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Article

Research on Model Reduction of AUV Underwater Support Platform Based on Digital Twin

by
Daohua Lu
1,2,*,
Yichen Ning
1,
Jia Wang
1,
Kaijie Du
1 and
Cancan Song
1
1
School of Mechanical Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China
2
Marine Equipment and Technology Institute, Jiangsu University of Science and Technology, Zhenjiang 212003, China
*
Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2024, 12(9), 1673; https://doi.org/10.3390/jmse12091673
Submission received: 12 August 2024 / Revised: 9 September 2024 / Accepted: 17 September 2024 / Published: 19 September 2024
(This article belongs to the Special Issue Theories and Techniques in Intelligent Digital Twins in Marine Science and Engineering)
Browse Figures
Figure 1
<p>Digital twin architecture of subsea support platform.</p> "> Figure 2
<p>Digital twin construction process of subsea support platform.</p> "> Figure 3
<p>Subsea support platform digital twin.</p> "> Figure 4
<p>Singular value decomposition principle.</p> "> Figure 5
<p>Space–time relationship of singular value decomposition.</p> "> Figure 6
<p>Data fusion flow chart.</p> "> Figure 7
<p>Flow chart of the AUV subsea support platform model reduced-order.</p> "> Figure 8
<p>Support structure 3D model.</p> "> Figure 9
<p>Stress change curve of sling. (<b>a</b>) The force diagram of point C in the x, y and z directions respectively. (<b>b</b>) The force diagram of point E in the x, y and z directions respectively. (<b>c</b>) The force diagram of point D in the x, y and z directions respectively. (<b>d</b>) The force diagram of point G in the x, y and z directions respectively.</p> "> Figure 10
<p>C, E, D, and G are the forces of the sling changing over time, and A, B, H, and F are the vertical downward forces, respectively.</p> "> Figure 11
<p>Eigenvalue distribution.</p> "> Figure 12
<p>Trend of eigenvalue proportion of displacement field.</p> "> Figure 13
<p>Cloud image of deformation of Class 5 sea state support structure.</p> ">
Versions Notes

Abstract

:
Digital twin technology, as a data-driven and model-driven innovation means, plays a crucial role in the process of digital transformation and intelligent upgrading of the marine industry, helping the industry to move towards a new stage of more intelligent and efficient development. In order to solve the defects of the Autonomous Underwater Vehicle (AUV) underwater support platform structure deformation field, digital twin technology and model reduction technology are applied to an AUV underwater support platform, and a five-dimensional digital twin model of the AUV underwater support platform is studied, including five dimensions: physical world, digital world, twin data center, service application, and data connection. The digital twin of the subsea support platform is established by using the digital twin modeling technology. The POD method is used to calculate the deformation field matrix of the support structure of the subsea support platform under the 0–5 sea state, and the corresponding eigenvalues and eigenvectors are obtained. By intercepting the eigenvectors corresponding to the eigenvalues of the high energy proportion, the low-order equation is constructed, and the reduced-order model under each sea state can be quickly solved. The experimental results show that the model reduction technology can greatly shorten the model solving time, and the calculated results are highly consistent with the simulation results of the finite element full-order model, which can realize the rapid analysis of the deformation response of the subsea support platform structure, and provide a theoretical basis and technical support for the subsequent simulation, state evaluation, visual monitoring, and predictive maintenance.

