Uncertainty Theory-Based Structural Reliability Analysis and Design Optimization under Epistemic Uncertainty

Reliability analysis and trade-offs between safety and cost with insufficient data represent an inevitable problem during the early stage of structural design. In this paper, efficient uncertainty theory-based reliability analysis and a design method are proposed under epistemic uncertainty. The factors influencing the structure are regarded as uncertain variables. Based on this, a new metric termed uncertain measure is employed to define an uncertainty reliability indicator (URI) for estimating the reliable degree of structure. Two solving methods, namely, the crisp equivalent analytical method and uncertain simulation (US) method, are introduced to calculate the URI and acquire reliability. Thereafter, a URI-based design optimization (URBDO) model is constructed with target reliability constraints. To solve the URBDO model and obtain optimal solutions, crisp equivalent programming and a genetic-algorithm combined US approach are developed. Four physical examples are solved to verify the adaptability and advantage of the established model and corresponding solving techniques.


Introduction
Structural design optimization is a pivotal discipline in industrial engineering products. Generally, structural design aims at searching the best solution of design variables that minimizes the mass or total costs while satisfying several performance constraints [1]. Various uncertainties are intrinsically involved in the process of engineering structural design and optimization [2]. Traditional design optimization is performed under deterministic conditions that neglect the influence of uncertainties stemming from structure size imprecision, external loads fluctuation, and material property variation [3,4]. Those ineluctable uncertainties will lead to unreliable results and decrease confidence in deterministic design optimization. Consequently, reliability assessment and reliability-based design optimization (RBDO) have been widely employed in various engineering problems [5], such as vehicle engineering [3], aerospace engineering [6][7][8], offshore engineering [9], and civil engineering [10].
As a precondition for reliability assessment and RBDO, an appropriate mathematical framework has to be selected to describe the characterization of uncertainties [2,11]. Various methods have been used to determine the best designs according to the uncertainty type. The most accepted classification is to divide uncertainty into two categories, i.e., aleatory uncertainty and epistemic uncertainty [12].
Aleatory uncertainty is also referred to as statistical or variability uncertainty, which is inherent in any engineering product. It is supposed that the sample data of aleatory structural reliability. Subsequently, the crisp equivalent analytical and uncertain simulation (US) approaches are proposed to solve the URI. Based on the URI, a class of URI-based design optimization (URBDO) model is established to meet a specified level of uncertainty reliability. The URBDO model can provide an alternative methodology for the early design of structural problems with limited experimental data. Furthermore, crisp equivalent programming (CEP) and US combined with genetic algorithm (USGA) methods are constructed to solve the URBDO model.
The remainder of this research is structured as follows. The commonly used mathematical concepts of uncertainty and uncertain programming are reviewed in Section 2. A new URI and corresponding two solving methods for structural reliability analysis under epistemic uncertainty are explained in Section 3. The presented general URBDO model is introduced in Section 4. Then, CEP and USGA strategies are provided to acquire the optimal results of the URBDO model. Several engineering applications are investigated to demonstrate the significance of the proposed approaches in Section 5. The application summary and future research directions of the proposed method are discussed in Section 6. Finally, the concluding remarks are summarized in Section 7.

Theoretical Background
In this section, some basic definitions and theorems in uncertainty theory, which represent the mathematical foundation of structural uncertainty reliability, are briefly introduced. Moreover, uncertain programming and crisp mathematical programming methods that guide the establishment of the uncertainty reliability based design optimization model are reviewed.

Fundamental Concepts of Uncertainty Theory
As a vital concept of uncertainty theory, uncertain measure is interpreted as the human belief degree of an event that may occur.
Definition 1 (Uncertain measure [31]). Let Γ be a nonempty set and L be a σ -algebra over Γ. A set function M : L → [0, 1] is defined as an uncertain measure if it satisfies normality, subadditivity, and duality axioms.
Definition 2 (Uncertain variable [31]). The uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M)to the set of real numbers, i.e., {ξ ∈ B} is an event for any Borel set B.
In general, regular uncertainty distribution is defined as a continuous and strictly increasing uncertainty function Φ(x) with respect to x with 0 < Φ(x) < 1. For example, a normal uncertain variable ξ ∼ N (m, σ) has a normal uncertainty distribution: where m and σ > 0 are the expected value and standard deviation, respectively. A linear uncertain variable ξ ∼ L(a, b) has a linear uncertainty distribution: where a and b are real numbers with a < b. Appl Definition 4 (Inverse uncertainty distribution [4]). Let ξ be an uncertain variable with regular uncertainty distribution Φ(x). The inverse function Φ −1 (α) is called the inverse uncertainty distribution of ξ.

