Hydrofoil multi-objective hydrodynamic optimization method and device
By constructing a surrogate model using Latin hypercube sampling and deep neural networks, and combining it with a multi-objective optimization algorithm, the problems of long cycles and local optima in the multi-objective hydrodynamic optimization of hydrofoils were solved, achieving an efficient and reliable optimization process and results.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SUN YAT SEN UNIV
- Filing Date
- 2026-04-07
- Publication Date
- 2026-07-24
AI Technical Summary
Existing multi-objective hydrodynamic optimization methods for hydrofoils rely on high-fidelity CFD simulations, resulting in long optimization cycles, high costs, and a lack of data-driven guidance capabilities. They also struggle to effectively utilize existing simulation information, and traditional multi-objective evolutionary algorithms are prone to getting trapped in local optima, affecting engineering practicality and decision-making credibility.
Sample data were obtained using the Latin hypercube sampling method, a surrogate model based on a deep neural network was constructed, and iterative optimization was performed using a multi-objective optimization algorithm. The hydrodynamic performance indicators of individuals in the population were predicted using the surrogate model, and the optimal solution set was verified using CFD simulation to ensure the engineering reliability of the optimization results.
It significantly improves the efficiency of multi-objective hydrodynamic optimization for hydrofoils, reduces computational costs, enhances global search efficiency and the reliability of optimization results, and ensures the engineering feasibility and practicality of the optimization process.
Smart Images
Figure CN122452301A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of hydrodynamic optimization technology, and in particular to a multi-objective hydrodynamic optimization method and apparatus for hydrofoils. Background Technology
[0002] Currently, hydrofoils, by virtue of the lift generated by the hydrofoil, can effectively reduce the wetted area of the hull, thereby significantly increasing speed, reducing drag, and improving seakeeping. Therefore, they have broad application prospects in high-speed passenger ships, government vessels, and military ships. Hydrodynamic shape optimization of the hydrofoil system is the core means to improve the speed, seakeeping, and stability of hydrofoils. Currently, existing hydrodynamic optimization methods are mainly divided into three categories: optimization based on design of experiments and response surface models, direct numerical optimization based on computational fluid dynamics (CFD), and multi-objective optimization based on evolutionary algorithms. Among these, the most mainstream technical approach is to combine CFD with multi-objective evolutionary algorithms. This involves parametrically modeling the hydrofoil geometry, using CFD to perform high-fidelity numerical simulations on each candidate design, calculating performance indicators such as lift and drag, and then iteratively optimizing based on this to finally obtain the optimal solution set.
[0003] Existing multi-objective hydrodynamic optimization methods for hydrofoils rely solely on high-fidelity CFD simulation as the only means of performance evaluation. However, a single CFD calculation is extremely time-consuming, resulting in an optimization cycle that can last for weeks or even months, leading to high computational costs and extremely low optimization efficiency. Secondly, existing methods only use CFD as a deterministic performance evaluation tool, failing to effectively construct intelligent proxy models from historical computational data. This results in a lack of data-driven guidance in the optimization process, high blindness in the search process, and difficulty in fully utilizing existing simulation information. Furthermore, traditional multi-objective evolutionary algorithms are prone to getting trapped in local optima when dealing with high-dimensional nonlinear, multi-peak complex optimization problems such as hydrofoil hydrodynamics, affecting the engineering practicality and decision-making credibility of the final design. Summary of the Invention
[0004] This invention provides a method and apparatus for multi-objective hydrodynamic optimization of hydrofoils, so as to improve the efficiency of multi-objective hydrodynamic optimization of hydrofoils and improve the practicality and reliability of the optimization scheme.
[0005] To address the aforementioned technical problems, this invention provides a multi-objective hydrodynamic optimization method for hydrofoils, comprising: Obtain an optimization variable set, which includes several optimization variables and their corresponding feasible value ranges; Several sample data points are collected within the range of values of the optimization variables. Based on the sample data, CFD simulation is performed on the preset hydrofoil parameter model to generate training data. Obtain a preset agent model, which is trained based on a preset deep neural network and the training data; The optimization iteration is performed based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, the hydrodynamic performance index of individuals in the population constructed by the multi-objective optimization algorithm is predicted based on the surrogate model. Based on the prediction results, the multi-objective optimization algorithm is driven to perform population evolution until the preset termination condition is met, and the Pareto optimal solution set is output. The schemes in the Pareto optimal solution set are verified based on the preset CFD simulation, and the target scheme is determined based on the verification results.
[0006] This invention clarifies the design space and boundary conditions for multi-objective optimization of hydrofoils by obtaining the set of optimization variables and their corresponding feasible value ranges. Sample data is then collected within the value range of the optimization variables, and CFD simulations are performed based on the hydrofoil parameter model to generate training data, providing a high-fidelity, structured dataset for the subsequent construction of the surrogate model. Next, a surrogate model trained using a deep neural network and the training data is obtained, achieving accurate fitting of the complex nonlinear mapping between design variables and multi-objective performance. Further optimization iterations are performed based on the multi-objective optimization algorithm and the surrogate model, and the surrogate model is used to predict the hydrodynamic performance indicators of individual populations to drive population evolution, significantly reducing computational costs and improving global search efficiency. Finally, CFD simulations are used to verify the solutions in the Pareto optimal solution set and determine the target solution, ensuring the engineering reliability and practical applicability of the optimization results and improving the efficiency of multi-objective hydrodynamic optimization for hydrofoils.
[0007] Furthermore, the process involves collecting several sample data points within the value range of the optimization variables, performing CFD simulations on a preset hydrofoil parameter model based on the sample data, and generating training data, including: Based on the preset Latin hypercube sampling method, several sets of sample data are uniformly extracted within the value range of the optimization variable. Each set of sample data includes specific values of the front hydrofoil sweep angle, hydrofoil span, and hydrofoil spacing. Based on the sample data of each group, the corresponding hydrofoil parameter models are constructed respectively, and the hydrofoil parameter models are meshed to obtain the computational mesh model of each hydrofoil parameter model. CFD simulations were performed on each computational grid model to obtain the hydrodynamic performance indicators of the hydrofoil corresponding to each set of sample data, wherein the hydrodynamic performance indicators include drag, pitch and heave. Training data is obtained by correlating each group of sample data with the corresponding drag, pitch, and heave values.