1. Introduction

As the hub of reality and virtual, digital twin technology can map the whole spatial scale and life cycle of physical entities to the digital information world. At present, digital twin technology is mainly applied to fault warning, health status management, equipment product design and development, collaborative control, and intelligent operation and maintenance in the ship industry [1,2,3]. As a key technology for equipment performance analysis, the global operation parameter distribution can be obtained through the physical field calculation model, which can be applied to drive the digital twin model, and the real-time monitoring of equipment can be realized. At present, traditional numerical simulation methods, such as the finite element method and finite volume method, have been well studied to solve the computational media problem of physical fields. However, due to the high degree of freedom of traditional finite element models and the need to discretize the spatial domain and the time domain at the same time, the calculation amount is huge and the calculation time is as long as several days, which cannot meet the real-time requirements of digital twin technology [4,5,6,7,8].
Digital twins are used in many disciplines to support the engineering, monitoring, control, and optimization of cyber–physical systems [9]. Digital twin technology is a digital and technical concept, which means the integration and fusion of cyber and physical realms based on data and models. It accurately constructs physical objects in real time in the digital space, and simulates, validates, predicts, and controls the entire life cycle process of physical entities based on data fusion and analytical prediction [10,11]. Digital twins can capture the combined use of heterogeneous models and respective evolution data throughout their life cycle [12]. For example, Byungmo Kim et al. [13] analyzed the cause of the difference between the dynamic characteristics of the actual structure and the existing simulation model, and found the necessity of modeling non-structural quality and environmental factors. Tofte et al. [14] described a digital twin simulation system that encapsulates virtual humans during the installation of a ship’s pipes, which has been praised for its effectiveness in hardware testing and operational training. Schirmann et al. [15] established a digital twin system for fatigue estimation of ship structures based on measured wave response data as an assessment tool for cumulative damage sustained by ships. Majewska et al. [16] proposed a method using a fiber Bragg grating (FBG) sensor to carry out simultaneous strain/stress monitoring on the foremast of sailing ships. Tessler and Spangler [17,18] at NASA Langley Research Center developed a breakthrough shape and stress sensor algorithm called IFFM. The practicability and accuracy of the IFFM framework in evaluating structural deformation fields have been fully verified [19,20,21,22]. Hambli et al. [23] predicted the real-time deformation of tennis balls and rackets during collision by applying neural networks, and users could feel the impact load of rackets through tactile gloves in a virtual reality environment. Although the above research achieves a fast response to the predicted results by training the agent model, it is still limited by the size of the finite element model and may produce unreliable results for untrained inputs. Simplifying decryption methods for interacting and propagating finite element data in practical applications remains a huge challenge [24,25,26].
One of the main purposes of using digital twin models for ship structure monitoring is to extract and utilize effective data to the maximum extent. In the response analysis of structural deformation fields, a complex system solution is involved, which violates the real-time characteristic of digital twins. In order to reduce the computing time, the model reduction technique based on the Proper orthogonal decomposition can reduce the high-order model to a low-order model while retaining most of the features of the original model. For example, in [27], in order to evaluate the influence of different bending dimensions on the flow characteristics of fuel assemblies, POD coefficients and modes of the snapshot matrix were determined by the proper orthogonal decomposition (POD) method, which can quickly predict the pressure and flow field in the assembly with a given flow rate and bending shape. In [28], in order to find the high-dimensional and complex solution of the special valve model, the calculation time of which is relatively long, the proper orthogonal decomposition (POD) method is applied to the mathematical model of the valve, by constructing a snapshot POD matrix and establishing a low-dimensional model. The calculation speed and efficiency are improved, and the effectiveness of the POD method is verified by experiments. In [29], the problem of linear vibration calculation of prestressed elastic structures is studied, and the natural frequency of elastic structures under static load parameters is evaluated based on proper orthogonal decomposition (POD). The results show that the combination of a limited number of POD modes and linear prestressed characteristic modes can effectively calculate the frequency function of evolutionary parameters. Zhai Yu-Jia et al. [30] studied the model reduction problem and dynamic characteristic analysis of the bolted flange structure, simplified the linear finite element model, and compressed the degree of freedom of the node interface by coordinate condensation, greatly reducing the calculation cost.
In this paper, to solve the defects of AUV underwater support platform structure deformation fields, which require a large amount of calculations and a long calculation time, a digital twin-based AUV underwater support platform model and model order reduction technology are proposed. This method can satisfy the calculation accuracy and solve the deformation field with a reduced-order model with a small amount of calculations, so as to realize the rapid response of the structure deformation field. Compared with the full-order model, the calculation time of the reduced-order model is reduced from hours to seconds, and the accuracy and timeliness of the reduced-order model are verified by an example, which can ensure the high precision of the solution of the digital twin model and improve the solution efficiency to the maximum extent. The rapid solution and real-time response of the AUV underwater support platform can effectively detect the deformation of the structure, and provide a theoretical basis and technical support for subsequent simulation, state assessment, visual monitoring, predictive maintenance, etc.

