A method for optimizing aerodynamic shape of flat steel box girder wind fairing considering initial disturbance
By combining deep neural networks and the Northern Eagle optimization algorithm with CFD simulation, the influence of initial disturbances on the shape of the wind nozzle in the design of long-span bridges was solved, achieving high-precision flutter performance prediction and global optimization, thus improving the wind resistance of the bridge.
Patent Information
- Application Number
- CN202611058286.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-07-16
- Publication Date
- 2026-11-06
AI Technical Summary
Existing technologies struggle to account for the impact of initial disturbances on the shape of the wind turbine in the design of long-span bridges, leading to inaccurate flutter performance predictions. Furthermore, traditional methods neglect the complex mapping relationship between nonlinear flutter characteristics and initial disturbances.
By employing a deep neural network surrogate model combined with the Northern Eagle optimization algorithm, and through CFD fluid-structure interaction numerical simulation and optimal Latin hypercube sampling, an initial disturbance is precisely applied to establish a high-precision nonlinear mapping relationship between the nozzle geometric parameters and the flutter divergence wind speed, thereby achieving global optimization design.
It improves the accuracy and robustness of flutter performance prediction, significantly enhances the wind resistance of long-span bridges, and has a high degree of automation in the optimization process, overcoming the limitations of traditional methods.
Smart Images
Figure CN123049697A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of wind-resistant bridge design, and in particular to a method for optimizing the aerodynamic shape of a flat steel box girder wind nozzle considering initial disturbance. Background Technology
[0002] Flutter is a core threat to the safety of long-span bridges. With the completion of the Akashi Kaikyo Bridge in Japan (main span 1991m) and the Çanakkale Bridge in Turkey (main span 2023m), the span of suspension bridges has exceeded 2000m. The Zhangjinggao Yangtze River Bridge and the Shiziyang Bridge in China are currently challenging the main span records of 2300m and 2180m, respectively. Research shows that flutter stability has become a key factor limiting further increases in span. Among existing flutter control strategies, aerodynamic measures, by improving the flow characteristics around the main girder cross-section, offer the advantages of both significant effectiveness and low cost. The air nozzle, as a key aerodynamic component at the leading edge of the main girder, directly affects the aerodynamic performance of the main girder through its geometric parameters.
[0003] Due to strong aerodynamic nonlinearity, subcritical bifurcation may occur in bridge cross-sections, and the vibration state is closely related to initial conditions. Wind tunnel tests on rectangular cross-sections, truss beams, and streamlined box girders have all confirmed that initial disturbances can induce flutter at relatively low wind speeds, and the critical flutter wind speed decreases as the initial amplitude increases. During service, bridges face numerous sources of disturbance, including vortex-induced vibration, strong turbulent buffeting, ship collisions, waves, ocean currents, and earthquakes. Aerodynamic interference between adjacent main girders can also induce large-amplitude vibrations. The initial amplitudes generated by these disturbances can weaken the flutter performance of the main girders. Ignoring the influence of initial disturbances in the design will overestimate the actual wind resistance of the bridge, posing a potential risk to structural safety. Therefore, considering the effect of initial disturbances in the wind-resistant design of long-span bridges is of great significance.
[0004] However, achieving a reasonable design for the nozzle shape under initial disturbance remains challenging. On one hand, there is a lack of precise means to apply the initial disturbance; manual excitation in wind tunnel tests can only obtain approximate initial amplitudes, and CFD simulation frameworks have not yet achieved precise application of the initial disturbance. On the other hand, existing aerodynamic optimization frameworks are generally based on linear flutter theory under the assumption of small amplitude, neglecting both the nonlinear flutter characteristics of the main beam and the influence of initial disturbance on flutter performance. Furthermore, the traditional surrogate models used have limited accuracy in handling the complex mapping relationship between nozzle geometry and nonlinear flutter performance. Summary of the Invention
[0005] Therefore, the purpose of this invention is to provide a method for optimizing the aerodynamic shape of a flat steel box girder nozzle that takes into account initial disturbances, which can effectively consider the influence of initial disturbances on flutter performance and achieve global optimization design of the nozzle shape.