[0008] This invention employs the Latin hypercube sampling method to uniformly extract several sets of sample data within the range of optimization variables, ensuring the uniform distribution and statistical representativeness of the samples in the design space. By constructing hydrofoil parameter models based on each set of sample data and performing mesh generation to obtain computational grid models, an accurate and reliable geometric foundation for numerical simulation is established. Furthermore, CFD simulations are performed on each computational grid model to obtain drag, trim, and heave values, acquiring high-fidelity multi-objective hydrodynamic performance indicators. Finally, each set of sample data is correlated with its corresponding drag, trim, and heave values to form training data, providing a well-structured and complete input-output paired dataset for training the surrogate model.
[0009] Furthermore, obtaining the preset proxy model includes: An initial proxy model is constructed based on a preset deep neural network, wherein the number of input layer nodes of the initial proxy model is the same as the number of optimization variables, and the number of output layer nodes is the same as the number of hydrodynamic performance indicators. The deep neural network is trained based on the training data and a preset backpropagation algorithm until the preset conditions are met, and then a proxy model is output.
[0010] This invention provides independent sample support for the training process and hyperparameter tuning of the surrogate model by dividing the training data into training and validation sets, effectively preventing overfitting. By constructing an initial surrogate model based on a deep neural network and setting the number of input layer nodes to be the same as the number of optimization variables, and the number of output layer nodes to be the same as the number of hydrodynamic performance indicators, the dimensionality of the model structure and the optimization problem is ensured. By training the deep neural network based on the training set and the backpropagation algorithm until the preset conditions are met, a surrogate model with high prediction accuracy is obtained, achieving an efficient replacement for high-fidelity CFD simulations.
[0011] Furthermore, the optimization iteration is performed based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm are predicted based on the surrogate model. The prediction results drive the multi-objective optimization algorithm to perform population evolution until a preset termination condition is met, outputting a Pareto optimal solution set, including: Initialize the population of the multi-objective red-billed blue finches optimization algorithm, and encode each individual in the population as a set of optimization variable values; The optimization variable values of each individual are input into the surrogate model, and the predicted values of the hydrodynamic performance index of each individual are output based on the surrogate model. The predicted values are used as the fitness of the individual. Pareto non-dominated individuals in the population are determined by non-dominated ranking based on the fitness of all individuals. Population evolution is performed based on the Pareto non-dominated individuals until a preset termination condition is met, and the Pareto optimal solution set is output.
[0012] This invention establishes an initial solution space and a unified encoding representation for the optimization search by initializing the population of a multi-objective red-billed bluefin finches optimization algorithm and encoding each individual as a set of optimization variable values. By inputting the optimization variable values of each individual into a surrogate model and outputting the predicted values of hydrodynamic performance indicators as the fitness of that individual, rapid evaluation of the performance of individuals in the population is achieved, avoiding large-scale direct CFD computation. Furthermore, non-dominated sorting is performed based on the fitness of all individuals to identify Pareto non-dominated individuals in the population, effectively identifying the dominant solution set in the current population. Thus, population evolution is carried out based on Pareto non-dominated individuals until the termination condition is met and the Pareto optimal solution set is output, ensuring the global convergence of the optimization process and the diversity and uniformity of the final solution set in the objective space.
[0013] Furthermore, the verification of each scheme in the Pareto optimal solution set based on the preset CFD simulation, and the determination of the target scheme based on the verification results, includes: One or more candidate solutions are selected from the Pareto optimal solution set, wherein each candidate solution corresponds to a set of optimization variable values; For each candidate scheme, CFD simulations were performed to obtain the verification hydrodynamic performance indicators corresponding to each candidate scheme. The prediction error is calculated based on the verification hydrodynamic performance indicators and the predicted values of the hydrodynamic performance indicators for each candidate scheme. The target scheme is determined based on the prediction error.
[0014] This invention provides a representative sample of design alternatives for the verification process by selecting one or more candidate schemes from the Pareto optimal solution set. CFD simulations are then performed on the optimization variable values of each candidate scheme to obtain corresponding verification hydrodynamic performance indices, acquiring high-fidelity real performance data independent of the surrogate model. Furthermore, the prediction error is calculated based on the verification hydrodynamic performance indices of each candidate scheme and the predicted values of the surrogate model, quantitatively evaluating the prediction accuracy and generalization ability of the surrogate model at different design points. Finally, by determining the target scheme based on the prediction error, the final selected design scheme is ensured to have engineering-acceptable accuracy, achieving effective verification of the optimization results and reliable decision-making.
[0015] Secondly, an embodiment of the present invention provides a multi-objective hydrodynamic optimization device for a hydrofoil, comprising: a variable determination module, a simulation module, a model acquisition module, an optimization module, and a scheme verification module; The variable determination module is used to obtain an optimization variable set, which includes several optimization variables and their corresponding feasible value ranges. The simulation module is used to collect several sample data within the value range of the optimization variable, and perform CFD simulation on the preset hydrofoil parameter model based on the sample data to generate training data. The model acquisition module is used to acquire a preset proxy model, which is trained based on a preset deep neural network and the training data. The optimization module is used to perform optimization iterations based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, it predicts the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm based on the surrogate model, and drives the multi-objective optimization algorithm to perform population evolution based on the prediction results until the preset termination condition is met, and outputs the Pareto optimal solution set. The scheme verification module is used to verify each scheme in the Pareto optimal solution set based on a preset CFD simulation, and to determine the target scheme based on the verification results.
[0016] Furthermore, the simulation module is used to collect several sample data points within the value range of the optimization variables, perform CFD simulation on a preset hydrofoil parameter model based on the sample data, and generate training data, including: Based on the preset Latin hypercube sampling system, several sets of sample data are uniformly extracted within the range of the value of the optimization variable. Each set of sample data includes the specific values of the front hydrofoil sweep angle, hydrofoil span and hydrofoil spacing. Based on the sample data of each group, the corresponding hydrofoil parameter models are constructed respectively, and the hydrofoil parameter models are meshed to obtain the computational mesh model of each hydrofoil parameter model. CFD simulations were performed on each computational grid model to obtain the hydrodynamic performance indicators of the hydrofoil corresponding to each set of sample data. Training data is obtained by associating each group of sample data with the corresponding hydrodynamic performance indicators.