2. Methodology

2.1. AUV Subsea Support Platform Digital Twin Architecture

According to the technical connotation and functional definition of the digital twin, the architecture of the subsea support platform digital twin is summarized, as shown in Figure 1:
(1)
The physical world covers the real structure of the subsea support platform and the surrounding environment, and in order to build a digital twin, various sensors need to be used to accurately analyze the subsea support platform and effectively collect relevant data. In the operation phase, accurate analysis refers to the definition of the structure and material characteristics of the subsea support platform, and effective data refer to the equipment data and environmental data collected by the sensor in real time, which is the basic premise of building the digital twin and the key to ensure the normal operation of the subsea support platform.
(2)
The digital world refers to the actual reproduction of the physical world. In the digital field, the real state of the equipment can be simulated with multi-scale and multi-dimensional conditions so as to accurately present the actual state of the underwater support platform. The original information collected from external sensors is input into the digital twin model to make it run synchronously with the real system, and the dynamic evolution of the subsea support platform twin model is continuously promoted through virtual–real mapping, so that the real structural response can be intuitively reflected on the digital platform.
(3)
Digital twin data centers, as the core elements, include the test data of the structural simulation of the subsea support platform in the experimental stage, the process data generated in the manufacturing process, and the data recorded during operation and maintenance, which can show the geometric shape, material characteristics, deformation characteristics, and constraint relations of the subsea support platform from multiple dimensions. By using common algorithms such as empirical formulas and industry knowledge, data models can be constructed and numerically computed to generate a derived database that can be used to predict the structural response of subsea support platforms.
(4)
The service application layer is the ultimate goal, which is the packaged application after all data integration, simulation, and deduction to meet the different needs of the industry. Addressing the differences in the service process can produce a variety of application functions and provide the external customer with a customized standard interface, so as to reduce the requirements for professional technical levels, so that users in the industry can more conveniently use these services.
(5)
Data connectivity is the key driver to make all parts work together, and the realization of interconnection mainly relies on high-performance sensors and high-speed data transmission networks. Thanks to advanced distributed sensing technology and the rapid development of “5G” communication technology, digital twins can obtain richer and more accurate physical data, and are no longer affected by the bandwidth of large data transfers, thus ensuring efficient access to databases, applications, and real-time computing power.

2.2. Digital Twin Modeling

The digital twin model of the subsea support platform plays a crucial role in the accuracy of subsequent simulation analysis, data simulation, and virtual and real interaction. According to the object-oriented modeling idea, the subsea support platform can be divided into several categories: the mechanical system, electrical system, control system, and hydraulic system. Then, the subsystem class, sub-module class, sub-component class, and sub-part of each class according to different categories are determined until each category is subdivided layer by layer.
Taking the mechanical system as an example, based on the continuity hypothesis, uniformity hypothesis, and isotropy hypothesis of material mechanics, physical factors such as gravity and elastic deformation are not considered in the construction process of the geometric model, and only the components of the underwater support platform are regarded as rigid bodies. In the process of constructing the digital twin mechanical geometric model, according to the idea of incremental modeling, the overall framework of the subsea support platform geometric model is designed from top to bottom, from simple to complex, and then the components are designed and defined according to the overall framework structure. Standard parts can be imported from the standard parts library, such as bolts, nuts, bearings, etc. Custom parts need to be generated by professional modeling software features, such as stretching, rotation, scanning, array, etc. The complexity and accuracy of the model can be better controlled by gradually adding model details, and the structure and details of the model can be easily adjusted at any time according to requirements, as shown in Figure 2.
According to the idea of fine modeling, it is necessary to combine geometric modeling with feature modeling. Feature modeling refers to the manufacturing features of geometric models such as holes, slots, and threads. Tolerance, chamfer, roughness, and other shape characteristics, as well as image features, such as colors and textures, include as much detail as possible to ensure that the model more realistically reflects real-world objects or systems. After the construction of the geometric elements and feature elements of the parts, the parts are assembled into subsystems, and then the subsystems are assembled into the digital twin geometric model of the subsea support platform, as shown in Figure 3.

2.3. Model Reduction Technique

After elaborating on the basic framework of the subsea support platform digital twin, this paper studies the simulation model reduction technology in this field in depth, aiming to provide a solid theoretical foundation and technical support for the construction of a subsea support platform digital twin system. In this method, the snapshot matrix is constructed by using the deformation of N nodes of the structure obtained from the measured data or the finite element simulation data under L samples. Based on the proper orthogonal decomposition (POD) method, the orthogonal basis of L features is extracted from the constructed snapshot matrix, and then the orthogonal basis of the first r orders that can reflect most of the model features is extracted to construct the low-order equations. Finally, the corresponding coefficient matrix α is obtained by solving the lower-order equations, and the approximate value of the deformation field is calculated from the coefficient matrix α . This method not only ensures the accuracy of the calculation results within a reasonable error range, but also reduces the calculation time and scale of the original model to a great extent, thus greatly improving the calculation efficiency.

