Optimization Design of Fairings for VIV Suppression Based on Data-Driven Models and Genetic Algorithm

Vortex induced vibration (VIV) is a challenge in ocean engineering. Several devices including fairings have been designed to suppress VIV. However, how to optimize the design of suppression devices is still a problem to be solved. In this paper, an optimization design methodology is presented based on data-driven models and genetic algorithm (GA). Data-driven models are introduced to substitute complex physics-based equations. GA is used to rapidly search for the optimal suppression device from all possible solutions. Taking fairings as example, VIV response database for different fairings is established based on parameterized models in which model sections of fairings are controlled by several control points and Bezier curves. Then a data-driven model, which can predict the VIV response of fairings with different sections accurately and efficiently, is trained through BP neural network. Finally, a comprehensive optimization method and process is proposed based on GA and the data-driven model. The proposed method is demonstrated by its application to a case. It turns out that the proposed method can perform the optimization design of fairings effectively. VIV can be reduced obviously through the optimization design.


Introduction
Many structures are used in areas of marine currents and winds. When a fluid flows around structures, flow separates from structures and vortices form. Unsteady loads due to the flow separation are exerted on structures and cause vibration. This is called vortex-induced vibration (VIV). VIV is a challenge in many ocean engineering areas, such as risers, pipelines, hydrodynamic energy harvesters and other hydrodynamic applications (Li et al., 2017;Xu et al., 2017;Liu et al., 2020). Circular cylinders are the most frequently used structures in the above fields. If no suppression devices are taken application, cylinders are likely to be destroyed when suffering continuous VIV. Therefore, many VIV suppression devices have been proposed to reduce VIV, such as rigid splitter plates, fairings and helical strakes. These VIV suppression devices have been introduced and discussed by researchers (Rashidi et al., 2016;Wang et al., 2020). Every VIV suppression device is no longer to be reviewed in detail in this paper.
Several types of VIV suppression devices have been developed in the previous research works. However, it appears that the literature lacks of an optimization design method for VIV suppression devices. The main reason is that complicated fluid field model problems need to be solved repeatedly in an optimization process. Models for analyzing VIV can be grouped into two categories, physicsbased models and data-driven models (Bourdeau et al., 2019;Tang and Zhang, 2019). In physics-based models, VIV is often modeled based on the computational fluid dynamics (CFD) due to its clear explanation of VIV. Yet, the direct implementation of CFD for the optimization design of VIV suppression devices is difficult due to the high compu-tational cost. Data-driven models have been vigorously discussed in past years due to their relatively low computational costs. Besides, data-driven models can simulate accurately the complex relationships between input and output variables without a physical model (Zendehboudi et al., 2019). Consequently, data-driven models have been widely applied in various fields, such as modeling of industrial processes and energy prediction of buildings (Sadati et al., 2018;Wang et al., 2018). In view of this, it may be feasible that data-driven models are used to substitute physics-based CFD models in the optimization design of VIV suppression devices.
Among suppression devices, fairings are frequently employed in engineering applications due to their low-drag performance during the recent two decades (Baarholm et al., 2015;Pontaza et al., 2012;Wang et al., 2020). Therefore, fairings are chosen as the research object. Problems that involve the optimization design of fairings for VIV suppression based on data-driven models are addressed in this study. The remainder of this paper is organized as follows. Section 2 describes the methodology for the optimization design of fairings based on data-driven models. Section 3 introduces parameterized CFD models and a database for model training. Section 4 presents a data-driven model and its training and verification. Section 5 introduces an optimization design for fairings based on data-driven models, and Section 6 provides the conclusion.

Methodology
The goal of the shape design of fairings is to minimize VIV. The optimization model can be written as follows: where X is the shape of fairings, f is the VIV model, g and h are inequality constraints and equality constraints, respectively.
The optimization problem shown in Eq. (1) can be further divided into two subproblems. The first one is to propose an optimization method searching for the optimal shape of fairings. The optimization problem, as shown in Eq. (1), can be solved by using enumeration and random methods. Genetic algorithm (GA) serves as a random method of searching the optimal solution by simulating the natural evolutionary process. GA has been proven to be high in efficiency and accuracy and applied in various optimization problems (Liu et al., 2018). Therefore, GA is chosen for the optimization design of fairings. The second one is to establish the VIV model. Physics-based methods such as CFD are commonly used in developing models for predicting VIV based on physical equations. However, physical equations such as Navier−Stokes (N−S) equations in physicsbased models need to be solved repeatedly in the optimiza-tion design of fairings while the solving of N−S equations is difficult and time-consuming. Therefore, data-driven models with low computational cost are suitable for the optimization design of fairings. A set of observed data need to be prepared before the training of data-driven model. The database and the training of data-driven model will be introduced in detail in Sections 3 and 4, respectively.