Uncertain Measure-Based Structural Reliability Assessment
Probabilistic methods are well developed for the structural reliability problem with aleatory uncertainty. In classical reliability analysis approaches, the structural performance Appl. Sci. 2022, 12, 2846 is characterized by a LSF g(η). Moreover, all input factors of the structure are considered as random variables η = (η 1 , η 2 , · · · , η n ) T . Nevertheless, epistemic uncertainties cannot be accurately explained by probability theory or statistical methods under insufficient sample data. In this section, input factors containing epistemic uncertainties are treated as uncertain variables. Then, a new metric based on uncertainty theory is defined to quantify structural reliability. Definition 6. Let (Γ, L, M) be an uncertainty space. Input factors of LSF g(ξ) are uniformly characterized by an uncertain vector ξ = (ξ 1 , ξ 2 , · · · , ξ n ) T , in which ξ i are uncertain variables. The performance state space of structure is divided by the LSF into two domains: the safety domain {g(ξ) > 0} and failure the domain {g(ξ) ≤ 0}. Then, the uncertainty reliability indicator (URI) can be defined as the uncertain measure of the safety domain: Due to the self-duality of the uncertain measure, the corresponding uncertainty failure indicator (UFI) can be calculated as follows: The URI is a meaningful tool for quantifying structural reliability under epistemic uncertainties. A higher URI ∈ [0, 1] indicates that structural engineers believe the safety event is more possible to happen.
In many circumstances, it is often relatively difficult for structural engineers to directly obtain the uncertainty distribution Ψ g(ξ) (x) of LSF g(ξ). To address this issue, two URI solving approaches are proposed for engineering structural reliability problems in the following two subsections.
However, the monotonicity of LSF with respect to corresponding uncertain variables may not be determined, and Equation (10) sometimes has no solution. To overcome this problem, the decision-maker can estimate URI according to the US algorithm described in Section 3.2.

Uncertain Simulation Algorithm for Calculating the URI
In this section, the uncertain simulation (US) algorithm for URI is explored by combining the philosophy of uncertain simulation [41] and Definition 6.
Step 2. Generate a set of uncertain variables ξ k = (ξ Step 3. Rank uncertain variables ξ (i) k from small to large: ξ Step 5. If m 1 > 0 and m 2 > 0, calculate Step 8. Return the estimated value URI = L.
The US algorithm does not depend on the condition that LSF is a strictly monotone function for uncertain variables. Moreover, this algorithm avoids solving complex equations when the LSF is highly nonlinear or high-dimensional are present.

URI-Based Structural Design Optimization
Generally speaking, reliability analysis and design of engineering structures often face the problem of epistemic uncertainty. To handle this epistemic uncertainty problem with insufficient sample data, a new URBDO model based on URI and chance-constrained uncertain programming model is presented in this section. Due to the lack of sample data, uncertain variables are employed in structural design optimization to uniformly describe input factors. To search for the lowest cost or lightest mass by changing design variables while maintaining the safety requirements of structures, a general URBDO model combines uncertainty reliability demands into design optimization constraints.
The LSF g(d, x; ξ) of engineering structures depends on deterministic design variables, uncertain design variables, and uncertain variables, which are represented by d = (d 1 , d 2 · · · , d n d ) T , x = (x 1 , x 2 · · · , x m ) T , and ξ = (ξ 1 , ξ 2 , · · · , ξ n ) T , respectively. Structural engineers tend to ensure that uncertainty reliability is higher than the target reliability level R U . Uncertainty reliability constraint combined with the URI definition can be expressed as follows: In practical engineering structures, design problems may contain multiple objectives that require to be simultaneously considered. Uncertain multi-objective optimization belongs to uncertain programming in which the constraints and objectives contain uncertain variables. In such instances, typical formulation of the multi-objective URBDO model is defined as follows.