2.3.1. Construct the Physical Field Matrix

POD constructs a set of orthogonal bases by using the simulation results of the full-order model, and intercepts the optimal pre-r basis from it, so that it can describe the physical characteristics of the original system within the allowable error range. In the first step, the snapshot matrix S = A i | i = 1 L is created with the full-order model data, where A i can be the digital simulation result of the theoretical model or the measured data of the physical process. Each element A b a in S represents the numerical result of node b in the a snapshot, so the snapshot matrix can reflect the spatial distribution of physical fields in different samples.
S = A 1 1 A 1 2 ⋯ A 1 L A 2 1 A 2 2 ⋯ A 2 L ⋮ ⋮ ⋱ ⋮ A N 1 A N 2 ⋯ A N L
In Formula (1), L represents the number of selected snapshots, and N refers to the number of grid nodes.

2.3.2. Solve the Orthogonal Basis of the Matrix

The core of POD model reduction technique is to find a set of orthogonal bases:
Φ = ϕ 1 ϕ 2 ⋯ ϕ L
The optimal condition is that the error norm of the sampled data T i and its projection is equivalent to a constrained maximum problem:
G = m a x φ   ∑ i = 1 n   ( T i , ϕ ) 2 ϕ 2
Φ T Φ = I
In Formula (2), the vector ϕ represents the POD orthonormal basis, and matrix Φ is the set of POD orthonormal bases, which is a matrix containing N rows and L columns ( N ≫ L ) and conforms to the orthogonality shown in Formula (4).
Formula (3) can be used to solve the maximum problem with constraints by the Lagrange multiplier method, and the eigenvalue problem of Formula (5) is used to replace the original optimization problem.
S S T x = λ x
In order to solve the eigenvalue problem of Equation (5), the optimal orthogonal basis set can be obtained by the singular value decomposition of matrix S . Therefore, the matrix S ϵ R N × L needs to be decomposed by a singular value.
S = U σ V T
S 1 , S 2 , S 3 , ⋯ , S N ⏟ N   sets   of   simulation   data = U 1 U 2 … U M · σ 1 0 ⋱ 0 σ n · V 1 V 2 ⋯ V N T
where U ∈ R N × N and V T ∈ R L × L are orthogonal matrices with dimensions N × N and L × L , and σ is an approximate diagonal matrix consisting of several singular values of the matrix S , where each element of σ represents a singular value, arranged in order from largest to smallest, σ ∈ R N * L .
As shown in Figure 4, according to the mathematical characteristics of singular value decomposition, the singular value σ i of matrix σ decreases gradually with an increase in i , and the lower the order of the singular value, the lower its influence on the whole matrix. By truncating the σ singular value matrix within σ reasonable error range, both matrix U and matrix V T can reduce dimensionality on a large scale, thus achieving the effect of an exponential reduction in data volume.
As shown in Figure 5, from the perspective of space–time, matrix S can be decomposed into a linear superposition matrix of the first r data combinations. U and V T represent the orthonormal basis of the original domain and the orthonormal basis after transformation, respectively, and U i and V i in either case represent the spatiotemporal combination of the matrix S in a physical sense. The dimension of U i is the same as the solution of S i under specific load conditions, which mainly represents the numerical information of the result file in the matrix and is related to the structure of the monitoring object, i.e., it expresses the spatial relationship. The dimension of V i is related to the number of solutions, and different load combinations are mainly manifested as the gradient change in time in the monitoring object, i.e., the relationship in time is expressed.
Because the number of nodes N in the grid of the numerical calculation model is much larger than the number L ( N ≫ L ) of the selected snapshot samples, the dimension of the matrix S S T is much larger than that of the matrix S T S , but their non-zero eigenvalues are exactly the same. Therefore, the eigenvalue λ i of the matrix S T S with smaller dimensions can be solved first and the eigenvector φ i afterwards.
S T S φ i = λ i φ i , i = 1,2 , ⋯ , l , l ⩽ L
Then, the eigenvector ϕ i of the matrix S S T with higher dimensions can be calculated.
ϕ i = 1 λ i S φ i , i = 1,2 , ⋯ , l , l ⩽ L
Finally, a set of orthonormal bases Φ can be obtained, which can describe the physical properties of the full-order system within a reasonable margin of error, and can express the numerical response of any node on the model through a linear combination of spatiotemporal relations, that is,
A = ∑ i = 1 l   ϕ i α i
where α i is the coefficient of ϕ i in each basis.