CFD model and database
3.1 CFD model A CFD model for VIV simulation of a circular cylinder is established in COMSOL Multiphysics, as shown in Fig. 1. Parameters in the present study are the same as those in the numerical experiment of Tang et al. (2013), as listed in Table 1. The element number is set as 17 856 to keep the CFD model mesh independence. The calculated VIV response in the transverse direction based on the CFD model are compared with the numerical simulation results of Tang et al. (2013), as shown in Fig. 2. The results show that the CFD simulation results are in good agreement with those in the previous literature.  3.2 Database Traditional drop-shaped fairings generally have a tail by simply converging two shaped sides of a fairing at a point on the centerline of the fairing (Liang et al., 2018;Wang et al., 2015), as shown in Fig. 3 Fig. 3. In this study, the detailed coordinate variables are listed in Table 2. By combining different values of v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 and v 8 , 288 parameterized models of fairings can be generated. Variables A cylinder with the circular diameter of 0.02 m is chosen for CFD analysis and database generation. The fluid density is 1 000 kg/m 3 , the fluid viscosity is 0.001 Pa·s, the Reynolds number is 800, the coefficient of spring stiffness is 5.16 N/m, and the damping coefficient is 0.001 N·s/m. The structural natural frequency of a circular cylinder with different sizes of fairings is assumed to be the same to focus on the influence of fairing shape on the VIV response (Lou et al., 2016). The element number of the fairing CFD model is set to 22 490 to keep mesh independence. 288 parameterized models of fairings are generated according to Table 2 and analyzed in COMSOL. The final established database is composed of different sets of control points and their corresponding VIV amplitude ratios.

Data-driven model
Artificial neural networks (ANN) modeling inspired from neural networks in the human brain is one of the most applied data-driven methods (Bourdeau et al., 2019). A well-known training approach in ANN is the so-called backpropagation (BP) neural network (Sun and Gao, 2019). It does not require users to input accurate mathematical models, has strong ability to deal with nonlinear problems, and has strong fitting ability and stable performance. Therefore, BP neural network is chosen for the training of data-driven models. A basic form of BP neural network is composed of three layers, as shown in Fig. 4. The input layer is used to get prediction from input data while the output layer is set for giving the final results. The hidden layer whose number and structure can be modified depending on the needs of the modeling is to bridge between inputs and outputs (Bourdeau et al., 2019).
For this study, the inputs are coordinates of control points v 1 −v 8 , and the output is VIV amplitude ratio. Therefore, there are 8 neurons in the input layer, 1 neuron in the output layer, as shown in Fig. 4. The training number is 500, the learning rate is 0.01, and the convergence error is 0.0001. Other key parameters such as the number of hidden layers and the neuron number of each hidden layer should  be analyzed further to determine a proper value. The database established in Section 3.2 are used to train the BP neural network. Before the training, the database is randomly divided into training data and test data according to the ratio of 19:1. The training data are used to train the model based on BP neural network while the test data are set to verify the accuracy of the data-driven model.

