Mathematical Problems in Engineering

Volume 2015, Article ID 484615, 11 pages

http://dx.doi.org/10.1155/2015/484615

## Structural Response Analysis under Dependent Variables Based on Probability Boxes

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China

Received 29 October 2014; Revised 16 March 2015; Accepted 7 April 2015

Academic Editor: Alessandro Palmeri

Copyright © 2015 Z. Xiao and G. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper considers structural response analysis when structural uncertainty parameters distribution cannot be specified precisely due to lack of information and there are complex dependencies in the variables. Uncertainties in parameter are quantified by probability boxes (-boxes) and dependence among uncertain parameters is modeled by copula. To calculate uncertainty structural response, a sampling-based method is proposed. In this method, a sampling strategy is used to sample random intervals from dependent -boxes according to the copula theory and the metamodel-based optimization method is applied to solve a range of structural interval response problems. Two types of errors are presented to evaluate the error of different -boxes. Four numerical examples are investigated to demonstrate the effectiveness of the present method.

#### 1. Introduction

Traditional quantification of the uncertainties existing in a system and the corresponding calculation of uncertainty propagation are generally based on the probability model, in which random distributions are used to describe the uncertainty. Unfortunately, for practical engineering problems, sufficient experimental samples are not always available or sometimes very expensive to obtain. Thus alternative imprecise probabilities have been proposed, including the interval theory [1], the evidence theory [2], and probability boxes (-boxes) [3]. The focus of this paper is on -boxes, which are more expressive generalization of both traditional probability distributions and interval representations. The -box incorporates facilities from probability theory for modeling correlations and dependencies and from interval analysis expressing ignorance by expressing interval bounds on the cumulative distribution function (CDF) for a random variable [4]. On the other hand, the -box has a clear behavioral interpretation and therefore is easy acceptable for practicing engineers [5].

In the field of uncertainty analysis based on -boxes, a lot of exploratory work has been reported in theoretical and application aspects. For example, Williamson and Downs [6] proposed an algorithm to compute binary arithmetic operations (addition, subtraction, multiplication, and division) on pairs of independent discrete -boxes for risk assessment. Karanki et al. [7] used the two-phase Monte Carlo simulation to calculate the multiplication of two -boxes for the safety assessment. Bruns [8] compared three uncertainty propagation methods to find the appropriate computational methods for propagating -boxes through the black box engineering models. Zhang et al. [9–11] combined the interval finite element analysis (FEA) and the interval sampling method and proposed an interval Monte Carlo (IMC) method for structural analysis with -boxes which are defined by distributions with interval parameters. Bai et al. [12] developed a numerical method to compute the linear elastic static and dynamic response of structures with epistemic uncertainty represented by evidence variables, which integrated the moment concept and finite element method and also suited to -boxes variables. Ghosh and Olewnik [13] improved the performance of the two-phase Monte Carlo simulation by replacing the outer loop by optimization algorithms and replacing the inner loop by the sparse grid numerical integration.

Among the above-mentioned methods, they focused on discussing uncertainty propagation of independent -boxes. However, the complex dependencies are common in physical systems and can have profound impacts on the numerical results of calculations [14–16]. Ferson et al. [17] illustrated several methods which are used to model the dependence among variables that are cannot be generalized easily for use with -boxes and proposed that copulas could represent easily the dependence in -boxes. Based on the analysis, they presented an approach known as the dependency bounds convolution (DBC) method. The DBC method could calculate the propagation of dependent -boxes whose dependence is expressed by given parametric copula. Despite the fact that DBC is useful for a sequence of basic arithmetic operations or elementary function with a small number of uncertain variables, it must overcome some obstacles for application in engineering design. Among these problems, the low efficiency difficulty seems to be the severest one. The Cartesian product method is applied in the DBC, which can impose significant computational burden.

In this paper, a computational method is proposed to propagate dependent -boxes through structural model, where dependence in -boxes is described by copula. By the copula sampling technique, the input epistemic uncertainties represented by -boxes are transformed into a range of intervals. The metamodel-based optimization is employed as the solver to calculate the structural response with input intervals. A mathematical function has been tested to show the procedure and the efficiency of the proposed method and then three structural numerical examples are investigated by the present method.

#### 2. Probability Boxes Theory under Dependence

##### 2.1. Probability Boxes

A -box is a class of distribution functions delimited by an upper and a lower bound which collectively represent the epistemic uncertainty about the distribution function of a random variable. Let denote the space of distribution functions on the real numbers . A -box will be defined aswhere , , and are the lower and upper bounds of a -box, which is also written as .

There are several ways to construct -boxes, depending on the type of information available. In this paper, -boxes are constructed based on 95% confidence intervals on the parameters of a known distribution type. The distribution types are determined on the basis of the theoretical knowledge or previous experience.

##### 2.2. Describing Dependence in Probability Boxes with Copula

Now, consider the dependence in -boxes which can be described by copulas. A copula is a multivariate probability distribution for which the marginal probability of each variable is uniformly distributed. By Sklar’s theorem [18], copulas are simply the dependence functions that knit together marginal distributions to form their joint distribution. Consider a random vector with CDFs which are represented by -boxes in this paper, the multivariate cumulative distribution function can be written aswhere is a multivariate copula.

