Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
6
5
2016
AN IMPROVED SCORE FUNCTION FOR RANKING NEUTROSOPHIC SETS AND ITS APPLICATION TO DECISION-MAKING PROCESS
377-385
Nancy
School of Mathematics, Thapar University Patiala 147004, Punjab, India
Harish
Garg
School of Mathematics, Thapar Institute of Engineering and Technology (Deemed University)
Patiala 147004, Punjab, India
The neutrosophic set (NS) is a more general platform which generalizes the concept of crisp, fuzzy, and intuitionistic fuzzy sets to describe the membership functions in terms of truth, indeterminacy, and false degree. Under this environment, the present paper proposes an improved score function for ranking the single as well as interval-valued NSs by incorporating the idea of hesitation degree between the truth and false degrees. Shortcomings of the existing function have been highlighted in it. Further, the decision-making method has been presented based on proposed function and illustrates it with a numerical example to demonstrate its practicality and effectiveness.
A UNIFIED FRAMEWORK FOR RELIABILITY ASSESSMENT AND RELIABILITY-BASED DESIGN OPTIMIZATION OF STRUCTURES WITH PROBABILISTIC AND NONPROBABILISTIC HYBRID UNCERTAINTIES
387-404
Shu-Xiang
Guo
College of Science, Air Force Engineering University, Xi'an 710051, China
Zhen-Zhou
Lu
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
Yan-Yu
Mo
College of Science, Air Force Engineering University, Xi'an 710051, China
Cheng
Tang
College of Science, Air Force Engineering University, Xi'an 710051, China
In the reliability assessment of structures, a situation we frequently encounter is that by means of available information some parameters involved can be depicted accurately by their probability distributions and others can only be described by the bounds or ranges of variations. So, it is meaningful to construct a reliability model by which the probabilistic and nonprobabilistic uncertainties can be treated reasonably in an integrated framework. The main purpose of this paper is to establish a strictly mathematical foundation and a unified framework for reliability assessment and reliability-based design optimization (RBDO) of structures in the presence of both probabilistic and nonprobabilistic (bounded) hybrid uncertainties. The input uncertain parameters are divided into two different groups and treated respectively as random variables and interval variables, and the traditional probability and convex set models are adopted to describe the probabilistic and bounded uncertainties, respectively. In the reliability measuring system developed in the paper, dimensionless hybrid reliability indices are defined in different situations by adopting a similar method as for the traditional probabilistic reliability method for structures. A computational procedure for performing the RBDO of structures with hybrid uncertainties is presented. Two numerical examples are investigated to demonstrate the effectiveness and feasibility of the presented method.
A PARAMETER SUBSET SELECTION ALGORITHM FOR MIXED-EFFECTS MODELS
405-416
Kathleen
Schmidt
Lawrence Livermore National Laboratory
Ralph C.
Smith
Department of Mathematics, North Carolina State University, Raleigh, NC 27695
Mixed-effects models are commonly used to statistically model phenomena that include attributes associated with a population or general underlying mechanism as well as effects specific to individuals or components of the general mechanism. This can include individual effects associated with data from multiple experiments. However, the parameterizations used to incorporate the population and individual effects are often unidentifiable in the sense that parameters are not uniquely specified by the data. As a result, the current literature focuses on model selection, by which insensitive parameters are fixed or removed from the model. Model selection methods that employ information criteria are applicable to both linear and nonlinear mixed-effects models, but such techniques are limited in that they are computationally prohibitive for large problems due to the number of possible models that must be tested. To limit the scope of possible models for model selection via information criteria, we introduce a parameter subset selection (PSS) algorithm for mixed-effects models, which orders the parameters by their significance. We provide examples to verify the effectiveness of the PSS algorithm and to test the performance of mixed-effects model selection that makes use of parameter subset selection.