Definition 7.
Assume that g j (d, x; ξ) is the jth LSF of a structure. If the design optimization problem has N f objective functions f i (d, x; ξ), i = 1, 2, · · · , N f that conflict with each other, then the general URBDO model subjected to uncertainty reliability constraint can be established by weighting the objective functions: where d is the vector of deterministic design variables, x is the vector of uncertain design variables, ξ is the vector of uncertain variables, ω 1 , ω 2 , · · · , ω N f are positive weights with∑ N f i=1 ω i = 1, and R U j denotes the prescribed reliability level for the jth constraint. Since the objective functions are also uncertain variables, it is relatively difficult to directly minimize f i (d, x; ξ). Therefore, minimizing its uncertain mean value E[ f i (d, x; ξ)] is reasonable.
To solve the constructed URBDO model, the theorem described in Section 4.1 provides a crisp equivalent programming model for some special cases.
After the general URBDO model is converted into a crisp equivalent model, structural engineers can solve it by employing any typical numerical methods or metaheuristic algorithms. Nevertheless, sometimes the LSF in practical structural problems is not a strictly monotonic function. In addition, when too many input factors and constraints are present in the design problem, Model (14) may not have an optimal solution. Therefore, a hybrid intelligent algorithm that combines US and genetic algorithm (USGA) is developed to solve the proposed URBDO model (13).

Hybrid Intelligent Algorithm for Solving the URBDO Model
A hybrid intelligent algorithm is employed when uncertainty reliability constraint in the URBDO model cannot be processed by the CEP model. According to the uncertain simulation philosophy, the uncertain measure can be calculated by the US algorithm introduced in Section 3.2. To find an optimal solution, the GA is used because of its ability of global search, expansion capabilities, convenient toolbox, and outstanding effectiveness and robustness in solving structural optimization problems. Therefore, the US algorithm is embedded into GA to construct the USGA algorithm.
Algorithm 2. USGA for solving the URBDO model step 1. Initialize Pop_size chromosomes and set the mutation probability P m , crossover probability P c , and iteration times N of GA.
Step 10. The feasibility of chromosomes is checked by the US (from Step 2 to Step 9).
Step 11. Chromosomes are updated by mutation and crossover operations. The feasibility of offspring is also checked by the US.
Step 12. Calculate the uncertain expected values E[ f i (d, x; ξ)], i = 1, 2, · · · , N f of f i (d, x; ξ) for all chromosomes and compute the fitness functions.
Step 13. Select chromosomes by spinning the roulette wheel.
Step 14. Repeat steps 2 and 13 N times and obtain the most suitable chromosome as the optimal result.
The significance of this hybrid intelligent algorithm is that it does not require LSF to be a strictly monotone function. Additionally, this hybrid intelligent algorithm can address high-dimensional and nonlinear challenges. Therefore, to process the general URBDO model (13) under epistemic uncertainty, structural designers only need to know the uncertainty distribution of uncertain variables.
Chromosome evaluation Selection process

Crossover operation
Mutation operation Figure 1. Flowchart of the proposed hybrid intelligent algorithm.

Example Analysis and Discussion
To demonstrate the applicability and rationality of the proposed URI formulation and URBDO model, four examples are provided in this section: the first is a latch lock mechanism of hatch, the second is a passive vehicle suspension mechanical model, the third is a vehicle disc brake system, and the last is a welded beam. All the calculations in these problem are performed in MATLAB 2020a.

