Reliability Analysis and Neural-network Modeling of Ship Radar Mast Structure by Adopting Quasi-random Sampling

2.1 Quasi-Random Sampling

The quasi-random sampling method is a statistical reliability analysis technique that can evaluate the uncertainty of a probabilistic performance function because of random variables at a low computational cost, provided that an appropriate number of samples consistent with the target reliability probability is applied. The Sobol sequence method is a quasi-random sampling method developed to enable uniform sampling with a small number of samples in the design space (Sobol and Levitan, 1999; Saltelli et al., 2010; Song, 2020). A Sobol sequence is generated by combining binary fractions of bit length w and direction numbers vij, and the direction numbers are generated from the following primitive polynomial and recursive polynomial:

where ⊕ denotes the bitwise XOR operator. The Sobol sequence in the j^th dimension is defined as follows:

Eq. (3) is organized as follows by applying the primitive polynomial in each dimension:

In the Sobol sequence method, the number of samples is determined using the following equation:

where m is the number of random variables. The number of samples is determined by the complexity of the problem and the number of random variables. Therefore, it is necessary to review a reasonable number of samples according to the characteristics of the problem being evaluated for reliability using the Sobol sequence method.

2.2 Neural-Network Model

An approximate model was used to examine the design space because it can maximize computational efficiency when applied to design optimization or reliability analysis, which require high computational costs. In addition, the approximate model accuracy was determined by the generation conditions of the sampling data in the design space and the approximation method. This study used a neural network model as the approximation method to compare the accuracy of design space exploration under quasi-random sampling conditions. The neural network model was constructed using the elliptical basis function (EBF). The EBF neural network model (Luo et al., 2004) is a class of artificial neural networks that employ an elliptical basis function as an activation function. The structure of the EBF neural network model includes an input layer, a hidden layer with a non-linear activation function, and a linear output layer. An EBF neural network model with N hidden neurons can be expressed as follows:

where x is the input vector; w_i are the weights; b is the bias term; ϕ(x, c_i, σ_i) represents the elliptical basis function centered at c_i with a covariance matrix σ_i . The elliptical basis function can be defined as follows:

The EBF allows the neural network model to handle data with anisotropic properties and varying scales along different dimensions, enhancing flexibility and performance in complex pattern recognition and function approximation tasks.

3.1 Vibration and Strength Performance

A ship radar mast is commonly installed on crude oil tankers with a dead weight tonnage (DWT) exceeding 100,000 and bulk carriers exceeding 180,000 DWT. Table 1 lists the main ship particulars.

Considering the high cost and significant time investment required for statistical reliability analysis and approximation processes, it is appropriate to evaluate the vibration and strength performance through numerical analysis using the FEM. Fig. 1 presents the design specifications and FEM model of the ship radar mast. The ship radar mast is represented by a three-dimensional FEM model, produced using shell elements and concentrated mass elements, comprising 129,428 nodes and 131,513 elements (Fig. 1). The material properties of ASTM A283-C steel, with a density of 7,850 kg/m³, elastic modulus of 210 GPa, Poisson’s ratio of 0.3, and yield strength of 313 MPa, were applied to the FEM model to ensure that the weight of the FEM model matched the actual weight.

The boundary conditions applied in the vibration and strength analysis were implemented by constraining all degrees of freedom at the lowermost nodes of the radar mast of ships. The vibration analysis was conducted via a normal mode analysis with fixed boundary conditions and without additional loads to determine resonance with the major excitation frequencies. The strength analysis for the structural safety was performed by calculating the von Mises stress for each load case, derived from the wind load and ship motion acceleration as specified by the classification society regulations (KR, 2023), and comparing these stresses with the allowable yield stress. The wind load, F, was calculated using the following equation:

where P is the wind pressure; A (m²) is the area sum of structural members under wind pressure in projection in respective wind direction; C_h is the height factor; C_s is the shape factor to be determined depending on the shapes of main parts; g (m/s²) is the acceleration due to gravity; V (m/s) is the design wind velocity. The wind load calculated with C_h = 1.3, C_s = 0.7, and V = 51.5 m/s for the ship radar mast using Eq. (8) was 585 kN, and the calculated wind load was applied to the FEA model in the form of pressure on the projection area against the sailing direction of the ship as shown in Fig. 1(b). The accelerations used to calculate the dynamic loads due to ship motion are the combination of ±1.0 g and ±0.5 g in the translational direction. Eight load cases were constructed considering the wind load and ship motion accelerations; Table 2 lists the maximum von Mises stress for each load case.

The vibration and strength analyses were performed using the general-purpose FEM program MSC.NASTRAN (Hexagon, 2023). Fig. 2 presents representative results, including the first normal mode shape and the stress contour of LC2, which exhibited the highest von Mises stress. The location of the maximum stress was the same in all load cases at the connection between the lower part of the main body and the upper part of the lower bracket. The first normal mode was determined to be a transverse mode at 3.6 Hz (Fig. 2(a)). The potential for resonance between the first normal mode frequency of the ship radar mast and major excitation frequencies was assessed using a Campbell diagram (Campbell, 1924), as shown in Fig. 3. The 3.6 Hz normal mode frequency does not resonate within the NCR to MCR range, where most voyages occur (Fig. 3). Considering various uncertainties, the resonance is predicted to be avoided within the NCR to MCR range if the design target for the first normal mode frequency of the ship radar mast is set in the range of 3.6 to 4.0 Hz. Higher upper-frequency limits can improve the stiffness, but if the upper-frequency limit is higher than 4.0 Hz, resonance with the propeller first or main engine fourth excitation frequency may occur near the NCR. The maximum stress of LC2, calculated at 213.3 MPa, occurred at the connection between the lower part of the main body and the upper part of the lower bracket, as shown in Fig. 2(b). The allowable stress was set to 85% of the material yield stress, according to the regulations of the classification society, resulting in an allowable stress of 266.05 MPa. Although the maximum stress was below the allowable stress of 266.05 MPa, the safety margin was only 19.8%, suggesting that the strength performance was not sufficiently high. Based on the vibration and strength performance evaluation, the current ship radar mast requires a detailed review of the design reliability through an appropriate statistical reliability analysis.

