A comprehensive model for structural non-probabilistic reliability and the key algorithms

It is very difficult to know the exact boundaries of the variable domain for problems with small sample size, and the traditional convex set model is no longer applicable. In view of this, a novel reliability model was proposed on the basis of the fuzzy convex set (FCS) model. This new reliability model can account for different relations between the structural failure region and variable domain. Key computational algorithms were studied in detail. First, the optimization strategy for robust reliability is improved. Second, Monte Carlo algorithms (i.e., uniform sampling method) for hyper-ellipsoidal convex sets were studied in detail, and errors in previous reports were corrected. Finally, the Gauss-Legendre integral algorithm was used for calculation of the integral reliability index. Three numerical examples are presented here to illustrate the rationality and feasibility of the proposed model and its corresponding algorithms.


Introduction
Engineering of practical structures involves many uncertain factors. Identifying, measuring, and controlling these uncertainties are very important for ensuring the reliability and comprehensive performance of the engineered structures. The traditional method of addressing uncertainties is based on probability theory, and its development is very mature. However, probability models depend on a large amount of objective statistical data, which are used to determine the appropriate probability density function (PDF). In practical engineering, statistical data are often insufficient due to limitations of experimental conditions and costs. In addition, some parameters, e.g., geometries, do not exhibit any randomness, and their uncertainties stem from measurement or subjective understanding of people. This is the so-called unascertained type of uncertainty [Liu, Wu and Wang (1997)]. In summary, probability theory has inherent limitations for problems with small sample size. To overcome the limitations of probabilistic methods, Ben-Haim et al. [Ben-Haim and Elishakoff (1990)] opened a new way to deal with structural uncertainty using a non-probabilistic convex set model, which does not require a large amount of statistical data. Ben-Haim first proposed using a non-probabilistic concept for structural reliability [Ben-Haim (1994)] and subsequently presented a new index of non-probabilistic reliability [Ben-Haim (1995)]. Elishakoff [Elishakoff (1995)] later presented another reliability index based on the interval safety factor. Guo et al. [Guo, Lu and Feng (2001)] presented a non-probabilistic reliability model based on interval analysis and a state function describing the structural limit, and this reliability index is an infinite norm. In recent years, robust reliability theory based on convex set uncertainties has been extensively studied [Ni, Jiang and Han (2016); Yang, Zhang, Meng et al. (2017); Gou, Li, Luo et al. (2016); Liu, Yu, Li et al. (2016); Yi, Shi, Dhillon et al. (2016); Chen, Fan and Bian (2017)]. Wang et al. [Wang, Wang, Su et al. (2017)] studied the time-dependent non-probabilistic reliability problem for structures with fatigue cracks. There are many studies on non-probabilistic reliability-based optimization problems for engineering design [Meng, Hu and Zhou (2018); Meng, Zhou, Li et al. (2016); Wang, Fan and Hu (2018); Chen, Wang, Qiu et al. (2018); Hao, Wang, Liu et al. (2017); Zheng, Luo, Jiang et al. (2018); Saad, Chateauneuf and Raphael (2018)]. However, robust reliability theories are no longer applicable when the variable domain interferes with the structural failure region. Many scholars have studied the reliability problem in this case. Wang et al. [Wang, Qiu and Wu (2007)] proposed a non-probabilistic reliability model based on interval interference, and this reliability index represents the volume ratio of the safety domain to the entire variable domain. Zhou et al. [Zhou, An and An (2009)] successfully applied the volume ratio-based reliability model to reliability analysis of supercavitating bodies. Qiao et al. [Qiao, Qiu and Kong (2009)] further studied volume ratio-based reliability based on a hyper-ellipsoidal convex set model. Wang et al. [Wang, Wang, Wang et al. (2016)] studied the computational method of non-probabilistic reliability for linear structural systems. There are many other aspects of these topics. For example, Wang et al. [Wang, Zhou, Gao et al. (2018)] studied non-probabilistic sensitivity analysis and its application in optimization of aeronautical hydraulic pipelines. Fang et al. [Fang, Su, Xiao et al. (2017)] studied the non-probabilistic reliability problem for structures with an implicit performance function. Zhang et al. [Zhang, Jiang and Fang (2016)] and Khairul et al. [Khairul, Norhisham and Hong (2017)] applied non-probabilistic reliability theories to several practical engineering problems. All the aforementioned non-probabilistic reliability methods are based on interval or hyper-ellipsoidal models, which are