Uncertainty Quantification in Metamodel-Based Reliability Prediction

2016 ◽  
Author(s):  
Saideep Nannapaneni ◽  
Zhen Hu ◽  
Sankaran Mahadevan
Keyword(s):  
Design Optimization ◽  
Reliability Analysis ◽  
Limit State ◽  
Random Variables ◽  
Point Load ◽  
System Model ◽  
Robust Design Optimization ◽  
Statistical Uncertainty ◽  
Input Variables ◽  
Metamodel Uncertainty

Optimization under uncertainty has been studied in two directions — (1) Reliability-based Design Optimization (RBDO), and (2) Robust Design Optimization (RDO). One of the crucial elements in an RBDO problem is reliability analysis. Reliability analysis is affected by different types of epistemic uncertainty, due to inadequate data and modeling errors, along with aleatory uncertainty in input random variables. When the original physics-based model is computationally expensive, a metamodel has often been used in reliability analysis, introducing additional uncertainty due to the metamodel. This work presents a framework to include statistical uncertainty and model uncertainty in metamodel-based reliability analysis. Inadequate data causes uncertainty regarding the statistics (distribution types and distribution parameters) of the input variables, and regarding the system model parameters. Model errors include model form errors, solution approximation errors, and metamodel uncertainty. Two types of metamodels have been considered in literature for reliability analysis: (1) metamodels that compute the system model output over the desired ranges of the input random variables; and (2) metamodels that concentrate only on modeling the limit state. This work focuses on the latter type, using Gaussian process (GP) metamodels for performing both component reliability (single limit state) and system reliability (multiple limit states) analyses. A systematic procedure for the inclusion of model discrepancy terms in the limit-state metamodel construction is developed using an auxiliary variable approach. An efficient single-loop sampling approach using the probability integral transform is used for sampling the input variables with statistical uncertainty. The variability in the GP model prediction (metamodel uncertainty) is also included in reliability analysis through correlated sampling of the model predictions at different inputs. Two mechanical systems — a cantilever beam with point-load at the free end and a two-bar supported panel with point load at its center, are used to demonstrate the proposed techniques.

scholarly journals An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties

Algorithms ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 229
Author(s):  
Fangyi Li ◽  
Yufei Yan ◽  
Jianhua Rong ◽  
Houyao Zhu
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Random Variables ◽  
Equivalent Model ◽  
Independent Random Variables ◽  
Problem Analysis ◽  
Analysis Method ◽  
Calculation Algorithm ◽  
Reliability Problem ◽  
Interval Variables

In practical engineering, due to the lack of information, it is impossible to accurately determine the distribution of all variables. Therefore, time-variant reliability problems with both random and interval variables may be encountered. However, this kind of problem usually involves a complex multilevel nested optimization problem, which leads to a substantial computational burden, and it is difficult to meet the requirements of complex engineering problem analysis. This study proposes a decoupling strategy to efficiently analyze the time-variant reliability based on the mixed uncertainty model. The interval variables are treated with independent random variables that are uniformly distributed in their respective intervals. Then the time-variant reliability-equivalent model, containing only random variables, is established, to avoid multi-layer nesting optimization. The stochastic process is first discretized to obtain several static limit state functions at different times. The time-variant reliability problem is changed into the conventional time-invariant system reliability problem. First order reliability analysis method (FORM) is used to analyze the reliability of each time. Thus, an efficient and robust convergence hybrid time-variant reliability calculation algorithm is proposed based on the equivalent model. Finally, numerical examples shows the effectiveness of the proposed method.

