Pelaez inverse formulae and on using the approximation by discrete Fourier transform and the fast Fourier transform (FFT) algorithm for computing PDF/CDF of the univariate continuous random variables. As illustrated here, for typical metrological applications based on linear measurement models the suggested methods are an efficient alternative to the standard Monte Carlo methods.</span></p>

Постановка задачi. З розвитком технологiй радiозв’язку, особливо, мобiльного, усе бiльш важливу
\nроль вiдiграє розробка моделей радiоканалiв. У данiй
\nстаттi розглядається одна з популярних нинi моделей,
\nа саме, двопроменевий радiоканал з розсiюванням потужностi сигналу, яку в лiтературi прийнято позначати
\nабревiатурою TWDP (вiд Two-Wave with Diffuse Power).
\nДля цiєї моделi вiдомi як точнi, так i наближенi вирази
\nдля густини розподiлу iмовiрностей (ГРI) i кумулятивної функцiї розподiлу (КФР) обвiдної сигналу. Однак,
\nзагальним їх недолiком є висока складнiсть з точки зору практичного застосування. У зв’язку iз цим у данiй
\nроботi розглядається можливiсть значно бiльш простої
\nапроксимацiї ГРI та КФР обвiдної сигналу на основi
\nметоду Накагами.
\nРозробка методу. Апроксимацiя розподiлiв по методу Накагами зводиться до обчислення другого та
\nчетвертого початкових моментiв обвiдної сигналу на
\nвиходi каналу. Такi обчислення й виконанi в роботi,
\nщо дозволяє представити зазначенi моменти у виглядi
\nфункцiй вiд двох загальноприйнятих параметрiв каналу
\nTWDP: К-фактора та параметра Д, який визначається
\nспiввiдношенням амплiтуд радiо променiв.
\nАналiз похибок апроксимацiї. Похибки оцiнювалися шляхом зiставлення розрахункiв ГРI i КФР за
\nточними формулами та розрахункiв iз використанням
\nапроксимацiї Накагами. Паралельно також проводилися розрахунки вiдповiдно до формул апроксимацiї, що
\nвiдомi у лiтературi. Аналiз показав, що похибка методу
\nНакагами при обчисленнi КФР є цiлком задовiльною
\nдля практичних потреб майже у всьому дiапазонi змiн
\nпараметрiв каналу. У той же час точнiсть апроксимацiї
\nГРI у порiвняннi з iншими вiдомими методами апроксимацiї виявилася не дуже задовiльна.
\nВисновки. Проведенi дослiдження пiдтвердили можливiсть апроксимацiї КФР обвiдної сигналу в каналi TWDP розподiлом Накагами. Результати такої ж
\nапроксимацiї для ГРI виявилися менш задовiльними.
\nТакож вiдзначається кориснiсть отриманих результатiв
\nдля трактування даних натурних експериментiв.

Oil recovery forecast relies on predictions of saturation fields from reservoir simulations. These predictions, in return, are dependent on the quality of the input data available, such as the reservoir permeability and porosity fields. However, because of inherent uncertainties in these input data, due to sparse and possibly inaccurate measurements, a stochastic framework is usually adopted to describe their variability. Probability distributions are then assigned to the input fields. In this stochastic context, predicting the saturation fields consists in determining the saturation probability distributions. For nonlinear immiscible two-phase flow in highly heterogeneous porous media, we propose an efficient distribution method to estimate the saturation cumulative distribution function (CDF) and probability density function (PDF). While high heterogeneity is traditionally regarded as an obstacle to designing stochastic methods, this distribution method exploits the physics of highly heterogeneous porous media. The method identifies hidden smooth random fields (RFs) that explain the saturation distribution, together with the nonlinear mapping from these hidden RFs to the saturation. We show that the statistics and distributions of these hidden RFs can be estimated very efficiently and simply, leading to a method that is computationally inexpensive compared to full Monte Carlo (MC) simulations. We provide numerical evidence of the agreement between MC-based and distribution-based saturation CDF, PDF and moments, in high heterogeneous cases such as high variance lognormal permeability fields, anisotropic layered fields and channelized systems. Finally, we show how the saturation CDF can be used for risk assessment by relying not only on the mean and standard deviation of the saturation, but primarily on saturation quantiles, obtained from probability of exceedance.

This paper proposes a first-order geometric autoregressive model based on a new thinning operator. The parameters of the model are estimated by the method of Two-Stage Conditional Least Squares and Saddlepoint Approximation. Simulation studies demonstrate that the saddlepoint approximation estimator performs well, even in the case of small sample sizes. To illustrate the significance of the model, a real data analysis of the number of downloads of a TeX editor is presented.