so-called rigid convex set models. However, when the sample size is very small, it is difficult to know the exact boundary of the variable domain. The robust reliability and volume ratio-based reliability methods cannot take into account all cases where the variable domain intersects with or is separated from the failure region, thus these methods do not have universal applicability in engineering. Sun et al. [Sun, Yang and Chen (2018)] proposed a new uncertainty model named the fuzzy convex set (FCS) model. The results presented in this paper stem from this uncertainty model. FCS has the following features: (1) the FCS model is used to establish non-probabilistic reliability model to overcome the limitations of traditional rigid convex models; (2) the new reliability model can take into account all kinds of location relationships between the failure region and the variable domain; the index is continuous, comparable, and clear in its physical meaning; (3) the location of the robust reliability index is discussed, and a previous conclusion is corrected. Then, an improved optimization algorithm for a robust reliability index is proposed. The uniform sampling method for a hyper-ellipsoidal convex set is prepared, which revises the method in previous research reports and lays the foundation for application of volume ratio-based reliability theory. Finally, we present the numerical integration method for solving FCS-based reliability problems. All key algorithms of this new model will be presented.
2 The comprehensive structural non-probabilistic reliability model Assume the structural limit state equation is where x=(x1, x2,…, xn) is the uncertain variable vector. The surface G(x)=0 divides the variable space into two parts: the failure region Ωf and the safe region Ωs.
Assume the multiple FCS of the structural parameters is  ( , , ) U  x θ φ . According to Cor.2 in Sun et al. [Sun, Yang and Chen (2018)], the cut set of  ( , , ) U  x θ φ under any level is certainly a convex set, and the following equation can be obtained: interferes with the failure region Ωf. In this case, we can define the non-probabilistic reliability index as Then, the comprehensive index for non-probabilistic reliability is defined as The reliability index defined above is based on the ordinary convex set model. Even if the nominal value point of the convex set is located in the failure region, the index is still valid. Moreover, continuity of the reliability index is guaranteed for all real numbers. The integral reliability index based on FCS is When the nominal value point of the convex set is located in the failure region, and the convex set interferes with the safe region, the robust reliability index is less than 0, but the comprehensive reliability index in Eq. (5) is greater than 0. Thus, the robust reliability model cannot reasonably reflect the structural reliability in this case. This case is illustrated in Fig. 1. Regarding the reliability problem shown in Fig.1, the robust reliability degree is η(λ)<0, but the comprehensive reliability index is κ(λ)>0. Although η(λ)<0 according to Fig. 1, the shaded failure region is far smaller than the safety region. Therefore, the comprehensive reliability model can more rationally measure the interference level between the failure region and the variable domain. Furthermore, the proposed integral reliability index can account for uncertainties in the structural parameters and the inaccuracy of the ordinary convex set model. Thus, this index is a more rational and comprehensive structural reliability model for problems with small sample size.

Normalization of the multiple convex set
The non-probabilistic reliability under any-level cut set can be obtained by spreading the original convex set or the interference between the failure region and the original convex set. However, it is often impractical to use the original convex set directly, as the designed algorithm has many difficulties. Thus, we first need to normalize the original convex set model. Let us assume that the multiple convex set model for uncertain parameters includes p interval models and m hyper-ellipsoidal models. The interval vector X can be converted to a normalized interval variable vector with the following equation: where, c X is the central vector of X, 1 δ is the p-dimensional normalized interval vector, ∆X is the deviation vector, and p denotes the dimensionality of the interval variable vector. Let us assume that the i-th hyper-ellipsoidal model can be expressed by According to Eq. (8), we can obtain The original hyper-ellipsoidal convex set has been converted to a unit hyper-sphere through the above conversion. According to Eq. (10), we can obtain ( ) After substitution of Eqs. (7) and (13) into Eq. (1), we can obtain the following normalized limit state function: To make the expression more convenient, the normalized limit state function will be written as where n is the total number of dimensions in the non-deterministic variable space, and i ∆u is the normalized variable vector in the i-th submodel.