Reliability Analysis of a Latch Lock Mechanism
A structural reliability problem for a latch lock mechanism of hatch is considered, as shown in Figure 2a [42]. The latch lock mechanism can be visualized as a crank slider schematically shown in Figure 2b. According to the kinematic analysis of the crank slider mechanism, the horizontal distance between the fulcrum and the endpoint of the latch lock is expressed as: where α 1 is the angle between the horizontal direction and the crank, r, L 1 , L 2 , and e are dimensional parameters.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 21 assessment of the latch lock mechanism can be implemented based on the two uncertainty reliability methods presented in Section 3. To verify the credibility of the proposed structural reliability analysis methods, the traditional FORM-based on probability theory is provided as a reference. In this case, all input factors are regarded as random variables with normal probability distribution shown in Table 1. Comparative reliability results of the aforementioned three analysis methods in 3 [286, 291] L ∈ are shown in Figure 3.  As presented in Figure 3, with an increase 3 L , both URI and probabilistic reliability indicator (PRI) decrease. It is worth noting that the uncertainty theory-based URI trend is consistent with the probability theory-based PRI trend. When 3 L increases, the URI decreases, and the belief degree of the mechanism reliability becomes lower. Therefore, the URI introduced in Section 3 is effectively describes the trend of mechanism reliability with epistemic uncertainties. It clearly shows that the reliability estimated by the presented method is lower than the one calculated by the classical probabilistic reliability method.
Furthermore, as shown in Figure 3, the URI estimated based on the equivalent analytical model is highly consistent with the one calculated by the US approach. Therefore, it can be inferred that the US approach meets the requirements of prediction accuracy. The URI can be used to help structural engineers make the most reasonable prediction when the input factors are obtained from epistemic uncertainty with insufficient sample data. When L is not less than L 3 (mm), the latch lock can be normally locked. Otherwise, the latch lock cannot normally operate. Therefore, the LSF of the latch lock mechanism can be established as follows: In this work, it is assumed that the input factors ξ = (α 1 , r, L 1 , L 2 , e) T are uncertain variables with independent normal uncertainty distributions under insufficient sample data. Relevant distribution parameters of input factors are listed in Table 1. The reliability assessment of the latch lock mechanism can be implemented based on the two uncertainty reliability methods presented in Section 3. Table 1. Distribution parameters of input factors. To verify the credibility of the proposed structural reliability analysis methods, the traditional FORM-based on probability theory is provided as a reference. In this case, all input factors are regarded as random variables with normal probability distribution shown in Table 1. Comparative reliability results of the aforementioned three analysis methods in L 3 ∈ [286, 291] are shown in Figure 3.

Input Factors
As presented in Figure 3, with an increase L 3 , both URI and probabilistic reliability indicator (PRI) decrease. It is worth noting that the uncertainty theory-based URI trend is consistent with the probability theory-based PRI trend. When L 3 increases, the URI decreases, and the belief degree of the mechanism reliability becomes lower. Therefore, the URI introduced in Section 3 is effectively describes the trend of mechanism reliability with epistemic uncertainties. It clearly shows that the reliability estimated by the presented method is lower than the one calculated by the classical probabilistic reliability method.

Reliability Analysis of a Vehicle's Suspension Mechanics
To exhibit the advantage of the US approach, a passive vehicle suspension problem is investigated in this subsection [43]. The simplified model is shown in Figure 4. The considered LSF is the road-holding ability of the vehicle, which can be expressed as follows:   Table 2.  Furthermore, as shown in Figure 3, the URI estimated based on the equivalent analytical model is highly consistent with the one calculated by the US approach. Therefore, it can be inferred that the US approach meets the requirements of prediction accuracy. The URI can be used to help structural engineers make the most reasonable prediction when the input factors are obtained from epistemic uncertainty with insufficient sample data.