2.3.3. Intercept the Optimal Orthogonal Basis

The full-order model is reduced by truncating a certain number of feature vectors. Since each POD canonical vector can evaluate its contribution to describing the entire full-order model with the eigenvalue it represents, this truncation process is based on the principle of energy proportion, i.e., we need to select the pre-r-order basis with the highest energy proportion for the construction of the low-order model. The total energy ratio of the subspace after order reduction to the full-order model should be close to 1, and the expression is
r = argmin ⁡ I r : I r ⩾ γ 100
Among them,
I r = ∑ i = 1 r   σ i ∑ i = 1 l   σ i
Generally, the value of γ is 99.99, and the optimal orthogonal basis after truncation can be expressed as
Φ ^ ∈ R r × r , r < l
The unknown quantity to be solved is
A ≈ Φ ^ α

2.3.4. Reduced-Order Method for Structural Analysis

The stress of the underwater support platform is nonlinear and transient with the change in the sea state during the hoisting process. According to the idea of POD model reduction technology, the transient force can be regarded as the static force at a certain moment, and the snapshot matrix at a certain moment can be obtained by solving it according to the static method.
In the static analysis of the subsea support platform structure, the static equation can be expressed as Equation (15):
K u = F
where K represents the stiffness matrix of the system, u represents the deformation vector of the node to be found, and F is the load vector of the node. We substitute Equation (14) into Equation (15) and multiply it by Φ ^ T left at both ends of the equation to obtain Equation (16).
K ^ α = F ^
Among them,
K ^ = Φ ^ T K Φ ^
F ^ = Φ ^ T F
Equation (18) is the reduced-order model of the node deformation problem in structural analysis, and the unknown quantity to be solved is transformed from node deformation vector u to coefficient matrix α . Compared with the original problem, the degrees of freedom of the unknowns are reduced from N to r ( N ≫ r ), which greatly reduces the amount of computation required to solve the computational model. By substituting the initial condition α o = Φ ^ T u o of the reduced-order model, the coefficients α i corresponding to each sample point after selecting the optimal orthogonal basis can be calculated one by one. Accordingly, we obtain the corresponding coefficient matrix A e for a particular sample α e , whose deformation prediction response value in the reduced-order model is
A e ≈ Φ ^ α e

2.4. Correction of the Digital Twin Models

Because the finite element model has been simplified to a certain extent during modeling, there is a certain degree of deviation between the simulation data and the real data, which are prone to sudden changes or a jagged distribution. Therefore, between the twin data source with error and the sensor data with measurement error, the optimal data fusion screening should be carried out according to the optimal relationship between the multivariate data. In this paper, the Kalman filter algorithm is used for multivariate data fusion.
According to the definition of the space state equation, the equation of the dynamic system is shown in Equation (20).
x k = A x k − 1 + B u k − 1 + ω k − 1 z k = H x k + v k
where A represents the state transition matrix; x k represents a state variable; B represents the control input matrix; u k − 1 represents the input signal; ω k − 1 represents prediction error; z k represents the observed variable; H represents the state observation matrix; and v k represents measurement error.
According to the operating mechanism of the Kalman filtering algorithm, the calculation flow and expression after filtering are shown in Equation (21). The running processes are prior prediction, prior error covariance prediction, Kalman gain correction, posterior estimation, and error covariance update, respectively.
x ^ − = A x ^ k − 1 + B u k − 1 p k − = A p k − 1 A T + Q k k = p k − H T H p k − H T + R x ^ k = x ^ k − + k k ( z k − H x ^ k − ) p k = ( I − k k H ) p k −
where x ^ − represents a prior estimate; p k − represents the prior estimation covariance; and k k represents the Kalman gain, which is a number from 0 to 1. When it approaches 0, the expectation of the estimated value is the largest; when it approaches 1, the expected value of the measured value is the lowest; x ^ k represents the actual estimate; p k represents the error covariance at time k .
The data source in this paper includes three parts, which are digital bending data, sensor data, and Kalman estimates. According to the research process of the Kalman filtering algorithm for multiple data sources, the data fusion process of this paper is shown in Figure 6.