Optimization algorithm for robust reliability 3.2.1 Discussion of the robust reliability index position
Jiang et al. [Jiang and Chen (2007)] has pointed out that the robust reliability index under the interval model can only be located at one of the intersection points between the i Ω ultra-rays (passing through the origin and the vertices of the normalized interval set) and the normalized failure surface, and a detailed proof was presented. Some applications related to this conclusion have been reported [Zhou, An and Jia (2011)]. However, the proof was not rigorous and the conclusion was somewhat biased. In order to save space, this proof will not be discussed here in detail, but the problem of the conclusion will be illustrated with a counter-example. A 2-dimensional case is shown in Fig. 2, where the limit state surface denoted by M1 monotonically decreases in the 2-dimensional space. The robust reliability index is indeed located at an intersection point. However, the robust reliability for M2 is not located on the ultra-rays passing through the origin and the vertices of the interval sets. In fact, one can prove that the conclusion in Jiang et al. [Jiang and Chen (2007)] is correct if the normalized limit state surface is monotonic in the normalized interval variable space. However, when the limit state surface is not monotonic, this conclusion is not always true. When the limit state surface is monotonic, the conclusion can be proven as follows: Proof: Assume that the n-dimensional normalized limit state surface M=G(δ1,δ2,…,δn)=0 is a monotonic surface, and the robust reliability index point is . This conclusion will be proven by contradiction.
does not stand, and there is a relation . Because the surface M=0 is monotonic, any two dimensions of this surface must be monotonic.
is located on the normalized limit state surface G(δ)=0.
Case 2: Assume that 1 δ monotonically decreases with 2 δ . Then, not be the minimum infinite norm point. In other words, it must not be the location of the robust reliability index.
In summary, must not be the location of the robust reliability index.
must be true. This concludes the proof.
When the uncertain variables are described by a hyper-ellipsoidal model, the model can be converted into a unit hyper-sphere. In this hyper-sphere space, the robust reliability index point will not certainly satisfy the condition 1 2 n δ δ δ ′ = ′ = = ′  , regardless of whether the limit state surface is monotonic or non-monotonic. As shown in Fig. 3, the limit state surface G′(δ)=0 is monotonic in the normalized hyper-sphere space, but the components of the robust reliability index point are not equal. Figure 3: Location of the ellipsoidal robust reliability index According to the above discussion, the following two corollaries can be obtained when we use a multiple convex set model to express the variables' uncertainties: (1) When the limit state surface is monotonic in the normalized space, the following relation for the robust reliability point holds: (2) When the limit state surface is non-monotonic in the normalized space, any one of the following relations for the robust reliability index point may not exist: Therefore, it is very difficult to reduce the scope of the feasible solution by adding constraints when the limit state surface is non-monotonic, or if it is difficult to judge whether the surface is monotonic.

