1. Introduction
In the era of the fourth industrial revolution, digital transformation has become an essential element in the shipbuilding and marine industry. In particular, structural health management is becoming increasingly important for the safe operation and extended lifespan of ships, thereby attracting interest in structural health management for ships using digital twin technology. A digital twin is a technology that replicates a physical object or system in a virtual space for real-time monitoring and simulation. Implementing a digital twin for the structural health assessment of a ship enables structural responses to be predicted under various environmental conditions through a virtual model identical to the actual ship and proactive maintenance by diagnosing structural damage in advance. The digital twins of ships for structural integrity can be implemented as virtual models constructed through various simulation techniques that reflect the physical characteristics of the actual ships, enabling the prediction of the stress, deformation, and vibration of ship structures with high accuracy. Additionally, the structural integrity of ships can be monitored, and any damage can be detected by collecting sensor data in real time and comparing them with digital twin models.
Because ships and marine structures typically have millions of degrees of freedom (DOFs), a considerable computational cost is required. Thus, limitations exist in implementing digital twin models with numerical models that consider the total DOFs. Therefore, studies that implemented digital twins through machine-learning models (e.g., artificial neural networks) were conducted to predict the response characteristics of entire structures with high reliability while reducing the computational costs of simulations. Cho et al. (2021) generated data through nonlinear time-series numerical analysis of spar-type floating wind turbines and classified six types of failures for the blade pitch system using recurrent neural network (RNN) machine-learning models. Shin et al. (2024) constructed an artificial neural network that adjusted hyper parameters using Bayesian optimization to detect and predict missing data from ocean buoys. They also presented machine-learning techniques with the highest predictive performance for the significant wave height, peak wave period, and heading angle. Park and Kang (2024) proposed a gated recurrent unit-based machine-learning model for predicting highly nonlinear waves in marine and coastal areas. They optimized hyper parameters and window sizes that constituted the machine-learning model through various wave data learning and predicted the significant wave height and modal period within errors of up to 0.062 m and 0.152 s, respectively. Xie and Tang (2024) proposed a domain generalization (DG) machine-learning model to improve predictive performance under new marine environmental conditions in addition to those used as training data. They obtained a high predictive performance compared with existing artificial neural networks in predicting the mooring line tension of a floating body. Wang et al. (2024) extracted damage pattern identification-related features over time by applying a convolutional neural network (CNN), an image processing-based machine-learning model, to fixed jacket marine structures and implemented a structural damage detection digital twin with an accuracy of over 95% through a particle swarm optimization algorithm. In addition, studies were conducted on the implementation of physics-based digital twin models that enable high-accuracy structural response simulations by creating a reduced system that reflects the characteristics of the structure by reducing the order of the entire system’s DOFs. Sharma et al. (2018) applied the reduced basis finite-element analysis technique and component mode synthesis to semi-submersible drilling rigs to create a reduced system that considers the material properties, loads, and boundary conditions of a structure. They evaluated structural integrity in the time domain through this system. Truong et al. (2024) researched the analysis of the frequency of vibration signals from the engine of a fishing boat from a perspective of structural integrity. Ko and Boo (2022) presented a transformation matrix that reduces the finite element matrix for the entire system of a structure considering only the stiffness and inertial effects of key nodes, and they created a reduced matrix for the stiffened plate used in ships. They verified it by conducting time-efficient modal, frequency response, and transient response analyses and comparing the results with the responses of the entire system. Sim and Lee (2024a) developed a distortion base mode that represents the deformation state of a structure under irregular wave load conditions based on fluid–structure coupled analysis data for multi-linked floating offshore structures composed of multiple floating bodies and beams connecting them; they implemented a conversion matrix-based digital twin model that can predict structural responses at unmeasured locations using measured data as input. They evaluated the performance of the conversion matrix through model test. They also explored a sensor array with the highest structural response prediction accuracy at unmeasured locations by performing genetic algorithm-based optimization and improved the performance of the conversion matrix for multi-linked floating offshore structures (Sim and Lee, 2024b). Regarding machine learning, machine-learning models and data scenarios used for training must be selected, and hyper parameters that constitute the machine-learning model must be analyzed to ensure a high structural response prediction performance. In scenarios other than training, achieving a high structural response prediction performance can be difficult. However, the implementation of a physics-based digital twin model through model-order reduction enables the prediction of structural responses under various environmental conditions with high accuracy because the physical characteristics of the structure are reflected. For ships and offshore structures, in particular, wave loads that change irregularly over time act on the structures in real time. For a highly accurate prediction of resulting structural responses, a digital twin that reflects the dynamic characteristics of a structure should be implemented.