2.5. AUV Underwater Support Platform Model Reduction Process

The core idea of the proper orthogonal decomposition (POD) method is to find a set of optimal orthogonal bases that can fully express the main characteristics of the full-order system, and the degree of mode represented by each POD base is determined by the corresponding eigenvalue, so only the corresponding POD base that accounts for the majority of the proportion can be retained to truncate and reduce the order of the full-order model. The space formed by the intercepted POD base modes is the reduced-order space, and the linear combination with the projection coefficient of the full-order model on this space can reconstruct the response state of the original system at each sample point of the design space, and realize the reduced-order model calculation method of solving the full-order response with the low-order base. The specific program process and steps are shown in Figure 7:
(1)
Construct the AUV underwater support platform digital twin.
(2)
Based on the digital twin of the AUV underwater support platform, the full-order model is solved by the finite element analysis method, and the deformation field sample database is generated.
(3)
Multivariate data fusion is carried out based on the Kalman filter algorithm, and the optimal data fusion screening is carried out according to the optimal relationship between multivariate data.
(4)
Eigenvalues and eigenvectors are obtained by calculating the sample database matrix based on the proper orthogonal decomposition (POD) method.
(5)
The reduced-order model is constructed by truncating the eigenvector corresponding to the eigenvalue with a high energy proportion.
(6)
The calculation accuracy and calculation time of the full-order model and the reduced-order model are compared.

3. Case

In this paper, the above model simplification method is adopted to carry out digital twin practice on the support structure of the underwater support platform of the underwater vehicle, and quickly calculate the deformation response of the key areas of the support structure during the hoisting process under various sea conditions. By means of fluid simulation software, the stress variation of the sling under 0~5 sea conditions is simulated, and the stress variation curve of the sling with time is obtained. We load the sling force on the four lifting points in the support structure, and obtain the component force of the x axis, y axis, and z axis of each lifting point. The deformation values of support structures under different sea conditions are calculated by the finite element method with the database, and the optimal data fusion screening is carried out based on the Kalman filter algorithm according to the optimal relationship between the multi-source data. The POD method is used to calculate the orthogonal basis of the matrix, and the first five bases that can express 99.9832% of the main characteristics are intercepted to form a low-order equation. Finally, the deformation estimation of the reduced-order model is solved. The results show that the maximum deformation error of the proposed model reduction method is only 0.245% compared with the full-order finite element model, but the calculation efficiency is increased by tens of times, and the rapid deformation response of the support structure can be achieved under different sea conditions.

3.1. Support Structure 3D Model

In this paper, the support structure of the underwater support platform is selected as the research object. First, a complete three-dimensional model of the support structure of the underwater support platform is established, which includes the support structure, beam, column, plate, and other components, as shown in Figure 8. According to the national standard GB/T 702-2004 [31], 80 mm square steel is selected for this model, and the material is structural steel. The main parameters and material information of the model are shown in Table 1.

3.2. Distribution and Loading of Forces

In this study, fluid simulation technology is used to study the stress changes experienced by slings over time in the range of 0 to 5 sea states through accurate simulation. During the simulation process, the stress of the sling in each level of sea state was recorded in detail, and its dynamic response at different time points was observed. The stress change curve of the sling with time is shown in Figure 9.
Through the above sling simulation analysis, the components of the lifting points C, E, D, and G in the direction of the x axis, y axis, and z axis were obtained, and the forces varying with time were loaded into the finite element simulation model. In addition, four constant forces of 8000 were added at A, B, H, and F to simulate the gravity of AUV, and the distribution of forces is shown in Figure 10.

