1. Introduction
Wind energy has been playing an increasingly important role lately regarding the successful transition from fossil fuels to renewable energy [
1]. Offshore wind energy has advantages such as a higher quality of wind resources, a larger suitable free area to develop [
2,
3], and a smaller influence on the environment (particularly wind turbine noise [
4,
5,
6]) compared with onshore wind energy. In the last few decades, a rapid growth in wind energy development has been witnessed throughout the world [
7,
8,
9]. As in the ocean environment, an offshore wind turbine (OWT) structure is always excited by various random excitations, such as wind, waves, and currents, which all pose a great threat to the structure. Hence, it is a challenge for designers and manufacturers to construct an offshore wind farm [
10,
11].
Traditionally, code-based designs such as IEC [
12], DNVGL [
13], and CCS [
14] utilize deterministic methods to predict the structural dynamic response. In this regard, the stochastic dynamic response of the OWT can be obtained by one or several deterministic loads or excitations. Moreover, the structural dynamic responses calculated by deterministic methods are normally faster than those by probabilistic (stochastic) methods. However, in many practical structures, a perfect deterministic behavior cannot be guaranteed, not only because of unpredictable excitation, but also because of various uncertainties in the structures. Therefore, dynamic response analysis was implemented in recent studies based on the assumption that some randomness is present in the excitation part [
15] and it is of importance for the normal operation and maintenance to ensure the safety of the OWT. To obtain a more accurate dynamic response of the OWT under external excitation, a probability approach can be used to calculate the probability density function (PDF) of the state variables and evaluate the reliability of the OWT.
For dynamic systems, the joint PDFs and joint moments of the state variables can be computed by probabilistic methods [
16]. Normally, it is hard to attain the precise joint response PDFs and the transient evolutionary PDF under stochastic excitation [
15]. To solve this problem, many approximate methods, such as equivalent linearization [
17], the perturbation method [
18], and stochastic averaging [
19], have been adopted and these replace the original nonlinear system in a probabilistic sense. The methods mentioned above simplify the problem, in which the important nonlinear features of systems were often neglected. Even though these methods can provide stationary or non-stationary response results with an acceptable level of accuracy, for more practical and complex problems, they fail to obtain an analytical solution [
20]. In addition, the Monte Carlo simulation (MCS) method is commonly regarded to obtain accurate results and can be used to verify the calculation results extracted from the other approximate methods. Nevertheless, when high dimensions and long-period simulations are encountered, the MCS seems to be impractical because of the large demand of the computational resources and efforts [
16]. The probabilistic properties of stochastic dynamic systems are governed by the Fokker–Planck (FP) equation when systems are excited by white noise or filtered white noise [
15]. Two research problems on the system’s variable distribution and the corresponding reliability evaluation can be solved based on an accurate joint response PDF, which can be obtained directly by solving the FP equation [
20]. Analytical solutions of the FP equation are only obtainable for some linear or limited nonlinear systems, while direct numerical solutions, e.g., via the finite element method [
21] and the finite different method [
22], suffer from the “curse of dimensionality.” As an alternative, the path integration (PI) method is assumed to be an effective numerical method for the accurate solution of the FP equation [
23]. For the PI method, which is based on an iterative method to compute the response PDFs for systems that satisfy Markov properties, the response PDFs are computed by means of a step-by-step solution technique according to the total probability law. Due to the advantage of the PI method in stochastic dynamic analysis, it has already been effectively adjusted for Markov processes to obtain response PDF and system reliability [
24,
25].
In recent years, many studies have been conducted on the stochastic dynamic response and the reliability of low-dimensional systems with linear or nonlinear restoring forces or damping in order to obtain 2D to 6D state spaces. Alevras and Yurchenko [
20] applied the PI approach to analyze high-dimensional dynamic systems (4D to 6D) and accelerate the computation speed of obtaining the joint response PDF by means of a graphics processing unit (GPU). Iourtchenko et al. [
26] presented a reliability analysis of strongly nonlinear single-degree-of-freedom (SDOF) systems by the PI approach. Naess and Johnsen [
27] applied a 3D PI method to estimate the response PDF of nonlinear and compliant offshore structures. Zhu and Duan [
28] evaluated the nonlinear ship-rolling driven by random wave load in random seas by a 4D PI method and verified by the MCS technique. Currently, some 6D problems can be resolved by the PI method with the help of a GPU, while 4D or lower dimension problems can be evaluated by this method at satisfactory computational efforts. Two different main streams are available to obtain the results at an acceptable cost, i.e., the acceleration of computational speed and the reduction of execution efforts. GPU acceleration [
20] is put forward to accelerate the computation speed, while techniques such as fast Fourier transform (FFT) [
29], decoupling [
30], and decomposition [
14] can be adapted to lessen the extensive computational resources on the center processing unit (CPU).