[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:
[0007] A method for optimizing the aerodynamic shape of a flat steel box girder wind nozzle considering initial disturbance, comprising the following:
[0008] The geometric design parameters of the flat steel box girder wind nozzle are selected as optimization variables, and their upper and lower bounds constitute constraints. The optimization objective is to maximize the flutter divergence wind speed under the target initial disturbance condition.
[0009] An aerodynamic shape optimization model for a flat steel box girder nozzle is established and trained to obtain a trained aerodynamic shape optimization model for a flat steel box girder nozzle. The trained aerodynamic shape optimization model for a flat steel box girder nozzle takes samples within the range of geometric design parameters of the flat steel box girder nozzle as input and takes the optimal geometric design parameters of the flat steel box girder nozzle that maximize the flutter divergence wind speed under the target initial disturbance condition as output.
[0010] Within the range of the constraints, n sets of geometric design parameters for flat steel box girder nozzles are randomly generated and input into the trained flat steel box girder nozzle aerodynamic shape optimization model to obtain the geometric design parameters for flat steel box girder nozzles that maximize the flutter divergence wind speed under the target initial disturbance condition as the optimal nozzle shape.
[0011] Furthermore, the geometric design parameters of the flat steel box girder air nozzle include the horizontal coordinate x1, the vertical coordinate x2, and the platform length x3 of the air nozzle vertex. The origin of the three coordinates is the intersection of the centroidal axis of the main beam and the vertical joint surface of the air nozzle. The three parameters are independent of each other and together represent the shape of the air nozzle.
[0012] Furthermore, in step 1, the three geometric parameters are used as optimization variables, and the objective function is to maximize the flutter divergence wind speed under the target disturbance condition:
[0013]
[0014] The constraints are:
[0015]
[0016] In the formula, U cr The flutter divergence wind speed under the target disturbance condition; The nonlinear mapping relationship represented by the surrogate model; and These are the upper and lower bounds of the range of values for each optimization variable.
[0017] Furthermore, the training process of the aerodynamic shape optimization model for the flat steel box girder nozzle is as follows:
[0018] A deep neural network surrogate model is obtained and trained. A mapping relationship is established between the optimization variables and the flutter divergence wind speed under the target initial disturbance condition. The trained deep neural network surrogate model is obtained. The trained deep neural network surrogate model is embedded into the Northern Eagle optimization algorithm to obtain the trained flat steel box girder aerodynamic shape optimization model of the nozzle.
[0019] Furthermore, the training process of the deep neural network proxy model is as follows:
[0020] The optimal Latin hypercube sampling method is used to generate n sets of optimization variables as a sample set under the constraints; where n is a natural number greater than or equal to 1.
[0021] A fluid-structure interaction numerical simulation framework based on computational fluid dynamics is constructed to apply initial torsional excitation. This framework is used to obtain the structural aerodynamic response of different nozzle shapes in the sample set under the initial disturbance of the target, and to obtain the corresponding real flutter divergence wind speed.
[0022] The optimization variables and corresponding real flutter divergence wind speeds in the sample set are used as labels and input into the deep neural network surrogate model to construct a multi-layer neural network and establish a mapping relationship between the optimization variables and the flutter performance under the target disturbance; the mean square error loss function is used in the training process.
[0023] Furthermore, the specific process by which the trained flat steel box girder aerodynamic shape optimization model obtains the optimal nozzle shape is as follows:
[0024] The geometric design parameters of n sets of flat steel box girder wind nozzles are defined as the initial population of the Northern Eagle optimization algorithm, and the algorithm termination criterion is set.
[0025] The trained deep neural network proxy model is invoked to obtain the flutter divergence wind speed corresponding to each individual in the initial population, which is used as the fitness value; the optimal position of the individual and the global optimal position are updated according to the fitness value, and the search is iterated accordingly;
[0026] If the preset termination criterion is met, the optimized variable corresponding to the globally optimal individual is output, which is the optimal nozzle shape; otherwise, return to the previous steps to continue iterating until the optimal nozzle shape is output.