Improved PSO optimization strategy
In order to calculate the non-probabilistic comprehensive reliability index in Eq. (5), we first need to calculate the robust reliability index under the corresponding cut set. According to Eq. (3), the physical meaning of η(λ) is the shortest distance measured by the uncertainty level parameter χ from the centric point x of the convex set model can be obtained from the following constrained optimization problem: where ρ(δ) is the distance from any point δ to the origin derived from the normalized convex set model of the multiple convex set According to the analysis in Section 3.2.1, it is difficult to reduce the feasible region for robust reliability index when the limit state surface is not monotonic. In order to develop a universally applicable method, the algorithm for determining the robust reliability index based on particle swarm optimization (PSO) is presented here.
In this paper, the basic theory of PSO will not be thoroughly discussed. PSO is suitable for continuous and discrete functions. Li [Li (2012)] presented the following unconstrained optimization problem by introducing a penalty term: sgn (·) is sign function, and C1 is the penalty factor. The penalty factor only needs to be greater than 10 because the non-probabilistic reliability index is usually a small number. According to Eq. (20), the former constrained optimization problem has been converted to an unconstrained optimization problem after introducing the penalty factor. In order to improve the performance and efficiency of the PSO algorithm, another penalty factor has been introduced into the fitness function. This approach is equivalent to adding a constraint on the opposite side of the nominal value of the convex set about the limit state surface. The optimization problem remains an unconstrained problem with the following form ξ is the parameter controlling the scale of the search field, and C2 is the second penalty factor. The principle of the value of C2 is similar to that of C1. C2 can meet the requirement when it is slightly larger than or equal to C1. Because the PSO algorithm has no requirement for differentiability of the fitness function, introducing the penalty function defined in Eq. (23) is feasible and can effectively improve the efficiency of the PSO algorithm. This algorithm is shown in Fig. 4. The steps of the improved PSO algorithm are as follows: Step 1: Determine a particle's fitness function denoted by f(δ), defined as where the values of δ represent the location of the particles, and ρ(δ) is defined in Eq.
. Redefine the problem as a minimization problem.
Step 2: Determine the search range and maximum velocity. According to the practical problem, make a substantial expansion of the normalized multiple convex set model and use the smallest external ultra-cuboid L U [ , ] n = I δ δ as the search range. In addition, define the maximum velocity as Step 3: Determine the parameters of the particle swarm.
Step 4: Solve by iteration. Randomly select l samples in In as the initial location of the particle swarm. Then, the historical best position of the particle (δb), global best position (δgb), and neighborhood optimum position (δnb) can be obtained. The position of the next step can be determined using update formulas. Then, the values of the fitness function can be calculated, and iteration can be carried out until that the termination criteria are reached.
Step 5 The robust reliability index can be obtained from Eq. (18). The improved PSO algorithm has better performance and efficiency. The robust reliability under any cut set level can be obtained from this algorithm and has desirable accuracy.

Monte Carlo algorithm for the hyper-ellipsoidal convex set model
According to the non-probabilistic comprehensive reliability definition in Eq. (5) Monte Carlo samples in the orthogonal space can be obtained after performing the above transformation. However, after performing the transformation, uniform samples within the spherical coordinates will no longer follow a uniform distribution in the orthogonal space. The samples gradually become sparse as the radius increases. A 2-dimensional unit sphere with 900 samples is shown in Fig. 5, and one can find that the samples become increasingly sparse as the radius increases. In fact, the cause of this phenomenon is not complicated. The samples are uniformly distributed along the radial direction in spherical coordinates. In other words, as long as the widths of the rings are equal, the number of samples contained in each ring tend to be equal. However, the areas of the rings increase with radius, and the samples gradually become sparse. In order to obtain a uniform distribution of samples in a unit hyper-sphere in orthogonal coordinates, we need to change the sampling density function along the radial direction. This problem will be studied in the following section, and we will find that the density functions for hyper-spheres with different dimensions are different.
For an n-dimensional sphere, the area of the n-1 dimensional spherical surface is Take two hyper ring infinitesimals at radii r1 and r2 with differential widths dr1 and dr2, respectively. The volumes of the two infinitesimals are Now that we have obtained the PDF of r, generating pseudo-random numbers is another problem that must be discussed. There are only some pseudo-random number generators for typical probability distributions in common software platforms (e.g., MATLAB). For an atypical probability distribution (e.g., f(r)) or even for any type of probability distribution, we need to develop additional algorithms and programs. In this study, the required pseudo-random numbers are sampled using the Metropolis method. This method is one of the Markov chain Monte Carlo (MCMC) algorithms and can be used to draw samples from any complex distribution. The principle of Metropolis sampling is to simulate a Markov chain in a state space of random variables whose stationary distribution is the target distribution. Assuming an arbitrary distribution f(r), the Metropolis sampling algorithm is as follows: (1) At t=0, choose an initial value r0 that meets the requirement of f(r0) ≥0.
(2) At the (t+1) th iteration, sample the candidate value r c from the proposal distribution q(r|rt), which should be a symmetric distribution, such as a normal distribution or uniform distribution.
The above sampling yields a Markov chain with stationary distribution f(r), i.e., the random numbers of f(r) are obtained. Different PDFs for r should be used for hyper-spheres with different dimensions. The other components should be sampled from uniform distributions. Convert the points sampled from spherical coordinates to orthogonal coordinates, and the uniform Monte Carlo samples can be obtained. Fig. 6 shows a 2-dimensional unit sphere including 1000 samples obtained via the modified method. When the variable domain interferes with the failure region, we can use the above Monte Carlo algorithm to calculate the non-probabilistic reliability index for a high dimensional non-linear complex limit state function as follows: where qall is the sum of all simulation calculation times, and qs is the times corresponding to