##### 2.3. Dependency Bounds Convolution Method

Ferson et al. [17] firstly used dependency bounds convolution method to calculate binary arithmetic operations on pairs of dependent discrete Dempster-Shafter structures. This method mainly contains discretization of the -boxes and Cartesian product. Consider that , are two dependent -boxes connected by copula function , and is the response quantity. The -boxes and are discretized into focal elements, which are represented by and , respectively. By Yager’s Cartesian product, the response quantity in the th focal element can be defined by an interval , where . The probability mass associated with the th focal element can be calculated bywhere and are the cumulative masses, and . The result is written as focal elements which can be transferred into a -box bywhere and are upper bound and lower bound, respectively.

#### 3. Structural Response Analysis

Using -boxes to quantify the structural parameters, the structural response will be also a -box variable. In this section, we will develop an efficient response analysis method for structures by combining -box theory and the optimization method.

##### 3.1. Sampling Random Intervals from the Dependent -Boxes

By virtue of Sklar’s theorem [18], pseudorandom samples can be generated from (2). That is, given a procedure to generate a sample from the copula , the required sample can be constructed aswhere are quasi-inverse function of CDF and are a sample from a random vector . So the random intervals in accordance with the dependent probability boxes can be generated by following two steps.

The first is to generate a sample from copula. There are lots of algorithms which can generate a sample from a specified copula function [19]. As conditional method may be applied for every chosen copula, it is used as the sample method and illustrated in this section. Just to explain this method in a simple way, let us assume a bivariate copula in which all of its parameters are known and pairs need to be generated from bivariate copula function . Two independent uniform (0, 1) variates and are generated, and then , where denotes a quasi-inverse of which equals the partial derivative of the copula .

The other step is to compute dependent random intervals from -boxes. Suppose that are a sample from the copula and , are their CDFs. For each and corresponding to , a random interval is generated:where is random interval from . The superscript represents the interval distribution parameters and the superscripts and , respectively, represent the lower and upper bounds of interval. and denote the quasi-inverse function of upper and lower bounds of a -box. Thus using will simulate a range of dependent intervals .

Zhang et al. [9] presented an IMC method for sampling intervals from the independent -boxes. When random variables contain both dependent and independent -boxes, the present sampling method and the IMC method can be used to perform random sampling for dependent and independent -boxes, respectively.

##### 3.2. Metamodel-Based Interval Analysis

Through above section sampled treatments, a range of interval vectors is acquired from -boxes. Consider the vector of uncertain parameters defined to be contained within an interval vector (or hypercube) . Interval vectors sampling times from -boxes are represented as (). For a practical engineering problem, the input-output relationship between these parameters and the output quantity of interest is represented by the function applied on these parameters. The interval analysis for the function is numerically equivalent to solving the following equation:where contains all which are obtained from applying the function on all possible vectors within the interval vector . In many cases, an exact description of is extremely difficult to find. Therefore, an interval is used to approximate the exact solution . For the black box model in the practical engineer problem, solution of will make the computational cost extremely expensive [20]. Besides, a great number of repeating interval analyses will be conducted, which will generate unaffordable computational burden. So metamodeling techniques are used to approximate for improving the computational cost.

To improve the accuracy of metamodel, design of experiments (DOE) should be applied. An experimental design domain can be acquired by truncating the -box at a probability of ( is level of significance), which are represented as . The vector form is . In this paper, the Latin hypercube design (LHD) method is adopted, which is a space-filling design with constrainedly stratified sampling method. The sample of this method does not increase exponentially with the number of variables. Radial basis function (RBF) [21] is used to approximate the function , which can be expressed as where is the sampling point selected through LHS, is the number of sampling points, is mass coefficient, and is the basis function. The common basis functions include thin-plate spline, Gaussian, and multiquadrics. The metamodel-based solution of can be transferred into the following optimization problems: The gradient-based optimization techniques such as sequential quadratic programming can be used to solve this problem, and the initial point can be selected on the bound point of intervals.

##### 3.3. Computation of Output Quantity

Repeating times interval analysis, we will acquire intervals , which can be transformed into a -box by [9] with and , where is the indicator function. Summarizing the above procedure, the present method can be summarized as follows.

*Step 1. *Collect all -boxes and sample interval vectors from by the method in Section 3.1.

*Step 2. *Conduct the design of experiments with LHS in the experimental design domain .

*Step 3. *Construct a RBF response surface for approximating (7).

*Step 4. *Conduct optimization analysis over each and acquire interval vector .

*Step 5. *Generate the -box of the output quantity by (10).

#### 4. Error Analysis

The absolute error and the relative error are popular to measure the accuracy between the experimental data and the true value, which is extended to analysis errors of different -boxes. In this paper, the absolute errors contain the area error and the boundary error. Suppose that the two -boxes and ; then the left area error and the right area error between two -boxes are, respectively, calculated by

The area error between two -boxes is defined as the sum of the left area error and the right area error, which is written as . As the area error cannot reflect the discrepancy of boundary between two -boxes, the discrepancy of the boundary is used as the boundary error. Plotting a line with the cumulative probability will produce two intersection points with the boundary of a -box. For , two intersection points are and . Two intersection points of are and . Thus the boundary error iswhere and . The illustration of the area error and the boundary area is shown in Figure 1.