[0027] Furthermore, the fluid-structure interaction numerical simulation framework obtains the real flutter divergence wind speed through CFD free vibration simulation, and its judgment criterion is: the lowest wind speed corresponding to the system aerodynamic response changing from a convergent state or a constant amplitude oscillation state to a state with continuously increasing amplitude.
[0028] The initial torsional excitation is applied as follows: within a very short time after the calculation starts, a user-defined function is used to assign a preset velocity condition to the main beam, which is then released immediately after reaching the specified initial torsional amplitude, triggering free vibration.
[0029] Furthermore, the deep neural network surrogate model establishes a high-precision nonlinear mapping relationship between the optimization variables and the flutter divergence wind speed under the target disturbance condition through nonlinear transformation of multiple hidden layers.
[0030] Furthermore, the loss function of the deep neural network surrogate model is the mean squared error:
[0031]
[0032] Where N is the number of training samples; y i Let be the true value of the flutter divergence wind speed for the i-th sample; Let be the predicted flutter divergence wind speed for the i-th sample.
[0033] Furthermore, a Dropout layer is added after each hidden layer of the deep neural network agent model to prevent overfitting; a segmented decay strategy is adopted for the learning rate during training.
[0034] The present invention has the following beneficial effects: (1) It can consider the nonlinear flutter characteristics of the main beam. The aerodynamic response time history under different nozzle shapes is obtained by CFD free vibration simulation. The wind speed corresponding to the change of the response from convergence or constant amplitude oscillation to continuous divergence is taken as the flutter divergence wind speed, which overcomes the limitation of the traditional method based on linear flutter theory. (2) It can consider the influence of initial disturbance on flutter performance. By using a user-defined function in CFD to accurately apply the initial torsional excitation, the real flutter response under disturbance conditions is obtained, which solves the problem that the traditional method is difficult to consider the effect of initial disturbance. (3) The surrogate model has high prediction accuracy. The deep neural network is used to accurately establish the mapping relationship between the geometric parameters of the nozzle and the flutter divergence wind speed under disturbance conditions through multi-layer nonlinear transformation. (4) The optimization results are robust. Combined with the global optimization capability of the Northern Eagle optimization algorithm, the optimized section can maintain a significant improvement in flutter performance under different initial disturbance conditions. (5) It has a high degree of automation. By employing optimal Latin hypercube sampling, CFD automatic calculation, deep neural network automatic learning, and swarm intelligence algorithm automatic optimization, the entire process is highly automated, significantly improving optimization efficiency. Attached Figure Description
[0035] Figure 1 Flowchart for optimizing the nozzle shape considering initial disturbances;
[0036] Figure 2 A parameterized schematic diagram of the streamlined box girder cross-section and the shape of the air nozzle;
[0037] Figure 3 This is a schematic diagram of a two-degree-of-freedom aeroelastic model;
[0038] Figure 4A schematic diagram showing the setup of the computational domain and boundary conditions;
[0039] Figure 5 This is a schematic diagram illustrating the watershed grid division in a specific embodiment;
[0040] Figure 6 This is a schematic diagram of a DNN network structure;
[0041] Figure 7 This is a comparison chart of the predicted values from the DNN surrogate model and the calculated values from the CFD in a specific embodiment;
[0042] Figure 8 Flowchart for the NGO algorithm to search for the optimal nozzle shape;
[0043] Figure 9 This is a comparison diagram of the original cross-section and the optimized cross-section in a specific embodiment. Detailed Implementation
[0044] The present invention will now be described in further detail with reference to the accompanying drawings and specific preferred embodiments.
[0045] In the description of this invention, it should be understood that the terms "left side," "right side," "upper part," "lower part," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. "First," "second," etc., do not indicate the importance of the components, and therefore should not be construed as a limitation of this invention. The specific dimensions used in this embodiment are only for illustrating the technical solution and do not limit the scope of protection of this invention.
