Reliability Analysis for Structure Design of Automatic Ocean Salt Collector Using Sampling Method of Monte Carlo Simulation

2.1 MCS

MCS is a practical reliability analysis method to determine the uncertainties associated with random variables in a probability performance function for a suitable sampling frequency that satisfies the target confidence level in the evaluation of the probabilities of reliability or failure. However, because the MCS involves a simple random sampling method in which the sampling frequency increases with the target confidence level and accuracy, the efficiency of reliability analysis may be lower. Therefore, numerical calculation experiments, rather than physical experiments, are applied in MCS-based reliability analysis (Siddall, 1983). As shown in Fig. 1, an MCS is performed in three stages, i.e., sampling of random variables, numerical experimentation, and statistical analysis, for random variables X = (X₁, X₂, …, X_n) and a probability performance function Y = g(X).

Fig. 1.

Procedure of MCS

In the statistical analysis stage, the probability performance function, Y, is calculated as many times as the sampling frequency, N, to perform reliability analysis by calculating the mean Ȳ, variance σY2, and probability of failure p_f, as follows:

The probability of failure, p_f, in Eq. (1) can be expressed using an indicator function, I(x), as follows:

The probability of failure is arranged as follows using the mean value of the indicator function:

Here, N_f is the sampling frequency when g ≤ 0. The probability of reliability, R, is defined as follows:

The probability of failure is obtained in terms of a cumulative distribution function (CDF), F_Y(y), and a probability density function (PDF), f_Y(y), as shown below:

f_Y(y) = [F_Y(y)]′

As shown in Eq. (5), the PDF is a differential of the CDF. The accuracy of the MCS can be assessed as a function of error rate, e, based on the 95% confidence level as follows:

As shown in Eq. (6), once the target probability of failure and error rate are defined, the sampling frequency required for the reliability analysis can be determined. The accuracy of the probability of failure evaluated in the MCS may be improved if the sampling frequency requirement is satisfied; however, the sampling frequency should be determined based on the characteristics of the analysis subject and the efficiency of numerical calculations in engineering design.

2.2 SSM

SSM is a quasi-MCS method designed to allow a consistent sampling with a low sampling frequency of multidimensional variables in a design space (Sobol & Levitan, 1999; Saltelli et al., 2010). In the SSM, a sampling position is selected based on the previous sampling point, thereby minimizing the gap with a cluster. A Sobol sequence is generated from a combination of binary fractions with a length of w beats and the direction numbers of vij, [i = 1, 2, …, w; j = 1, 2, …, d]. To generate directional numbers in the j dimension, the following primitive and recursive polynomials of directional numbers are defined:

In Eq. 7, ⊕ refers to the XOR bitwise operation. The Sobol sequence of the j dimension, xqj(n=∑i=0wbi2i,bi∈{0,1}), is generated using primitive polynomials at each dimension, as follows:

The sampling frequency, N_T, required in the SSM is defined as follows:

m in Eq. (9) refers to the number of random variables. The sampling frequency in the SSM is not fixed and should be determined based on the complexity of the analysis subject and the number of random variables.

2.3 Kriging Metamodel

Because metamodels are generated from the sampled data in the experimental planning or design space, the accuracy of the generated metamodels can be quantitatively reviewed to verify the approximation of the design space based on the sampling method (Lee & Song, 2013). Creating a highly accurate metamodel is one of the most important tasks in the exploratory research of the design space as it can significantly reduce the numerical inefficiencies by utilizing it for multipurpose or reliability-based design optimizations that involve a high number of calculation iterations.

A Kriging model is expressed as the sum of the global model of the actual design space function to be approximated and the local model that corresponds to the difference between the actual function and the global model (Cho et al., 2009).

Here, A_K = {a₁, a₂, …, a_p}^T is an unknown coefficient vector, and Z(x) = {z₁(x), z₂(x), …, z_p(x)}^T is a global model vector defined with a design variable, x ∈ E^n_d. The global model can be defined as p = 1 in a constant case, p = n_d + 1 in a linear case, and p = (n_d + 1) (n_d + 2)/2 in a second-order polynomial case. E(x)  represents an independent normal distribution, and the response vector obtained from n experimental positions is defined as follows:

E(x) is a spatial correlation of the design data and is generally defined as a Gaussian correlation function, expressed as follows:

Here, θ, x ∈ E^n_d and the correlation matrix is a positive definite matrix with diagonal elements of 1. The correlation coefficient, θ_k, is a parameter that expresses the correlation of the response values in the x_k-direction and determines the curvature of the Kriging model in the x_k-direction. The correlation decreases with increasing θ_k, as shown in Eq. (12), and the approximation of the Kriging metamodel in the direction of the input variables shows nonlinear characteristics. The correlation becomes more evident with decreasing θ_k, as shown in Eq. (12), and the approximation of the Kriging metamodel in the direction of the input variables shows linear characteristics. The correlation coefficient, θ, with the highest probability can be calculated through the maximum likelihood estimation, which maximizes the likelihood function.