[0017] Furthermore, the model acquisition module is used to acquire a preset proxy model, including: An initial proxy model is constructed based on a preset deep neural network, wherein the number of input layer nodes of the initial proxy model is the same as the number of optimization variables, and the number of output layer nodes is the same as the number of hydrodynamic performance indicators. The deep neural network is trained based on the training data and a preset backpropagation algorithm until the preset conditions are met, and then a proxy model is output.
[0018] Furthermore, the optimization module is used to perform optimization iterations based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, it predicts the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm based on the surrogate model, and drives the multi-objective optimization algorithm to perform population evolution based on the prediction results until a preset termination condition is met, outputting a Pareto optimal solution set, including: Initialize the population of the multi-objective red-billed blue finches optimization algorithm, and encode each individual in the population as a set of optimization variable values; The optimization variable values of each individual are input into the surrogate model, and the predicted values of the hydrodynamic performance index of each individual are output based on the surrogate model. The predicted values are used as the fitness of the individual. Pareto non-dominated individuals in the population are determined by non-dominated ranking based on the fitness of all individuals. Population evolution is performed based on the Pareto non-dominated individuals until a preset termination condition is met, and the Pareto optimal solution set is output.
[0019] Furthermore, the scheme verification module is used to verify each scheme in the Pareto optimal solution set based on a preset CFD simulation, and to determine the target scheme based on the verification results, including: One or more candidate solutions are selected from the Pareto optimal solution set, wherein each candidate solution corresponds to a set of optimization variable values; For each candidate scheme, CFD simulations were performed to obtain the verification hydrodynamic performance indicators corresponding to each candidate scheme. The prediction error is calculated based on the verification hydrodynamic performance indicators and the predicted values of the hydrodynamic performance indicators for each candidate scheme. The target scheme is determined based on the prediction error. Attached Figure Description
[0020] Figure 1 This is a schematic flowchart of a multi-objective hydrodynamic optimization method for a hydrofoil provided in an embodiment of the present invention; Figure 2 A schematic diagram of a hydrofoil model provided in an embodiment of the present invention; Figure 3 This is a schematic diagram of a multi-objective hydrodynamic optimization device for a hydrofoil provided in an embodiment of the present invention. Detailed Implementation
[0021] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.
[0022] The terms "first" and "second," etc., in the specification, claims, and drawings of this application are used to distinguish different objects, not to describe a specific order. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or apparatus that includes a series of steps or units is not limited to the listed steps or units, but may optionally include steps or units not listed, or may optionally include other steps or units inherent to such processes, methods, products, or apparatus.
[0023] In this document, the term "embodiment" means that a particular feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment mutually exclusive with other embodiments. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described herein can be combined with other embodiments.
[0024] Example 1 See Figure 1 , Figure 1 This is a schematic flowchart illustrating a multi-objective hydrodynamic optimization method for a hydrofoil provided in an embodiment of the present invention. The embodiment of the present invention provides a multi-objective hydrodynamic optimization method for a hydrofoil, including steps 101 to 104, as detailed below: Step 101: Obtain the set of optimization variables, which includes several optimization variables and their corresponding feasible value ranges; Please refer to Figure 2 , Figure 2 This is a schematic diagram of a hydrofoil model provided in an embodiment of the present invention.
[0025] In this embodiment, when obtaining the set of optimization variables, the key geometric parameters affecting the core hydrodynamic performance of the hydrofoil are first determined as optimization variables, specifically including the front hydrofoil sweep angle. θ The hydrofoil span (d) and hydrofoil spacing (l) are considered. For each optimization variable, a feasible value range is set according to the hydrofoil boat's design specifications, operating conditions, and structural constraints. The feasible value range for the forward hydrofoil sweep angle is determined based on the adjustable range of the angle between the hydrofoil and the hull centerline; the feasible value range for the hydrofoil span is determined based on the hull's lateral space and the hydrofoil structural strength requirements; and the feasible value range for the hydrofoil spacing is determined based on the longitudinal arrangement space between the forward and aft hydrofoils and the hydrodynamic interference effects. All optimization variables and their corresponding feasible value ranges together constitute the optimization variable set, serving as the design space boundary conditions for subsequent optimization design.
[0026] In this embodiment, during subsequent optimization design, the multi-objective optimization problem is clearly defined as follows: the optimization objectives are set as three indicators: minimum heave, pitch, and drag, and the feasible value range of each design variable is determined, thereby fully defining the optimization design space. The specific range of hydrofoil parameters is as follows: (1) Changing the hydrofoil spacing involves adjusting the front and rear hydrofoil along their surfaces. S Symmetrical movement, surface S 1570mm from the stern. The optimization function is as follows: (2) in, This is the resistance value. This is the dip value. It is an increasing value.
[0027] As a specific example of an embodiment of the present invention, a certain type of hydrofoil is selected as the optimization object, and the obtained set of optimization variables includes three optimization variables. The feasible range of the front hydrofoil sweep angle is set to 15 degrees to 45 degrees. The feasible range of the hydrofoil span is set to 600 mm to 1000 mm. The feasible range of the hydrofoil spacing is set to 200 mm to 500 mm, wherein the specific definition of the hydrofoil spacing is to move the front and rear hydrofoils symmetrically along a reference plane perpendicular to the bottom of the hull, which is 1570 mm away from the stern. The above three optimization variables and their respective value ranges together form the optimization variable set of the hydrofoil, providing a clear search boundary for subsequent sample collection and optimization iteration.