[0046] like Figure 1 As shown, this embodiment provides a method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance, including the following steps:
[0047] Step 1: Determine the optimization objective and constraints
[0048] The geometric design parameters of the air nozzles for flat steel box girders are selected as optimization variables. Specifically, such as... Figure 2 As shown, with the intersection of the main beam's centroidal axis and the vertical joint surface of the nozzle as the coordinate origin, the nozzle's shape is characterized by three independent geometric parameters: the horizontal coordinate x1 of the nozzle's vertex, the vertical coordinate x2, and the platform length x3. These three parameters are not coupled. In this embodiment, the feasible regions for the three parameters are x1∈[0, 130] mm, x2∈[-20, 20] mm, and x3∈[0, 40] mm, respectively.
[0049] The optimization objective is to maximize the flutter divergence wind speed under the target disturbance condition. In this embodiment, an 8° torsion is selected as the target disturbance condition. The objective function is:
[0050]
[0051] The constraints are:
[0052]
[0053] Among them, U cr For flutter dispersion wind speed; This represents the nonlinear mapping relationship characteristic of the surrogate model.
[0054] Step 2: Generate sample points
[0055] The optimal Latin hypercube sampling method is used to generate sample points within the design space. Stratified sampling and distance optimization ensure uniform filling of the design space by the samples. In this embodiment, the number of training samples is 100 and the number of test samples is 10.
[0056] Step 3: Establish a numerical model for aerodynamic evaluation
[0057] A numerical simulation framework for fluid-structure interaction can be constructed based on computational fluid dynamics, allowing for the application of initial torsional excitation. For example... Figure 3 As shown, the motion of the main beam is described by a two-degree-of-freedom aeroelastic model, and its equations of motion are:
[0058]
[0059]
[0060] In the formula, m and I represent the equivalent mass and moment of inertia per unit length, respectively; h and α represent the vertical and torsional displacements, respectively; F L (t) and M L (t) represents the aerodynamic lift and torque, respectively. Damping coefficient c h c α and stiffness coefficient k h k α The equations of motion are determined by the system's dynamic characteristics. The structural equations of motion are solved using the Newmark-β method.
[0061] The fluid domain is described by the incompressible, unsteady Reynolds-averaged Navier-Stokes equations. The SSTk-ω model is used as the turbulence model, and the pressure-velocity coupling is performed using the SIMPLEC algorithm. Momentum, turbulent kinetic energy, and dissipation rate are all discretized using a second-order upwind scheme. The computational domain and boundary conditions are set as follows: Figure 4 As shown, the computational domain dimensions are: center distance from inlet 10B, distance from outlet 20B, height 10B, where B is the width of the main beam. The watershed mesh is as follows: Figure 5 As shown, the mesh around the main beam is refined, and 35 boundary layers are laid to ensure the wall surface y + <1, the total number of grids is approximately 43,000, and the transient calculation step size is set to 0.001s.
[0062] The initial torsional excitation is applied as follows: within a very short time after the calculation starts, a user-defined function (UDF) is used to assign a preset velocity condition to the main beam, causing it to reach a specified initial torsional amplitude and then immediately release it, triggering free vibration. In this embodiment, the applied initial torsional amplitude is 8°.
[0063] The criterion for determining flutter divergence wind speed is the lowest wind speed corresponding to the change in system aerodynamic response from a convergent state or a constant amplitude oscillation state to a state of continuously increasing amplitude.
[0064] Based on the above CFD numerical simulation method, aeroelastic analysis is performed on the shape of the nozzle corresponding to each sample point generated in step 2 to obtain its flutter divergence wind speed under an initial disturbance of 8°, thus forming a training sample set.
[0065] Step 4: Train the deep neural network proxy model
[0066] Using three optimization variables x1, x2, and x3 as inputs, and the flutter divergence wind speed U under an initial perturbation of 8°, cr To output the model, construct a deep neural network proxy model. For example... Figure 6 As shown, the network input layer contains 3 nodes, the output layer contains 1 node, and there are 5 hidden layers, each containing 80 neurons. A Dropout layer with a dropout rate of 0.1 is added after each hidden layer to prevent overfitting. Training uses the Adam optimizer with a maximum of 500 iterations and an initial learning rate of 1×10⁻⁶. -3 A segmented decay strategy is implemented during training.