Key parameters
The number of hidden layers and neurons in each hidden layer still needs to be determined. Nevertheless, there is no proven method for selecting the numbers and sizes of hidden layers. It is a trial-and-error process to determine the optimum structure of BP neural network (Wong and Kim, 2018;Zendehboudi et al., 2019). In this section, BP neural networks with 1 hidden layer, 2 hidden layers and 3 hidden layers were established respectively and compared with each other. The influence of the number of neurons in each hidden layer is also studied. Each model is trained through training database and then verified by test database to determine the calculation accuracy of each training model. The mean deviation and standard deviation for each datadriven model are listed in Table 4. It turns out that the optimal number of hidden layer is 1 and the optimal number of neurons in the hidden layer is 34. Under this condition, the prediction average deviation and standard deviation of amplitude ratio are 5.71% and 7.86%, respectively. Furthermore, it only needs less than one second to calculate the VIV response of a cylinder with fairing. Therefore, the datadriven model based on BP neural network can be used for the following VIV prediction and optimization design of fairings. 5 Optimization design 5.1 Optimization process As mentioned previously, GA is selected for the optimization design of fairings because of its efficient global search ability. A data-driven model is introduced to simulate the VIV response. A comprehensive optimization process solving the optimization problem shown in Eq. (1) is further proposed based on GA and the data-driven model, as shown in Fig. 5.
The optimization process is divided into five parts, namely, encoding module, GA module, filtering module, VIV module, and decoding module, as shown in Fig. 5. The encoding module is used to transfer the shape of fairings to genetic strings while the decoding module is set for transfer-ring generic strings back to the shape of fairings. The GA module is designed for GA calculations, including the generation, reproduction, crossover, and mutation of fairings encoded in genetic strings. The filtering module is designed for filtering possible solutions according to constraints. The VIV module of data-driven model is designed for calculating the fitness value during the optimization process. In actual optimization process, fairings are firstly represented as genetic strings to perform subsequent configuration evolution based on GA. Then the genetic strings go through the GA module, filtering module and VIV module cyclically until the fitness value satisfies termination criteria. Finally, the optimal generic string is transferred back to the actual fairing which is the optimal fairing. An optimization program based on MATLAB is developed to conduct the optimization design of fairings. The optimal fairing is a finlike fairing, as shown in Fig. 6.

Optimization verification
An optimal fin-like faring has been obtained through the optimization design. The VIV suppression effect of the optimal fairing needs to be further verified. Therefore, three CFD models including models of a bare cylinder, a cylinder with traditional drop-shape fairing and a cylinder with optimal fin-like fairing are analyzed and compared with each other, as shown in Figs. 7−9. Compared with the bare cylinder, cylinders with fairings can extend the shear layer from the cylinder body to farther into the wake, as shown in Figs. 7a, 8a and 9a. The extension of the shear layer delays the formation of vorticities and decreases the magnitude of vorticities. Furthermore, the optimal fin-like fairing can suppress the VIV more effectively compared with the traditional drop-shape fairing, as shown in Figs. 7b, 8b and 9b. The detailed VIV amplitude ratios are listed in Table 5. Compared with VIV amplitude ratios of the bare cylinder and the cylinder with traditional drop-shape fairing, the amplitude ratio of the cylinder with optimal fin-like faring decreases by 87% and 80.4%, respectively. Besides, the drag coefficient of the cylinder with optimal fin-like faring also decreases obviously compared with drag coefficients of two other cylinders. In summary, the optimal fin-like fairing obtained based on the proposed optimization design method can reduce the VIV response of a cylinder effectively.

Conclusions
In this paper, a challenging problem has been addressed, i.e. the optimization design of fairings for VIV suppression.   LIU Xiu-quan et al. China Ocean Eng., 2021, Vol. 35, No. 1, P. 153-158 Several VIV suppression devices including fairings have been developed to reduce VIV. However, an optimization design method for VIV suppression devices has not been proposed. Fairings are chosen as the research object in this paper. An effective method is presented to solve the optimization design problem of fairings. The contributions of this approach for the optimization design of fairings are summarized as follows: A data-driven model substituting complicated physicsbased CFD models is introduced to predict VIV response. A parameterized CFD model is firstly proposed to represent fairings with different shapes based on Bezier curves. A database including the relationship between fairings and their VIV performance is then established for model training. BP neural network is chosen for the training of the data-driven model which achieves low computational cost and good accuracy for predicting VIV response.
GA is selected for the optimization design of fairings because of its efficient global search ability. A comprehensive analysis algorithm for optimization design of fairings is proposed based on GA and the data-driven model. The relevant optimization program is also developed based on MATLAB. Then the optimization design of fairings can be conducted by taking the full advantage of the two methods.
The proposed method is applied for the optimization design of a fairing. The results show that the proposed approach can perform the optimization design of fairings for VIV suppression effectively. The optimal fin-like fairing has a high VIV suppression effect compared with the traditional drop-shaped faring. The methodology proposed in this paper provides a practical engineering method for the optimization design of fairings and can also be extended to the optimization design of other VIV suppression devices.