In this study, Krylov subspace-based model-order reduction was applied to the KRISO Container Ship (KCS), a 6,750 twenty-foot equivalent unit (TEU) container ship developed by the Korea Research Institute of Ships and Ocean Engineering (KRISO), as shown in Fig. 1, to create a reduced-order model (ROM) that considers irregular wave loads acting on the ship. First, numerical modeling was performed for the KCS full-order model (FOM). To consider various marine environments, we generated irregular wave loads for each heading angle and ship speed and calculated structural responses through fluid–structure coupled analysis. In addition, the ROM was calculated by outputting the system matrix and irregular wave loads that constituted the FOM of KCS and projecting them into the Krylov subspace. The performance of the ROM was verified by calculating the relative root mean squared error (rRMSE) between the structural responses of the FOM and the results predicted through the ROM under various irregular wave load conditions. In addition, the structural response prediction performance of the ROM was analyzed according to the reduced number of DOFs (nDOF).
2. Numerical Simulation for the KRISO Container Ship
2.1 Numerical Model
To create the ROM of KCS, we constructed a numerical model for the entire system based on the properties listed in Table 1. We first calculated the wave loads acting on the ship to determine the structural response of KCS. The wave loads acting on KCS were calculated through motion analysis using the panel model shown in Fig. 2(a). Thereafter, the structural responses for FOM could be calculated through a numerical analysis by transmitting the wave loads to the finite-element model of Fig. 2(b), and the model-order reduction technique was applied to the finite-element model of KCS. For the panel model, the external hull below the waterline was modeled. It consisted of 2,000 nine-node high-order boundary elements. The finite-element model consisted of 4,224 four-node shell elements and 1,866 one-dimensional spring elements. While the interior of an actual ship is composed of various structural members, the KCS numerical model consisted of only the external hull and bulkhead. The mass and stiffness of the actual structure by length were matched by modeling using components with the properties of 15 shell elements in Table 2, as shown in Fig. 3. The buoyancy caused by the hydrostatic pressure below the waterline of KCS was implemented using spring elements in the water depth direction with stiffness that considered the horizontal area for each depth by connecting all nodes and virtual nodes that constituted the external hull below the waterline, as shown in Fig. 4.
2.2 Fluid-Structure Coupled Analysis
A fluid–structure coupled analysis was conducted to calculate the structural response of KCS using the numerical model constructed in Section 2.1. The wave loads acting on KCS were calculated through motion analysis based on the potential for wave flow (Choi and Hong, 2002). For the wave loads acting on KCS, the incident wave potential φI of Eq. (1), reflection of the incident wave, the scattering potential φS by diffraction, and radiation potential φR by the motion of the floating body were considered. Discrete linear Eqs. (2) and (3) were derived through the higher order boundary element method (HOBEM) and calculated using Eq. (4). Finally, the equation of motion for the floating body in the frequency domain was derived using Eq. (5).
where [G] in Eqs. (2) to (4) is the green function matrix, [H] is the derivative matrix of the green function, n is the normal vector, ω is the wave frequency, ρ is the seawater density, and {un} is the displacement vector at the node of the panel model. In Eq. (5), [MFB] and [Madd] are the mass and additional mass matrices of the KCS floating body, respectively. [CFB] is the hydrodynamic damping matrix, [KFB] is the stiffness matrix, {q} is the motion vector of the floating body, and {fwave(ω)} is the wave load vector according to the wave frequency ω.
The wave loads calculated in the frequency domain were converted into the time domain through a convolutional integral and applied to the finite element model as load conditions. Finally, structural analysis in the time domain was conducted through the equation of motion in Eq. (6). [M], [C], and [K] are the mass, damping, and stiffness of the structural analysis model of KCS, whereas u(t), u̇(t), and ü(t) are the displacement, velocity, and acceleration in the time domain, respectively. The hydrodynamic damping matrix of the numerical model was calculated from the memory function R of Eq. (7), which represents the response of the floating body according to the unit impulse velocity. In addition, the structural damping matrix was expressed as a linear combination of the mass and stiffness matrices using the reyleigh damping model as shown in Eq. (8). The values of coefficients α and β used were 0.0120 and 0.0061, respectively. The Newmark algorithm, a representative implicit method, was applied to the equation of motion in Eq. (6) to conduct transient response analysis in the time domain (Sim et al., 2023; Lee and Sim, 2024).
A container ship operates under various sea conditions. To consider this, we performed a simulation under irregular wave load conditions using ship speed and heading angle, as shown in Table 3. The performance of the ROM was evaluated. The significant wave height and modal period of irregular waves were set in the same manner as the wave load conditions of the numerical simulation by Kim et al. (2023). The time step of numerical simulation was 0.1 s, and the simulation time was three hours (10,800 s). The performance of the ROM was also evaluated based on the structural response results for the simulation time. Before creating the ROM, we compared the displacement results at the center of gravity of the floating body to validate the FOM, as shown in Fig. 5. For the irregular waves with a heading angle of 180°, the rRMSE according to the simulation time was calculated to be less than 0.01%, confirming that the entire model of this study had almost the same response as the numerical model of the previous study.