[0067] Deep neural networks establish a high-precision nonlinear mapping relationship between optimization variables and flutter divergent wind speeds under target disturbance conditions through nonlinear transformations across multiple hidden layers. The loss function for network training is the mean squared error.
[0068]
[0069] In the formula, N is the number of training samples; y i Let be the true value of the flutter divergence wind speed for the i-th sample; Let be the predicted flutter divergence wind speed for the i-th sample.
[0070] To verify the prediction accuracy of the deep neural network surrogate model, the DNN predictions and CFD calculations of the test set samples were compared. The results are as follows: Figure 7 As shown, the two match well.
[0071] Step 5: Embed the trained surrogate model into the Northern Eagle optimization algorithm for global optimization.
[0072] The trained deep neural network surrogate model was used as the fitness function and embedded into the Northern Eagle Optimization Algorithm. The optimization objective was to maximize the flutter divergence wind speed under an 8° perturbation. The algorithm population size was set to 50, and the maximum number of iterations was set to 50. The optimization process of the NGO algorithm is as follows: Figure 8 As shown, the specific steps are as follows:
[0073] Step 5-1: Randomly distribute the initial population individuals within the design space and set the algorithm termination criteria;
[0074] Step 5-2: The Northern Eagle optimization algorithm calls the proxy model to predict the flutter divergence wind speed corresponding to each individual in the population, and uses it as the fitness value.
[0075] Step 5-3: Update the individual optimal position and the global optimal position based on the fitness value, and iterate the search accordingly;
[0076] Step 5-4: If the preset termination criterion is met, output the optimization variable corresponding to the globally optimal individual, which is the optimal nozzle shape; otherwise, return to step 5-2 to continue iterating.
[0077] To verify the global optimality of the optimization results, the DBO algorithm, WOA algorithm and GWO algorithm were used for cross-validation. All algorithms converged to the same optimal solution.
[0078] Step 6: Verify the flutter performance of the optimized section.
[0079] Substitute the optimal nozzle shape obtained in step 5 into the aerodynamic evaluation numerical model in step 3, and calculate its flutter divergence wind speed under initial disturbance conditions of 0°, 4° and 8°, respectively, to verify the flutter performance of the optimized section under initial disturbance.
[0080] In this embodiment, the cross-sections before and after optimization are as follows: Figure 9 As shown, after optimization, the nozzle angle was reduced from 77° to 45°, the lower web inclination angle was reduced from 11° to 5°, and the overall cross-section became more streamlined. The optimized cross-section showed an approximately 10% increase in flutter divergence wind speed under initial disturbance conditions of 0°, 4°, and 8°, verifying the optimization effect and robustness.
[0081] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.
Claims
1. A method for aerodynamic shape optimization of a flat steel box girder nose considering initial disturbance, characterized in that: Includes the following: The geometric design parameters of the flat steel box girder air nozzle are selected as optimization variables, and their upper and lower bounds constitute the constraints. The optimization objective is to maximize the flutter divergence wind speed under the initial disturbance condition. An aerodynamic shape optimization model for a flat steel box girder nozzle is established and trained to obtain a trained aerodynamic shape optimization model for a flat steel box girder nozzle. The trained aerodynamic shape optimization model for a flat steel box girder nozzle takes samples within the range of geometric design parameters of the flat steel box girder nozzle as input and takes the optimal geometric design parameters of the flat steel box girder nozzle that maximize the flutter divergence wind speed under the target initial disturbance condition as output. Within the range of the constraints, n sets of geometric design parameters for flat steel box girder nozzles are randomly generated and input into the trained flat steel box girder nozzle aerodynamic shape optimization model to obtain the geometric design parameters for flat steel box girder nozzles that maximize the flutter divergence wind speed under the target initial disturbance condition as the optimal nozzle shape.
2. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 1, characterized in that: The geometric design parameters of the flat steel box girder air nozzle include the horizontal coordinate x1, the vertical coordinate x2, and the platform length x3 of the air nozzle vertex. The origin of the three coordinates is the intersection of the centroidal axis of the main beam and the vertical joint surface of the air nozzle. The three parameters are independent of each other and together characterize the shape of the air nozzle.
3. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 2, characterized in that: In step 1, the three geometric parameters are used as optimization variables, and the objective function is to maximize the flutter divergence wind speed under the target disturbance condition. ； The constraints are: ； In the formula, U cr is the flutter divergence wind speed under the target perturbation condition; is a nonlinear mapping relationship represented by the surrogate model; and are the upper and lower bounds of the value range of each optimization variable, respectively.
4. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 1, characterized in that: The training process of the aerodynamic shape optimization model for the flat steel box girder nozzle is as follows: A deep neural network surrogate model is obtained and trained. A mapping relationship is established between the optimization variables and the flutter divergence wind speed under the target initial disturbance condition. The trained deep neural network surrogate model is obtained. The trained deep neural network surrogate model is embedded into the Northern Eagle optimization algorithm to obtain the trained flat steel box girder aerodynamic shape optimization model of the nozzle.
5. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 4, characterized in that: The training process of the deep neural network proxy model is as follows: The optimal Latin hypercube sampling method is used to generate n sets of optimization variables as a sample set under the constraints; where n is a natural number greater than or equal to 1. A fluid-structure interaction numerical simulation framework based on computational fluid dynamics is constructed to apply initial torsional excitation. This framework is used to obtain the structural aerodynamic response of different nozzle shapes in the sample set under the initial disturbance of the target, and to obtain the corresponding real flutter divergence wind speed. The optimization variables and the corresponding real flutter divergence wind speeds in the sample set are used as labels and input into the deep neural network proxy model to construct a multi-layer neural network and establish a mapping relationship between the optimization variables and the flutter performance under the target disturbance. The training process uses the mean squared error loss function.
6. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 5, characterized in that: The specific process of obtaining the optimal nozzle shape from the trained flat steel box girder aerodynamic shape optimization model is as follows: The geometric design parameters of n sets of flat steel box girder wind nozzles are defined as the initial population of the Northern Eagle optimization algorithm, and the algorithm termination criterion is set. The trained deep neural network proxy model is invoked to obtain the flutter divergence wind speed corresponding to each individual in the initial population, which is used as the fitness value; the optimal position of the individual and the global optimal position are updated according to the fitness value, and the search is iterated accordingly; If the preset termination criterion is met, the optimized variable corresponding to the globally optimal individual is output, which is the optimal nozzle shape; otherwise, return to the previous steps to continue iterating until the optimal nozzle shape is output.
7. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 1, characterized in that: The fluid-structure interaction numerical simulation framework obtains the real flutter divergence wind speed through CFD free vibration simulation. The criterion is: the lowest wind speed corresponding to the system aerodynamic response changing from a convergent state or a constant amplitude oscillation state to a state with continuously increasing amplitude. The initial torsional excitation is applied as follows: within a very short time after the calculation starts, a user-defined function is used to assign a preset velocity condition to the main beam, which is then released immediately after reaching the specified initial torsional amplitude, triggering free vibration.
8. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 5, characterized in that: The deep neural network surrogate model establishes a high-precision nonlinear mapping relationship between the optimization variables and the flutter divergence wind speed under the target disturbance condition through nonlinear transformation of multiple hidden layers.
9. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 5, characterized in that: The loss function of the deep neural network surrogate model is the mean squared error: ； wherein N is the number of training samples; y i is the real value of the flutter divergence wind speed of the i-th sample; is the predicted value of the flutter divergence wind speed of the i-th sample.
10. The method for optimizing the aerodynamic shape of a flat steel box girder nozzle considering initial disturbance as described in claim 5, characterized in that: A Dropout layer is added after each hidden layer of the deep neural network surrogate model to prevent overfitting; a segmented decay strategy is used for the learning rate during training.