Reliability Analysis of a Vehicle's Suspension Mechanics
To exhibit the advantage of the US approach, a passive vehicle suspension problem is investigated in this subsection [43]. The simplified model is shown in Figure 4. The considered LSF is the road-holding ability of the vehicle, which can be expressed as follows: where A = 1.0 cm 2 / cycle m, b 0 = 0.27, V = 10.0 m/s, M = 3.2633 kg·s 2 /cm, G = 981 cm/s 2 , and m = 0.8158 kg·s 2 /cm are deterministic parameters. The considered input factors are the spring stiffness coefficient c (kg/cm), tire stiffness c k (kg/cm), and shock absorber damping coefficient k (kg/cm s). In this paper, all input factors are regarded as uncertain variables ξ = (c, c k , k) T . Distribution types and parameters of input factors are represented in Table 2.
The LSF g(ξ) of this real engineering problem is not a strictly monotonic function for c. Therefore, the URI cannot be calculated based on the equivalent analytical model. Hence, the US method is used to estimate the reliability of the vehicle suspension. The classical MCS method with a sample size of 10 7 is provided as a reference. Within the MCS method, all input factors are considered as random variables with normal probability distribution shown in Table 2. The corresponding reliability estimation results of the aforementioned two approaches are shown in Figure 5 under the assumption that the mean value µ k fluctuates within µ k ∈ [0, 30].  Table 2.     The LSF ( ) g ξ of this real engineering problem is not a strictly monotonic f for c. Therefore, the URI cannot be calculated based on the equivalent analytical Hence, the US method is used to estimate the reliability of the vehicle suspensi classical MCS method with a sample size of 10 7 is provided as a reference. Within t method, all input factors are considered as random variables with normal probabi tribution shown in Table 2. The corresponding reliability estimation results of th mentioned two approaches are shown in Figure 5 under the assumption that th value k μ fluctuates within . Therefore, the proposed methods can help engineering ers estimate structural reliability when input factor information is insufficient. T plied engineering problem shows that the US method can be more widely employ the equivalent analytical model. According to Figure 5, both URI and PRI increase with µ k . It clearly shows that the structural reliability estimated by the US method is higher than the one calculated by the traditional MCS method when µ k ∈ [0, 17.4819]. In addition, the structural reliability calculated by the proposed method is lower than the one estimated by the MCS method when µ k ∈ [17.4819, 30]. Therefore, the proposed methods can help engineering designers estimate structural reliability when input factor information is insufficient. This applied engineering problem shows that the US method can be more widely employed than the equivalent analytical model.