[0028] Step 102: Collect several sample data within the range of values of the optimization variables, and perform CFD simulation on the preset hydrofoil parameter model based on the sample data to generate training data; In this embodiment, several sample data points are collected within the value range of the optimization variable. Based on the sample data, a CFD simulation is performed on a preset hydrofoil parameter model to generate training data, including: Based on the preset Latin hypercube sampling method, several sets of sample data are uniformly extracted within the value range of the optimization variable. Each set of sample data includes specific values of the front hydrofoil sweep angle, hydrofoil span, and hydrofoil spacing. Based on the sample data of each group, the corresponding hydrofoil parameter models are constructed respectively, and the hydrofoil parameter models are meshed to obtain the computational mesh model of each hydrofoil parameter model. CFD simulations were performed on each computational grid model to obtain the hydrodynamic performance indicators of the hydrofoil corresponding to each set of sample data, wherein the hydrodynamic performance indicators include drag, pitch and heave. Training data is obtained by correlating each group of sample data with the corresponding drag, pitch, and heave values.
[0029] In this embodiment, when collecting sample data, the optimal Latin hypercube sampling method is used to perform uniform sampling within the design space. First, the feasible value range of each optimization variable is divided into several equally spaced sub-intervals, with the number of sub-intervals being the same as the required number of samples. Then, a value is randomly selected from each sub-interval, ensuring that the combinations of values for each variable in different sub-intervals have a uniform distribution characteristic, thereby generating several sets of sample data.
[0030] In this embodiment, each set of sample data includes specific values for the front hydrofoil sweep angle, hydrofoil span, and hydrofoil spacing. Based on each set of sample data, a corresponding hydrofoil parameter model is constructed. This hydrofoil parameter model is a three-dimensional geometric model of the hydrofoil, and an unstructured mesh is generated for this geometric model. The boundary layer mesh is fined, with a focus on the near-wall region, to ensure the analytical accuracy of the boundary layer flow, thereby obtaining the computational mesh model corresponding to each sample.
[0031] In this embodiment, when refining the boundary layer mesh in the near-wall region, firstly, all solid wall boundaries in the hydrofoil geometry model are extracted, including the hydrofoil surface and the wetted surface of the hull. For these solid wall boundaries, a boundary layer mesh is constructed using prism or triangular prism layer mesh generation techniques. The number of boundary layer mesh layers is set, and the thickness of each layer increases progressively outward from the wall surface along the normal direction at a preset growth ratio, forming a gradual distribution from dense to sparse. The thickness of the first layer is determined based on the turbulence model used and the expected wall treatment method, ensuring that the dimensionless wall distance is within a reasonable range applicable to the turbulence model. The total thickness of the boundary layer mesh is set based on the boundary layer thickness predicted by boundary layer theory, ensuring coverage of the near-wall region with significant velocity gradients. After generating the prism layer mesh, the mainstream region far from the wall is filled with tetrahedral or hexahedral meshes, and a buffer zone with a smooth size transition is set between the boundary layer mesh and the outer mesh to avoid numerical dissipation caused by abrupt changes in mesh size. After mesh generation, the mesh quality of the boundary layer region is checked, including indicators such as mesh skewness, aspect ratio, and orthogonality, to ensure that it meets the convergence and accuracy requirements of computational fluid dynamics simulation. Through the above processing, a refined boundary layer mesh capable of accurately analyzing the velocity gradient and shear stress distribution near the wall is obtained, thus yielding the computational mesh model corresponding to each sample.
[0032] As a specific example of an embodiment of the present invention, the wall boundaries of the hydrofoil's geometric model are extracted, including the hydrofoil surface and the wetted surface of the hull. For these wall boundaries, a prism layer mesh generation technique is used to refine the boundary layer mesh. The number of boundary layer mesh layers is set to ten, and the thickness of the first layer is determined according to the wall function requirements, specifically estimated based on the Reynolds number and turbulence model, ensuring that the dimensionless wall distance y+ value is between 30 and 300. The growth ratio of the boundary layer mesh is set to 1.2, that is, the thickness of each layer is 1.2 times that of the previous layer, to achieve a smooth mesh distribution that gradually transitions from the wall surface outwards.
[0033] In this embodiment, the total thickness of the boundary layer mesh is estimated based on boundary layer theory to ensure coverage of the entire boundary layer region. After generating the prism layer mesh, the region far from the wall is filled with tetrahedral or hexahedral meshes, and a smooth transition is set between the boundary layer mesh and the outer mesh to avoid computational distortion caused by abrupt changes in mesh size. Finally, the computational mesh model corresponding to each sample is obtained. This model can accurately resolve the velocity gradient and shear stress distribution near the wall, thereby ensuring the computational accuracy of the boundary layer flow.
[0034] In this embodiment, high-fidelity computational fluid dynamics simulations are performed on each computational grid model. Reasonable turbulence models, boundary conditions, and solver parameters are set during the simulation. After convergence, the hydrodynamic performance indicators of the hydrofoil corresponding to each set of samples are extracted, specifically including drag, pitch, and heave values. Finally, the sample data of each set are structurally correlated with their corresponding drag, pitch, and heave values according to a one-to-one correspondence, forming a training dataset for subsequent surrogate model training.
[0035] In this embodiment, a Latin hypercube sampling method is used to uniformly extract several sets of sample data within the range of values for the optimization variables, ensuring the uniform distribution and statistical representativeness of the samples in the design space. A computational grid model is obtained by constructing a hydrofoil parameter model based on each set of sample data and performing mesh generation, thus establishing an accurate and reliable geometric foundation for numerical simulation. CFD simulations are then performed on each computational grid model to obtain drag, trim, and heave values, yielding high-fidelity multi-objective hydrodynamic performance indicators. Finally, each set of sample data is correlated with its corresponding drag, trim, and heave values to form training data, providing a well-structured and complete input-output paired dataset for training the surrogate model.
[0036] Step 103: Obtain a preset proxy model, which is trained based on a preset deep neural network and the training data; In this embodiment, obtaining the preset proxy model includes: An initial proxy model is constructed based on a preset deep neural network, wherein the number of input layer nodes of the initial proxy model is the same as the number of optimization variables, and the number of output layer nodes is the same as the number of hydrodynamic performance indicators. The deep neural network is trained based on the training data and a preset backpropagation algorithm until the preset conditions are met, and then a proxy model is output.