Moreover, many pieces of research on the stochastic dynamic characteristics of OWT have also been explored recently from three aspects, including physical model experiments, the finite element method, and analytical solutions. In order to make up for the deficiencies of DNV or API code, which are used to obtain the dynamic response under cyclic loading excitation, Domenico et al. [
31] proposed a small scaled model of a monopile wind turbine in kaolin clay soil subjected to cyclic horizontal loading to study the long-term behavior of a system and to quantify the changes in natural frequency and the damping of models using dimensionless parameters, i.e., the length-to-diameter ratio and the cyclic stress ratio. It was concluded based on the experimental results that higher strain levels lead to higher reductions in the natural frequency of the model. However, when the cyclic stress ratio is less than 0.02%, there is practically no degradation in natural frequency. Considering the wind and wave loads, the soil stiffness, and the geometric size of the structure, a comprehensive study on the dynamic behavior of an OWT supported by a monopile in the time domain was investigated by Swagata et al. [
32]. It showed that the soil–monopole–tower interaction and soil nonlinearity can increase the responses of the OWT system. Simultaneously, the rotor frequency was found to play a more dominant role than the blade passing frequency and the wave frequency. A new method was proposed to calculate the mean degradation index based on the derivatives of the degradation functions by Woochul et al. [
33]. It can be used to significantly decrease the computational effort considering the degradation of the soil modulus of the foundation under stochastic loading conditions. Further, the evolution of the dynamic response should be considered in the design process to secure the serviceability of OWTs and substructures. Considering the soil stiffness and geometric size affecting the dynamic response in clay, Swagata and Sumanta [
34] established a dynamic analysis system of the OWT using a beam on a nonlinear Winkler foundation model to address the feasibility of soft–soft and soft–stiff design approaches. It was shown that the main control standard for a wind turbine design in hard clay was the fatigue limit state and fatigue load is a key factor in wind turbine design and stable operation. Damgaard et al. [
35] evaluated the extent to which changes in soil properties affect the fatigue loads of one OWT installed on the monopile under parked conditions. More than 30% changes of the soil stiffness, the soil damping, and the presence of sediment transportation at the seabed may occur and were shown to be critical for the fatigue damage equivalent moment at the mudline. In order to accurately estimate the dynamic response of the OWT tower under wind excitation, Feyzollahzadeh et al. [
36] proposed an analytical transfer matrix method to determine the wind load response based on Euler–Bernoulli’s beam differential equation. This new method can be used to maintain a higher accuracy in wind-induced vibration analysis compared with conventional numerical methods. Laszlo et al. [
37] presented a simplified design procedure for OWT foundations to simplify the design steps and the calculation process based on the site characteristics, the turbine characteristics, and the ground profile. The research example showed that the simplified method arrived at a similar foundation to the one actually used in the London Array wind farm project. The state of practice in seismic design of the OWT and the existing design codes was firstly reviewed by Kaynia [
38]. It was indicated that the vertical earthquake excitation makes an obvious influence on the OWTs due to their rather high natural frequencies in the vertical direction. Further, the earthquake loads can be considerably reduced by radiation damping. In order to avoid 1P frequency close to the natural frequency, Saleh et al. [
39] proposed analytical solutions to predict the eigenfrequencies of the OWTs supported by the jacket using the finite element method. The ratio of the super-structure stiffness to the vertical stiffness of the foundation and the aspect ratio of the jacket governed the rocking frequency of a jacket. These results have an impact on the choice of foundations for jackets.
This paper focuses on the numerical investigation of the stochastic dynamic analysis of the OWT under horizontal stochastic wind excitation. In
Section 2, the dynamic system of the OWT is expressed as a linear SDOF system and the stochastic wind excitation is described as filtered Gaussian white noise by means of the second-order filter technique. Therefore, the 4D PI method is adjusted to obtain the joint PDF of the system’s response, while the marginal PDF of the response variables is obtained by the FFT-based acceleration method to reduce computational efforts. Meanwhile, the results are verified with the generally-used MCS method in
Section 3. Furthermore, the influences of horizontal mean wind speed and turbulence intensity on the dynamic response PDF and the reliability of the OWT are explored. It is shown that the FFT-based PI method for the OWT not only provides accurate response statistics distribution at acceptable computation efforts, but also offers an evaluation for reliability under normal operation.