3. Krylov Subspace-based Model-order Reduction for the KRISO Container Ship
3.1 Krylov Subspace-based Model-order Reduction
Krylov subspace-based model-order reduction can reduce the simulation time and computational cost by generating the Krylov subspace using the matrices and force vectors of the FOM and reducing the order by projecting the FOM. It is suitable for creating a ROM of a dynamic system under time-varying loads because basis vectors that constitute the Krylov subspace reflect the dynamic characteristics of the system. Therefore, it was also applied to KCS under time-varying irregular wave loads in this study, and the system matrix of the equation of motion (Eq. (6)) for KCS was defined as Eq. (9). Finally, FOM was defined as Eq. (10) and the Krylov subspace-based ROM was created.
When the Laplace transform is applied to Eq. (10) for which all of the initial conditions are zero, the transfer function H that represents the relationship between the input vector U and output vector Y can be calculated, as shown in Eq. (11). This can be expressed as the product of KFOM, MFOM, and irregular wave unit force vector F through the series expansion shown in Eq. (12). Here, m is the moment of the transfer function. V, which represents the Krylov subspace vector, can be calculated through the mass and stiffness matrices and force vectors of the FOM expressed from the transfer function moment. Basis vectors that constitute V should maintain orthogonality, and k basis vectors can be effectively calculated through the Arnordi process, as shown in Fig. 6. In this instance, if the load acting on the structure is three-dimensional, it can be decomposed into three directions (x, y, and z) based on the global coordinate system to generate basis vectors for each load component, and the size of the Krylov subspace-based transformation matrix is also threefold (Han, 2010).
3.2 Generation of Reduced Order Model for KCS
Unlike general machines and onshore structures, ships and offshore structures are subjected to irregular wave loads over time. This should be considered when creating basis vectors during the generation of the Krylov subspace. Basically, Krylov subspace basis vectors are calculated using unit force vectors by direction that normalize the magnitude of the load acting on the structure. However, for ships, there are limitations in constructing unit force vectors by direction because wave loads act as pressure on three-dimensional surfaces, as shown in Fig. 7 (Han and Kim, 2021). For regular waves, unit force vectors can be created because they have consistently repeated pressure distributions according to the wave period. However, wave loads that occur during actual operations have irregular wave heights and periods, making it impossible to consider the marine environmental conditions of the actual ship. In addition, because irregular waves are expressed as the linear superposition of regular waves in various sizes and frequencies based on the wave spectrum of the target sea area, unit force vectors for numerous regular waves should be used if the Krylov subspace is constructed using regular wave-based unit force vectors. This is inefficient in reducing the order as the size of the Krylov subspace increases exponentially and may violate the orthogonality between basis vectors by unit force vector. Therefore, in this study, unlike the conventional Krylov subspace-based model-order reduction technique, the Krylov subspace was created by selecting snapshots at random time points from irregular wave loads over time as unit force vectors as shown in Fig. 8. Here, irregular wave snapshots refer to the irregular wave pressure distributions acting on the hull plate of KCS at each time point calculated in the numerical simulation. Because irregular wave snapshots include numerous regular wave pressure data that constitute the wave spectrum, a ROM that can effectively calculate structural responses to random irregular wave loads can be created using only a single force vector. Therefore, when irregular wave loads are applied, all snapshots over time can be applied to generate the reduced order space. In this study, irregular wave snapshots at 50 s were used.
A simulation was performed based on the procedure in Fig. 9 to create and validate ROM of KCS. First, motion analysis was conducted using a KRISO in-house code, and wave loads were exported to the created numerical model through ANSYS MAPDL, a commercial structural analysis software program. Through various analyses, the matrices and vectors for the FOM were derived. After generating the Krylov subspace and creating the ROM using the matrices of FOM through the in-house code using MATLAB, a commercial numerical calculation software program, we performed simulation and result verification.