Robust Design Optimization of Ball Grid Array Packaging

2011 ◽  
Author(s):  
Jae Chang Kim ◽  
Joo-Ho Choi ◽  
Yeong K. Kim
Keyword(s):  
Design Optimization ◽  
Robust Design ◽  
Strain Energy ◽  
Reduction Method ◽  
Ball Grid Array ◽  
Robust Design Optimization ◽  
Molding Compound ◽  
Materials Modeling ◽  
Solder Balls ◽  
Input Variables

In this paper, comparisons of the design optimization of ball grid array packaging geometry based on the elastic and viscoelastic material properties are made. Six geometric dimensions of the packaging are chosen as input variables. Molding compound and substrate are modeled as elastic and viscoelastic, respectively. Viscoplastic finite element analyses are performed to calculate the strain energy densities (SED) of the eutectic solder balls. Robust design optimizations to minimize SED are carried out, which accounts for the variance of the parameters via Kriging dimension reduction method. Optimum solutions are compared with those by the Taguchi method. It is found that the effects of the packaging geometry on the solder ball reliability are significant, and the optimization results are different depending on the materials modeling.

STRUCTURAL RELIABILITY ANALYSIS OF STEEL TRUSS ELEMENTS ON BUCKLING USING PBOX APPROACH

2021 ◽  
pp. 45-53
Author(s):  
A.A. Solovyova ◽  
◽  
S.A. Solovyov ◽  
Keyword(s):  
Reliability Analysis ◽  
Structural Reliability ◽  
Statistical Data ◽  
Limit State ◽  
Random Variables ◽  
Distribution Functions ◽  
Structural Safety ◽  
Random Load ◽  
Load Models ◽  
Steel Trusses

Abstract. The reliability of load-bearing structural elements is one of the indicators of structural safety. The article presents methods for steel trusses bars reliability analysis according to the buckling criterion using p-boxes. A p-box consists of two boundary probability distribution functions that form the area of possible distribution functions. Such model used for modeling random variables in conditions of incomplete statistical data by quantity or quality. An algorithm for summing p-boxes of random load models is demonstrated on the example of a probabilistic estimate of the force in the truss bar. The result of reliability analysis using p-boxes is presented in interval form. The use of p-boxes makes it possible to obtain a more cautious assessment of reliability in case of incomplete statistical data. To increase the informativity of the reliability analysis result, it is necessary to obtain more statistical data about random variables in design mathematical models of limit state, which will allow forming p-boxes with narrower boundary distribution functions.

High-Dimensional Reliability Method Accounting for Important and Unimportant Input Variables

2021 ◽  
Author(s):  
Jianhua Yin ◽  
Xiaoping Du
Keyword(s):  
Dimension Reduction ◽  
Reliability Analysis ◽  
Limit State ◽  
Physical Models ◽  
Reliability Prediction ◽  
High Dimensional ◽  
Core Element ◽  
Reliability Method ◽  
Input Variables ◽  
Reduced Space

Abstract Reliability analysis is usually a core element in engineering design, during which reliability is predicted with physical models (limit-state functions). Reliability analysis becomes computationally expensive when the dimensionality of input random variables is high. This work develops a high dimensional reliability analysis method by a new dimension reduction strategy so that the contributions of both important and unimportant input variables are accommodated by the proposed dimension reduction method. The consideration of the contributions of unimportant input variables can certainly improve the accuracy of the reliability prediction, especially where many unimportant input variables are involved. The dimension reduction is performed with the first iteration of the first order reliability method (FORM), which identifies important and unimportant input variables. Then a higher order reliability analysis, such as the second order reliability analysis and metamodeling method, is performed in the reduced space of only important input variables. The reliability obtained in the reduced space is then integrated with the contributions of unimportant input variables, resulting in the final reliability prediction that accounts for both types of input variables. Consequently, the new reliability method is more accurate than the traditional method, which fixes unimportant input variables at their means. The accuracy is demonstrated by three examples.

scholarly journals Hybrid Reliability Analysis for Spacecraft Docking Lock with Random and Interval Uncertainty

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Jianguo Zhang ◽  
Jiwei Qiu ◽  
Pidong Wang
Keyword(s):  
Reliability Analysis ◽  
Structural Reliability ◽  
Limit State ◽  
Random Variables ◽  
Simple Problem ◽  
State Function ◽  
Limit State Function ◽  
Interval Variables ◽  
Reliability Method ◽  
Hybrid Reliability