3.2 Statistical Reliability Analysis

The Sobol sequence method was applied in the statistical reliability analysis to identify a suitable quasi-random sampling process for evaluating the design reliability related to the vibration and strength performance of the ship radar mast. The limit state for the probabilistic performance function concerning vibration performance was set in the range of 3.6 to 4.0 Hz, and it was set at the allowable yield stress of 266.05 MPa for strength performance. The reliability of the defined probabilistic performance functions was assessed based on a 1.0% failure probability. The probability of reliability was estimated with an accuracy of ±5% at a 95% confidence level, which is the reliability of the lower and upper bounds. The design uncertainty of the ship radar mast was considered as the manufacturing tolerance of the seven major component thicknesses shown in Fig. 1(a) and was adopted as a random variable. The standard deviation of the random variable was set by referring to the manufacturing tolerance standards (ASTM, 2008; KS, 2019), and the probability distribution function was defined to follow a normal distribution (Navidi, 2008; NIST/SEMATECH, 2003), as listed in Table 3.

In the quasi-random sampling process, the number of samples was set in five cases by sequentially varying the integer k in Eq. (5) from 5 to 9 considering the ±2 range for the number of random variables, resulting in 192 times (Case-I), 384 times (Case-II), 768 times (Case-III), 1,536 times (Case-IV), and 3,072 times (Case-V). The probability of reliability for each case was compared with the results of the Monte Carlo Simulations (MCSs) (Siddall, 1983), which require relatively large-scale sampling but offer high accuracy in reliability evaluation. The required number of MCS samples (N_pf) for estimating the probability of reliability was calculated using the following equation:

where p_f is the target probability of failure and e is the error ratio of accuracy based on the 95% confidence level. From Eq. (9), the number of MCS samples was set to 792,000. Table 4 lists the results of the statistical reliability analysis for the vibration and strength performance of the ship radar mast based on the application of quasi-random sampling and MCS considering the 0.00001 convergence tolerance.

The quasi-random sampling conditions of Case-V yielded the same probability of reliability results as the MCS, as listed in Table 4. For cases with fewer samples than Case-III, however, the error compared to the MCS increased, and the probability of reliability tended to be overestimated. The probability of reliability for the lower limit of vibration was lower than that of the other probabilistic performance functions. This result can be attributed to the initial value of the first normal mode frequency being actively close to the lower limit of vibration. The effective probability of reliability for the lower limit of vibration, the upper limit of vibration, and the strength of the ship radar mast were estimated to be 90.5%, 99.5%, and 98.3%, respectively. Figs. 4 and 5 present the statistical frequency histogram (SFH) results for each probabilistic performance function for Case-V and MCS.

The distribution characteristics of the SFH for Case-V and MCS were almost identical (Figs. 4 and 5). In the SFH, the height of the histogram was proportional to the frequency of occurrence of the characteristic values of the corresponding probabilistic performance function. A quasi-random sampling number of approximately 0.4% of the MCS sampling number was sufficient for the statistical reliability analysis of the ship radar mast, as shown in Table 4 and Figs. 4 and 5. Hence, applying the Sobol sequence method-based quasi-random sampling to the statistical reliability analysis of the ship radar mast can achieve high numerical computational efficiency.

3.3 Approximation

This section reports the correlation between the sampling conditions applied in the statistical reliability analysis and the accuracy of the EBF neural network model. Fig. 6 presents the structure of the EBF neural network model used in this study, including the input, hidden, and output layers.

The seven design variables were placed in the input layer (Fig. 6); three hidden layers were used, and the probabilistic performance functions of vibration and strength were applied to the output layer. The convergence was set as a learning rate of 0.7 and ten training iterations. In addition, the model aims to identify the most suitable conditions for approximating the vibration and strength performance of the radar mast of ships. The approximation accuracy of the EBF neural network model was determined using the R² value, which is expressed as follows:

where t_i represents the actual values; y_i represents the predicted values from the EBF neural network model; t̄_i represents the mean of the actual values. An R² value of 1.0 suggests that the predictions from the EBF neural network model perfectly match the actual values across the entire design space. Table 5 lists the approximation fidelity of the EBF neural network model.

Case-IV, Case-V, and MCS exhibited R² values closest to 1.0, indicating the highest approximation accuracy across the design space (Table 5). The approximation results for Case-I and Case-II showed negligible differences, with an error rate of approximately 6% compared to the averaged R² results of MCS, while Case-III exhibited an error rate of 5.4%. Although the differences in approximation accuracy were observed across the sampling conditions, all cases showed sufficient accuracy for use as approximate models to estimate the design space. Thus, the increase in sampling frequency applied in statistical reliability analysis showed a significant correlation with the accuracy of the EBF neural network model. Based on the results in Table 5, an effective sampling frequency for the EBF neural network model-based approximation of the ship radar mast, using quasi-random sampling, was approximately 0.2% of the MCS sampling frequency.