4. Results of the Reduced-order Model for the KRISO Container Ship
4.1 Structural Response Evaluation for the Reduced-order Model
The total nDOF for KCS was 25,566. To evaluate the performance of the ROM according to the nDOF, we compared the structural response prediction results using the nDOF with the simulation results of the FOM. When the ROM was created by increasing the nDOF from 2 to 50, as shown in Fig. 10, and the rRMSE for displacement response using the axial direction at node 1,536 in Fig. 11(a) compared with the FOM, the results converged within 0.02% from an nDOF of 12. The rRMSE, an indicator that can compare the predicted and actual values, as shown in Eq. (13), can compare the performance of the ROM between different scenarios without relying on the scale of the data. Therefore, it is useful in identifying how similar the model is to the actual FOM response as the nDOF increases. When the displacement by axial direction was examined at node 1,536 located on the topside at the center of the hull with the largest displacement, as shown in Fig. 11, a large error was observed from the response of the FOM at an nDOF of 2, but very similar responses were observed as the nDOF increased. The simulation time was less than 0.1 s for all ROMs calculated in Fig. 11, confirming that it decreased by more than 99% compared with the computation time of five minutes for the FOM. Even under irregular wave load conditions that considered the ship speed, as shown in Fig. 12, the displacement response by ship speed for the ROM with nDOF = 20 was almost identical to that of the FOM.
4.2 Stress Recovery for Reduced Order Model
To examine the stress results and displacement response of KCS, we expanded the displacement response calculated through the ROM to the FOM response and performed stress recovery. The displacement response of the ROM can be expanded to the total DOFs of the FOM through multiplication by the previously calculated Krylov subspace vector, and the stress response can be finally calculated using the displacement and stress relationships of the shell elements in Eqs. (14) and (15). σx, σy, τxy, τyz, and τzx represent each stress component of the shell elements, whereas and denote the stress–strain and strain–displacement relationship matrices, respectively. As shown in Table 4, the maximum bending stress for irregular wave loads by heading angle and ship speed was calculated for the FOM and ROM. Under all irregular wave load conditions, the ROM exhibited almost identical structural response to the FOM with an rRMSE of less than 0.1%. The maximum bending stress was 127 MPa at a speed of 5.14 m/s (10 kn) and heading angle of 0°. As shown in Fig. 13, almost identical stress distributions were observed under irregular wave load conditions with the maximum bending stress.
Finally, as shown in Table 5, the performance of the ROM was compared with that of the FOM for the accuracy and computational time using the nDOF. The results of the FOM could be approximated with high accuracy while significantly reducing the complexity of the model through model-order reduction. Specifically, while the FOM had 25,566 DOFs, the ROM effectively reduced the complexity of the model using 2 to 50 DOFs. Overall, the rRMSE for the displacement and maximum bending stress in the Ux, Uy, and Uz directions exhibited a tendency to decrease as the DOFs of the ROM increased. In particular, when the DOFs of ROM were 20 or higher, very low rRMSE values were confirmed compared with the FOM, and some items exhibited almost perfect approximation performance. The computation time was also significantly reduced from 458 s for the FOM to 0.01 to 0.84 s for the ROM, indicating a significant improvement in computational efficiency. Considering the prediction accuracy and computation time, we concluded that a ROM that uses 20 to 30 DOFs achieves the optimal balance between accuracy and computational efficiency.
5. Conclusion
In this study, a model-order reduction technique was developed and validated by creating a ROM for a container ship and comparing it with the simulation of a FOM. For container ships, wave loads that change irregularly act in real time in the form of pressure distributions on hull surfaces. Thus, limitations exist in calculating Krylov subspace basis vectors using conventional unit force vectors by axial direction. Therefore, Krylov subspace basis vectors were calculated using snapshots for real-time irregular wave loads to consider the marine environment experienced during the operation of a container ship. In the transient response analysis results for irregular wave loads using heading angle and ship speed, the ROM predicted structural responses with a relative root mean squared error (rRMSE) of 0.02% compared with the FOM when the reduced number of degrees of freedom (nDOF) was 12 or higher. In addition, the computational time for the calculation of 1,000 transient responses was less than 0.1 s for ROM with 2 to 50 DOFs, thereby reducing the computation cost by more than 99% compared with FOM. We observed that the ROM is very effective in predicting the structural response of the numerical model for KRISO container ship. The model-order reduction technique can approximate the results of the original model with high accuracy while significantly reducing the complexity of the model. For KCS, the use of 20 to 30 DOFs achieved the optimal balance between accuracy and computational efficiency and reduced the computation time by up to thousands of times compared with the FOM. These are important results that show that the computational time and cost can be significantly reduced by applying the model-order reduction technique to the analysis of complex structures. In addition, the tendency that the rRMSE decreases as the nDOF increases show that an appropriate nDOF can be selected based on the required accuracy.
The model-order reduction technique of this study can be applied to ships and offshore structures with various forms, enabling digital twins to be implemented with high accuracy and low computation costs if information on wave loads acting on ships and offshore structures can be obtained in real time. Therefore, the Krylov subspace-based model-order reduction technique is a very useful tool in predicting the structural responses of large structures under irregular wave loads, such as container ships, in real time. In the future, it can be used in the digital twins of ships and offshore structures for structural health assessment, including structural damage diagnosis and fatigue strength evaluation in stress concentration regions.
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