3.1 Evaluation of AOSC Strength Performance

The system configuration of the entire AOSC and the thickness of the major structural components for FEM modeling are shown in Fig. 2. As shown in Fig. 2(a), the AOSC has a salt collector and a towing unit system for automatic salt collection; a drive and power system for delivering electric drive power; a rail frame for transferring collected salt; and a collector frame, rail frame, and a main frame for the system installation and load support. As shown in the FEM analysis model in Fig. 2(b), the main frame structure was created as a shell element, the weight of each system was realized as a lumped mass, and the rigid link element was used to connect each structural component and apply the lumped mass and design load. The thicknesses of the main frame, rail frame, collector frame, and bracket were designed to be 6.0, 1.0–3.0, 1.0, and 4.0 mm, respectively. The FEM model contained 112,896 elements and 114,431 nodes with the following material properties applied: a density of 8,000 kg/m³, elastic modulus of 193 GPa, the Poisson’s ratio of 0.29, and a yield strength of 216 MPa, as in austenitic stainless steel (SUS316).

Fig. 2.

System configuration and structure analysis model

During the initial design phase, the design load conditions for the AOSC strength performance evaluation were determined by combining the system weight and operating load to reflect the worst-case scenario and by selecting the maximum loading conditions as well as the operating and braking conditions under the maximum load, as listed in Table 1. The inertial load in the direction of gravity, the acceleration of operating conditions in the transfer direction, and the acceleration of the braking conditions in the opposite direction of transfer were applied. The design load conditions listed in Table 1 were applied in the FEM analysis, and all degrees of freedom were restricted for the boundary conditions, except for the degree of freedom associated with the rotation of each wheel mounted at the bottom of the rail. As shown in Table 2, the general-purpose finite element analysis program, Abaqus/Implicit (Simulia, 2018), was used in the FEM analysis to calculate the maximum von Mises stress for each design load condition and evaluate the structural safety based on the yield stress of the material. The structure was considered to be safe if the maximum stress under each design load condition was less than 85% of the yield stress, or 183 MPa (DNV-GL, 2015). As outlined in Table 2, the structural safety was adequate in all of the design load conditions, with the highest stress of 173.8 MPa occurring at the bracket part in LC3. The structural components with the highest stress were the main frame part in LC1, rail frame part in LC2, and bracket part in LC3, whereas the collector frame part in all of the load conditions indicated the lowest stress level.

Table 1.

Design load cases

Table 2.

Structure analysis results

The stress distribution result for each design load condition is shown in Fig. 3. The maximum stress in LC1 occurred at the main frame part connected to the rail frame, the maximum stress in LC2 was at the rail frame connection, and the maximum stress in LC3 was at the center of the bracket. The maximum stress occurred at the center of the bracket part in LC3 because the weight of the salt collector part and the weight of the collected salt concentrated at the center, resulting in a high level of load under the braking conditions. Whereas the strength performance of the AOSC structural design satisfied the structural safety standard, a safety margin of less than 5% before reaching the performance limit was indicated in LC3; therefore, a quantitative reliability review is necessary to address uncertainties in the design factors.

Fig. 3.

Stress contour results (unit: MPa)

3.2 Reliability Analysis of AOSC Structural Design

Unlike marine structures that have design regulations regarding corrosion allowance, an AOSC is a fishery equipment with insufficient design regulations. Because AOSCs are used in marine salt ponds and their material thickness is the design factor, corrosion can be considered as a key uncertainty. Therefore, in the reliability evaluation of AOSCs, the thickness of the major structural components was set as a random variable to account for corrosions that may be caused by the nature of the work environment, whereas the strength performance was considered as a probability performance function. To address the uncertainty around the thickness of the structural components as a random variable, the mean was set to the initial design thickness, and the relative standard deviation (RSD) was set as 10% based on the experimental study results pertaining to the corrosion properties of austenitic stainless steel (Han and Park, 2012); furthermore, they were defined to follow a normal distribution because the variability in the strength performance with corrosion was reviewed for a normal operation of the AOSC in this study. The settings of the random variables used in the reliability analysis are listed in Table 3.