3.3. Support Structure Reduction Model Construction

Fluid simulation software was used to generate the stress sample points of the sling under different sea conditions in the range of 0–5 sea conditions over time, and the stress sample points under each sea state were loaded onto the support structure. The deformation response of the support structure was calculated by finite element simulation software, and the deformation of M nodes in each group of sea states was recorded. The optimal data fusion screening was carried out based on the Kalman filter algorithm. The snapshot matrix S is constructed using the filtered deformed data set, and the POD basis and corresponding eigenvalues of the matrix S are calculated using the POD method above. As shown in Figure 11, a bar chart of the first 20 order eigenvalue sizes is made. From the distribution values, it can be seen that after the eigenvalues corresponding to each order of the POD base are arranged in descending order, their value sizes decay significantly, which proves that a small part of the eigenvalues of the previous order account for a large proportion of energy.
Figure 12 shows the ratio of eigenvalues and energy of each order in Table 2. It can be seen from the analysis that the energy ratio of the first R order eigenvalues presents a sharp rise and then a gentle trend. When r   = 5, the energy ratio of the first five order eigenvalues can be calculated from Equation (12): I (5) = 99.9832%. It is proven that the truncation space at this time retains most of the features of the original model, and has achieved the principle of energy similarity between the truncation space and the full-order model required by Equation (11). Therefore, the first five POD bases Φ = φ 1 , φ 2 , ⋯ , φ 5 are intercepted.

3.4. Support Structure Reduction Model Analysis

Using the intercepted eigenvalues and corresponding eigenvectors, a reduced-order model of the support structure is constructed, and the deformation response results are solved by using the reduced-order model and compared with the finite element analysis results of the original model. Class 1, Class 2, Class 3, Class 4, and Class 5 sea conditions were selected for the deformation response verification. The maximum deformation and error of the two methods are shown in Table 3. The model reduced-order simulation cloud image and the finite element simulation cloud image at the level 5 sea state are compared, as shown in Figure 13.
According to the comparative analysis of Table 3 and Figure 13, in these five typical sea states, the cloud image calculated by the reduced-order model is close to the cloud image calculated by the traditional finite element method, and there is a small difference in the maximum deformation amount, which proves that the reduced-order model can describe the deformation situation well.

3.5. Computational Efficiency of Support Structure Reduction Model

As shown in Table 4, the time required for the model reduction method described in this paper and the finite element method to solve the deformation under different sea states is compared. It can be seen from the table that the computational efficiency of the model reduction method is much higher than that of the traditional finite element method, and it can be used for rapid response analysis of the deformation of transient forces. The time required by the model reduction method includes the total time required for solving the POD base, truncating the eigenvalues, and building the reduced model, excluding the preparation time for solving the sample set, which can be composed of completed simulation data sets or historical data sets.

4. Conclusions

In this paper, the digital twin of the subsea support platform is established based on the five-dimensional digital twin model, and the deformation response of the typical four-point support structure under different wave classes (0–5) is studied. In this paper, the model reduction technique in the field of digital twins is discussed, and the bending deformation behavior of the support structure of the underwater support platform is analyzed. By using the POD method, the optimal POD bases of the first five orders are obtained. The lower-order equations are set up with these bases and the eigenvalues of each base are calculated. The calculated values of the original descending order model are obtained by linear combination. The results show that the model reduction method in this paper is highly consistent with the finite element method, and the error is small and the calculation efficiency is significantly improved, which can be used to realize the real-time deformation response analysis of the subsea support platform under different sea conditions.
In spite of this, this paper only discusses the digital twin modeling technology and model reduction technology in structural deformation analysis. The ultimate goal of digital twins is to realize the management application of the whole life cycle of physical entities from birth to death. At present, there are still many unknowns about the structural deformation response of digital twins, most of which only stay in the future planning phase, so more theoretical exploration and innovative practice are needed to promote the comprehensive application of digital twin technology.

Author Contributions

Conceptualization, J.W. and Y.N.; methodology, J.W. and Y.N.; software, Y.N. and K.D.; validation, Y.N. and K.D.; formal analysis, J.W. and D.L.; investigation, Y.N. and C.S.; data curation, Y.N. and K.D.; writing—original draft preparation, Y.N.; writing—review and editing, J.W. and D.L.; visualization, Y.N. and C.S.; supervision, J.W. and D.L.; project administration, J.W. and D.L.; funding acquisition, J.W. and D.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was made possible by the Key Research and Development Program of Jiangsu Province (BE2022062).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Acknowledgments

Thanks to each of the authors for their dedication to the experiment and technical support.

Conflicts of Interest

The authors confirm that they have no conflicts of interest to report in relation to the present study.