This paper presents a novel procedure based on first-order reliability method (FORM) for structural reliability analysis with hybrid variables, that is, random and interval variables. This method can significantly improve the computational efficiency for the abovementioned hybrid reliability analysis (HRA), while generally providing sufficient precision. In the proposed procedure, the hybrid problem is reduced to standard reliability problem with the polar coordinates, where an n-dimensional limit-state function is defined only in terms of two random variables. Firstly, the linear Taylor series is used to approximate the limit-state function around the design point. Subsequently, with the approximation of the n-dimensional limit-state function, the new bidimensional limit state is established by the polar coordinate transformation. And the probability density functions (PDFs) of the two variables can be obtained by the PDFs of random variables and bounds of interval variables. Then, the interval of failure probability is efficiently calculated by the integral method. At last, one simple problem with explicit expressions and one engineering application of spacecraft docking lock are employed to demonstrate the effectiveness of the proposed methods.

General Robust Design by Semi-Second-Order Taylor Expansion

2008 ◽  
Author(s):  
Xiaoping Du
Keyword(s):  
Design Optimization ◽  
Robust Design ◽  
Taylor Expansion ◽  
Random Variables ◽  
Second Order ◽  
Robust Design Optimization ◽  
Integration Strategy ◽  
Interval Variables ◽  
Standard Deviations ◽  
Robustness Assessment

The purpose of robust design optimization is to minimize variations in design performances and therefore to make the design insensitive to uncertainties. Current robust design methods fall into two types — probabilistic robust design and worst-case (interval) robust design. The former method is used when random variables are involved. In this method, robustness is measure by standard deviations of design performances. The later method is used when uncertainties are represented by intervals. The widths of design performances are then used to measure robustness. In many engineering application, both random variables and interval variables may exist simultaneously. In this paper, a general approach to robust design optimization is proposed. The generality comes from the ability to handle both random and interval variables. To alleviate the computational burden, we employ a previously developed general robustness assessment method — semi-second-order Taylor expansion method, to evaluate the maximum and minimum standard deviations of a design performance. An efficient integration strategy of the general robustness assessment and optimization is proposed. With the integration strategy, the number of function calls can be reduced while good accuracy is maintained. A robust shaft design problem is given for demonstration.

Reliability Based Design Optimization With Correlated Input Variables Using Copulas

2007 ◽  
Author(s):  
Kyung K. Choi ◽  
Yoojeong Noh ◽  
Liu Du
Keyword(s):  
Reliability Analysis ◽  
Random Variables ◽  
Performance Measure ◽  
Gaussian Copula ◽  
Joint Probability Distribution ◽  
Random Input ◽  
Marginal Distributions ◽  
Wide Range ◽  
Input Variables ◽  
Nataf Transformation

For the performance measure approach (PMA) of RBDO, a transformation between the input random variables and the standard normal random variables is necessary to carry out the inverse reliability analysis. For reliability analysis, Rosenblatt and Nataf transformations are commonly used. In many industrial RBDO problems, the input random variables are correlated. However, often only limited information such as the marginal distribution and covariance could be practically obtained, and the input joint probability distribution function (PDF) is very difficult to obtain. Thus, in literature, most RBDO methods assume all input random variables are independent. However, in this paper, it is found that the RBDO results can be significantly different when the input variables are correlated. Thus, various transformation methods are investigated for development of a RBDO method for problems with correlated input variables. It is found that Rosenblatt transformation is impractical for problems with correlated input variables due to difficulty of constructing a joint PDF from the marginal distributions and covariance. On the other hand, Nataf transformation can construct the joint CDF using the marginal distributions and covariance, and thus applicable to problems with correlated random input variables. The joint CDF is Nataf model, which is called a Gaussian copula in the copula family. Since the Gaussian copula can describe a wide range of the correlation coefficient, Nataf transformation can be widely used for various types of correlated input variables. In this paper, Nataf transformation is used to develop a RBDO method for design problems with correlated random input variables. Numerical examples are used to demonstrate the proposed method. Also, it is shown that the correlated random input variables significantly affect the RBDO results.

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