[0037] In this embodiment, when constructing the deep neural network surrogate model, the dataset generated by the computational fluid dynamics simulation is first randomly divided into three non-overlapping subsets according to a preset ratio: a training set, a validation set, and a test set. The training set is used for fitting and learning the parameters of the deep neural network model, the validation set is used for hyperparameter tuning and overfitting monitoring during the training process, and the test set is used for independent evaluation of the final model accuracy.
[0038] In this embodiment, an initial surrogate model is constructed based on a deep neural network, which includes an input layer, multiple hidden layers, and an output layer. The number of nodes in the input layer is set to be the same as the number of optimization variables, with each input node corresponding to one optimization variable. The number of nodes in the output layer is set to be the same as the number of hydrodynamic performance indicators, with each output node corresponding to one hydrodynamic performance indicator. The number of hidden layers and the number of nodes in each hidden layer are pre-set according to the complexity of the design problem and are optimized on a validation set.
[0039] In this embodiment, after defining the network structure, the deep neural network is trained using a training set. During training, a mini-batch gradient descent strategy is employed, randomly selecting a batch of samples from the training set each time to input into the network, and calculating the predicted values of the output layer through forward propagation. The predicted values are compared with the true values in the training set, and the prediction error is calculated using a preset loss function. Based on the loss function value, the gradient of the loss with respect to each network parameter is calculated layer by layer using the backpropagation algorithm, and the network parameters are updated using an optimizer according to the gradient direction to reduce the loss function value. Each iteration of all training samples is called a training cycle. After each training cycle, the prediction error of the current network is calculated using a validation set to monitor the training process. When the validation set error stops decreasing or begins to increase for several consecutive cycles, training is terminated early to prevent overfitting; or training is stopped when the preset maximum number of training cycles is reached. After training, the final surrogate model is independently evaluated for accuracy using a test set. When the prediction accuracy on the test set meets the preset requirements, the output of the deep neural network model is used as the final surrogate model for rapid performance prediction in subsequent optimization iterations.
[0040] As a specific example of this invention, there are three optimization variables: the front hydrofoil sweep angle, the hydrofoil span, and the hydrofoil pitch. Therefore, the number of nodes in the input layer of the deep neural network is set to three. There are three hydrodynamic performance indicators: drag, pitch, and heave. Therefore, the number of nodes in the output layer is set to three. The hidden layers are set to four, each containing sixty-four neurons, and the activation function is a linear rectified function. The network parameters are randomly initialized using a normal distribution.
[0041] In this embodiment, the determination coefficient of the surrogate model is calculated using a test set to evaluate the model accuracy. When the model accuracy is greater than a preset threshold, it indicates that the surrogate model can replace high-fidelity computational fluid dynamics simulation.
[0042] In this embodiment, the model accuracy is evaluated on the test set using the coefficient of determination R² (Equation 3). When R² > 0.95, the surrogate model can replace the CFD for rapid performance evaluation in subsequent optimization, i.e., the input fore-hydrofoil sweep angle ( θ ), hydrofoil span ( d ) and hydrofoil spacing ( l By doing so, we can obtain the hydrofoil's resistance, trim, and lift values.
[0043] (3) in, For the first One measured value; The predicted value corresponding to the model; This is the average of the measured values; The total number of samples.
[0044] In this embodiment, an initial surrogate model is constructed based on a deep neural network, and the number of input layer nodes is set to be the same as the number of optimization variables, and the number of output layer nodes is set to be the same as the number of hydrodynamic performance indicators, ensuring that the model structure matches the dimension of the optimization problem. By training the deep neural network based on the training set and the backpropagation algorithm until the preset conditions are met, a surrogate model with high prediction accuracy is obtained, achieving an efficient replacement for high-fidelity CFD simulation.
[0045] Step 104: Perform optimization iteration based on the preset multi-objective optimization algorithm and the preset surrogate model. During the optimization iteration, predict the hydrodynamic performance index of individuals in the population constructed by the multi-objective optimization algorithm based on the surrogate model. Drive the multi-objective optimization algorithm to perform population evolution based on the prediction results until the preset termination condition is met, and output the Pareto optimal solution set. In this embodiment, the optimization iteration is performed based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm are predicted based on the surrogate model. The prediction results drive the multi-objective optimization algorithm to perform population evolution until a preset termination condition is met, and a Pareto optimal solution set is output, including: Initialize the population of the multi-objective red-billed blue finches optimization algorithm, and encode each individual in the population as a set of optimization variable values; The optimization variable values of each individual are input into the surrogate model, and the predicted values of the hydrodynamic performance index of each individual are output based on the surrogate model. The predicted values are used as the fitness of the individual. Pareto non-dominated individuals in the population are determined by non-dominated ranking based on the fitness of all individuals. Population evolution is performed based on the Pareto non-dominated individuals until a preset termination condition is met, and the Pareto optimal solution set is output.
[0046] In this embodiment, when performing optimization iterations based on the multi-objective red-billed blue finch optimization algorithm and the surrogate model, a population initialization operation is first performed. The population size is set, that is, the number of individuals in the population, and each individual corresponds to a candidate solution to the hydrofoil optimization problem.
[0047] In this embodiment, a real-number encoding method is used to encode each individual as a set of optimization variable values. The number of optimization variables is the same as the number of design variables in the optimization problem, and the value of each optimization variable is randomly generated within its corresponding feasible range. After the population initialization is completed, the optimization iteration loop begins.
[0048] In this embodiment, in each iteration, the optimization variable values of each individual in the population are sequentially input into the trained surrogate model. The surrogate model quickly outputs the predicted values of the hydrodynamic performance indicators corresponding to that individual through forward computation, including drag, pitch and heave values, and uses these predicted values as the fitness vector of that individual.