Reliability-Based Optimization of a Vehicle Disc Brake System
A vehicle disc brake system that contains a brake disc and a pair of brake pads is depicted in Figure 6. The brake pads are made of back plates and friction materials. According to Xia et al. [44], a continuum 3D finite element model of the vehicle disc brake system includes 26,125 elements and 37,043 nodes. Since the lightweight structure is a critical requirement in the preliminary design stage of a vehicle, the mass of the back plate needs to be minimized. Therefore, back plate thickness h 3 serves as the objective function, and the general URBDO model of vehicle disc brake can be formulated as follows: where h 1 is the friction material thickness, h 2 the disc thickness, h 1 and h 2 are assumed as linear uncertain variables with h 1 ∼ L(14.5 mm, 15.5 mm) and h 2 ∼ L(19 .5 mm, 20.5 mm), respectively. µ is the friction coefficient, p indicates the brake pressure, µ and p follow the normal uncertainty distribution with µ ∼ N (0.35, 0.01) and p ∼ N (0.5 MPa, 0.02 MPa) respectively, and R U is the prescribed reliability level according to the engineering requirement.  In this problem, the URBDO model of the brake system can be implemented by the presented CEP method for different reliability levels. To confirm the optimal results of the proposed approach, the classical FORM-based RBDO (reliability index approach, RIA) method is also applied to the brake system problem when all input factors are considered to follow normal or uniform probability distributions. In other words,  99 0.991 0.992 0.993 0.994 0.995 0.996 0.  In this problem, the URBDO model of the brake system can be implemented by the presented CEP method for different reliability levels. To confirm the optimal results of the proposed approach, the classical FORM-based RBDO (reliability index approach, RIA) method is also applied to the brake system problem when all input factors are considered to follow normal or uniform probability distributions. In other words, h 1 ∼ U(14.5 mm, 15.5 mm) , h 2 ∼ U(19 .5 mm, 20.5 mm), µ ∼ N 0.35, 0.01 2 , and p ∼ N 0.5 MPa, 0.02 2 MPa) . The optimal results of the RBDO and URBDO model are plotted in Figure 7.
According to Figure 7, with an increase in predetermined reliability requirements, more material is needed to fabricate the brake system. According to the optimal results, variation trends of back plate thickness computed by URBDO and RBDO models are similar. When the prescribed reliability R pre is in R pre ∈ [0.50, 0.99335], the back plate thickness estimated by URBDO model is lower than the one calculated by the RBDO model. However, the back plate thickness calculated by the URBDO model is higher than the one estimated by the RBDO model when the prescribed reliability is in R pre ∈ [0.99335, 0.999]. This is because the impact of epistemic uncertainty is omitted in the RBDO model. Consequently, the RBDO model is not suitable when human subjective judgment appears in structural design problems. The URBDO model based on the uncertain measure can evaluate the design of engineering structure more accurately and credibility under epistemic uncertainty.
In this problem, the URBDO model of the brake system can be implemented presented CEP method for different reliability levels. To confirm the optimal resul proposed approach, the classical FORM-based RBDO (reliability index approac method is also applied to the brake system problem when all input factors are con to follow normal or uniform probability distributions. In other  The CEP method and the USGA algorithm are simultaneously applied to the brake disc system for the desired reliability level R U = 0.9987 to discuss the performance of two proposed approaches for solving the URBDO model. Parameters in the USGA algorithm are set as follows: Pop_size = 30, P c = 0.8, P m = 0.2, and N = 12. In Figure 8, the iterative convergence process of back plate thickness h 3 generated by URBDO (CEP), URBDO (USGA), and RBDO (RIA) method for R U = 0.9987 is displayed. A detailed comparison of the optimal solutions of the three methods is summarized in Table 3. The number of function calls of constraint is expressed as F-evaluations. Furthermore, the CPU time is used to compare the efficiency of the CEP and USGA algorithms in processing the URBDO model.
Appl. Sci. 2022, 12, x FOR PEER REVIEW According to Figure 7, with an increase in predetermined reliability requir more material is needed to fabricate the brake system. According to the optimal variation trends of back plate thickness computed by URBDO and RBDO models ilar. When the prescribed reliability pre , the back pla ness estimated by URBDO model is lower than the one calculated by the RBDO However, the back plate thickness calculated by the URBDO model is higher than estimated by the RBDO model when the prescribed reliability is in . This is because the impact of epistemic uncertainty is omitted in the RBDO mod sequently, the RBDO model is not suitable when human subjective judgment ap structural design problems. The URBDO model based on the uncertain measure c uate the design of engineering structure more accurately and credibility under ep uncertainty.
The CEP method and the USGA algorithm are simultaneously applied to th disc system for the desired reliability level U 0.9987 R = to discuss the performanc proposed approaches for solving the URBDO model. Parameters in the USGA al are set as follows: is displayed. A detailed com of the optimal solutions of the three methods is summarized in Table 3. The nu function calls of constraint is expressed as F-evaluations. Furthermore, the CPU used to compare the efficiency of the CEP and USGA algorithms in processing the U model.    As shown in Figure 8, the optimal values of the three approaches can be obtained if the iterations reach 10 times. According to Table 3, the CEP method is more efficient than the USGA algorithm when both approaches are simultaneously used to solve the URBDO model. This engineering problem confirms that both CEP method and the USGA algorithm can obtain almost equal optimal results.
Since the rank-sum test is a classical nonparametric test that is not limited by the type of population distribution [45], it is used to benchmark the impact of the genetic algorithm parameters on the performance of the USGA algorithm. In this work, different population sizes and mutation probabilities are used to test the performance of USGA. Assuming that the value of population size Pop_size = 20, two sets of sample data are carried out based on the USGA algorithm with mutation probability P m = 0.01 and P m = 0.1, respectively. Similarly, supposing that the value of P m is 0.2, another two sets of sample data are carried out based on the USGA algorithm with Pop_size = 15 and Pop_size = 30, respectively. The optimal results of back plate thickness corresponding to the aforementioned four sets at P c = 0.8 are summarized in Table 4. The following null hypothesis is designed for this problem under the significance level α = 0.05.
H 0 : There is no significant difference in the optimal solution of USGA algorithm (19)  As presented in Table 4, the sample size of each set is 8, and the rank-sum T for Pop_size = 20 and P m = 0.2 are 52 and 53.5, respectively. The function "ranksum" in the MATLAB is employed to calculate the p-value. Then, the P-values corresponding to Pop_size = 20 and P m = 0.2 are 0.0957 and 0.1383, respectively. Since both 0.0957 and 0.1383 are greater than 0.05, the null hypothesis H 0 is retained. Therefore, there is no significant difference in the back plate thickness optimized based on the USGA algorithm. In other words, the setting of genetic algorithm parameters has no significant effect on the performance of the proposed USGA algorithm.