MONTE CARLO BASED UNCERTAINTY ANALYSIS FOR VARIABLE PROPERTY MIXED CONVECTION FLOW IN A UNIFORMLY HEATED CIRCULAR TUBE
417-428
Biswadip
Shome
Global Technology and Engineering Center, Offices No. 501 & No. 502, D Block, Weikfield IT Citi Info Park, Pune-Nagar Road, Pune, India 411014
A Monte Carlo based stochastic numerical analysis of simultaneously developing mixed convection laminar flow and heat transfer with variable properties in a uniformly heated circular tube was carried out. The Monte Carlo simulations were performed for heating of a large Prandtl number liquid for a wide range of Prandtl numbers (50 ≤ Pr ≤ 1000), Rayleigh numbers (105 ≤ Ra ≤ 107), and nondimensional heat flux values (100 ≤ qwd/k ≤ 1000). The results show that the effect of uncertainties in the properties is greater on the friction factors uncertainty than it is on the Nusselt numbers uncertainty, and the uncertainty of the predicted friction factors and Nusselt numbers increases with increasing free convection and entrance effects. The maximum uncertainties in the predicted friction factors and Nusselt numbers were around 7.5% and 3.3%, respectively, for a 10% uncertainty value in the fluid properties. The uncertainties in the friction factors and Nusselt numbers were found to increase approximately linearly with increasing fluid property uncertainties.
A MULTIMODES MONTE CARLO FINITE ELEMENT METHOD FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS
429-443
Xiaobing
Feng
Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996, USA
Junshan
Lin
Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849, USA
Cody
Lorton
Department of Mathematics and Statistics, University of West Florida, Pensacola, Floridas 32514, USA
This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon a multimodes representation of the solution as a power series of the perturbation parameter, and the Monte Carlo technique for sampling the probability space. One key feature of the proposed method is that the governing equations for all the expanded mode functions share the same deterministic diffusion coefficient; thus an efficient direct solver by repeatedly using the LU decomposition of the discretized common deterministic diffusion operator can be employed for solving the finite element discretized linear systems. It is shown that the computational complexity of the algorithm is comparable to that of solving a few deterministic elliptic partial differential equations using the director solver. Error estimates are derived for the method, and numerical experiments are provided to test the efficiency of the algorithm and validate the theoretical results.
A NOVEL GLOBAL METHOD FOR RELIABILITY ANALYSIS WITH KRIGING
445-466
Zigan
Zhao
National University of Defense Technology
Xiaojun
Duan
Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha, Hunan, HN 731, People's Republic of China
Zhengming
Wang
College of Science, National University of Defense Technology, Changsha, Hunan Province 410000, China
In engineering applications, an important challenge of structural reliability analysis is to minimize the number of calls to the performance function, which is expensive to evaluate. In recent years, metamodels have been introduced to solve this problem and the methods combining the Kriging model and Monte Carlo simulation have widely gained attention, because they sample training points sequentially. Two essential issues should be considered in these kinds of methods: the selection of the next training point and the stopping criterion of the iteration. EGRA (efficient global reliability analysis) and AK-MCS (active learning reliability method combining Kriging and Monte Carlo simulation) are two representative approaches, whose strategy of sampling training points and stopping criterion are based on evaluations of each Monte Carlo sample under certain learning functions. However, these proposed learning functions are based on the individual performance of each Monte Carlo sample, causing the methods to focus more on the local optimization. As a result, the idea of globally solving the problem was proposed and applied to the GSAS method in 2015. In this paper, a new global reliability analysis method is proposed. Unlike GSAS, a new learning function called the uncertainty reduction quantification function (URQF) is put forward. Specifically, a new random variable proposed in GSAS that helps simplify the statistical feature of Monte Carlo samples is inherited from GSAS, and a weighting function that establishes the connection between any point and the next training point is proposed. These two functions are combined to build URQF, which quantifies the uncertainty reduction of predicted failure probability after a new training point is added. Meanwhile, a novel stopping criterion is proposed by calculating a prediction of an upper bound of failure probability's relative error; the iteration stops when this prediction reaches a preset bound. In the end, a series of examples are performed, which indicates that both URQF and the novel stopping criterion can improve the efficiency of the method, more or less.