[0049] In this embodiment, a fast non-dominated sort is performed based on the fitness vectors of all individuals. The non-dominated level of each individual is calculated, and Pareto non-dominated individuals in the current population are identified. These individuals constitute the optimal solution set for the current generation. Subsequently, population evolution is performed based on these Pareto non-dominated individuals. Specifically, Pareto non-dominated individuals are designated as the leader set, and the other individuals in the population are designated as followers. Each follower randomly selects a leader from the leader set, simulates the foraging and cooperative behavior of the Red-billed Bluefinch, and updates its own position based on the leader's location information and the group's cooperative strategy, i.e., updates its own optimization variable value, generating a offspring population. Then, the parent and offspring populations are merged, and the merged population undergoes another non-dominated sort and crowding distance calculation. Based on the principle of prioritizing non-dominated level and maximizing crowding distance, a new generation population with the same size as the parent population is selected.
[0050] In this embodiment, for each individual in the merged population, the number of individuals dominated by it and the set of other individuals it dominates are calculated. Through iterative selection, all individuals not dominated by any other individual are identified, forming the first non-dominated front. Then, from the remaining individuals, individuals not dominated by any remaining individuals are selected, forming the second non-dominated front. This process continues until all individuals are assigned to their respective fronts. After completing the non-dominated sorting, the crowding distance is calculated for individuals within the same front.
[0051] In this embodiment, for each objective function direction, individuals within the frontal plane are sorted in ascending order of objective function values. The crowding distance between the first and last individuals is set to infinity, while the crowding distance of the middle individuals is accumulated based on the difference between their two adjacent individuals on the objective. The total crowding distance of an individual is obtained by summing the distances obtained in all objective directions. This crowding distance reflects the density of solutions surrounding the individual in the objective space; a larger crowding distance indicates a sparser environment and better diversity around the individual.
[0052] In this embodiment, when selecting the next generation population, individuals from the entire frontier are first added to the next generation population in ascending order of non-dominance level. If adding individuals to a frontier causes the population size to exceed the preset parent population size, the individuals within that frontier are sorted by their crowding distance from largest to smallest, with individuals having larger crowding distances being added sequentially until the number of individuals in the next generation population reaches the size of the parent population. Through this method, the next generation population retains non-dominant individuals with good convergence while maintaining a uniform distribution of the solution set in the target space, thus ensuring a balance between diversity and convergence in the optimization process.
[0053] In this embodiment, the process of surrogate model prediction, non-dominated sorting, and population evolution is repeated until a preset termination condition is met. The termination condition can be reaching the maximum number of iterations or the population converging to a preset accuracy. After terminating the iteration, all Pareto non-dominated individuals in the final generation of the population are output as the Pareto optimal solution set. Each solution contains a set of optimized variable values and their corresponding predicted values of hydrodynamic performance indicators.
[0054] In this embodiment, by initializing the population of the multi-objective red-billed bluefin finches optimization algorithm and encoding each individual as a set of optimization variable values, an initial solution space and a unified encoding representation for the optimization search are established. By inputting the optimization variable values of each individual into the surrogate model and outputting the predicted values of hydrodynamic performance indicators as the fitness of that individual, rapid evaluation of the performance of individuals in the population is achieved, avoiding large-scale direct CFD computation. Furthermore, non-dominated sorting is performed based on the fitness of all individuals to determine the Pareto non-dominated individuals in the population, effectively identifying the dominant solution set in the current population. Thus, population evolution is carried out based on Pareto non-dominated individuals until the termination condition is met and the Pareto optimal solution set is output, ensuring the global convergence of the optimization process and the diversity and uniformity of the final solution set in the target space.
[0055] Step 105: Verify each scheme in the Pareto optimal solution set based on the preset CFD simulation, and determine the target scheme based on the verification results.
[0056] In this embodiment, the verification of each scheme in the Pareto optimal solution set based on a preset CFD simulation, and the determination of the target scheme based on the verification results, includes: One or more candidate solutions are selected from the Pareto optimal solution set, wherein each candidate solution corresponds to a set of optimization variable values; For each candidate scheme, CFD simulations were performed to obtain the verification hydrodynamic performance indicators corresponding to each candidate scheme. The prediction error is calculated based on the verification hydrodynamic performance indicators and the predicted values of the hydrodynamic performance indicators for each candidate scheme. The target scheme is determined based on the prediction error.
[0057] In this embodiment, during the verification phase, one or more candidate solutions are first selected from the Pareto optimal solution set output by the optimization iteration for high-fidelity verification. The selection strategy can be determined according to actual engineering needs. For example, the two endpoint solutions of the Pareto front and the compromise solutions in the middle region can be selected. The endpoint solutions correspond to extreme designs where one performance index is optimal but other performance indexes are relatively poor, and the compromise solutions correspond to designs where all performance indexes are balanced.
[0058] In this embodiment, for each candidate scheme, its corresponding optimization variable values are extracted, including the specific values of the front hydrofoil sweep angle, hydrofoil span, and hydrofoil spacing. Then, for the optimization variable values of each candidate scheme, the hydrofoil boat geometric model is reconstructed and high-fidelity computational fluid dynamics simulation is performed. During the simulation, the same mesh generation strategy, turbulence model, boundary conditions, and solver settings as in the sample generation stage are used to ensure the consistency between the verification results and the training data.
[0059] In this embodiment, after simulation convergence, the verification hydrodynamic performance indicators of the candidate scheme are extracted, including drag, pitch, and heave. The verified performance indicators are compared with the predicted values output by the surrogate model during the optimization process, and the prediction errors of each performance indicator are calculated. The prediction errors can be expressed as absolute or relative errors. Based on the prediction errors of each performance indicator, it is determined whether the prediction accuracy of the surrogate model on the candidate scheme meets the engineering requirements. If the prediction errors of all candidate schemes are within the preset engineering allowable range, the reliability of the surrogate model's predictions on the entire Pareto solution set is confirmed.
[0060] Finally, the target solution is determined based on the verification results. Specifically, a multi-criteria decision-making method can be adopted. For example, weights can be assigned to each performance index according to engineering preferences, the weighted comprehensive score of each candidate solution can be calculated, and the candidate solution with the highest score can be determined as the target solution; or when a certain performance index of a candidate solution is significantly better than other solutions and the prediction error is the smallest, it can be directly determined as the target solution.