Reliability-Based Optimization of a Welded Beam
A welded beam design problem shown in Figure 9 has four uncertain design variables and four uncertainty constraints [14]. In this problem, the optimization objective is to minimize the welding cost C(x) and tip deflection δ(x; ξ) and tip deflection of the beam. Uncertainty constraints are related to physical characteristics such as bending stress, shear stress, buckling, and displacement of the free end. The input factors are the beam length L = 355.6 mm, the Young's Modulus E, shear Modulus G = 82, 740 MPa, point load F = 26, 688 N, maximum normal stress σ, and maximum shear stress τ. All uncertain variables are independent and characterized by normal or linear uncertainty distributions. The URBDO model of the welded beam design is formulated as follows: (20) where x = [x 1 , x 2 , x 3 , x 4 ] T is the vector of uncertain design variables, ξ = [E, σ, τ] T is the vector of uncertain variables, and R U j = 0.99, j = 1, 2, 3, 4 is the predetermined reliability. Weights of objective functions f 1 and f 2 are ω 1 and ω 2 , respectively, i.e., ω 1 + ω 2 = 1.    Suppose that all uncertain design variables follow normal uncertainty distr due to insufficient sample data. Since the experimental data are imprecise, σ , τ can be characterized by linear uncertainty distribution combined with uncertain sion analysis [4]. For this design problem, as multiple uncertain variables and con are included, there is no solution using the CEP model. Hence, the USGA algo utilized to solve the URBDO model (20) of the welded beam design problem.
Parameters in the USGA method are set as follows:  Suppose that all uncertain design variables follow normal uncertainty distributions due to insufficient sample data. Since the experimental data are imprecise, σ, τ, and E can be characterized by linear uncertainty distribution combined with uncertain regression analysis [4]. For this design problem, as multiple uncertain variables and constraints are included, there is no solution using the CEP model. Hence, the USGA algorithm is utilized to solve the URBDO model (20) of the welded beam design problem.
Parameters in the USGA method are set as follows: Pop_size = 30, P c = 0.9, P m = 0.1, and N = 40. To investigate the effectiveness of the URBDO (USGA) method, the RBDO (RIA) method is also employed for this design problem when all the input factors are considered to follow normal or uniform probability distributions. Detailed distribution types and parameters for input factors are listed in Table 5. Design results of objective function values f 1 and f 2 generated by URBDO and RBDO model under different weights ω 1 = 0.4, ω 1 = 0.5, and ω 1 = 0.6 are displayed in Figure 10. Furthermore, specific optimal results are summarized in Table 6. Another issue that should be discussed is the LSF. It is clear that the LSF are assumed to be explicit in our proposed models. In many real engineering problems, however, these functions may not be that easy to obtain. Integrating surrogate models with uncertain variables, would be a rational way to solve the implicit function problems. Fortunately, because of the good adaptability of the USGA algorithm, the proposed method is able to handle this issue. Structural designers can verify this with some reliability benchmark problems elaborated in [46].
It should be emphasized that to use the URBDO model and corresponding solving techniques, the above processes should be carried out under the framework of uncertainty theory using uncertain variables, instead of random variables, since the sampling of uncertain variables is based on the inverse uncertainty distribution. People can calibrate and validate the proposed URI and URBDO models by adding the design optimization module to the UQLab framework. Inspired by the research shown in [46], another research direction is to reduce the computational burden of the USGA algorithm by introducing an active learning strategy.

