Eigenvector Dimension Reduction Method Research Articles

Eigenvector dimension-reduction method for reliability analysis of power systems under the time-dependent load uncertainty

ABSTRACT Power systems have become central pillars of modern society. Various uncertainties in power systems, however, make reliability analysis difficult. To date, the worst-case scenario was usually employed to tackle uncertainties in the system, which could make the overall system design prohibitively expensive especially when more and more uncertainty sources would be considered in modern power grids. This paper proposes reliability analysis of power systems under the time-dependent load uncertainty using an effective uncertainty quantification (UQ) method, i.e., the eigenvector dimension reduction (EDR) method. As such, reliability can be explicitly calculated in the probability manner for meeting the performance constraints without significantly overdesigning the system. Compared to the Monte Carlo simulation (MCS), the EDR method is much more efficient with satisfactory accuracy. Two case studies, including a 12-bus and a modified IEEE 118-bus power system, are used to demonstrate the feasibility of the proposed work.

Stacking Yield Prediction of Package-on-Package Assembly Using Uncertainty Propagation Analysis—Part II: Implementation of Stochastic Model

The stochastic model for yield loss prediction proposed in Part I is implemented for a package-on-package (PoP) assembly. The assembly consists of a stacked die thin flat ball grid array (TFBGA) as the top package and a flip chip ball grid array (fcBGA) as the bottom package. The top and bottom packages are connected through 216 solder joints of 0.5 mm pitch in two peripheral rows. The warpage values of the top and bottom package are calculated by finite element analysis (FEA), and the corresponding probability of density functions (PDFs) are obtained by the eigenvector dimension reduction (EDR) method. The solder ball heights of the top and bottom package and the corner pad joint heights are determined by surface evolver, and their PDFs are determined by the EDR method, too. Only 137 modeling runs are conducted to obtain all 549 PDFs in spite of the large number of input variables considered in the study (27 input variables). Finally, the noncontact open-induced staking yield loss of the PoP assembly is predicted from the PDFs.

Predictive model of dynamic properties of elastomers in thermal degradation environment

In dynamic systems, numerous structural components, such as mounts, bushings, and tires, deteriorate over time due to environmental conditions. In these elastomers, the typical environmental factor of aging is manifested by the oxidation of polymer chains due to exposure to air. The amount of oxidation in elastomers heavily depends on temperature. Elastomer components under thermal degradation show different dynamic characteristics over time. Predicting the mechanical properties of aged elastomers is crucial in structural design for enduring high-quality noise and vibration. In this work, we propose a procedure for predicting the dynamic properties of elastomers undergoing thermal degradation. This process is divided into 1) an inter-variability analysis in specific degradation states caused by uncertainties in the operational temperature environment, variation in the manufacturing process, and error of the material property model and 2) an analysis of the degradation variability caused by uncertainties in the degradation environment and the degradation model. The degradation variability includes the inter-variability and thus becomes a distribution of distributions. Therefore, the double-loop eigenvector dimension reduction method is proposed to predict degradation variability. The time–temperature superposition principle and an equivalent degradation time are introduced to accumulate the degradation states in case of thermal degradation at various temperatures. The proposed method is applied to a circular mount problem. The numerical results illustrate that the proposed method effectively predicts the aging of the dynamic elastomer properties that is due to the thermal environment.

Probabilistic Lifetime Prediction of Electronic Packages Using Advanced Uncertainty Propagation Analysis and Model Calibration

We propose a novel methodology for calibrating the physics-based lifetime models of the electronic packages using the eigenvector dimension-reduction (EDR) method and a censored data analysis. The methodology enables to overcome two challenges that are encountered in typical electronic packaging applications: 1) the minimum computational cost without sacrificing the prediction accuracy and 2) the proper handling of the censored data. The EDR method is first employed for uncertainty propagation for the computational efficiency when multiple unknown variables are to be used in nonlinear damage models. Next, the likelihood function is modified to handle the failure data as well as the censored data in the likelihood analysis, and thus establishes the correlation between the model response and the experimental result. Finally, through an unconstrained optimization process, a calibrated parameter set of statistical distributions for unknown input variables is obtained while maximizing the modified likelihood. The proposed statistical calibration approach is implemented for solder joint fatigue reliability. The results confirm the claimed computational effectiveness for an accurate physics-based lifetime model.