[0061] In this embodiment, one or more candidate schemes are selected from the Pareto optimal solution set, providing representative design alternatives for the verification stage. CFD simulations are then performed on the optimization variable values of each candidate scheme to obtain corresponding verification hydrodynamic performance indices, acquiring high-fidelity real performance data independent of the surrogate model. Furthermore, the prediction error is calculated based on the verification hydrodynamic performance indices of each candidate scheme and the predicted values of the surrogate model, quantitatively evaluating the prediction accuracy and generalization ability of the surrogate model at different design points. Finally, by determining the target scheme based on the prediction error, the final selected design scheme is ensured to have engineering-acceptable accuracy, achieving effective verification of the optimization results and reliable decision-making.
[0062] Please refer to Figure 3 , Figure 3 A schematic diagram of a multi-objective hydrodynamic optimization device for a hydrofoil provided in an embodiment of the present invention includes: a variable determination module 301, a simulation module 302, a model acquisition module 303, an optimization module 304, and a scheme verification module 305; The variable determination module 301 is used to obtain an optimization variable set, which includes several optimization variables and their corresponding feasible value ranges. The simulation module 302 is used to collect several sample data within the value range of the optimization variable, and perform CFD simulation on the preset hydrofoil parameter model based on the sample data to generate training data. The model acquisition module 303 is used to acquire a preset proxy model, which is trained based on a preset deep neural network and the training data; The optimization module 304 is used to perform optimization iteration based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, it predicts the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm based on the surrogate model, and drives the multi-objective optimization algorithm to perform population evolution based on the prediction results until the preset termination condition is met, and outputs the Pareto optimal solution set. The scheme verification module 305 is used to verify each scheme in the Pareto optimal solution set based on a preset CFD simulation, and to determine the target scheme based on the verification results.
[0063] In this embodiment, the simulation module is used to collect several sample data points within the value range of the optimization variable, perform CFD simulation on a preset hydrofoil parameter model based on the sample data, and generate training data, including: Based on the preset Latin hypercube sampling system, several sets of sample data are uniformly extracted within the range of the value of the optimization variable. Each set of sample data includes the specific values of the front hydrofoil sweep angle, hydrofoil span and hydrofoil spacing. Based on the sample data of each group, the corresponding hydrofoil parameter models are constructed respectively, and the hydrofoil parameter models are meshed to obtain the computational mesh model of each hydrofoil parameter model. CFD simulations were performed on each computational grid model to obtain the hydrodynamic performance indicators of the hydrofoil corresponding to each set of sample data. Training data is obtained by associating each group of sample data with the corresponding hydrodynamic performance indicators.
[0064] In this embodiment, the model acquisition module is used to acquire a preset proxy model, including: An initial proxy model is constructed based on a preset deep neural network, wherein the number of input layer nodes of the initial proxy model is the same as the number of optimization variables, and the number of output layer nodes is the same as the number of hydrodynamic performance indicators. The deep neural network is trained based on the training data and a preset backpropagation algorithm until the preset conditions are met, and then a proxy model is output.
[0065] In this embodiment, the optimization module is used to perform optimization iterations based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, it predicts the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm based on the surrogate model, and drives the multi-objective optimization algorithm to perform population evolution based on the prediction results until a preset termination condition is met, outputting a Pareto optimal solution set, including: Initialize the population of the multi-objective red-billed blue finches optimization algorithm, and encode each individual in the population as a set of optimization variable values; The optimization variable values of each individual are input into the surrogate model, and the predicted values of the hydrodynamic performance index of each individual are output based on the surrogate model. The predicted values are used as the fitness of the individual. Pareto non-dominated individuals in the population are determined by non-dominated ranking based on the fitness of all individuals. Population evolution is performed based on the Pareto non-dominated individuals until a preset termination condition is met, and the Pareto optimal solution set is output.
[0066] In this embodiment, the scheme verification module is used to verify each scheme in the Pareto optimal solution set based on a preset CFD simulation, and to determine the target scheme based on the verification results, including: One or more candidate solutions are selected from the Pareto optimal solution set, wherein each candidate solution corresponds to a set of optimization variable values; For the optimization variable values of each candidate scheme, CFD simulations were performed to obtain the verification hydrodynamic performance indicators corresponding to each candidate scheme. The prediction error is calculated based on the verified hydrodynamic performance indicators and the predicted values of the hydrodynamic performance indicators for each candidate scheme. The target scheme is determined based on the prediction error.
[0067] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the scope of protection of the present invention. In particular, it should be noted that any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention for those skilled in the art.
Claims
1. A multi-objective hydrodynamic optimization method for a hydrofoil, characterized in that, include: Obtain an optimization variable set, which includes several optimization variables and their corresponding feasible value ranges; Collect several sample data within the range of the optimization variables, and perform CFD simulation on the preset hydrofoil parameter model based on the sample data to generate training data; Obtain a preset agent model, which is trained based on a preset deep neural network and the training data; The optimization iteration is performed based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, the hydrodynamic performance index of individuals in the population constructed by the multi-objective optimization algorithm is predicted based on the surrogate model. Based on the prediction results, the multi-objective optimization algorithm is driven to perform population evolution until the preset termination condition is met, and the Pareto optimal solution set is output. The schemes in the Pareto optimal solution set are verified based on the preset CFD simulation, and the target scheme is determined based on the verification results.
2. The multi-objective hydrodynamic optimization method for a hydrofoil as described in claim 1, characterized in that, The process involves collecting several sample data points within the value range of the optimization variables, performing CFD simulations on a preset hydrofoil parameter model based on the sample data, and generating training data, including: Based on the preset Latin hypercube sampling method, several sets of sample data are uniformly extracted within the value range of the optimization variable. Each set of sample data includes specific values of the front hydrofoil sweep angle, hydrofoil span, and hydrofoil spacing. Based on the sample data of each group, the corresponding hydrofoil parameter models are constructed respectively, and the hydrofoil parameter models are meshed to obtain the computational mesh model of each hydrofoil parameter model. CFD simulations were performed on each computational grid model to obtain the hydrodynamic performance indicators of the hydrofoil corresponding to each set of sample data, wherein the hydrodynamic performance indicators include drag, pitch and heave. Training data is obtained by correlating each group of sample data with the corresponding drag, pitch, and heave values.