Table 3.

Random variables

In this study, the limit of the strength performance as a probability performance function was set to be 183 MPa, which is the standard for structural safety, whereas the reliability of the structural design was evaluated based on a probability of failure, p_f, of 0.01. Reliability analysis was performed based on the MCS and SSM. The sampling frequency in the MCS was set to 158,400 based on Eq. (6) with a 95% confidence level and an error rate of 5%. The frequency of sampling in the SSM was varied sequentially by changing the constant k in Eq. (9) from 10 to 4, for the following seven cases: 6,144 (Case 1), 3,072 (Case 2), 1,536 (Case 3), 768 (Case 4), 384 (Case 5), 192 (Case 6), and 96 (Case 7). The results of reliability analysis based on the MCS and SSM with varying sampling frequency are presented in Table 4, and the PDF results from the reliability evaluation of the strength performance in each case are illustrated in Fig. 4.

Table 4.

Comparison of reliability analysis results

Fig. 4.

Comparison of PDF results

As shown in Table 4, similar probabilities of reliability were calculated from Cases 1 to 3 when using the SSM and MCS. Furthermore, the PDF results from Cases 1 to 3 based on the SSM and MCS were similar, as shown in Fig. 4. However, the probability of reliability of Cases 4 through 7 when using the SSM reduced in proportion to the decrease in the sampling frequency, whereas the probability of reliability tended to be higher than based on the MCS, which can be considered as an indicator of reduced design safety. Additionally, the differences in the PDF results of Cases 4 through 7 obtained using the SSM compared with those obtained using the MCS increased significantly with decreasing sampling frequency. To apply the SSM in the reliability estimation of the AOSC’s structural design addressed in this study, it was discovered that at least 1% of the sampling frequency used in the MCS would be required. Therefore, the efficiency of numerical calculations in the SSM-based reliability analysis was high, and the analysis method would be effective for evaluating the structural design safety of fishery equipment, such as AOSCs.

3.3 Comparison of Kriging Metamodeling Characteristics

In this study, a Kriging metamodel was created for each sampling method used for the reliability analysis of the AOSC, and the correlation between the sampling method characteristics and metamodels was reviewed by analyzing the accuracy of the result in each metamodel. Because a Kriging metamodel can approximate multiple design variables and highly nonlinear problems in the design space by incorporating random sampling, Kriging metamodeling was applied to the data used in the random sampling methods, i.e., MCS and SSM (Cho et al., 2009). The accuracy in the design space exploration of the Kriging metal model created for each strength performance response in the design load conditions, i.e., LC1 through LC3, was reviewed based on each of the sampling methods used in the MCS and SSM for reliability analysis. The accuracy of the metamodel approximation was determined in terms of R², as shown in the following equation:

Here, t_i is an actual value, y_i is a predicted valued in the approximation model, and t̄_i is the mean of the actual values. An R² value of 1.0 indicates that the predicted values from the metamodel correspond exactly to the actual values in the entire design space. The accuracy of the Kriging metamodel result for each sampling method is summarized in Table 5. As shown in Table 5, the design space can be approximated with a 0.5% error rate in the MCS with a sampling frequency of 158,400 times, whereas the approximation accuracies from Cases 1 to 3 in the SSM were similar with 5% to 6% higher error rates than those in the MCS. The approximation accuracies of Cases 4 and 5 in the SSM with sampling frequencies of 768 and 384, respectively, tended to exhibit higher error rates of 7 to 9% than in the MCS, respectively. In Cases 6 and 7 with a sampling frequency of 192 or less, the approximation accuracy reduced significantly and was considered to be unsuitable for a metamodel. Although the correlation between the sampling frequency and the accuracy of the Kriging metamodel appeared to be proportional in general, a certain level of sampling frequency would be required to ensure an equivalent level of accuracy in Kriging metamodels, as demonstrated in Cases 1 through 3 in the SSM. Furthermore, as shown in the reliability analysis results in Table 4 and Fig. 4, the SSM incorporating more than 1% of the sampling frequency of the MCS is applicable not only to the reliability analysis, but also to the Kriging metamodeling of the structural design of fishery equipment, such as AOSCs.

Table 5.

Comparison of Kriging meta-modeling results