Variability analysis of vibrational responses in a passenger car considering the uncertainties of elastomers

Passenger cars have many elastomeric joints that are used to reduce the vibration transmission from the source to the cabin structure. In the design stage, the dynamic characteristics of the elastomeric joints are optimally determined in order to satisfy the design goals for interior noise and vibration. However, the material properties of the elastomeric joints have large variations due to production variability. In addition, operational conditions, e.g. environmental temperature, vary according to the locations of the car. As a result, the vibrational comfort of a passenger car exhibits large variations. In this paper, the amount of vibration fluctuation due to uncertain elastomeric joint parameters is estimated using a statistical approach. First, the dynamic characteristics of the elastomers are described using the fractional derivative model. The uncertainties of the fractional derivative model parameters of the elastomers are characterized using a statistical calibration approach based on specimen test data. Then, a finite element model for the elastomers and the passenger car structure are constructed in order to calculate the vibrational response. In order to estimate the variability of the vibrational response due to the uncertainties of the elastomers, uncertainty propagation analyses are conducted using the eigenvector dimension reduction (EDR) method. The operational conditions such as temperature are also included in the variability analysis. The performance of the EDR method is assessed through comparing the estimated response distribution with that of the Monte Carlo simulation (MCS) method in a simplified structure. The variability analysis results demonstrate that the vibrational response variation due to the uncertainties of the elastomers can be efficiently predicted using the proposed method. The proposed variability analysis procedure could be an effective tool in the design stage for quality control of passenger car comfort.

Predictive carbon nanotube models using the eigenvector dimension reduction (EDR) method

It has been reported that a carbon nanotube (CNT) is one of the strongest materials with its high failure stress and strain. Moreover, the nanotube has many favorable features, such as high toughness, great flexibility, low density, and so on. This discovery has opened new opportunities in various engineering applications, for example, a nanocomposite material design. However, recent studies have found a substantial discrepancy between computational and experimental material property predictions, in part due to defects in the fabricated nanotubes. It is found that the nanotubes are highly defective in many different formations (e.g., vacancy, dislocation, chemical, and topological defects). Recent parametric studies with vacancy defects have found that the vacancy defects substantially affect mechanical properties of the nanotubes. Given random existence of the nanotube defects, the material properties of the nanotubes can be better understood through statistical modeling of the defects. This paper presents predictive CNT models, which enable to estimate mechanical properties of the CNTs and the nanocomposites under various sources of uncertainties. As the first step, the density and location of vacancy defects will be randomly modeled to predict mechanical properties. It has been reported that the eigenvector dimension reduction (EDR) method performs probability analysis efficiently and accurately. In this paper, molecular dynamics (MD) simulation with a modified Morse potential model is integrated with the EDR method to predict the mechanical properties of the CNTs. To demonstrate the feasibility of the predicted model, probabilistic behavior of mechanical properties (e.g., failure stress, failure strain, and toughness) is compared with the precedent experiment results.

Random Field Characterization Considering Statistical Dependence for Probability Analysis and Design

The proper orthogonal decomposition method has been employed to extract the important field signatures of random field observed in an engineering product or process. Our preliminary study found that the coefficients of the signatures are statistically uncorrelated but may be dependent. To this point, the statistical dependence of the coefficients has been ignored in the random field characterization for probability analysis and design. This paper thus proposes an effective random field characterization method that can account for the statistical dependence among the coefficients for probability analysis and design. The proposed approach has two technical contributions. The first contribution is the development of a natural approximation scheme of random field while preserving prescribed approximation accuracy. The coefficients of the signatures can be modeled as random field variables, and their statistical properties are identified using the chi-square goodness-of-fit test. Then, as the paper’s second technical contribution, the Rosenblatt transformation is employed to transform the statistically dependent random field variables into statistically independent random field variables. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Hence, this paper proposes a novel procedure of determining an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity into the system response. The proposed random field characterization can be integrated with any advanced probability analysis method, such as the eigenvector dimension reduction method or polynomial chaos expansion method. Three structural examples, including a microelectromechanical system bistable mechanism, are used to demonstrate the effectiveness of the proposed approach. The results show that the statistical dependence in the random field characterization cannot be neglected during probability analysis and design. Moreover, it is shown that the proposed random field approach is very accurate and efficient.

Optimal Design of Constrained-Layer Damping Structures Considering Material and Operational Condition Variability

of a surface treatment is highly sensitive to the variability in operational temperature. This paper proposes a statistical approachto modelthe variabilityof viscoelastic damping material in aconstrained-layer dampinglayout, to predict variability in the dynamic responses of viscoelastic damping material, and to obtain an optimal robust layout that accounts for severe variability in operational temperature. The viscoelastic damping material property can be modeled as a sum of 1) a random complex modulus due to operational temperature variability and 2) experiment/model errors in the complex modulus. The eigenvector dimension reduction method is used in probability analysis to predict the variability in the dynamic responses of the viscoelastic damping material. It is concluded that temperature variability is strongly propagated to that in the dynamic responses of the damping material. This study also performs reliability-based design optimization for an optimal robust design of the constrained-layer damping structure. It is shown that reliability-based design optimization gives a more robust and reliable damping layout design amidst severe variability in operational temperature.