3. The multi-objective hydrodynamic optimization method for a hydrofoil as described in claim 2, characterized in that, The process of obtaining the preset proxy model includes: An initial proxy model is constructed based on a preset deep neural network, wherein the number of input layer nodes of the initial proxy model is the same as the number of optimization variables, and the number of output layer nodes is the same as the number of hydrodynamic performance indicators. The deep neural network is trained based on the training data and a preset backpropagation algorithm until the preset conditions are met, and then a proxy model is output.
4. The multi-objective hydrodynamic optimization method for a hydrofoil as described in claim 3, characterized in that, The optimization iteratively operates based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iterative process, the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm are predicted based on the surrogate model. The prediction results drive the multi-objective optimization algorithm to perform population evolution until a preset termination condition is met, outputting a Pareto optimal solution set, including: Initialize the population of the multi-objective red-billed blue finches optimization algorithm, and encode each individual in the population as a set of optimization variable values; The optimization variable values of each individual are input into the surrogate model, and the predicted values of the hydrodynamic performance index of each individual are output based on the surrogate model. The predicted values are used as the fitness of the individual. Pareto non-dominated individuals in the population are determined by non-dominated ranking based on the fitness of all individuals. Population evolution is performed based on the Pareto non-dominated individuals until a preset termination condition is met, and the Pareto optimal solution set is output.
5. The multi-objective hydrodynamic optimization method for a hydrofoil as described in claim 4, characterized in that, The verification of each scheme in the Pareto optimal solution set based on the preset CFD simulation, and the determination of the target scheme based on the verification results, includes: One or more candidate solutions are selected from the Pareto optimal solution set, wherein each candidate solution corresponds to a set of optimization variable values; For the optimization variable values of each candidate scheme, CFD simulations were performed to obtain the verification hydrodynamic performance indicators corresponding to each candidate scheme. The prediction error is calculated based on the verified hydrodynamic performance indicators and the predicted values of the hydrodynamic performance indicators for each candidate scheme. The target scheme is determined based on the prediction error.
6. A multi-objective hydrodynamic optimization device for a hydrofoil, characterized in that, include: The system includes a variable determination module, a simulation module, a model acquisition module, an optimization module, and a scheme verification module. The variable determination module is used to obtain an optimization variable set, which includes several optimization variables and their corresponding feasible value ranges. The simulation module is used to collect several sample data within the value range of the optimization variable, and perform CFD simulation on the preset hydrofoil parameter model based on the sample data to generate training data. The model acquisition module is used to acquire a preset proxy model, which is trained based on a preset deep neural network and the training data. The optimization module is used to perform optimization iterations based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, it predicts the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm based on the surrogate model, and drives the multi-objective optimization algorithm to perform population evolution based on the prediction results until the preset termination condition is met, and outputs the Pareto optimal solution set. The scheme verification module is used to verify each scheme in the Pareto optimal solution set based on a preset CFD simulation, and to determine the target scheme based on the verification results.
7. The hydrofoil multi-objective hydrodynamic optimization device as described in claim 6, characterized in that, The simulation module is used to collect several sample data points within the value range of the optimization variables, perform CFD simulation on a preset hydrofoil parameter model based on the sample data, and generate training data, including: Based on the preset Latin hypercube sampling system, several sets of sample data are uniformly extracted within the range of the value of the optimization variable. Each set of sample data includes the specific values of the front hydrofoil sweep angle, hydrofoil span and hydrofoil spacing. Based on the sample data of each group, the corresponding hydrofoil parameter models are constructed respectively, and the hydrofoil parameter models are meshed to obtain the computational mesh model of each hydrofoil parameter model. CFD simulations were performed on each computational grid model to obtain the hydrodynamic performance indicators of the hydrofoil corresponding to each set of sample data. Training data is obtained by associating each group of sample data with the corresponding hydrodynamic performance indicators.
8. The hydrofoil multi-objective hydrodynamic optimization device as described in claim 7, characterized in that, The model acquisition module is used to acquire a preset proxy model, including: An initial proxy model is constructed based on a preset deep neural network, wherein the number of input layer nodes of the initial proxy model is the same as the number of optimization variables, and the number of output layer nodes is the same as the number of hydrodynamic performance indicators. The deep neural network is trained based on the training data and a preset backpropagation algorithm until the preset conditions are met, and then a proxy model is output.
9. The hydrofoil multi-objective hydrodynamic optimization device as described in claim 8, characterized in that, The optimization module is used to perform optimization iterations based on a preset multi-objective optimization algorithm and a preset surrogate model. During the optimization iteration process, it predicts the hydrodynamic performance indicators of individuals in the population constructed by the multi-objective optimization algorithm based on the surrogate model, and drives the multi-objective optimization algorithm to perform population evolution based on the prediction results until a preset termination condition is met, outputting a Pareto optimal solution set, including: Initialize the population of the multi-objective red-billed blue finches optimization algorithm, and encode each individual in the population as a set of optimization variable values; The optimization variable values of each individual are input into the surrogate model, and the predicted values of the hydrodynamic performance index of each individual are output based on the surrogate model. The predicted values are used as the fitness of the individual. Pareto non-dominated individuals in the population are determined by non-dominated ranking based on the fitness of all individuals. Population evolution is performed based on the Pareto non-dominated individuals until a preset termination condition is met, and the Pareto optimal solution set is output.
10. The hydrofoil multi-objective hydrodynamic optimization device as described in claim 9, characterized in that, The scheme verification module is used to verify each scheme in the Pareto optimal solution set based on a preset CFD simulation, and to determine the target scheme based on the verification results, including: One or more candidate solutions are selected from the Pareto optimal solution set, wherein each candidate solution corresponds to a set of optimization variable values; For the optimization variable values of each candidate scheme, CFD simulations were performed to obtain the verification hydrodynamic performance indicators corresponding to each candidate scheme. The prediction error is calculated based on the verified hydrodynamic performance indicators and the predicted values of the hydrodynamic performance indicators for each candidate scheme. The target scheme is determined based on the prediction error.