Numerical integration algorithm and overall analysis procedures
In Eq. (6), the analytical form of κ(λ) is difficult to obtain. Thus, it is necessary to present a numerical method to calculate the integration in Eq. (6). Given n integral nodes, the Gaussian integral formula has the highest algebraic accuracy [Yan (2006)].
Substituting λ=(1+t)/2 into Eq. (6) yields ( ) Take the zero points of Ln(t) as integral nodes, and the following integral formula can be obtained: Eq. (38) is the Gauss-Legendre integral formula. The expression for Ai is Thus, the integration in Eq. (36) can be calculated with 1 1 1 The integral nodes and integral coefficients of Gauss-Legendre integral formula can be found in the numerical analysis monograph [Yan (2006)]. The overall analytical procedure for the proposed reliability model is as follows: a) Build the FCS model for the structural variables, and determine the possibility distribution function for the fuzzy extending parameter. b) n cut set levels λi can be obtained according to the n integral nodes ti of the Gauss-Legendre integral formula. c) Transform the multiple convex set models under n cut set levels into n unit convex set The extending parameters 1 θ  -5 θ  follow a semi-trapezoidal distribution with small bias.
The distribution functions decrease linearly from 1 to 0 and with extension parameters ranging from 1 to 1.5. The Gauss-Legendre integral formula including seven nodes is used here. The corresponding cut set levels can be obtained from the integral nodes and the relation  The robust reliability index η and the comprehensive reliability index κ change with the cut set level, as shown in Fig. 9. According to the Gauss-Legendre integral formula, the integral non-probabilistic reliability degree is If the robust reliability model is applied, the result is 0.95598. When the cut set level is high enough, the convex set does not interfere with the failure region, and the volume-ratio based non-probabilistic reliability degree is equal to 1. The difference between the results stems from the physical definitions of the various metrics. The proposed non-probabilistic reliability model takes into account the advantages of the robust reliability and volume ratio-based reliability methods. Moreover, it can account for the fuzziness of the structural parameters or the fuzziness of the ordinary convex set model. Thus, the proposed reliability model is a more comprehensive and realistic reflection of the structural reliability, but in a non-probabilistic way.

A ring-stiffened cylindrical shell example
The instability between ring ribs is one of the main failure modes of a cylindrical shell. The reliability of a ring-stiffened cylindrical shell in this failure mode will be analyzed. The critical pressure for instability between adjacent ribs is cr g s E