Conclusions
In this paper, a new structural reliability assessment and design optimization method based on uncertainty theory was introduced under epistemic uncertainties. To enable the engineering designers to estimate the structural reliability according to incomplete sample data, the URI and the corresponding two calculating approaches are provided. Engineering examples indicate that both calculating approaches are applicable, whereas the US approach is more effective because it does not need to know the monotonicity of LSF and can avoid unsolvable equations. In addition, a typical URI-based URBDO model that combines uncertainty reliability demands into structural design constraints was established to provide an optimal result with limited sample data. Two general methods were designed to solve the URBDO model. Actual engineering structural problems confirm that the USGA method is applicable in any situation. However, it is more time-consuming than the CEP method. Generally speaking, if the designer knows the monotonicity of LSF, the CEP method is more convenient. Otherwise, it is only suitable for the USGA method.
It should be further emphasized that in instances where sufficient information of input factors cannot be acquired due to limitations in technical, human, facility, and time resources, the results obtained by the presented methods can provide a valuable reference for decision-makers in the preliminary design stage of the structure. On the other hand, the probability-theory-based method is more accurate and realistic when the information regarding input factors is sufficient.  According to Figure 10 and Table 6, with an increase in ω 1 , the solution result of objective 2 increases, while the solution result of objective 1 decreases. In addition, the value of objective 1 calculated by the URBDO method is higher than the value estimated by the RBDO method. The value of objective 2 calculated by the URBDO method is lower than the value estimated by the RBDO method. This is because the RBDO method neglects the influence of epistemic uncertainties and negatively affects the credibility of optimal results. Hence, when the engineering designers only know the expected value and variance of input factors, it is more reasonable to deal with the input factors as uncertain variables. The presented URBDO model can be applied to help structural designers make the most suitable design when input factors are obtained from the subjective interpretation of insufficient observed data.

Discussion
Four practical engineering problems show that the URI based on uncertainty theory is a better indicator than PRI for quantifying epistemic uncertainties. The established URBDO model can meet structural design requirements of engineering applications with insufficient sample data. In general, when the CEP and USGA methods can solve the same optimal design problem, the computational efficiency of the CEP method is higher than that of the USGA method. Nevertheless, the CEP model sometimes has no solution when faced with high-dimensional nonlinear or non-monotonicity challenges. Consequently, the USGA algorithm can solve high-dimensional nonlinear and non-monotonic problems, although it is time-consuming.
Another issue that should be discussed is the LSF. It is clear that the LSF are assumed to be explicit in our proposed models. In many real engineering problems, however, these functions may not be that easy to obtain. Integrating surrogate models with uncertain variables, would be a rational way to solve the implicit function problems. Fortunately, because of the good adaptability of the USGA algorithm, the proposed method is able to handle this issue. Structural designers can verify this with some reliability benchmark problems elaborated in [46].
It should be emphasized that to use the URBDO model and corresponding solving techniques, the above processes should be carried out under the framework of uncertainty theory using uncertain variables, instead of random variables, since the sampling of uncertain variables is based on the inverse uncertainty distribution. People can calibrate and validate the proposed URI and URBDO models by adding the design optimization module to the UQLab framework. Inspired by the research shown in [46], another research direction is to reduce the computational burden of the USGA algorithm by introducing an active learning strategy.

Conclusions
In this paper, a new structural reliability assessment and design optimization method based on uncertainty theory was introduced under epistemic uncertainties. To enable the engineering designers to estimate the structural reliability according to incomplete sample data, the URI and the corresponding two calculating approaches are provided. Engineering examples indicate that both calculating approaches are applicable, whereas the US approach is more effective because it does not need to know the monotonicity of LSF and can avoid unsolvable equations. In addition, a typical URI-based URBDO model that combines uncertainty reliability demands into structural design constraints was established to provide an optimal result with limited sample data. Two general methods were designed to solve the URBDO model. Actual engineering structural problems confirm that the USGA method is applicable in any situation. However, it is more time-consuming than the CEP method. Generally speaking, if the designer knows the monotonicity of LSF, the CEP method is more convenient. Otherwise, it is only suitable for the USGA method.
It should be further emphasized that in instances where sufficient information of input factors cannot be acquired due to limitations in technical, human, facility, and time resources, the results obtained by the presented methods can provide a valuable reference for decision-makers in the preliminary design stage of the structure. On the other hand, the probability-theory-based method is more accurate and realistic when the information regarding input factors is sufficient.