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Figure 1. Digital twin architecture of subsea support platform.
Figure 1. Digital twin architecture of subsea support platform.
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Figure 2. Digital twin construction process of subsea support platform.
Figure 2. Digital twin construction process of subsea support platform.
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Figure 3. Subsea support platform digital twin.
Figure 3. Subsea support platform digital twin.
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Figure 4. Singular value decomposition principle.
Figure 4. Singular value decomposition principle.
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Figure 5. Space–time relationship of singular value decomposition.
Figure 5. Space–time relationship of singular value decomposition.
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Figure 6. Data fusion flow chart.
Figure 6. Data fusion flow chart.
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Figure 7. Flow chart of the AUV subsea support platform model reduced-order.
Figure 7. Flow chart of the AUV subsea support platform model reduced-order.
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Figure 8. Support structure 3D model.
Figure 8. Support structure 3D model.
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Figure 9. Stress change curve of sling. (a) The force diagram of point C in the x, y and z directions respectively. (b) The force diagram of point E in the x, y and z directions respectively. (c) The force diagram of point D in the x, y and z directions respectively. (d) The force diagram of point G in the x, y and z directions respectively.
Figure 9. Stress change curve of sling. (a) The force diagram of point C in the x, y and z directions respectively. (b) The force diagram of point E in the x, y and z directions respectively. (c) The force diagram of point D in the x, y and z directions respectively. (d) The force diagram of point G in the x, y and z directions respectively.
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Figure 10. C, E, D, and G are the forces of the sling changing over time, and A, B, H, and F are the vertical downward forces, respectively.
Figure 10. C, E, D, and G are the forces of the sling changing over time, and A, B, H, and F are the vertical downward forces, respectively.
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Figure 11. Eigenvalue distribution.
Figure 11. Eigenvalue distribution.
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Figure 12. Trend of eigenvalue proportion of displacement field.
Figure 12. Trend of eigenvalue proportion of displacement field.
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Figure 13. Cloud image of deformation of Class 5 sea state support structure.
Figure 13. Cloud image of deformation of Class 5 sea state support structure.
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Table 1. Main parameters of the model.
Table 1. Main parameters of the model.
NameNumerical ValueNameNumerical Value
Length5580 mmTensile yield strength250 MPa
Width4000 mmCompressive yield strength250 MPa
Height3908 mmModulus of elasticity200,000 MPa
Density7.81 g/cm3Poisson’s ratio0.3
Table 2. Displacement field eigenvalue and energy proportion.
Table 2. Displacement field eigenvalue and energy proportion.
RankEigenvalueEnergy Proportion
11,562,309.362299.5742%
26295.236199.9306%
3576.462399.9569%
4176.536299.9758%
5125.632899.9832%
627.563499.9913%
712.326599.9951%
83.269899.9957%
90.569899.9968%
100.321699.9983%
Table 3. Maximum displacement and error.
Table 3. Maximum displacement and error.
Level 1 Sea StateLevel 2 Sea StateLevel 3 Sea StateLevel 4 Sea StateLevel 5 Sea State
Finite element method/mm0.96311.56322.46383.02473.6192
Model reduction method/mm0.96221.56032.46113.02033.6103
Error0.093%0.185%0.109%0.145%0.245%
Table 4. Comparison of calculation time between model reduction method and finite element method.
Table 4. Comparison of calculation time between model reduction method and finite element method.
Level 1 Sea StateLevel 2 Sea StateLevel 3 Sea StateLevel 4 Sea StateLevel 5 Sea State
Finite element method/min263248256254267
Model reduction method/min0.20.30.30.20.3
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MDPI and ACS Style

Lu, D.; Ning, Y.; Wang, J.; Du, K.; Song, C. Research on Model Reduction of AUV Underwater Support Platform Based on Digital Twin. J. Mar. Sci. Eng. 2024, 12, 1673. https://doi.org/10.3390/jmse12091673

AMA Style

Lu D, Ning Y, Wang J, Du K, Song C. Research on Model Reduction of AUV Underwater Support Platform Based on Digital Twin. Journal of Marine Science and Engineering. 2024; 12(9):1673. https://doi.org/10.3390/jmse12091673

Chicago/Turabian Style

Lu, Daohua, Yichen Ning, Jia Wang, Kaijie Du, and Cancan Song. 2024. "Research on Model Reduction of AUV Underwater Support Platform Based on Digital Twin" Journal of Marine Science and Engineering 12, no. 9: 1673. https://doi.org/10.3390/jmse12091673

APA Style

Lu, D., Ning, Y., Wang, J., Du, K., & Song, C. (2024). Research on Model Reduction of AUV Underwater Support Platform Based on Digital Twin. Journal of Marine Science and Engineering, 12(9), 1673. https://doi.org/10.3390/jmse12091673

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