p C C p =
where pE denotes the Euler pressure for instability and Cg is the model correction factor that takes into account initial imperfections on the shell. Cs is the model correction factor taking into account the plastic and residual stress.
The uncertainties in the parameters X= (p, r, h, E, l, Cs, Cg) T are described by the following fuzzy convex set model:  If the reliability analysis is based on the rigid convex set model, i.e., the kernel of the fuzzy convex set, the obtained reliability is 1.9521. It can be found that the reliability result based on the ordinary rigid convex set is higher than the value based on the FCS model. Thus, when we cannot determine the exact boundaries of the parameters, the proposed reliability model can effectively reduce the risk of the reliability analysis based on traditional rigid convex set models. We could also use the Gauss-Legendre integral formula with more integral nodes in order to obtain higher accuracy. Fig. 10 shows a ten-bar structure. There are 15 independent fuzzy interval variables, including modulus of elasticity E, rod length L, cross section areas Ai (i=1, 2,…,10), and external loads P1, P2, and P3. The maximum allowable displacement in the vertical direction of node 2 is 0.06 m. The uncertainties in these variables are described using the following fuzzy interval models:  }   3  3  3  3  3 , , 100 15 kN

A ten-bar truss example
Here, we still use the Gauss-Legendre integral formula with seven integral nodes. The corresponding cut set levels and integral coefficients are equal to their values in Example 1. The robust reliability index η and the comprehensive reliability index κ both change with the cut set level, as shown in Fig. 11. If the reliability analysis is based on the kernel of the fuzzy convex sets which are traditional rigid convex set models, the obtained reliability is 1.9237, which is obviously higher than the result based on the FCS model. As discussed in Section 4.2, when statistical samples of parameters are scarce, the exact boundaries of these parameters cannot be determined accurately. In this case, the proposed reliability model based on FCS model can better reflect the actual engineering, and effectively reduce the risk of the reliability analysis based on traditional rigid convex set models.
Here, we offer the following discussion: (1) When the structural failure region and the variable domain do not intersect, the volume ratio-based non-probabilistic reliability index is equal to 1. When the two regions interfere, the robust reliability index cannot strictly reflect the relationship or comparison between the failure and safety domains. The proposed reliability model in this paper is a comprehensive index, which can overcome the above two problems, and it is more reasonable and scientific. In addition, the new index is smooth, continuous, comparable, and has a clear physical definition.
(2) For problems with small sample size, the traditional rigid convex model cannot strictly express the structural uncertainties, i.e., the boundaries of the variable domain are difficult to determine. The FCS model can take into account fuzzy boundary information. Therefore, the proposed reliability model based on FCS can take into account different position relations between the variable domain and the failure region, as well as the uncertain information regarding the variable boundaries.
( 3) The examples demonstrated the correctness and effectiveness of the presented algorithms in the proposed non-probabilistic reliability theory. The improved optimization strategy based on PSO is especially suited for high dimensional reliability problems and has high efficiency. The amended Monte Carlo method for the hyper ellipsoidal model (i.e., the uniform sampling algorithm) provides corrections for errors that were previously reported in the literature. The pseudorandom number sampling method based on the Metropolis method for the PDF of r is very clever and scientific. The numerical integral algorithm based on the Gauss-Legendre integral formula is suitable for use in the proposed reliability model, and the accuracy increases with the number of integration points.

Conclusions
In this paper, a comprehensive reliability model was presented based on a fuzzy convex set model, and the key computational algorithms were studied in detail. The proposed reliability model can account for the various position relations between the structural failure region and the variable domain, and the index has a better physical meaning. It is smooth, continuous, and comparable. In addition, this novel model can overcome the limitations of the traditional rigid convex set model, especially for problems with small sample size. The improved optimization strategy ensures the robust reliability computation is efficient and accurate. The uniform sampling strategy for use in the hyper ellipsoidal model and the pseudorandom number simulation method for an atypical probability distribution function lays the foundation for the volume ratio-based reliability analysis. Integration of the comprehensive reliability index with the cut set level was solved with the Gauss-Legendre integral formula. The correctness and efficiency of the novel reliability model and all of the presented algorithms were demonstrated by applying them to three numerical examples. Although this study focuses on structural reliability problems, some of the presented methods can be extended to other engineering fields, such as structural static and dynamic responses, structural fatigue prediction, buckling and post buckling, etc. These areas are interesting for future studies.
regarding the present study.