Complementary Intersection Method for System Reliability Analysis

Although researchers desire to evaluate system reliability accurately and efficiently over the years, little progress has been made on system reliability analysis. Up to now, bound methods for system reliability prediction have been dominant. However, two primary challenges are as follows: (1) Most numerical methods cannot effectively evaluate the probabilities of the second (or higher)–order joint failure events with high efficiency and accuracy, which are needed for system reliability evaluation and (2) there is no unique system reliability approximation formula, which can be evaluated efficiently with commonly used reliability methods. Thus, this paper proposes the complementary intersection (CI) event, which enables us to develop the complementary intersection method (CIM) for system reliability analysis. The CIM expresses the system reliability in terms of the probabilities of the CI events and allows the use of commonly used reliability methods for evaluating the probabilities of the second–order (or higher) joint failure events efficiently. To facilitate system reliability analysis for large-scale systems, the CI-matrix can be built to store the probabilities of the first- and second-order CI events. In this paper, three different numerical solvers for reliability analysis will be used to construct the CI-matrix numerically: first-order reliability method, second-order reliability method, and eigenvector dimension reduction (EDR) method. Three examples will be employed to demonstrate that the CIM with the EDR method outperforms other methods for system reliability analysis in terms of efficiency and accuracy.

Reliability-based robust design optimization using the eigenvector dimension reduction (EDR) method

This paper presents an effective methodology for reliability-based robust design optimization (RBRDO). The eigenvector dimension reduction (EDR) method plays a pivotal role in making RBRDO effective because the EDR method turns out to be very efficient and accurate for probability analysis. The use of the EDR method provides three benefits to RBRDO. First, an approximate response surface facilitates sensitivity calculation of reliability and quality where the response surface is constructed using the eigenvector samples. Thus, sensitivity analysis becomes very efficient and simple. Second, one EDR execution evaluates a set of quality (objective) and reliability (constraint) functions. In general, the EDR requires 2N + 1 or 4N + 1 simulation runs where N is the total number of random variables. The EDR execution does not require an iterative process, so the proposed RBRDO methodology has a single-loop structure. Moreover, the EDR execution time can be much shorter by taking advantage of a parallel computing power, and RBRDO can be far more efficient. Third, the EDR method allows solving problems with statistically correlated and non-normally distributed random inputs. Three practical engineering problems are used to demonstrate the effectiveness of the proposed RBRDO method using the EDR method.

Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis

This paper presents the eigenvector dimension reduction (EDR) method for probability analysis that makes a significant improvement based on univariate dimension reduction (DR) method. It has been acknowledged that the DR method is accurate and efficient for assessing statistical moments of mildly nonlinear system responses in engineering applications. However, the recent investigation on the DR method has found difficulties of instability and inaccuracy for highly nonlinear system responses while maintaining reasonable efficiency. The EDR method integrates the DR method with three new technical components: (1) eigenvector sampling, (2) one-dimensional response approximation, and (3) a stabilized Pearson system. First, 2N+1 and 4N+1 eigenvector sampling schemes are proposed to resolve correlated and asymmetric random input variables. The eigenvector samples are chosen along the eigenvectors of the covariance matrix of random parameters. Second, the stepwise moving least squares (SMLS) method is proposed to accurately construct approximate system responses along the eigenvectors with the response values at the eigenvector samples. Then, statistical moments of the responses are estimated through recursive numerical integrations. Third, the stabilized Pearson system is proposed to predict probability density functions (PDFs) of the responses while eliminating singular behavior of the original Pearson system. Results for some numerical and engineering examples indicate that the EDR method is a very accurate, efficient, and stable probability analysis method in estimating PDFs, component reliabilities, and qualities of system responses.

Bayesian reliability-based design optimization using eigenvector dimension reduction (EDR) method

In practical engineering design, most data sets for system uncertainties are insufficiently sampled from unknown statistical distributions, known as epistemic uncertainty. Existing methods in uncertainty-based design optimization have difficulty in handling both aleatory and epistemic uncertainties. To tackle design problems engaging both epistemic and aleatory uncertainties, reliability-based design optimization (RBDO) is integrated with Bayes theorem. It is referred to as Bayesian RBDO. However, Bayesian RBDO becomes extremely expensive when employing the first- or second-order reliability method (FORM/SORM) for reliability predictions. Thus, this paper proposes development of Bayesian RBDO methodology and its integration to a numerical solver, the eigenvector dimension reduction (EDR) method, for Bayesian reliability analysis. The EDR method takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM. Efficiency and accuracy of the Bayesian RBDO process are substantially improved after this integration.